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Introduction to Mathematical Reasoning
The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician "plays" with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler "plays" with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.
Requirements for Completion: In order to complete this course, you need to work through each unit and all of its assigned materials. Pay special attention to Units 1 and 2 as these lay the groundwork for understanding the more advanced, exploratory material presented in latter units. You will also need to complete:
Subunit 1.2 Activity
Subunit 1.3 Activity
Sub-subunit 1.4.1 Activity
Sub-subunit 1.4.2 Activity
Unit 1 Assessments
Sub-subunit 3.4.5.1 Activity
Sub-subunit 3.4.5.2 Activity
Sub-subunit 3.7.1 Activity
The Final Exam
Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through the assignments and assessment 112.25 approximately 31.5 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 4 hours) on Monday night; subunit 1.2, which is optional (a total of 4 hours) on Tuesday night; subunits 1.3 and 1.4 (a total of 4.25 hours) on Wednesday night; etc.
Tips/Suggestions: As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, definitions, and proofs that stand out to you. These notes will be useful to review as you study for the Final ExamIn this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku. You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers. You will also learn why giving names to mathematical ideas will enable you to think more effectively about concepts that are built upon several ideas. Then, you will learn that propositions are (English) sentences whose truth value can be established. You will see examples of self-referencing sentences which are not propositions. You will learn how to combine propositions to build compound ones and then how to determine the truth value of a compound proposition in terms of its component propositions. Then, you will learn about predicates, which are functions from a collection of objects to a collection of propositions, and how to quantify predicates. Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.
Instructions: Please click on the link above and read the entire article. Try not to get sidetracked looking at variations. Pay special attention to the growth of the number of Latin squares as the size increases. Note that if you want to look ahead at the type of problem you will be asked to solve, check the file "Logic.pdf" at the end of Unit 2.
Instructions: Please click on the link above. Scroll down the webpage to "Games," and select the "Sudoku" link to download the PDF. Read Tom Davis' paper, paying special attention to the way he names the cells and to his development of language. Next, if you have not done Sudoku puzzles before, Web Sudoku and Daily Sudoku and are two popular sites. Do one or two before moving on to Ken-Ken. Please note that this reading also covers the topics outlined in sub-subunits 2.1.2 and 2.1.3.
This will take you about 3 hours if you have not done Sudoku before, and about 2 hours if you have.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
Instructions: Please keep in mind that this activity is optional. After reading Harold et al.'s paper, click on the link above to access Ken-Ken puzzles, and attempt to complete one of these puzzles. Note that you can choose the level of difficulty (easier, medium, and harder). After a few practices, challenge yourself to attempt a Ken-Ken puzzle that is at the next level of difficulty. Do not allow yourself to get addicted!
You should dedicate no more than 1 hour to practicing Ken-Ken puzzles.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: This article is optional. If you have an interest in solving Ken-Ken problems, then you will find this section interesting. Otherwise, omitting it will not hinder your understanding of subsequent material. Should you choose to work through this section, please click on the link above, and read the paper by Harold Reiter, et al. for an introduction to Ken-Ken. Complete the exercises in the PDF. After you have completed the exercises, check Harold Reiter and John Thornton's "Solutions to Using Ken-Ken to Build Reasoning Skills." Please note that this reading covers the topics outlined in sub-subunits 1.2.1 through 1.2.3.
Reading this article and completing the exercises should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 3 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use parity in Ken-Ken puzzles.
1.2.2 Counting
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 4 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use counting.
1.2.3 Stacked Cages
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 5 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the idea of stacked cages.
1.2.4 X-Wing Strategy
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 6 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the X-wing strategy.
1.2.5 Pair Analysis
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 6 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the idea of stacked cages.
1.2.6 Parallel and Orthogonal Cages
Note: This topic is covered by the reading assigned below subunit 1.2. Read Section 7 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use parallel and orthogonal cages.
1.2.7 Unique Candidates
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 8 of "Using Ken-Ken to Build Reasoning Skills" to learn how to use the unique candidate rule.
1.2.8 Modular Arithmetic
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 9 of "Using Ken-Ken to Build Reasoning Skills," even though you have not yet studied modular arithmetic. When you get to this part of the course, you will be asked to come back and take another look at this section.
Instructions: Please note that this activity is optional. If you choose to work through this activity, please click on the link above, read the game rules by clicking on the "daily puzzle rules" link, and play a bit.
You should dedicate no more than 1 hour to exploring SET and pick out a few videos to watch on brain teasers. The puzzle will be introduced to you at the beginning of the video. You should pause the video and attempt to solve the puzzle before viewing the solution. Watch the solutions only if you absolutely cannot solve the puzzle; then, go back and reattempt the problem.
You should spend approximately 1 hour on this site, watching a few of these videos and attempting to solve the problems.
Instructions: Please click on the link above and work on the problems on this webpage: liars and truth-tellers puzzles, the Rubik's cube, knots and graphs, and arithmetic and geometry.
Solving these problems should take approximately 2 hours Problems about finding the counterfeit coin among a large group of otherwise genuine coins are quite abundant. Please click on the link above and attempt to solve the problem on this webpage. Solutions appear at the bottom of the webpage. If this type of logical thinking interests you, attempt to find similar problems to solve with an online search.
You should spend approximately 15 minutes attempting to solve this problem.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read the entire webpage. This text will enable you to see the very close connection between propositional logic and naïve set theory, which you will study in Unit 3. Please note that this lecture covers the topics outlined in sub-subunits 1.5.1 and 1.5.2 as well as any inclusive sub-subunits.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Click on the link above and watch the entire lecture. In particular, focus on the information provided from the 12-minute mark until the 18-minute mark. In this lecture, you will learn which sentences are propositions. Please note that this lecture covers the topics outlined in sub-subunits 1.5.1 and 1.5.2 as well as any inclusive sub-subunits.
Watching this video and pausing to take notes should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Note: This topic is covered by the video lecture assigned below subunit 1.1. In particular, focus on and review the lecture from the 25-minute mark to the 32-minute mark for a discussion on implication and other Boolean connectives.
Instructions: Please click on the link above and read this article, which covers the properties of connectives. While reading, pay special attention to the connection between the Boolean connective and its Venn diagram.
Reading this article and taking notes should take approximately 1 hour.
Instructions: Please click on the links above and read these four sections of Koehler's lectures on logic and set theory. These sections cover the topics outlined below subunit 1.5, including all the sub-subunits.
Reading these sections should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. Make sure to review the "Logical Operations and Truth Tables" section for an introduction that helps define truth tables.
1.5.2.2 The Boolean Algebra of Propositions
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. Make sure to review the section on "Boolean Algebra."
1.5.2.3 Tautologies, Contingencies, and Contradictions
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. Make sure to read the definitions of tautology and contradictions (terms highlighted in bold) in the opening paragraphs of the "Logical Operations and Truth Tables" section. Please note that a contingency is simply a proposition that is caught between tautology (at the top) and contradiction (at the bottom). In other words, it is a proposition which is true for some values of its components and false for others. For example "if it rains today, it will snow tomorrow" is a contingency, because it can be true or false depending on the truth values of the two component propositions.
1.5.2.4 Logical Equivalence
Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2. In particular, focus on the text after the heading "Equivalence" toward the end of the "Logical Operations and Truth Tables" section.
Instructions: Please click on the links above and watch these lectures in their entirety to learn about predicates and quantifiers. These videos will also cover the topics outlined in sub-subunit 1.6.1, including 1.6.1.1 and 1.6.1.2.
Watching these lectures and pausing to take notes should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on these webpages above.
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6.
1.6.1.1 Negating Existential and Universal Predicates
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6. Note that the negation of an existentially quantified predicate is a universally quantified one, and vice-versa.
1.6.1.2 The Algebra of Predicates
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6. The main idea here is that predicates can be manipulated in much the same way as numbers, sets, or propositions as we have seen already in the course.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of logical connectives, propositions, negations, quantifiers, truth tables, and counterexamples. When you are done, check your work against those provided in the accompanying solutions file, the Saylor Foundation's "Logic Homework Set Solutions" (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and read the following sections: "Introduction", "Definition and Theorems", "Disproving Statements", and "Types of Proofs". The types of proofs include Direct Proofs, Proof by Contradiction, Existence Proofs, and Uniqueness Proofs. You may stop the reading here; we will cover the sixth one, Mathematical Induction, later in the course.
Reading these sections should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read through the examples in the article. The problems are not difficult, but they do serve as clear illustrations of the various aspects of entry-level problem solving.
Reading this article should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and complete this 10-question quiz on logic and related conditionals. Once you choose an answer, a pop up will tell you if you have chosen correctly or incorrectly. You may also click on the drop down menu for an explanation.
Completing this quiz should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above to download the assignment. In order to solidify your problem solving and logical skills, work the eight problems. Then, check your answers against the Saylor Foundation's "Logic Problems Solutions" (PDF).
In this unit, you will explore the ideas of what is called 'naive set theory.' Contrasted with 'axiomatic set theory,' naive set theory assumes that you already have an intuitive understanding of what it means to be a set. You should mainly be concerned with how two or more given sets can be combined to build other sets and how the number of members (i.e. the cardinality) of such sets is related to the cardinality of the given sets.
Instructions: Please click on the links above and read these webpages in their entirety. These texts discuss the basics of set theory. Note that there are three ways to define a set. The third method, recursion, will come up again later in the course, but this is a great time to learn it.
Reading these webpages should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above to download the PDF. Please read pages SF-1 through SF-8 of the file for an introduction to sets, set notation, set properties and proofs, and ordering sets.
Reading this article should take approximately 1 hour to download the PDF. Please read pages SF-9 through SF-11 to learn about subsets of sets. This text also is useful for learning how to prove various properties of sets. If needed, review pages SF1-SF8, which were covered in sub-subunit 2.1.1 above. In particular, please focus on example 9.
Reading these sections should take approximately 30 minutes and read the entire webpage. It is important that you become aware that sets combine under union and intersection in very much the same ways that numbers combine under addition and multiplication. For example, AUB=BUA is a way to say union is commutative in the same way as x + y = y + x says addition is commutative. One difference, however, is that the properties of addition and multiplication are defined as part of the number system (in our development) whereas the properties of sets under the operations we have defined are provable and hence must be proved.
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above, scroll down the webpage to week 7, and click on the link for "Boolean Algebra" to download the lecture as a PDF. Please read this entire lecture, paying special attention to the definition of Boolean Algebra and to the isomorphism between the two systems of propositional logic and that of sets. Work the three exercises at the bottom of the PDF and then have a look at the solutions at the end of the document. Note that this reading also covers the topic outlined in sub-subunit 2.2.2.
Reading this lecture and completing these exercises should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read the entire webpage. This brief text will show you how to use characteristic functions to prove properties of sets. However, there are other reasons to learn how to do this. You will see later in the course that functions (not just characteristic functions play a critical role in the theory of cardinality (set size).
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read this webpage, paying special attention to the proof of proposition 3.3.3 at the end of the page. There is a nice proof of this using characteristic functions, which you will be asked to produce later in the course.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and read this webpage, which demonstrates the basic inclusion/exclusion equation outlined in the title of this subunit. The examples on this webpage are especially interesting; pay attention to example 2, which is about playing cards.
Reading this webpage and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of elementary set theory. When you are done, check your work against the answers provided in the accompanying solutions file, the Saylor Foundation's "Elementary Set Theory Homework Set Solutions" (PDF).
This exercise set should take between 2 and 3 hours to complete, depending on your comfort level with the material.
This unit is primarily concerned with the set of natural numbers N = {0, 1, 2, 3, . . .}. The axiomatic approach to N will be postponed until the unit on recursion and mathematical induction. This unit will help you understand the multiplication and additive structure of N. This unit begins with integer representation: place value. This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multi-digit integers. The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers. Then, you will learn about the multiplicative building blocks, the prime numbers. The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way. The Division Algorithm enables you to associate with each ordered pair of non-zero integers – a unique pair of integers – the quotient and the remainder. Another important topic is modular arithmetic. This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication. Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers. Such equations with integer solutions are sometimes called Diophantine Equations.
Instructions: Please click on the link above and read this essay, "Fusing Dots," paying special attention to the exercises at the end. Please note that this reading covers all of the subunits assigned below subunit 3.1. You may find the second half of this reading very difficult. Try to read through Laurie Jarvis' "Understanding Place Value" first (sub-subunit 3.1.12) and then come back to this more challenging paper. You can access the solutions for selected problems here (PDF). Don't worry about understanding all of the details your first time through the reading. Instead, concentrate on the material in the first five sections of the document, and then attempt to generally understand the subsequent sections on Fusing Dots. The supporting details will become more familiar as you work through the various subunits.
Reading this essay should take approximately 2 hours, and completing the exercises should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the reading assigned below subunit 3.1.In particular, work the problem that involves finding the product of two numbers both given in base 5 notation, without translating to decimal notation.
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.1 Using Repeated Subtraction
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.2 Using Repeated Multiplication
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.3 Translating between Representations
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.2.4 Representing Repeating Base b Numbers as Quotients
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of "Fusing Dots."
3.1.3 Other Interesting Methods of Representation
3.1.3.1 Base phi Notation
Note: The Base number here is the irrational number phi. This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 6 of "Fusing Dots" and see problem 6 at the end of Section 6.
3.1.3.2 Fibonacci Representation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 6 of "Fusing Dots" and see problem 7 at the end of Section 6.
3.1.3.3 Cantor's Representation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, focus on Section 6 of "Fusing Dots." Problems 12 through 15 at the end of Section 6 all deal with Cantor's representation, also known as factorial notation.
3.1.3.4 Base Negative 4 Notation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 7 of "Fusing Dots." Problem 2 at the end of section 10 is devoted to base negative 4 notation and arithmetic.
Instructions: Please click on the link above to "Prime Numbers" and read the webpage, which includes a good overview of prime numbers and also a list of unsolved problems. Pay special attention to the unsolved problems 1 and 2. Then click on the second link above and read the proof of Euclid.
Reading these webpages should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above and read the entire webpage on Euclid's proof of the infinitude of primes. Be sure you understand why the prime P is not already in the list of primes; if necessary, re-read this text a few times until you have fully grasped this concept.
Reading this article should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above and read this webpage. Take note of the definition of Brun's constant. Also note that this is related to the Intel's famous $475 million recall of Pentium chips. Please also feel free to click on the link to "Enumeration to 1e14 of the twin primes and Brun's constant" link at the end of the webpage to read associated content.
Reading this webpage should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read this brief article to learn a about the Goldbach conjecture. Problems like this are the subject of intense work by mathematicians around the world, and progress is made nearly every year towards solving them.
Reading this article should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above, scroll down to the "Fundamental Theorem of Arithmetic" heading, and read this brief introductory information. Then, click on the second link above and read the entire webpage for information about the Fundamental Theorem of Arithmetic (FTA). Please note that we are going to postpone the proof of FTA until the end of Unit 4. This reading covers the topics outlined in sub-subunits 3.3.1 and 3.3.2.
Reading these articles should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read this entire article on the fundamental theorem of arithmetic. The article may take more time to read than some others. Please note that this reading covers the topics outlined in sub-subunits 3.3.1 and 3.3.2.
Note: This topic is covered by the readings assigned below subunit 3.3. In particular, please focus on Bogomolny's "Euclid's Algorithm" reading and section 2 and 3, "Euclid's Algorithm" and "Alternative Proof", in the Wikipedia article.
3.3.2 Some Applications of FTA
Note: This topic is covered by the readings assigned below subunit 3.3. In particular, please focus on Section 1 "Applications" of the Wikipedia article. Nearly all the proofs of irrationality of the square root of a composite non-square number depend on FTA. Of course, there are also many other applications.
Instructions: Please click on the link above and read this webpage. In particular, focus on the exercise in the reading. Do not be intimidated by the notation in the essay, just read it down to the part on ideals.
Reading this webpage should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the links above and watch these lectures in sequential order. These videos address the concepts outlined in sub-subunits 3.4.1 through 3.4.4. Then, if you chose to work through the Ken-Ken material in subunit 1.2, go to section 9 of the paper "Using Ken-Ken to Build Reasoning Skills" from subunit 1.2, and re-read the section to recall how to use modular arithmetic as a strategy for Ken-Ken puzzles.
Watching these lectures and re-reading this section of "Using Ken-Ken to Build Reasoning Skills" should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above and watch this video, which will help you understand the divisibility rules for 3 and 9. Then click on the second link above and watch the entire lecture, which discusses divisibility by 11.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Please click on the link above and try to solve the problem before checking the solution. This problem asks: what is the units' digit of the 2012th Fibonacci number? See if you can work this using your understanding that remainders work perfectly with respect to addition. After you have attempted this problem, review the solution on this webpage.
Completing this assignment should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: By now, the following type of problem should be familiar: what is the units' digit of the expression 7^2012 X 13^2011? See if you can work this using your understanding that remainders work perfectly with respect to multiplication. In other words, if you know the remainder when N is divided by d, then you can find the remainder when N^3 is divided by d.
The solution to this question is mentioned below, but please only check it after you have attempted the problem. After you have completed this problem, click on the link above, and work to solve the problem on this webpage. After you have attempted the problem, click on the link to see the solution.
Solution: The solution to the initial problem mentioned above is that the remainder when N^3 is divided by d is the same as when the r^3 is divided by d, where r is the remainder when N is divided by d.
Completing this assignment should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above and then select the hyperlink to the lecture titled "The Floor or Integer Part Function" to access the PDF. Please read the entire lecture. Then click on the second link, and select the link titled "v.3.0 numberI.pdf" under "1995 Lectures" to access the PDF. Please read the entire lecture.
Reading these lectures and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read this article, paying special attention to Sections 3 and 4, where you will learn about geometry of the divisors of an integer. Complete the problems on the document above, and then check your answers against the Saylor Foundation's "Just the Factors, Ma'am Solutions" (PDF). The topics outlined for subunit 3.6, including sub-subunits 3.6.1 through 3.6.3, are covered by these sections of reading.
Reading this article and taking notes should take about 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages linked above.
Instructions: Please click on the link above and then select the "PDF" link next to "Lecture 6: Number Theory I" to download the file. Please read this lecture, which provides an introduction to decanting (see the Die Hard example on pages 5-7) and the Euclidean algorithm.
Reading this lecture and taking notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
Instructions: Please click on the link above to download the PDF version of the text and read this paper. This is an easier version of this technique. Solutions to selected problems can be found here (PDF).
Reading this paper should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
Instructions: Note that you have already read this essay in sub-subunit 3.5.1. Please click on the link above, and select the "v.3.0 numberI.pdf" link under "1995 Lectures" to download the PDF. Review the section on "Division Algorithm" again, and then attempt the 3 sample problems in the lecture.
Reviewing this section and attempting the sample problems should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and then click on "Schedule" on the left side of the webpage. Scroll down the webpage to the section "Integer Divisibility," and select the "Linear Diophantine Equations" link to download Lecture 5 as a PDF. Please read this student-friendly version of the lecture, which discusses solving an integer divisibility type of equation. You should focus on solving linear Diophantine equations. In particular, you should be able to find a single solution and then generate all solutions from the one you found.
Reading this lecture should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
In this unit, you will learn to prove some basic properties of rational numbers. For example, the set of rational numbers is dense in the set of real numbers. That means that strictly between any two real numbers, you can always find a rational number. The distinction between a fraction and a rational number will also be discussed. There is an easy way to tell whether a number given in decimal form is rational: if the digits of the representation regularly repeat in blocks, then the number is rational. If this is the case, you can find a pair of integers whose quotient is the given decimal. The unit discusses the mediant of a pair of rational fractions, and why the mediant does not depend on the values of its components, but instead on the way they are represented.
Instructions: Please click on the link above and read this article. Pay special attention to the five problems on rational numbers at the beginning of the paper. Problem 10 will enable you to appreciate the different between the value of a number and the numeral used to express it. Pay special attention to Simpson's Paradox in the paper. Try the practice problems at the end of the reading. After you have attempted these problems, please check the solutions against the Saylor Foundation's "Fractions Solutions" (PDF).
Please note that this reading covers the topics outlined for subunit 4.1, as well as inclusive sub-subunits 4.1.1 through 4.1.3, and subunit 4.2, as well as sub-subunits 4.2.1 through 4.2.3.
Reading this article and taking notes should take approximately 3 hours. You should also spend approximately 3 hours working on the problems provided in the text.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, please focus on first 2 paragraphs after the text "Fraction versus rational number," especially where numbers and numerals are in bold font as this will help you understand the relationship between the two.
4.1.2 The Mediant of Two Fractions
Note: This topic is covered by the reading assigned below the Unit 4 introduction. Make sure to work on problem 10 in the essay to help you better understand the mediant of two fractions.
4.1.3 Building New Rational Numbers from Given Ones
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, pay attention to problems 1-3 under "Rational Numbers."
Note: This topic is covered by the reading assigned below the Unit 4 introduction. An interesting property of the rational numbers is that between any two rational numbers we insert another rational number. This property is called density. We say the rational numbers are dense in the real numbers. The same property holds for irrational numbers. Try proving these propositions. Problem 6 in the essay discusses this property.
This topic is also covered in the reading assigned below subunit 4.3.2. In particular, focus on Section 6, "Density of Rational Numbers."
Instructions: Please click on the first link above and watch the video, which shows a proof of the irrationality of the square root of 2. Can you see how to use these ideas to prove that the square root of 3 and of 6 are also irrational?
Next, click on the second link above, and watch the video, which shows a proof that the square root of 3 is irrational. After watching this video, do you think you could prove how the square root of 6 is also irrational?
Watching these videos, pausing to take notes, and answering the questions above should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
Instructions: Please click on the link above to access Professor Tsang's webpage. Select the link to "HW1" to download the PDF. Please read pages 7 through 9, from "Density of Rational Numbers" through "Density of Irrational Numbers." Please note that this reading also covers the topic of Density of Rational Numbers outlined for sub-subunit 4.2.3.
Reading this article, taking notes, and reviewing the proofs several times should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above to access Professor Tsang's webpage. Select the link to "HW1" to download the PDF. Read pages 1-7 of the text. The first 6 pages discuss the field and order axioms for real numbers. The Completeness Axiom on page 6 is what distinguishes the rational numbers from the real numbers – the latter is COMPLETE, while the former is not. This resource covers the topics for sub-subunits 4.4.1 through 4.4.3.
Reading this article and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
In this unit, you will prove propositions about an infinite set of positive integers. Mathematical induction is a technique used to formulate all such proofs. The term recursion refers to a method of defining sequences of numbers, functions, and other objects. The term mathematical induction refers to a method of proving properties of such recursively defined objects.
Instructions: Please click on the links above and watch these brief videos. These videos provide informative discussions as to why the well-ordering principle of the natural numbers implies the principle of mathematical induction.
Watching these videos provides an informative discussion on the principle of mathematical induction and the well-ordering principle of the natural numbers. It specifically addresses the notion of strong mathematical induction.
Watching this video illustrates how the principle of strong mathematical induction can link above and watch the brief video, which illustrates using the principle of strong mathematical induction to links above and read both essays. Notice the similarities between using recursion to define sets and using recursion to define functions. Then answer the four questions at the end of the first essay.
In this type of definition, first a collection of elements to be included initially in the set is specified. These elements can be viewed as the seeds of the set being defined. Next, the rules to be used to generate elements of the set from elements already known to be in the set (initially the seeds) are given. These rules provide a method to construct the set, element by element, starting with the seeds. These rules can also be used to test elements for the membership in the set.
Reading these essays, taking notes, and completing the assignment should take about 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topic of proofs using mathematical induction. When you are done, check your work against those provided in the accompanying solutions file, The Saylor Foundation's "Mathematical Induction Homework Set Solutions" (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
In this unit, you will learn about binary relations from a set A to a set B. Some of these relations are functions from A to B. Restricting our attention to relations from a set A to the set A, this unit discusses the properties of reflexivity(R), symmetry(S), anti-symmetry(A), and transitivity(T). Relations that satisfy R, S, and T are called equivalence relations, and those satisfying R, A, and T are called partial orderings.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the properties of relations and interrelationships among them, as well as specific examples of relations. When you are done, check your work against the answers provided in the accompanying solutions file, The Saylor Foundation's "Relations Homework Set Solutions" (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and watch the video. It may be worth spending some time watching this video twice. Note that the lecturer spends some time discussing the definitions of the properties below for sub-subunits 6.1.1 through 6.1.5. The examples he provides exhibit several properties. These are the defining properties of an equivalence relation (see subunit 6.4) and Partial Ordering (see subunit 6.5). Note that this resource covers the topics outlined for sub-subunits 6.1.1 through 6.1.6.
Watching this video twice and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and watch the brief video, which illustrates the notions of relations and functions. This video also provides examples of relations that are functions and some that are not.
Watching this video and pausing to take notes should take approximately 15 minutes.
Note: This topic is covered by the video assigned below subunit 6.2.2. A surjective function is one for which every element in the codomain is mapped to by an element in the domain. For such functions, the codomain and range are equal.
6.2.4 Bijections
Note: This topic is covered by the video assigned below subunit 6.2.2. A bijection is a function that is both one-to-one and onto.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of verifying properties of given functions and equivalence relations, determining if relations are equivalence relations, and commenting on the structure of a relation by using equivalence classes. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation's "Elementary Functions and Equivalence Relations Homework Set – Solutions" (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and watch the last 10 minutes of this video again. It is especially important that you understand the relationship between an equivalence relation and the partition it induces.
Reviewing this section of the lecture and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
In this unit, you will study cardinality. One startling realization is that not all infinite sets are the same size. In fact, there are many different size infinite sets. This can be made perfectly understandable to you at this stage of the course. In Unit 7.4.3, section (d)iii, you learned about bijections from set A to set B. If two sets A and B have a bijection between them, they are said to be equinumerous. It turns out that the relation equinumerous is an equivalence relation on the collection of all subsets of the real numbers (in fact on any set of sets). The equivalence classes (the cells) of this relation are called cardinalities.
Instructions: Please click on the first link above to watch the video about injections (1 to 1 functions) and surjections (onto functions). Then click on the second link to watch the video, which will show the relationship between injections and functions that have an inverse. Finally, click on the link and watch the last lecture.
Instructions: Please click on the link above and study the proof of Cantor's Theorem. Even though the proof is only one page, this idea is new to you, and therefore is likely to be harder to understand; thus, you should take your time studying this proof carefully.
Studying this proof should take approximately to supplement the written proof of Cantor's Theorem. Then click on the second link and watch the brief video about counting finite sets and Cantor's Diagonalization Theorem.
Watching these videos and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and scroll down to "Handout 8." Select the PDF link to download the file. Read this entire document to learn about countable and uncountable sets. Focus on the several examples of uncountable subsets of R. Please note that this reading also covers topics outlined in subunit 7.2, including sub-subunits 7.2.1 and 7.2.2.
Reading this handout and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of functional properties involving images and inverse images of sets as well as computing images and inverse images of sets. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation's "Functional Properties Homework Set – Solutions" (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of recognizing cardinality properties, determining if sets are countable or uncountable, and determining whether two sets are equivalent. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation's "Cardinality Homework Set – Solutions" (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
Instructions: Please click on the link above and study the proof on this webpage, which shows that rational numbers are countable. Note that information on this topic is also found in the reading assigned below sub-subunit 7.1.3.
Reading this webpage, taking notes, and studying the proof should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read only this webpage; there is no need to click on the "next" or "previous" buttons at this time. Most of these topics have already been covered in the videos for Unit 7, so some of this will be a review.
Reading this webpage should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and select the link to "Course Notes 4.2: Property of Functions" to download the PDF. Please read the paper, paying special attention to examples 2.2.5, 2.2.6, and two functions ax and logax in the paragraph above example 2.7.1. Please note that this resource also covers the topic outlined in sub-subunit 7.3.1.
Reading this paper should take about for an overview of the development of the formulas for the number of permutations and the number of combinations of n objects. For a much more elaborate introduction to counting, click on the second link above, and watch the lecture. This resource covers the topics outlined in sub-subunits 8.1.2 and 8.1.4 below.
Watching these lectures and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the link above and read the PDF. Attempt problems 1-20, starting on page 3. Once you have attempted these problems, check your solutions at the Saylor Foundation's "Counting Solutions" (PDF). Please note that this reading and these exercises cover the topics outlined in sub-subunits 8.1.1 through 8.1.4.
Reading this essay and working on these problems should take approximately 5 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
Instructions: Please click on the first link above and watch the video for an introduction to the principle. Then, follow this up by clicking on the second link and watching an example of the principle. Notice that the problem is about As, Bs, and Cs, not As, Bs, and Os as the teacher describes at the start.
Watching these videos and taking notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
Instructions: Please click on the link above and watch the brief video. The first video provides some an application of the pigeon-hole principle to divisibility and modular arithmetic. The second video provides some applications of the pigeon-hole principle to operations involving integers.
Watching this video and pausing to take notes should take approximately 30 minutes | 677.169 | 1 |
The purpose of the this website is to inform parents and students about the Renewed Mathematics Curriculum in Saskatchewan. This site is designed for parents and students of the Greater Saskatoon Catholic School System. If you have any questions or concerns, please contact the GSCS High School Mathematics Teacher-on-Assignment, Shelda Hanlan Stroh, at [email protected] or (306) 659-7678.
Grade 10 Math Courses
Authored on March 24, 2010 6:46 PM
March 24, 2010 6:46 PM
Students are strongly encouraged by the Ministry and GSCS to take both of the Grade 10 math courses: Foundations and Pre-Calculus 10 & Workplace and Apprenticeship 10.
Rationale:
·Students will benefit as there is rigorous math learning in both courses. This will allow students to develop a deep foundational understanding of mathematics.
·Students should be exposed to the mathematics in all three of the pathways before deciding on a pathway.
·It delays the decision regarding which pathway to take until Grade 11.
·It keeps the students' options open for all three pathways.
About this Entry
This page contains a single entry published on March 24, 2010 6:46 PM. | 677.169 | 1 |
Algebra and Trigonometry Algebra and Trigonometry. This text presents the traditional content of the entire Precalculus series of courses in a manner that answers the age-old question of When will I ever use this? Highlighting truly relevant applications, this text presents the material in an easy to teach from/easy to learn from approach. This book presents the traditional content of Precalculus in a manner that answers the age-old question of "When will I ever use this?" Highlighting truly relevant applications, this book ... MOREpresents the material in an easy to teach from/easy to learn from approach.KEY TOPICS Chapter topics include equations, inequalities, and mathematical models; functions and graphs; polynomial and rational functions; exponential and logarithmic functions; trigonometric functions; analytic trigonometry; systems of equations and inequalities; conic sections and analytic geometry; and sequences, induction, and probability. For individuals studying Precalculus.
Quadratic Functions. Polynomial Functions and Their Graphs. Dividing Polynomials: Remainder and Factor Theorems. Zeros of Polynomial Functions. More on Zeros of Polynomial Functions. Rational Functions and Their Graphs. Modeling Using Variation.
The Law of Sines. The Law of Cosines. Polar Coordinates. Graphs of Polar Equations. Complex Numbers in Polar Form; DeMoivre's Theorem. Vectors. The Dot Product.
8. Systems of Equations and Inequalities.
Systems of Linear Equations in Two Variables. Systems of Linear Equations in Three Variables. Partial Fractions. Systems of Nonlinear Equations in Two Variables. Systems of Inequalities. Linear Programming.
9. Matrices and Determinants.
Matrix Solutions to Linear Systems. Inconsistent and Dependent Systems and Their Applications. Matrix Operations and Their Applications. Multiplicative Inverses of Matrices and Matrix Equations. Determinants and Cramer's Rule.
10. Conic Sections and Analytic Geometry.
The Ellipse. The Hyperbola. The Parabola. Rotation of Axes. Parametric Equations. Conic Sections in Polar Coordinates. | 677.169 | 1 |
Algebra 1
Instructor's Annotated Edition
ISBN/ISBN10
978-0-495-38988-0
0-495-38988-9
Order#
1090340
Price
$202.75
Quantity
This special version of the complete student text contains a Resource Integration Guide to using the ancillary teaching and learning resources with each chapter of the text, as well as answers printed next to all respective exercises. Graphs, tables, and other answers appear in a special answer section in the back of the text.
Titles marked with asterisk (*) indicate product is restricted from sale to individuals and may only be purchased by a registered institution. Go here if you are not already logged in or need to register. | 677.169 | 1 |
Brief Calculus And Its Applications - 8th edition
Summary: Once again, these extremely readable, highly regarded, and widely adopted texts present "tried and true" formula pairing substantial ...show moreamounts of graphical analysis and informal geometric proofs with an abundance of hands-on exercises has proven to be tremendously successful with both students and instructors. What would the benefit to your students be of using a text which blends practical applications with mathematical concepts? Features
NEW - Details the ways in which technology can be used to foster understanding of several topics while it facilitates computation.
NEW - Ends each chapter with a Review of Fundamental Concepts, helping students focus on the chapter's key points.
NEW - Places greater emphasis on the significance of differential equations in applications involving exponential functions.
NEW - Customized calculus software is available through the study guide.
NEW - Companion website supports and extends the materials presented in the text.
NEW - All graphs of functions have been redrawn using Mathematicia.
Reinforces class lessons with carefully designed exercise sets, and challenges students to make their own connections.
Minimizes prerequisites, allowing those who have forgotten much of their high school mathematics to start anew with this self-contained material.
Functions and Their Graphs. Some Important Functions. The Algebra of Functions. Zeros of Functions. The Quadratic Formula and Factoring. Exponents and Power Functions. Functions and Graphs in Applications.
1. The Derivative
The Slope of a Straight Line. The Slope of a Curve at a Point. The Derivative. Limits and the Derivative. Differentiability and Continuity. Some Rules for Differentiation. More About Derivatives. The Derivative as a Rate of Change.
2. Applications of the Derivative
Describing Graphs of Functions. The First and Second Derivative Rules. Curve Sketching (Introduction.) Curve Sketching (Conclusion.) Optimization Problems. Further Optimization Problems. Applications of Calculus to Business and Economics.
3. Techniques of Differentiation
The Product and Quotient Rules. The Chain Rule and the General Power Rule. Implicit Differentiation and Related Rates.
Antidifferentiation. Areas and Reimann Sums. Definite Integrals and the Fundamental Theorem. Areas in the xy-Plane. Applications of the Definite Integral.
7. Functions of Several Variables
Examples of Functions of Several Variables. Partial Derivatives. Maxima and Minima of Functions of Several Variables. Lagrange Multipliers and Constrained Optimization. The Method of Least Squares. Double Integrals.
8. The Trigonometric Functions
Radian Measure of Angles. The Sine and the Cosine. Differentiation of sin t and cos t. The Tangent and Other Trigonometric Functions.
9. Techniques of Integration
Integration by Substitution. Integration by Parts. Evaluation of Definite Integrals. Approximation of Definite Integrals. Some Applications of the Integral. Improper Integrals | 677.169 | 1 |
Welcome to Algebra 2! There are no secrets to success. It is the result of preparation, hard work and learning from failure -Colin Powell- Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures So enter the doors of --- and lets explore the Wide World of Algebra 2. No need to guess; know your current grade or what you have missed by logging on to PINNACLE. Need your graduation status then VIRTUAL COUNSELOR is your information gateway. Need the latest news from your school? Why not log on to Miramar High and be informed. | 677.169 | 1 |
SPI 3108.3.3 Describe algebraically the effect of a single transformation (reflections in the x- or y-axis, rotations, translations, and dilations) on two-dimensional geometric shapes in the coordinate plane.
CLE 3103.3.3 Analyze and apply various methods to solve equations, absolute values, inequalities, and systems of equations over complex numbers.
CLE 3103.3.4 Graph and compare equations and inequalities in two variables. Identify and understand the relationships between the algebraic and geometric properties of the graph.
CLE 3103.3.5 Use mathematical models involving equations and systems of equations to represent, interpret and analyze quantitative relationships, change in various contexts, and other real-world phenomena.
Checks for Understanding
3103.3.1 Perform operations on algebraic expressions and justify the procedures.
3103.3.2 Determine the domain of a function represented in either symbolic or graphical form.
3103.3.3 Determine and graph the inverse of a function with and without technology.
3103.3.4 Analyze the effect of changing various parameters on functions and their graphs. | 677.169 | 1 |
Student Resources
Services
The courses offered by the Developmental Mathematics Department are designed to provide a foundation in preparatory mathematics necessary for success in future college courses throughout a variety of disciplines as well as mathematics. The courses also aim for the development of critical thinking skills applicable to all aspects of academic life. | 677.169 | 1 |
Mathematical Literacy
Mathematical Literacy is offered as a Matriculation subject from Form III at St Mary's. The subject focuses on real-life situations and is perfect for the pupil who finds difficulty with the more abstract facets of core Mathematics. Since Mathematical Literacy deals with real-life situations, it enables pupils to approach problems with confidence and allows them to discover methods that work for them. It equips pupils with techniques, knowledge, skills, values and attitudes for self-fulfilment and growth so that they can be meaningful participants in society. Mathematical Literacy lends itself successfully to activities such as presentations, research, debates, interviews, questionnaires, project work, assignments and practical investigations. Mathematical Literacy is accepted at university faculties where core Mathematics is not a prerequisite. | 677.169 | 1 |
From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
This text addresses the need for a new mathematics text for careers using digital technology. The material is brought to life through several applications including the mathematics of screen and printer displays. The course, which covers binary arithmetic to Boolean algebra, is emerging throughout the country and may fill a need at your school. This unique text teaches topics such as binary fractions, hexadecimal numbers, and Venn diagrams to students with only a beginning algebra background.
Table of contents
1. Computation.
An Introduction.
Exponents and Their Properties.
Calculator Functions.
Scientific Notation.
An Introduction to Statistics and Error Analysis.
Dimensional Analysis.
2. Binary Numbers.
The Binary System.
Base Two Arithmetic.
Two's Complement.
Binary Fractions.
Computer Memory and Quantitative Prefixes.
3. Octal and Hexadecimal Numbers.
The Octal System.
Hexadecimal Representation.
Base 16 Arithmetic.
Elements of Coding.
4. Sets and Algebra.
The Language of Sets.
Set Operators.
Venn Diagrams.
Propositions and Truth Tables.
Logical Operators and Internet Searches.
5. Boolean Circuits.
Equivalent Boolean Expressions.
Logic Circuits Part I - Switching Circuits.
Truth Tables and Disjunctive Normal Form.
Logic Circuits Part II - Gated Circuits.
Karnaugh Maps.
6. Graphs.
Color Sets.
Hexadecimal RGB Codes.
Cartesian and Monitor Coordinates.
Elements of Computer Animation.
Features & benefits
Practice Problems Computer math students require a large number of exercises in order to practice the material that they have just learned. These exercises actively engage students in the learning process and reinforce newly learned concepts and skills.
Bits of History Scattered throughout the text, Bits of History let students in on the background and the origins of the concepts they are currently learning. This feature helps students better understand the theories behind their actions.
Definition Boxes Important definitions are boxed and highlighted throughout each chapter to emphasize their importance to students and to make them easy to find for review purposes.
Elementary Logic More than any other text offered at this level, this text emphasizes the application of logic to programming and circuitry. Logic is part of the foundation of what the text refers to as its "Rosetta Stone." By the end of the course, students see that a logic circuit can be represented by a truth table, a Boolean expression, or a Venn diagram.
Ample opportunity for Review At the end of each chapter is a comprehensive review section. Starting with a chapter Summary, students are given the main concepts of the chapter. The Glossary is a comprehensive listing of key terms from throughout the chapter. Review Exercises require students to solve problems without the help of section references. Additionally challenging problems are highlighted by a triangle symbol around the exercise number. Cumulative Review Exercises gather various types of exercises from the preceding chapters to help students remember and retain what they are learning throughout the course. | 677.169 | 1 |
Welcome to Math 251 - Online!
Below you will find answers to common questions about this course:
Learning in an Online Environment
The best part about online learning is it offers flexibility. You can choose when and where you want to learn and you don't have to fight traffic or find parking to get there. You just need your computer and a great attitude.
Online learning is not for everyone. Ask yourself if you are disciplined and motivated enough to take a few hours a day, 3 or 4 times a week, to succeed in learning algebra and to pass this class. There will be no class that you have to be in at a certain time and no professor right there in class to answer your question.
What you will have is MyMathLab. MyMathLab has taken your textbook and put it online. It has enhanced it with buttons you can click to get short audio and video clips, see animations, and try practice problems with guided solutions. Each section of the book begins with a video lesson explaining the material. While doing the homework if you get stuck, MyMathLab can help you. It can walk you through examples, give you solutions, generate new similar problems and even take you to the part of the video that explains the particular type of problem you are working on. MyMathLab also has an online tutoring center.
The only time you are required to come to campus is for the final exam. All other work is online.
What if I can't make the on-campus final exam?
If you are out of the Southern California area and coming to the final exam would create a hardship, you must arrange a proctor through another college, high school, library, or military educational officer. Proctors need to be arranged at least 2 weeks before the scheduled final exam date and your final exam must be completed by the scheduled final exam date. A list of colleges that proctor for a small fee is available at
Course Website/Online Textbook
For those of you who have taken online courses with the college before, this one is not via the Saddleback College Blackboard site. Instead, the course uses MyMathLab, a website maintained by the textbook's publisher. Intead of buying a hard copy of the textbook, I require each student to register for this site which includes the online textbook. The cost of registering is about 1/2 the price of the hard copy book. The orientation assignment will give you instructions on how to register. | 677.169 | 1 |
Hi, can anyone please help me with my math homework? I am not quite good at math and would be grateful if you could explain how to solve mathematical mcqs problems. I also would like to find out if there is a good website which can help me prepare well for my upcoming math exam. Thank you!
Can you please be more descriptive as to what sort of guidance you are expecting to get. Do you want to learn the fundamentals and solve your math questions by yourself or do you need a utility that would provide you a step-by-step solution for your math problems?
I must agree that Algebrator is a great thing and the best program of this kind you can get. I was amazed when after weeks of frustration I simply typed in binomial formula and that was the end of my problems with math. It's also so good that you can use the program for any level: I have been using it for several years now, I used it in Basic Math and in Pre Algebra also! Just try it and see it for yourself! | 677.169 | 1 |
This study examined the effectiveness of four types of review sessions given the day before a unit exam. Over a three week period, four Algebra 1 classes were taught the same unit by the principal investigator. At the end ...
Due to steady increases in students being diagnosed with disabilities, schools have transitioned to becoming more inclusive. As a result, children with disabilities are receiving more instruction within the general education ...
In this experiment two classes received instruction on integer operations. The first received instruction with the use of technology and the second class was instructed through a traditional approach. The study progressedBone Morphogenetic Protein 1 (BMP 1) functions in normal embryological development. The goal of this research was to obtain the sequence of salamander BMPl. Following sequence determination, an in situ probe for BMPJ ...
We used radio telemetry to determine the distribution and movements of paddlefish Polyadon spathula in the Allegheny Reservoir. Thirty-one adult and subadult paddlefish collected from spring congregation areas in the ...
This paper discusses a study that solicited data from teachers within two small city school districts. The study resulted from a five year federal education grant whose main objective was to provide intensive training and ...
This study explores the connection between student understanding of arithmetic and algebra through the evaluation of numeric expressions and the simplification of structurally comparable algebraic expressions. It isThis study examines the types of mistakes that students make solving multi-step linear equations. During this study, students completed a 15-problem test containing different types of multi-step linear equations appropriate ...
Understanding the concept of mathematical variables gives an opportunity to expand and work on high-level mathematics. This study examined college students' comprehension of variables as well as variable use in well-known ... | 677.169 | 1 |
Stock Status:In Stock Availability: Usually Ships in 5 to 10 Business Days
Product Code:9781741250107
About this book
Author Information
This book has been specifically designed to help Year 12 students thoroughly revise all topics in the HSC Mathematics course and prepare for class assessments, trial HSC and HSC exams. Together with the Year 11 Preliminary Revision & Exam Workbook, the whole senior Mathematics course is covered.
The book includes: - topics covering the complete HSC Mathematics course. - 200 pages of practice exercises, with topic tests for all chapters. - cross-references to relevant pages in the HSC Mathematics study guide. - topic tests for all chapters. - two sample examination papers. - answers to all questions. | 677.169 | 1 |
Re: Why is quiz part often harder than content in a maths textbook?
It seems a convention that maths teachers leave the hardest part in problems.
I am forced to agree. I have seen some where the problems are undoable even if you are familiar with the chapter | 677.169 | 1 |
This is a decision that should be made by students and their parents. It should be based on a realistic view of the student's skills and aspirations.
How many units of mathematics should you study?
This will usually be 2 to 8.
Which units of mathematics should you study?
It is recommended that students should choose a "Pathway" in VCE mathematics. Suggested Pathways are shown below.
Accelerated Mathematics Yr 10
What is this unitUnitsSemester 1: Acceleration Mathematics
Real and Complex Number Systems
Matrices
Sequences and Series
Variation
Semester 2: Acceleration Mathematics
Non-linear Graphs
Trigonometric Ratios
Non linear Relations and Equations
Data
What type of things will I do?
Work with numbers and surds
Substitute, Transpose and Solve Equations
Plot and sketch graphs
Use technology to help with learning
Application of Matrices
Display and Summaries data
Correlations and Regression of data
Applications of Sequences and Series such as financial arithmetic
Minimization in problems of time and distance
What can this lead to?
Specialist Mathematics 3 and 4
Mathematical Methods (CAS) 3 and 4
Further Mathematics 3 and 4
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Accelerated Mathematics
Year 11
Further Mathematics (3 and 4)
Year 12
Mathematical Methods CAS
Specialist Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Matrices
CAS
Foundation Mathematics Units 1-2
What's it all UnitsUnit 1: Foundation Mathematics
Space and Shape
Patterns in Number
Handling Data
Measurement and Design
Unit 2: Foundation Mathematics
Pattern and Number
Space and Shape
Measurement and Design
Handling Data
What type of things will I do?
Two dimensional plans
Diagrams incorporating scales
Practical problems involving decimals, fractions and percentages
Formulas and their use
Reading roads maps, time tables, flowcharts
Metric measurement problems
Recording and analyzing instrument readings
Ordering and weighing food items
Interpreting financial information
What can this lead to?
VCAL
VET
Apprenticeships
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 1 Mathematics
Year 11
Foundation Mathematics
Year 12
VCAL and VET
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Finance
General Mathematics (Further) Units 1-2
What's it all about?
This study is designed to provide access to worthwhile and challenging mathematical learning in a way which takes into account the needs and aspirations of a wide range of students Further Mathematics Units 3 and 4. A Computer Algebra System (CAS) will be used by students to assist them in their learning and understanding.
Assessment for satisfactory completion of Units 1 and 2 is by tests, analysis tasks, and students work during the year.
What will I learn?
Unit 1: General Mathematics (Further)
Number Theory
Number Patterns and Applications
Relations in Linear Equations
Linear Graphs
Unit 2: General Mathematics (Further)
Represent and Interpret types of Data
Describe and use Networks
Matrices and their Applications
What type of things will I do?
Work with schedules, time zones
Applications of Sequences and Series such as financial arithmetic
Formulate Equations
Plot and sketch graphs
Display and Summarise data
Correlations and Regression of data
Minimisation in problems of time and distance
Eulerian Paths and Circuits
Use a Computer Algebra System
What can this lead to?
Further Mathematics 3 and 4
VCAL
VET
Apprenticeships
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 1 Mathematics
Year 11
General Mathematics (Futher)
Year 12
Futher Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Matrices
CAS
General Mathematics (Methods) Units 1-2
What's it all about?
This course is designed for students intending to do tertiary studies that will involve complex and/or specialized mathematical calculations and skills. Students selecting these units should be able to manipulate algebraic expressions and solve equations. These skills are further developed in this course Mathematical Methods (CAS) and/or Specialist Maths Units 3 and 4. A Computer Algebra System will be used by students to assist them in their learning and understanding.
Assessment for satisfactory completion of Units 1 and 2 is by tests, analysis tasks, and student's work during the year.
What will I learn?
Unit 1: General Mathematics (Methods)
Matrices
Number Systems
2D and 3D Geometry
Linear Graphs and Relations
Unit 2: General Mathematics (Methods)
Trigonometry
Non-Linear Graphs and Relations
Co-ordinate Geometry
Vectors
What type of things will I do?
Work with numbers and surds
Substitute, Transpose and Solve Equations
Apply geometry to applications
Plot and sketch graphs
Use trig ratios, pythagoras and geometry to solve problems
Use technology to help with learning
Application of Matrices
What can this lead to?
Specialist Mathematics 3 and 4
Mathematical Methods(CAS) 3 and 4
Further Mathematics 3 and 4
VET
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 2 Mathematics
Year 11
General Mathematics (Methods)
Mathematical Methods
Year 12
Mathematical Methods(CAS)
Specialist Mathematics
Further Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Applications of Geometry
Interpreting Graphs
Matrices
CAS
Further Mathematics Units 3-4
What's it all about?
This course is designed for those students whose employment and/or further study aspirations do not require heavily algebra based mathematical skills.
Students will develop their mathematical knowledge and skills to be able to investigate, analyse and solve problems. They will be required to communicate mathematical ideas clearly and concisely.
A Computer Algebra System will be used by students to assist them in their learning and understanding.
Assessment is by satisfactory completion of three outcomes, judged by the student's results in Student Assessed Coursework (SACS).
What will I learn?
Unit 3: Further Mathematics
Networks and Decision Mathematics
Statistics
Unit 4: Further Mathematics
Number Patterns and Applications
Matrices
What type of things will I do?
Use statistical techniques
Model relationships between data
Matrix representation and arithmetic
Predicting ahead in situations involving number patterns
Correlations and Regression of data
Minimisation in problems of time and distance
Features of networks and their applications
Use a Computer Algebra System
What can this lead to?
Tertiary Education
Apprenticeship
General Employment
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 1 Mathematics
Year 11
General Mathematics (Further)
General Mathematics (Methods)
Year 12
Further Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Matrices
Networks
Mathematical Methods (CAS) Units 1-4
What's it all about?
Mathematical Methods consists of study in the areas of 'Co-ordinate Geometry', 'Trigonometric Functions', 'Calculus', 'Algebra', and 'Statistics and Probability'.
There are no prerequisites for entry to Mathematical Methods (CAS) Units 1 and 2. However, students attempting Mathematical Methods (CAS) are expected to have a sound background in number, algebra, function, and probability. Students wanting to do Mathematical Methods (CAS) Units 3 and 4 should have completed Mathematical Methods (CAS) Units 1 and 2 and General Mathematics (Methods) Units 1 and 2.
The appropriate use of CAS technology to support and develop the teaching and learning of mathematics, and in related assessments, is to be incorporated throughout the course.
Assessment is by satisfactory completion of three outcomes, judged by the student's results in Student Assessed Coursework (SACS) and exams.
What will I learn?
Unit 1: Maths Methods(CAS)
Functions and Graphs
Algebra
Probability
Rates of Change
Unit 2: Maths Methods(CAS)
Functions and Graphs
Algebra
Probability
Calculus
Unit 3: Maths Methods(CAS)
Functions and Graphs
Differential Calculus
Unit 4: Maths Methods(CAS)
Integral Calculus
Probability
What type of things will I do?
Problem solving
Substitute, Transpose and Solve Equations
Apply geometry to applications
Plot and sketch graphs
Calculate and Interpret Probabilities
Apply Algebra, Logarithmic and Trigonometric properties
Use CAS to assist with learning
What can this lead to?
Tertiary Education
Apprenticeship
General Employment
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 2 Mathematics
Acceleration Mathematics
Year 11
General Mathematics (Methods)
Mathematics Methods (CAS)
Year 12
Mathematics Methods (CAS)
Specialist Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Calculus
Geometry
Functions
Probability
CAS
Specialist Mathematical Units 3-4
What's it all about?
Specialist Mathematics consists of the following areas of study: 'Functions, relations and graphs' 'Algebra', 'Calculus', 'Vectors' and 'Mechanics'. Students are expected to be able to apply techniques, routines and processes, involving rational, real and complex arithmetic, algebraic manipulation, diagrams and geometric constructions, solving equations, graph sketching, differentiation and integration related to the areas of study, as applicable, both with and without the use of technology.
Enrolment in Specialist Mathematics Units 3 and 4 assumes a current enrolment in Mathematical Methods (CAS) Units 3 and 4.
Assessment is by satisfactory completion of three outcomes, judged by the student's results in Student Assessed Coursework (SACS). | 677.169 | 1 |
Arithmetic Sequences
In the first lesson of Sequences and Series, Dr. Eaton begins with Arithmetic Sequences. She starts with the general form of sequences and moves into the common difference between each term of the sequence. Then she will teach you the formula and equation for the nth term before finishing with arithmetic means. Four video examples round out this first lesson.
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Arithmetic Sequences
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. | 677.169 | 1 |
Product Description
Review
"This book will fill an interesting niche in a library collection...it should be used by browsing students interested in making sure that they are prepared for success in their graduate programs." Choice
"All the Mathematics You Missed...is a help for students going on to graduate school..Since many students beginning graduate school do not have the mathematical knowledge needed, All the Mathematics You Missed aims to fill in the gaps." Berkshire Eagle, Pittsfield, MA
"From the preface: 'The goal of this book is to give people at least a rough idea of many topics that beginning graduate students at the best graduate schools are assumed to know." Mathematical Reviews
"The writing is lucid mathematical exposition, at a level quite appropriate to beginning graduate students." The American Statistician
"Before classes began, I jump started my graduate career with the help of this book. Even though I didn't believe that I could have missed much math, it became clear that my belief was wrong during the first week of class. While proving a theorem, my professor asked if anyone remembered a previous result from calculus. While I did not remember it from my days as an undergraduate, I had read about the theorem and had even seen a sketch of the proof in Garrity's book...This will be one of the books that I keep with me as I continue as a graduate student. It has certainly helped me understand concepts that I have missed." Elizabeth D. Russell, Math Horizons
"Point set topology, complex analysis, differential forms, the curvature of surfaces, the axiom of choice, Lebesgue integration, Fourier analysis, algorithms, and differential equations.... I found these sections to be the high points of the book. They were a sound introduction to material that some but not all graduate students will need." Charles Ashbacher, School Science and Mathematics
Book Description
Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. This book will help readers to fill in the gaps in their preparation by presenting the basic points and a few key results of the most important undergraduate topics in mathematics: linear algebra, vector calculus, geometry, real analysis, algorithms, probability, set theory, and more. By emphasizing the intuitions behind the subject, the book makes it easy for students to quickly get a feel for the topics that they have missed and to prepare for more advanced courses.
This book has a very particular purpose: to recap some basic concepts from undergraduate mathematics so that you get the "big picture". In other words, for every math course you took as an undergrad, this book provides a good outline of the major ideas and how they fit together. But, it is only an outline; nothing more. If you actually missed out on some topic, or your knowledge of a subject is shaky, then this book won't help much. It will only help by providing a bibliography of some other references for that subject.
This book is meant to organize your undergraduate math knowledge, not to supplement it.
With that said, I'll mention a few words about the content of the book. It is quite well written and definitely extracts the essential ideas for your quick consumption. There are a few topics that I personally feel are missing, such as Gram-Schmidt and Jordan Canonical Forms for Linear Algebra, and UFDs and PIDs from Algebra. In general, it seemed like the book leaned a little more towards analysis than algebra, but the vast majority of important topics were indeed encapsulated in their synopsis.
Good for a very specific audience, but otherwise not wonderfully useful.
There's no doubt about it -- this book designed for people who want to learn some real math. It doesn't take, as the title and description might lead you to believe, a "Math for Engineers" approach.
Each chapter covers, in the span of 10 or 15 pages, what would normally be an entire semester's worth of material, and as a result, is quite dense -- there are alot of ideas crammed onto each page. But unlike traditional advanced math books (which are notoriously dense) the focus is more on developing intuitions than on long strings of equations.
An important strength is that every chapter ends with suggestions on textbooks in that chapter's subject. This turns out to be quite helpful, since one can't reasonably expect to learn everything important about any of these subjects from a brief chapter in any book.
I can envision three main ways in which this book might be useful: First, in combination with one or more of the books in listed in the bibliography for learning a new subject. Second, on its own for review of topics you've seen before. Third, as a reference for "basic" definitions and theorems, as in: "What's a Hilbert space again?"
Overall, this will be a good book to have around, but not a substitute for real study.
I used this book for an opposite purpose to the one the author intended. For me it served to review all the math I *had* learned long ago in school (both undergraduate and graduate), but was starting to forget. The author's informal style and rapid-fire delivery were just right for these topics. The subjects I had truly missed, mainly the more abstract parts of algebra and geometry, I found difficult to follow, though I did come away with some feeling for them. This is not a perfect book. The informal style extends to numerous typos in equations, and modern computer-oriented approaches get short shrift. Nevertheless, I found it a unique resource and a pleasure to read. | 677.169 | 1 |
Keith Devlin
Consortium for Mathematics and Its Applications
Topic: math Age Level: advanced | 677.169 | 1 |
Elementary student-focused text addresses individual learning styles through the use of a complete study system that starts with a learning styles inventory and presents targeted learning strategies designed to guide students toward success in this and future college-level courses.Students who approach math with trepidation will find that Elementary Algebra, Second Edition, builds competence and confidence. The study system, introduced at the outset and used consistently throughout the text, transforms the student experience by applying time-tested s... MOREtrategies to the study of mathematics. Learning strategies dovetail nicely into the overall system and build on individual learning styles by addressing students' unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the 'big picture' of algebra. While using Elementary Algebra, Second Edition, you will find that the text focuses on building competence and confidence. The authors present the concepts, show how to do the math, and explain the reasoning behind it in a language you can understand. The text ties concepts together using the Algebra Pyramid, which will help you see the big picture of algebra. The skills Carson presents through both the Learning Strategy boxes and the Study System, introduced in the Preface and incorporated throughout the text, will not only enhance your elementary algebra experience but will also help you succeed in future college courses. Book jacket. | 677.169 | 1 |
The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses visit MIT OpenCourseWare at ocw.mit.edu.
PROFESSOR STRANG: Finally we get to positive definite matrices. I've used the word and now it's time to pin it down. And so this would be my thank you for staying with it while we do this important preliminary stuff about linear algebra. So starting the next lecture we'll really make a big start on engineering applications. But these matrices are going to be the key to everything. And I'll call these matrices K and positive definite, I will only use that word about a symmetric matrix. So the matrix is already symmetric and that means it has real eigenvalues and many other important properties, orthogonal eigenvectors. And now we're asking for more. And it's that extra bit that is terrific in all kinds of applications. So if I can do this bit of linear algebra.
So what's coming then, my review session this afternoon at four, I'm very happy that we've got, I think, the best MATLAB problem ever invented in 18.085 anyway. So that should get onto the website probably by tomorrow. And Peter Buchak is like the MATLAB person. So his review sessions are Friday at noon. And I just saw him and suggested Friday at noon he might as well just stay in here. And knowing that that isn't maybe a good hour for everybody. So you could see him also outside of that hour. But that's the hour he will be ready for all kinds of questions about MATLAB or about the homeworks. Actually you'll be probably thinking more also about the homework questions on this topic.
Ready for positive definite? You said yes, right? And you have a hint about these things. So we have a symmetric matrix and the beauty is that it brings together all of linear algebra. Including elimination, that's when we see pivots. Including determinants which are closely related to the pivots. And what do I mean by upper left? I mean that if I have a three by three symmetric matrix and I want to test it for positive definite, and I guess actually this would be the easiest test if I had a tiny matrix, three by three, and I had numbers then this would be a good test. The determinants, by upper left determinants I mean that one by one determinant. So just that first number has to be positive. Then the two by two determinant, that times that minus that times that has to be positive. Oh I've already been saying that. Let me just put in some letters. So a has to be positive. This is symmetric, so a times c has to be bigger than b squared. So that will tell us. And then for two by two we finish. For three by three we would also require the three by three determinant to be positive.
But already here you're seeing one point about a positive definite matrix. Its diagonal will have to be positive. And somehow its diagonal has to be not just above zero, but somehow it has to defeat b squared. So the diagonal has to be somehow more positive than whatever negative stuff might come from off the diagonal. That's why I would need a*c > b squared. So both of those will be positive and their product has to be bigger than the other guy.
And then finally, a third test is that all the eigenvalues are positive. And of course if I give you a three by three matrix, that's probably not the easiest test since you'd have to find the eigenvalues. Much easier to find the determinants or the pivots. Actually, just while I'm at it, so the first pivot of course is a itself. No difficulty there. The second pivot turns out to be the ratio of a*c - b squared to a. So the connection between pivots and determinants is just really close. Pivots are ratios of determinants if you work it out. The second pivot, maybe I would call that d_2, is the ratio of a*c - b squared over a. In other words it's (c - b squared)/a. Determinants are positive and vice versa. Then it's fantastic that the eigenvalues come into the picture.
So those are three ways, three important properties of a positive definite matrix. But I'd like to make the definition something different. Now I'm coming to the meaning. If I think of those as the tests, that's done. Now the meaning of positive definite. The meaning of positive definite and the applications are closely related to looking for a minimum. And so what I'm going to bring in here, so it's symmetric. Now for a symmetric matrix I want to introduce the energy. So this is the reason why it has so many applications and such important physical meaning is that what I'm about to introduce, which is a function of x, and here it is, it's x transpose times A, not A, I'm sticking with K for my matrix, times x. I think of that as some f(x).
And let's just see what it would be if the matrix was two by two, [a, b; b, c]. Suppose that's my matrix. We want to get a handle on what, this is the first time I've ever written something that has x's times x's. So it's going to be quadratic. They're going to be x's times x's. And x is a general vector of the right size so it's got components x_1, x_2. And there it's transpose, so it's a row. And now I put in the [a, b; b, c]. And then I put in x again. This is going to give me a very nice, simple, important expression. Depending on x_1 and x_2. Now what is, can we do that multiplication? Maybe above I'll do the multiplication of this pair and then I have the other guy to bring in. So here, that would be ax_1+bx_2. And this would be bx_1+cx_2. So that's the first, that's this times this. What am I going to get? What shape, what size is this result going to be? This K is n by n. x is a column vector. n by one. x transpose, what's the shape of x transpose? One by n? So what's the total result? One by one. Just a number. Just a function. It's a number. But it depends on the x's and the matrix inside.
Can we do it now? So I've got this to multiply by this. Do you see an x_1 squared showing up? Yes, from there times there. And what's it multiplied by? The a. The first term is this times the ax_1 is a(x_1 squared). So that's our first quadratic. Now there'd be an x_1, x_2. Let me leave that for a minute and find the x_2 squared because it's easy. So where am I going to get x_2 squared? I'm going to get that from x_2, second guy here times second guy here. There's a c(x_2 squared).
So you're seeing already where the diagonal shows up. The diagonal a, c, whatever is multiplying the perfect squares. And it'll be the off-diagonal that multiplies the x_1, x_2. We might call those the crossterms. And what do we get for that then? We have x_1 times this guy. So that's a crossterm. bx_1*x_2, right? And here's another one coming from x_2 times this guy. And what's that one? It's also bx_1*x_2. So x_1, multiply that, x_2 multiply that, and so what do we have for this crossterm here? Two of them. 2bx_1*x_2. In other words, that b and that b came together in the 2bx_1*x_2. So here's my energy. Can I just loosely call it energy? And then as we get to applications we'll see why.
So I'm interested in that because it has important meaning. Well, so now I'm ready to define positive definite matrices. So I'll call that number four. But I'm going to give it a big star. Even more because it's the sort of key. So the test will be, you can probably guess it, I look at this expression, that x transpose Ax. And if it's a positive definite matrix and this represents energy, the key will be that this should be positive. This one should be positive for all x's. Well, with one exception, of course. All x's except, which vector is it? x=0 would just give me-- See, I put K. My default for a matrix, but should be, it's K. Except x=0, except the zero vector. Of course. If x_1 and x_2 are both zero.
Now that looks a little maybe less straightforward, I would say, because it's a statement about this is true for all x_1 and x_2. And we better do some examples and draw a picture. Let me draw a picture right away. So here's x_1 direction. Here's x_2 direction. And here is the x transpose Ax, my function. So this depends on two variables. So it's going to be a sort of a surface if I draw it. Now, what point do we absolutely know? And I put A again. I am so sorry. Well, we know one point. It's there whatever that matrix might be. It's there. Zero, right? You just told me that if both x's are zero then we automatically get zero.
Now what do you think the shape of this curve, the shape of this graph is going to look like? The point is, if we're positive definite now. So I'm drawing the picture for positive definite. So my definition is that the energy goes up. It's positive, right? When I leave, when I move away from that point I go upwards. That point will be a minimum. Let me just draw it roughly. So it sort of goes up like this. These cheap 2-D boards and I've got a three-dimensional picture here. But you see it somehow? What word or what's your visualization? It has a minimum there. That's why minimization, which was like, the core problem in calculus, is here now. But for functions of two x's or n x's. We're up the dimension over the basic minimum problem of calculus. It's sort of like a parabola It's cross-sections cutting down through the thing would be just parabolas because of the x squared.
I'm going to call this a bowl. It's a short word. Do you see it? It opens up. That's the key point, that it opens upward. And let's do some examples. Tell me some positive definite. So positive definite and then let me here put some not positive definite cases. Tell me a matrix. Well, what's the easiest, first matrix that occurs to you as a positive definite matrix? The identity. That passes all our tests, its eigenvalues are one, its pivots are one, the determinants are one. And the function is x_1 squared plus x_2 squared with no b in it. It's just a perfect bowl, perfectly symmetric, the way it would come off a potter's wheel.
Now let me take one that's maybe not so, let me put a nine there. So I'm off to a reasonable start. I have an x_1 squared and a nine x_2 squared. And now I want to ask you, what could I put in there that would leave it positive definite? Well, give me a couple of possibilities. What's a nice, not too big now, that's the thing. Two. Two would be fine. So if I had a two there and a two there I would have a 4x_1*x_2 and it would, like, this, instead of being a circle, which it was for the identity, the plane there would cut out a ellipse instead. But it would be a good ellipse. Because we're doing squares, we're really, the Greeks understood these second degree things and they would have known this would have been an ellipse.
How high can I go with that two or where do I have to stop? Where would I have to, if I wanted to change the two, let me just focus on that one, suppose I wanted to change it. First of all, give me one that's, how about the borderline. Three would be the borderline. Why's that? Because at three we have nine minus nine for the determinant. So the determinant is zero. Of course it passed the first test. One by one was ok. But two by two was not, was at the borderline. What else should I think? Oh, that's a very interesting case. The borderline. You know, it almost makes it. But can you tell me the eigenvalues of that matrix? Don't do any quadratic equations.
How do I know, what's one eigenvalue of a matrix? You made it singular, right? You made that matrix singular. Determinant zero. So one of its eigenvalues is zero. And the other one is visible by looking at the trace. I just quickly mentioned that if I add the diagonal, I get the same answer as if I add the two eigenvalues. So that other eigenvalue must be ten. And this is entirely typical, that ten and zero, the extreme eigenvalues, lambda_max and lambda_min, are bigger than, these diagonal guys are inside. They're inside, between zero and ten and it's these terms that enter somehow and gave us an eigenvalue of ten and an eigenvalue of zero.
I guess I'm tempted to try to draw that figure. Just to get a feeling of what's with that one. It always helps to get the borderline case. So what's with this one? Let me see what my quadratic would be. Can I just change it up here? Rather than rewriting it. So I'm going to, I'll put it up here. So I have to change that four to what? Now that I'm looking at this matrix. That four is now a six. Six. This is my guy for this one. Which is not positive definite.
Let me tell you right away the word that I would use for this one. I would call it positive semi-definite because it's almost there, but not quite. So semi-definite allows the matrix to be singular. So semi-definite, maybe I'll do it in green what semi-definite would be. Semi-def would be eigenvalues greater than or equal zero. Determinants greater than or equal zero. Pivots greater than zero if they're there or then we run out of pivots. You could say greater than or equal to zero then. And energy, greater than or equal to zero for semi-definite. And when would the energy, what x's, what would be the like, you could say the ground states or something, what x's, so greater than or equal to zero, emphasize that possibility of equal in the semi-definite case.
Suppose I have a semi-definite matrix, yeah, I've got one. But it's singular. So that means a singular matrix takes some vector x to zero. Right? If my matrix is actually singular, then there'll be an x where Kx is zero. And then, of course, multiplying by x transpose, I'm still at zero. So the x's, the zero energy guys, this is straightforward, the zero energy guys, the ones where x transpose Kx is zero, will happen when Kx is zero. If Kx is zero, and we'll see it in that example.
Let's see it in that example. What's the x for which, I could say in the null space, what's the vector x that that matrix kills? , right? The vector . gives me . That's the vector that, so I get 3-3, the zero, 9-9, the zero. So I believe that this thing will be-- Is it zero at three, minus one? I think that it has to be, right? If I take x_1 to be three and x_2 to be minus one, I think I've got zero energy here. Do I? x_1 squared will be at the nine and nine x_2 squared will be nine more. And what will be this 6x_1*x_2? What will that come out for this x_1 and x_2? Minus 18. Had to, right? So I'd get nine from there, nine from there, minus 18, zero.
So the graph for this positive semi-definite will look a bit like this. There'll be a direction in which it doesn't climb. It doesn't go below the base, right? It's never negative. This is now the semi-definite picture. But it can run along the base. And it will for the vector x_1=3, x_2=-1, I don't know where that is, one, two, three, and then maybe minus one. Along some line here the graph doesn't go up. It's sitting, can you imagine that sitting in the base? I'm not Rembrandt here, but in the other direction it goes up. Oh, the hell with that one. Do you see, sort of? It's like a trough, would you say? I mean, it's like a, you know, a bit of a drainpipe or something. It's running along the ground, along this direction and in the other directions it does go up. So it's shaped like this with the base not climbing. Whereas here, there's no bad direction. Climbs every way you go. So that's positive definite and that's positive semi-definite.
Well suppose I push it a little further. Let me make a place here for a matrix that isn't even positive semi-definite. Now it's just going to go down somewhere. I'll start again with one and nine and tell me what to put in now. So this is going to be a case where the off-diagonal is too big, it wins. And prevents positive definite. So what number would you like here? Five? Five is certainly plenty. So now I have [1, 5; 5, 9]. Let me take a little space on a board just to show you. Sorry about that. So I'm going to do the [1, 5; 5, 9] just because they're all important, but then we're coming back to positive definite. So if it's [1, 5; 5, 9] and I do that usual x, x transpose Kx and I do the multiplication out, I see the one x_1 squared and I see the nine x_2 squareds. And how many x_1*x_2's do I see? Five from there, five from there, ten. And I believe that can be negative. The fact of having all nice plus signs is not going to help it because we can choose, as we already did, x_1 to be like a negative number and x_2 to be a positive. And we can get this guy to be negative and make it, in this case we can make it defeat these positive parts.
What choice would do it? Let me take x_1 to be minus one and tell me an x_2 that's good enough to show that this thing is not positive definite or even semi-definite, it goes downhill. Take x_2 equal? What do you say? 1/2? Yeah, I don't want too big an x_2 because if I have too big an x_2, then this'll be important. Does 1/2 do it? So I've got 1/4, that's positive, but not very. 9/4, so I'm up to 10/4, but this guy is what? Ten and the minus is minus five. Yeah. So that absolutely goes, at this one I come out less than zero. And I might as well complete.
So this is the case where I would call it indefinite. Indefinite. It goes up like if x_2 is zero, then it's just got x_1 squared, that's up. If x_1 is zero, it's only got x_2 squared, that's up. But there are other directions where it goes downhill. So it goes either up, it goes both up in some ways and down in others. And what kind of a graph, what kind of a surface would I now have for x transpose for this x transpose, this indefinite guy? So up in some ways and down in others. This gets really hard to draw. I believe that if you ride horses you have an edge on visualizing this. So it's called, what kind of a point's it called? Saddle point, it's called a saddle point. So what's a saddle point? That's not bad, right? So this is a direction where it went up. This is a direction where it went down. And so it sort of fills in somehow.
Or maybe, if you don't, I mean, who rides horses now? Actually maybe something we do do is drive over mountains. So the path, if the road is sort of well-chosen, the road will go, it'll look for the, this would be-- Yeah, here's our road. We would do as little climbing as possible. The mountain would go like this, sort of. So this would be like, the bottom part looking along the peaks of the mountains. But it's the top part looking along the driving direction. So driving, it's a maximum, but in the mountain range direction it's a minimum. So it's a saddle point. So that's what you get from a typical symmetric matrix. And if it was minus five it would still be the same saddle point, would still be 9-25, it would still be negative and a saddle.
Positive guys are our thing. Alright. So now back to positive definite. With these four tests and then the discussion of semi-definite. Very key, that energy. Let me just look ahead a moment. Most physical problems, many, many physical problems, you have an option. Either you solve some equations, either you find the solution from our equations, Ku=f, typically. Matrix equation or differential equation. Or there's another option of minimizing some function. Some energy. And it gives the same equations. So this minimizing energy will be a second way to describe the applications.
Now can I get a number five? There's an important number five and then you know really all you need to know about symmetric matrices. This gives me, about positive definite matrices, this gives me a chance to recap. So I'm going to put down a number five. Because this is where the matrices come from. Really important. And it's where they'll come from in all these applications that chapter two is going to be all about, that we're going to start. So they come, these positive definite matrices, so this is another way to, it's a test for positive definite matrices and it's, actually, it's where they come from. So here's a positive definite matrix. They come from A transpose A. A fundamental message is that if I have just an average matrix, possibly rectangular, could be a square but not symmetric, then sooner or later, in fact usually sooner, you end up looking at A transpose A. We've seen that already. And we already know that A transpose A is square, we already know it's symmetric and now we're going to know that it's positive definite. So matrices like A transpose A are positive definite or possibly semi-definite. There's that possibility. If A was the zero matrix, of course, we would just get the zero matrix which would be only semi-definite, or other ways to get a semi-definite.
So I'm saying that if K, if I have a matrix, any matrix, and I form A transpose A, I get a positive definite matrix or maybe just semi-definite, but not indefinite. Can we see why? Why is this positive definite or semi-? So that's my question. And the answer is really worth, it's just neat and worth seeing. So do I want to look at the pivots of A transpose A? No. They're something, but whatever they are, I can't really follow those well. Or the eigenvalues very well, or the determinants. None of those come out nicely. But the real guy works perfectly. So look at x transpose Kx. So I'm just doing, following my instinct here.
So if K is A transpose A, my claim is, what am I saying then about this energy? What is it that I want to discover and understand? Why it's positive. Why does taking any matrix, multiplying by its transpose produce something that's positive? Can you see any reason why that quantity, which looks kind of messy, I just want to look at it the right way to see why that should be positive, that should come out positive. So I'm not going to get into numbers, I'm not going to get into diagonals and off-diagonals. I'm just going to do one thing to understand that particular combination, x transpose A transpose Ax. What shall I do? Anybody see what I might do? Yeah, you're seeing here if you look at it again, what are you seeing here? Tell me again. If I take Ax together, then what's the other half? It's the transpose of Ax. So I just want to write that as, I just want to think of it that way, as Ax. And here's the transpose of Ax. Right? Because transposes of Ax, so transpose guys in the opposite order, and the multiplication--
This is the great. I call these proof by parenthesis because I'm just putting parentheses in the right place, but the key law of matrix multiplication is that, that I can put (AB)C is the same as A(BC). That rule, which is just multiply it out and you see that parentheses are not needed because if you keep them in the right order you can do this first, or you can do this first. Same answer. What do I learn from that? What was the point? This is some vector, I don't know especially what it is times its transpose. So that's the length squared. What's the key fact about that? That it is never negative. It's always greater than zero or possibly equal.
When does that quantity equal zero? When Ax is zero. When Ax is zero. Because this is a vector. That's the same vector transposed. And everybody's got that picture. When I take any y transpose y, I get y_1 squared plus y_2 squared through y_n squared. And I get a positive answer except if the vector is zero. So it's zero when Ax is zero. So that's going to be the key. If I pick any matrix A, and I can just take an example, but chapter, the applications are just going to be full of examples. Where the problem begins with a matrix A and then A transpose shows up and it's the combination A transpose A that we work with. And we're just learning that it's positive definite.
Unless, shall I just hang on since I've got here, I have to say when is it, have to get these two possibilities. Positive definite or only semi-definite. So what's the key to that borderline question? This thing will be only semi-definite if there's a solution to Ax=0. If there is an x, well, there's always the zero vector. Zero vector I can't expect to be positive. So I'm looking for if there's an x so that Ax is zero but x is not zero, then I'll only be semi-definite. That's the test. If there is a solution to Ax=0.
When we see applications that'll mean there's a displacement with no stretching. We might have a line of springs and when could the line of springs displace with no stretching? When it's free-free, right? If I have a line of springs and no supports at the ends, then that would be the case where it could shift over by the vector. So that would be the case where the matrix is only singular. We know that. The matrix is now positive semi-definite. We just learned that. So the free-free matrix, like B, both ends free, or C. So our answer is going to be that K and T are positive definite. And our other two guys, the singular ones, of course, just don't make it. B at both ends, the free-free line of springs, it can shift without stretching. Since Ax will measure the stretching when it just shifts rigid motion, the Ax is zero and we see only positive definite. And also C, the circular one. There it can displace with no stretching because it can just turn in the circle. So these guys will be only positive semi-definite.
Maybe I better say this another way. When is this positive definite? Can I use just a different sentence to describe this possibility? This is positive definite provided, so what I'm going to write now is to remove this possibility and get positive definite. This is positive definite provided, now, I could say it this way. The A has independent columns. So I just needed to give you another way of looking at this Ax=0 question. If A has independent columns, what does that mean? That means that the only solution to Ax=0 is the zero solution. In other words, it means that this thing works perfectly and gives me positive. When A has independent columns.
Let's just remember our K, T, B, C. So here's a matrix, so let me take the T matrix, that's this one, this guy. And then the third column is . Those three columns are independent. They point off. They don't lie in a plane. They point off in three different directions. And then there are no solutions to, no x's that's go Kx=0. So that would be a case of independent columns. Let me make a case of dependent columns. So and I'm going to make it B now. Now the columns of that matrix are dependent. There's a combination of them that give zero. They all lie in the same plane. There's a solution to that matrix times x equal zero. What combination of those columns shows me that they are dependent? That some combination of those three columns, some amount of this plus some amount of this plus some amount of that column gives me the zero vector. You see the combination. What should I take? again. No surprise. That's the vector that we know is in the everything shifting the same amount, nothing stretching.
Talking fast here about positive definite matrices. This is the key. Let's just ask a few questions about positive definite matrices as a way to practice. Suppose I had one. Positive definite. What about its inverse? Is that positive definite or not? So I've got a positive definite one, it's not singular, it's got positive eigenvalues, everything else. It's inverse will be symmetric, so I'm allowed to think about it. Will it be positive definite? What do you think? Well, you've got a whole bunch of tests to sort of mentally run through. Pivots of the inverse, you don't want to touch that stuff. Determinants, no. What about eigenvalues? What would be the eigenvalues if I have this positive definite symmetric matrix, its eigenvalues are one, four, five. What can you tell me about the eigenvalues of the inverse matrix? They're the inverses. So those three eigenvalues are? 1, 1/4, 1/5, what's the conclusion here? It is positive definite. Those are all positive, it is positive definite. So if I invert a positive definite matrix, I'm still positive definite.
All the tests would have to pass. It's just I'm looking each time for the easiest test. Let me look now, for the easiest test on K_1+K_2. Suppose that's positive definite and that's positive definite. What if I add them? What do you think? Well, we hope so. But we have to say which of my one, two, three, four, five would be a good way to see it. Would be a good way to see it. Good question. Four? We certainly don't want to touch pivots and now we don't want to touch eigenvalues either. Of course, if number four works, others will also work. The eigenvalues will come out positive. But not too easy to say what they are. Let's try test number four. So K_1. What's the test? So test number four tells us that this part, x transpose K_1*x, that that part is positive, right? That that part is positive. If we know that's positive definite. Now, about K_2 we also know that for every x, you see it's for every x, that helps, don't let me put x_2 there, for every x this will be positive.
And now what's the step I want to take? To get some information on the matrix K_1+K_2. I should add. If I add these guys, you see that it just, then I can write that as, I can write that this way. And what have I learned? I've learned that that's positive, even greater than, except for the zero vector. Because this was greater than, this is greater than. If I add two positive numbers, the energies are positive and the energies just add. The energies just add. So that definition four was the good way, just nice, easy way to see that if I have a couple of positive definite matrices, a couple of positive energies, I'm really coupling the two systems. This is associated somehow. I've got two systems, I'm putting them together and the energy is just even more positive. It's more positive either of these guys because I'm adding.
As I'm speaking here, will you allow me to try test number five, this A transpose A business? Suppose K_1 was A transpose A. If it's positive definite, it will. Be And suppose K_2 is B transpose B. If it's positive definite, it will be. Now I would like to write the sum somehow as, in this something transpose something. And I just do it now because I think it's like, you won't perhaps have thought of this way to do it. Watch. Suppose I create the matrix [A; B]. That'll be my new matrix. Say, call it C. Am I allowed to do that? I mean, that creates a matrix? These A and B, they had the same number of columns, n. So I can put one over the other and I still have something with n columns. So that's my new matrix C. And now I want C transpose. By the way, I'd call that a block matrix. You know, instead of numbers, it's got two blocks in there. Block matrices are really handy.
Now what's the transpose of that block matrix? You just have faith, just have faith with blocks. It's just like numbers. If I had a matrix [1; 5] then I'd get a row one, five. But what do you think? This is worth thinking about even after class. What would be, if this C matrix is this block A above B, what do you think for C transpose? A transpose, B transpose side by side. Just put in numbers and you'd see it. And now I'm going to take C transpose times C. I'm calling it C now instead of A because I've used the A in the first guy and I've used B in the second one and now I'm ready for C. How do you multiply block matrices? Again, you just have faith. What do you think? Tell me the answer. A transpose, I multiply that by that just as if they were numbers. And I add that times that just as if they were numbers. And what do I have? I've got K_1+K_2. So I've written K_1, this is K_1+K_2 and this is in my form C transpose C that I was looking for, that number five was looking for. So it's done it. It's done it. The fact of getting A, K_1 in this form, K_2 in this form. And I just made a block matrix and I got K_1+K_2. That's not a big deal in itself, but block matrices are really handy. It's good to take that step with matrices. Think of, possibly, the entries as coming in blocks and not just one at a time.
Well, thank you, okay. I swear Friday we'll start applications in all kinds of engineering problems and you'll have new applications | 677.169 | 1 |
It's a pilot project, so it remains to be seen if this approach makes a difference in improving the pass rates for students in lower-level math courses like College Algebra, which have been at around 60… | 677.169 | 1 |
Summary
4-color hardback text w/complete text-specific instructor and student print/media supplement package AMATYC/NCTM Standards of Content and Pedagogy integrated in Exercise Sets, Sourced-Data Applications (students are also asked to generate and interpret data), Scientific and Graphing Calculator Explorations Boxes, Mental Math exercises, Conceptual and Writing exercises, geometric concepts, Group Activities, Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews 6 step Problem-Solving Approach introduced in Chapter 2 and reinforced throughout the text in applications and exercises helps students tackle a wide range of problems Early and intuitive introduction to the concept of graphing reinforced with bar charts, line graphs, calculator screens, application illustrations and exercise sets. Emphasis on the notion of paired data in Chapters 1 and 2 leads naturally to the concepts of ordered pair and the rectangular coordinate system introduced in Chapter 3. Graphing and concepts of graphing linear equations such as slope and intercepts reinforced through exercise sets in subsequent chapters, preparing students for equations of lines in Chapter 7
Table of Contents
Preface
ix
Review of Real Numbers
1
(72)
Symbols and Sets of Numbers
2
(9)
Fractions
11
(8)
Exponents and Order of Operations
19
(6)
Introduction to Variable Expressions and Equations
25
(5)
Adding Real Numbers
30
(6)
Subtracting Real Numbers
36
(5)
Multiplying and Dividing Real Numbers
41
(7)
Properties of Real Numbers
48
(7)
Reading Graphs
55
(18)
Group Activity: Creating and Interpreting Graphs
62
(1)
Highlights
63
(4)
Review
67
(3)
Test
70
(3)
Equations, Inequalities, And Problem Solving
73
(90)
Simplifying Algebraic Expressions
74
(7)
The Addition Property of Equality
81
(7)
The Multiplication Property of Equality
88
(6)
Solving Linear Equations
94
(10)
An Introduction to Problem Solving
104
(8)
Formulas and Problem Solving
112
(10)
Percent and Problem Solving
122
(9)
Further Problem Solving
131
(8)
Solving Linear Inequalities
139
(24)
Group Activity: Calculating Price Per Unit
149
(1)
Highlights
150
(6)
Review
156
(3)
Test
159
(1)
Cumulative Review
160
(3)
Graphing
163
(64)
The Rectangular Coordinate System
164
(11)
Graphing Linear Equations
175
(10)
Intercepts
185
(10)
Slope
195
(13)
Graphing Linear Inequalities
208
(19)
Group Activity: Financial Analysis
217
(1)
Highlights
218
(4)
Review
222
(2)
Test
224
(1)
Cumulative Review
225
(2)
Exponents And Polynomials
227
(54)
Exponents
228
(10)
Adding and Subtracting Polynomials
238
(8)
Multiplying Polynomials
246
(6)
Special Products
252
(5)
Negative Exponents and Scientific Notation
257
(9)
Division of Polynomials
266
(15)
Group Activity: Making Predictions Based on Historical Data
273
(1)
Highlights
274
(2)
Review
276
(2)
Test
278
(1)
Cumulative Review
279
(2)
Factoring Polynomials
281
(62)
The Greatest Common Factor and Factoring by Grouping
282
(7)
Factoring Trinomials of the Form x2 + bx + c
289
(6)
Factoring Trinomials of the Form ax2 + bx + c
295
(9)
Factoring Binomials
304
(5)
Choosing a Factoring Strategy
309
(5)
Solving Quadratic Equations by Factoring
314
(10)
Quadratic Equations and Problem Solving
324
(19)
Group Activity: Choosing Among Building Options
333
(1)
Highlights
334
(3)
Review
337
(2)
Test
339
(1)
Cumulative Review
340
(3)
Rational Expressions
343
(68)
Simplifying Rational Expressions
344
(7)
Multiplying and Dividing Rational Expressions
351
(5)
Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator
356
(7)
Adding and Subtracting Rational Expressions with Unlike Denominators
363
(6)
Simplifying Complex Fractions
369
(7)
Solving Equations Containing Rational Expressions
376
(7)
Ratio and Proportion
383
(7)
Rational Equations and Problem Solving
390
(21)
Group Activity: Comparing Formulas for Doses of Medication
399
(1)
Highlights
400
(5)
Review
405
(2)
Test
407
(1)
Cumulative Review
408
(3)
Further Graphing
411
(42)
The Slope-Intercept Form
412
(6)
The Point-Slope Form
418
(7)
Graphing Nonlinear Equations
425
(8)
Functions
433
(20)
Group Activity: Matching Descriptions of Linear Data to Their Equations and Graphs | 677.169 | 1 |
Mathematics
Mathematics Lab
10
11
12
2 Semesters / 1 Credit(s)
This course is required for students who did not meet the State standard on the End of Course Assessment for Algebra I. Curriculum will focus on strengthening basic mathematical skills, conceptualization and communication of mathematical ideas and reinforcement of skills necessary for success in Algebra I.
NOTE: THIS COURSE COUNTS AS AN ELECTIVE TOWARD GRADUATION BUT DOES NOT FULFILL MATHEMATICS GRADUATION REQUIREMENTS.
Pre-Algebra
9
2 Semesters / 2 Credit(s)
This course is designed to help students who have previously struggled in mathematics. It is meant to help freshmen students prepare for taking and succeeding in Algebra I the following year. Topics covered in the course include simplifying expressions, solving equations, working with fractions and decimals, proportions, graphing equations, spatial thinking, and introductory geometry. Students will also be exposed to the basics of many Algebra 1 topics. This course counts as a mathematics course for the General Diploma ONLY.
Algebra I (1 & 2)
10
11
1 Semester / 2 Credit(s)
This course is designed for those students who did not receive a C- average or better in Algebra I or did not pass the ISTEP+ Algebra I Graduation Exam. This course will meet everyday.
Algebra I
9
10
2 Semesters / 2 Credit(s)
Algebra Idevelops traditional principles such as: solving equations and inequalities, performing operations with real numbers and polynomials, working with integer exponents and factoring polynomials, doing exercises with relations and functions, graphing linear equations and inequalities, graphing and algebraically solving linear systems, solving quadratic equations, and introducing topics from probability and statistics. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
Honors Algebra I
9
2 Semesters / 2 Credit(s)
The same topics as in Algebra I are covered with more emphasis on problem solving and critical thinking skills in order to challenge the mathematically talented student. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
Prerequisite: A High School Placement test Basic Skills Math Grade Equivalent of 10.0 and higher along with a Reading and Language Grade Equivalent above 8.0
Geometry
9
10
2 Semesters / 2 Credit(s)
The purpose of Geometry is to use logical thought processes to develop spatial skills. Students work with figures in one, two- and three-dimensional Euclidean space. The interrelationships of the properties of figures are studied through visualization, using computer drawing programs and constructions, as well as through formal proof and algebraic applications. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
This Geometry course is designed for those students who did not receive a C- average or better in Geometry. This course will meet everyday.
Honors Geometry
9
10
2 Semesters / 2 Credit(s)
This course covers the same topics as Geometry, but with greater emphasis on complex direct deductive proof and indirect proof and on utilization of more advanced algebraic techniques. Content is extended to include topics such as analytic geometry and the interrelationships of inscribed polyhedra. A GRAPHING CALCULATOR IS REQUIRED.
Prerequisite: B- or better in Honors Algebra I; Freshman enrollment is based on a math readiness test given in June.
Algebra II
11
12
2 Semesters / 2 Credit(s)
This course further develops the topics learned in Algebra I with extensive work on learning to graph equations and inequalities in the Cartesian coordinate system. Topics include: relations and functions, systems of equations and inequalities, conic sections, polynomials, algebraic fractions, logarithmic and exponential functions, sequences and series, and counting principles and probability. A GRAPHING CALCULATOR IS REQUIRED.
This course expands and develops the topics learned in Honors Algebra I. Content areas include the topics listed for Algebra II with greater emphasis on preparation for upper level mathematics content. The course is required for students who plan to take AP Calculus, and it is recommended that this course be taken at the same time as Honors Geometry unless Honors Geometry was taken as a freshman. A GRAPHING CALCULATOR IS REQUIRED.
Prerequisite: B or better in Honors Geometry; Freshman enrollment is based on a math readiness test given in June.
Pre-Calculus
11
12
1 Semester / 1 Credit(s)
This course continues the foundation concepts necessary for college level mathematics. Topics studied include: relations and functions, polynomials, rational and algebraic functions, logarithmic and exponential functions, analytic geometry, and data analysis. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
Knowledge of trigonometry is necessary for successful performance in college level mathematics. Topics covered in this course include: trigonometry in triangles, trigonometric functions, trigonometric identities and equations, and polar coordinates. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
This course covers the same topics as Pre-Calculus/Trigonometry listed above. Greater emphasis is placed on applications and developing the depth of understanding and skills necessary for success in AP Calculus. This course is required for students who plan to take AP Calculus. A GRAPHING CALCULATOR IS REQUIRED.
This course is intended for students who have a thorough knowledge of college preparatory mathematics. It covers both the theoretical basis for and applications of differentiation and integration. Concepts and problems are approached graphically, numerically, analytically and verbally. All students enrolled in this course will take the AP Calculus (AB) Exam. A GRAPHING CALCULATOR IS REQUIRED.
This course introduces and examines the statistical topics that are applied during the decision-making process. Topics include: descriptive statistics, probability, and statistical inference. Techniques investigated include: data collection through experiments or surveys, data organization, sampling theory and making inferences from samples. Computers are used for data analysis and data presentation. This course should not be taken as a replacement for Pre-Calculus/Trigonometry in a college preparatory course of study. A SCIENTIFIC CALCULATOR IS REQUIRED.
This course expands students' mathematical reasoning and problem-solving skills as they cover topics such as logic, graph theory, matrices, social choice, game theory, sequences, series and patterns. The course will encourage students to make mathematical connections from the classroom to the world after high school, while learning the importance of mathematics in everyday life. This course is offered as an addition to Pre-Calculus/Trigonometry, not a replacement. A SCIENTIFIC CALCULATOR IS REQUIRED | 677.169 | 1 |
Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
Richard Aufmann is Professor of Mathematics at Palomar College in California. He is the lead author of two best-selling developmental math series, a best-selling college algebra and trigonometry series, as well as several derivative math texts. The Aufmann name is highly recognized and respected among college mathematics faculty.Joanne Lockwood is co-author with Dick Aufmann and Vernon Barker on the hardback developmental series, Business Mathematics, Algebra with Trigonometry for College Students, and numerous software ancillaries that accompany Aufmann titles. She is also the co-author of Mathematical Excursions with Aufmann.
List price:
$74.95
Edition:
3rd
Publisher:
Brooks/Cole
Binding:
Trade Paper
Pages:
432
Size: | 677.169 | 1 |
Saxon Math 87 Or Lifepac 7?
Question:I currently have the lifepac 7th grade kit, but he will have used saxon 65, 76, so would it be better to just try and get saxon 87 before going into saxon 1/2?
Answers:
It sounds like you are currently using the Saxon math books, but if you are not, I HIGHLY recommend you use their placement test (see placement test link below).
Saxon used to recommend the following sequence:
Math 6/5
Math 7/6
Alg. 1/2 (OR Math 8/7)
Alg. 1
Math 8/7 was only recommended if a student was weak in basic arithmetic, as it had more arithmetic practice.
With the newer editions, they recommend the following sequence:
Math 6/5
Math 7/6
Math 8/7
(Alg. 1/2)
Alg. 1
With the newer Math 8/7 book, they have included more pre-algebra work, so the Alg. 1/2 is now only used if the student needs more of an intro to algebra before beginning Alg. 1. They do recommend skipping Math 8/7 (and using Alg. 1/2 instead) for an accelerated math student (see recommended sequence link below).
From the Saxon website:
What do you recommend for students after they have completed Math 7/6?
The recommended path after finishing Math 7/6 is to take Math 8/7. If your child finishes Math 8/7 with at least 80% mastery, skip ahead to Algebra 1. In previous editions, many people skipped Math 8/7 because they found it to be a weaker text than Algebra ½. In the newer third edition, pre-algebra has been added to Math 8/7, making it a much stronger book. Saxon 87 covers a lot I was going to skip it and I'm so glad i took it. I t covers review and algebra. It was very helpful. We were tested and the kids who took 87 scored higher then the kids who skipped and took algebra half. Given those two choices I'd have to say Saxon 8/7
However, 8/7 is an optional level, and if you have done the two previous books, you might actually be ready for Saxon Algebra 1/2.
Honestly though, what I'd go into is Teaching Textbooks Pre-Algebra.
have to agree with the other posters. While Saxon 8/7 is theoretically an optional book, it contains a LOT of fabulous review material that will ensure you are completely ready for Algebra 1/2. Good Luck!
This article contents is post by this website user, EduQnA.com doesn't promise its accuracy. | 677.169 | 1 |
Saxon Teacher, Algebra 2 is designed to supplement the 3rd edition homeschool kit for Algebra 2. Using this set of CDs without the textbook will lead to an incomplete understanding of the concepts. The program is written with the assumption that the textbook is being used while working on the computer CDs.
This program consists of 5 CDs and works on both Windows and Mac computer systems. The grade level is 8th to Adult and the cost including the textbook is reasonable. This set includes over 110 hours of Algebra 2 content, including instruction for every part of every lesson, as well as complete solutions for every example problem, practice problem, problem set, and test problem. There are two types of CDs included in this program. Lessons which include practice and problems sets, then the instruction CD which includes tutorials of each lesson and the answer key for teachers. Some of the following lessons are taught by a professional teacher and the teacher works through each set of problems. The practice sets are on one CD and are continuous videos; however, at the end of each set there are references given at the end of each lesson for students or teachers needing additional help. I found this to be a very helpful feature in returning to the text for help.
The CDs cover a wide range of problems from Polygons, Trinomials, Negative Exponents, Geometry, Trigonometry, Rounding, Factoring, and Formatting. All the problems and practice sets are equivalent to one full semester of Algebra 2 and the student has enough information and training to be able to do Pre-Calculus work when the CDs have been completed. The mastery aim is for 80% at completion.
The CD format offers students helpful navigation tools that are easy to access and are within a customized player and is compatible with both Windows and Mac.
The CDs are very well planned and have a general understanding of Algebra 2. One can easily follow the step-by-step instructions of the teacher. This program along with the text would take a student through to a successful completion of the Algebra 2 course. | 677.169 | 1 |
Math Solver
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Document Description
To
Math problems made Easy
One
Solving Math problems is not easy! A lot of students have difficulty with Math questions but employing some of these techniques will help you to solve Math problems easily :
Read it carefully - Math problem solving involves reading the problem slowly and carefully in order to understand what is it that you need to solve. At times you miss out important information when you give it a quick reading.
The following steps are genrally followed to solve Math problems:
Break
Change it into an equation - It is important to convert what you read in words into an equation you can solve. So basically you need to change the English into numbers!
Always cross check - Once you get the answer to your Math problem you should always go back and recheck. Sometimes you might miss out on small details and going over the problem and solution again helps.
Ask new topic is introduced you should write it down, review it and in case of any doubts bring it up with your tutor. This process of continuously revising what you learn will gear you up to solve math problems online.
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In mathematics, linear equation means ...
To Math solve problems quickly and accurately you need an understanding of various math concepts and solving math problems is not an easy task. TutorVista has a team of expert online Math tutors toDifferential Equation is a type of equation which contains derivatives in it. The
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Math Solver To Math problems made Easy One Solving Math problems is not easy! A lot of students have difficulty with Math questions but employing some of these techniques will help you to solve Math problems easily : Read it carefully - Math problem solving involves reading the problem slowly and carefully in order to understand what is it that you need to solve. At times you miss out important information when you give it a quick reading. The following steps are genrally fol owed to solve Math problems: Break Change it into an equation - It is important to convert what you read in words into an equation you can solve. So basical y you need to change the English into numbers! Always cross check - Once you get the answer to your Math problem you should always go back and recheck. Sometimes you might miss out on small details and going over the problem and solution again helps. Ask
new topic is introduced you should write it down, review it and in case of any doubts bring it up with your tutor. This process of continuously revising what you learn will gear you up to solve math problems online. Help with Math Topics TutorVista's expert tutors will make solving Math Solver very easy. Our expert tutors will work with you in a personalized one-on-one environment to help you understand Math questions better thereby ensuring that you are able to solve the problems. Solve problems in topics like: * Algebra * Geometry * Calculus * Pre-Algebra * Trigonometry * Discrete Mathematics Students frequently need help with fractions, solving algebra expressions, geometry problems, equations, ratios, probability and statistics measurements and calculus. Each of these topics has its own approach for solving problems. TutorVista's online tutoring in math can help students understand the methods for solving Math Solver in each of these categories. | 677.169 | 1 |
Questions About This Book?
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any CDs, lab manuals, study guides, etc.
Summary
These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs. New to the Third Edition are exercises that provide guided practice for the textbook's Problem-Solving Strategies, focusing in particular on working symbolically. | 677.169 | 1 |
:: 2012 AP* Summer Institutes at UT Austin
Use of a multi-representational approach to examine problems algebraically, graphically, numerically and verbally
Use of technology to reinforce these relationships, to experiment and to develop concepts
Examination of topics covered in both the Algebra II and PreCalculus classroom such as transformations, inverses, and various other aspects of functional analysis
Improving assessments to better prepare students for AP mathematics
Sharing best practices and designing Pre-AP problems and lessons in subject-area breakout sessions
Exploring online resources in a lab setting
Classroom-based activities that tie together multiple concepts
What participants should bring:
* Laptop computer
* Notepad or spiral notebook
* Graphing calculator
* 30 copies of a favorite lesson or activity
* One test given in the past year
* Current textbook(s)
* Post-it notes
Lead Consultant: Jill Bell
Jill Bell currently teaches Pre-AP Algebra II GT, Pre-Calculus and AP Calculus AB at Ronald Reagan High School in San Antonio, Texas. A 20+ year veteran of the classroom, Ms. Bell has been a College Board consultant for seven years and has presented various workshops across the country on Pre-AP Mathematics, AP Calculus, and SAT Math preparation. She is active in Rotary and has been honored by "Who's Who Among America's Teachers." She holds with a BS in Secondary Education from Baylor University and secondary certifications in Mathematics, Spanish and Gifted/Talented.
* Trademark Notice: College Board, AP, Advanced Placement Program, AP Vertical Teams, Pre-AP, and the acorn logo are registered trademarks of the College Board. Used with permission. | 677.169 | 1 |
This well-respected text gives an introduction to the theory and application of modern numerical approximation techniques for students taking a one- or two-semester course in numerical analysis. With ...
This book presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The point of departure is mathematical but the exposition strives ...
* This book is an introduction to numerical analysis and intends to strike a balance between analytical rigor and the treatment of particular methods for engineering problems* Emphasizes the earlier ... | 677.169 | 1 |
Abstrakt der Vorlesung
Optimization is a vibrant growing area of Applied Mathematics. Its many successful applications depend on efficient algorithms and this has pushed the development of theory and software. In recent years there has been a resurgence of interest to use 'non-standard' techniques to estimate the complexity of computation and to guide algorithm design. New interactions with fields like algebraic geometry, representation theory, number theory, combinatorial topology, algebraic combinatorics, and convex analysis have contributed non-trivially to the foundations of computational optimization. This course will be an introduction to the new techniques used in Optimization that have foundation in algebra (number theory, commutative algebra, real algebraic geometry, representation theory) and geometry (convex and differential geometry, combinatorial topology, algebraic topology, etc). | 677.169 | 1 |
Math Essentials: No-Nonsense Algebra and Geometry
We're back again to share about another resource for teaching your child math. For many, this particular subject is scary for them to teach, yet parents know just how important it is to learn for pretty much any career option out there. High school level math is where it typically gets scarier for the homeschool parent.
Math Essentials, the brain child of Rick Fisher, has a product to take the sting out of math at the high school level, No-Nonsense Algebra.
Each lesson in this book is presented in a straight forward manner. Users of the book also have access to online video lessons, which is a boon for many students who need to see someone work through problems. The lessons also include exercises for the student to practice similar problems as well as review problems. The chapters are wrapped up with a review to assess mastery of the material.
Unlike some books out there, No-Nonsense Algebra does not add fluff to distract or pile on busy work for the student. They even offer a money back guarantee that states using this product 20 minutes per day will give you improvements in test scores.
I have to admit that I had a hard time having my high school student spend much time on this over the past few weeks. He was more focused upon finishing his main algebra program for this recent school year. So, rather than relying mostly upon his feedback of the material, I spent a bit of time refreshing my own algebra skills. The video lessons have audio explanation with a handwritten white board look to them. I appreciate how straight forward his approach to presenting the material is in the book as well as being able to cover all essential topics for an algebra I level course.
The other product we were sent to review is Mastering Essential Math Skills: Geometry. This series of smaller booklets focus upon specific skill sets and offer further practice with the concepts previously introduced. These books are aimed at upper elementary/ middle school levels to lay a firm foundation for high school math.
Geometry is one area that the boys seem to be weak with regard to the annual state tests. Each page in this book is designed to be used within 20 minutes and should help shore up any deficiencies in the individual topics presented. Rather than instruction, there are 'helpful hints' meant to refresh their memory on what is required to work the problems.
You can pick and choose just the areas that need refreshing for your student or work through the entire book. I suspect spending a bit more time with this title will better acquaint them with the vocabulary and skills they need to boost that area of their scores come next spring.
No Nonsense Algebra sells for $27.95 for the print book plus online video lesson access. The Mastering Essential Math Skills book series titles retail for $11.95. You can find these and other products at the Math Essentials website which includes sample pages for perusal | 677.169 | 1 |
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In my seven years as math tutor, I've probably worked with twenty algebra books. Hands down, no contest, this is the absolute best I have used: Algebra: Structure and Method, Book 1. (Brown, Richard G. et al. McDougal Littell, Evanston, Illinois: 2000.)
This book doesn't have a ton of frills—there are barely any pictures or "extras." But
what makes this book exceptional is its GREAT sequencing. It does an excellent job of breaking the math down without dumbing it down. The problems get harder very incrementally. There are so many practice problems to choose from that you can really practice until each procedure becomes second nature. And the book only introduces new concepts once you've already mastered the prerequisite skills.
For example, when this book introduces factoring trinomials, it introduces each pattern that you might encounter one at a time. You practice that pattern extensively before facing a new pattern. Once you've practiced all the different patterns separately, THEN it mixes all the different patterns together in one problem set. But by now you know how to recognize the different patterns and what to do differently for each pattern. So when faced with a page full of different types of factoring patterns, you can just think, "OH—difference of squares!" or "OH—perfect squares!" instead of having to do trial and error until you erase a hole in your paper!!
The students I've used this book with acquire very, very strong algebra skills without getting bored or frustrated. And I think it's because the sequencing forces students to learn how to "chunk," a concept I learned from Daniel T. Willingham's book, Why Don't Students Like School?
For example, take two algebra students. One is still a little shaky on the distributive property, the other knows it cold. When the first student is trying to solve a problem and sees a(b + c), he's unsure whether that's the same as ab + c, or b + ac, or ab + ac. So he stops working on the problem and substitutes small numbers into a(b + c) to be sure he's got it right. The second student recognizes a(b + c) as a chunk and doesn't need to stop and occupy working memory with this subcomponent of the problem. Clearly the second student is more likely to complete the problem successfully. (p 31)
13 Comments on "The best Algebra book in the world?"
Julie on January 13th 1:50 pm
Hi Rebecca,
I saw your post on "The Best Algebra book in the World." I am looking for a book that will simply explain each step in an algebra function. I am in an algebra class for the first time in 15 years and I am scared speechless. I hate this stuff. The instructor said as long as I know how to do what is on the reviews for the test than I should be okay. Learning what is on the reviews is where I have problems. Thanks!
Rebecca Zook on January 13th 2:41 pm
Julie, thanks for stopping by! I am glad to help. I also highly, highly recommend Danica McKellar's math books. You can get them on amazon or any library. A lot of adults find them really helpful, and she's great at breaking things down and working things through step by step. Plus they are fun to read! Lots of people are scared speechless about math–you are not alone! I believe in you!!
Edgar on June 21st 5:11 am
I have heard this is a good book, but I doubt that it tops Paul Foerster's Algebra 1 book. Are you familiar with it?
Rebecca Zook on June 21st 8:02 pm
Edgar, it's nice to see you here! I haven't worked with that book yet. Thanks for the suggestion!
Edgar on June 21st 8:33 pm
Mathematically Correct has ranked a series of Algebra 1 books, and Brown's book scores very high. In fact, Brown's book scores in 2nd place, with only one book topping it. Can you guess which book? Yes, you guessed it – Foerster's! Haha, I hated math until I discovered Foerster! I like to say that I preach the good news of Paul Foerster.
Rebecca Zook on June 22nd 1:41 pm
I will have to check that out! I haven't heard of Mathematically Correct. I'm really glad you found a book that you like so much!!
[...] be able to check his answers without having to wait to see me. So, as a supplemental text, we added another algebra textbook that had better sequencing and more practice problems. In the end, we relied on it more than the [...]
Cricket on August 28th 12:53 pm
Thank you for this review. I have used Foerster's algebra, and he does skip a couple of steps. I do not know if it is corrected in later editions, but he does make an assumption that the student knows to divide the fraction in a chapter 2 problem.
I think an algebra text should be so thorough in the explanations that no answer key or solutions manual is necessary.
Rebecca Zook on August 29th 7:45 pm
Hey Cricket, it's great to meet you here! Thanks for your comment! I think every book has its strengths and weaknesses, but I have used this with many students. Some students need more preparation for it in terms of being really comfortable with the prerequisites like fractions and decimals. I'd love to hear more about resources you recommend!
Ashlee on January 30th 8:46 am
I need help. My son is in 9th grde, is very intelligent, but struggles a lot with math. He is in Algebra and is frustrated and barely getting by. Is your book a good book to help him? He needs something that explains each step,& would be helpful if there are tests or actually problems to solve at the end of each part. He especially struggles with word problems. I have purchased Danica McKeller's books as well as a "Dr Math" book. I dont know what to do to help him.
Rebecca Zook on January 30th 3:10 pm
Ashlee, Thanks so much for your comment, it's great to "meet" you here! Based on what you described, I would highly recommend Danica McKellar's books for your situation. This particular textbook might not do the trick for what it sounds like your son is going through. Your question is actually making me think it's time for an updated post about more algebra resources!
Also, while I haven't used the Algebra book yet myself, some of my students really like Teaching Textbooks. Here's the link to their algebra textbook: They are really excellent with the step-by-step teaching and having solutions to all the problems so you can check your work.
If you're feeling like your son really just needs personal attention and feedback and you're interested in him being tutored, I would be happy to set up a time for us to talk and explore whether or not it would make sense for us to work together. All you would need to do is give me a call at 617-888-0160 or email me at [email protected] and we would set up a time for us to have a complimentary conversation, just so I could learn more about your situation.
Tammy on February 6th 11:50 pm
Rebecca,
I am hoping you can make some suggestions. I have a 7th grader who is in pre-algebra. In looking at some of your previous recommendations, it appears that his pre-algebra book actually combines some topics covered in pre-algebra with some that are in algebra 1. Specifically, the name of his book is Big Ideas Math (blue book) by Ron Larson. I find it very difficult to understand, and it is not easy to learn the concepts from the book alone. My son has always been a strong math student; however, he is having some difficulty this 2nd semester. For example, he is having trouble grasping some of the concepts surrounding linear and nonlinear functions and how to determine which type of function it is by an equation or table. Another example of a type of problem he is struggling with: Y+ 1/3x + 1 (With the instructions: a line with slope of 1/3 contains the point (6,1). What is the equation of the line?) What textbook, on-line videos, etc would you recommend for thoroughly EXPLAINING every concept in a simple, easy to understand manner (whereby a student could learn everything they need to be extremely successful without needing classroom instruction)? We're not looking for a workbook of extra problems; we're looking for a resource that would TEACH him in very basic (easy to understand), yet thorough method on how to understand the concepts and figure out the problems. So, something that goes into very clear detail on how to solve each of the problems a student would need to know in each section of content. Ideally, we would love to have a video series as well that would demonstrate the concepts and serve as a virtual classroom. Please respond at your very earliest convenience. We need some help right away; he has a test mid-week, next week.
Thanks so much. | 677.169 | 1 |
Problem Sets
Suggestions to the Student
Our problems are a bit different from the usual calculus textbook problems.
They are not intended to be harder although some may well be.
They are intended, instead, to help you better understand the
concepts of calculus and how to apply them. None of these problems
asks simply for a computation, and some ask for no computation at all.
Instead, they may ask you to do one of the following: Apply a
concept or technique you have just learned in a mildly novel context;
combine concepts or techniques that you have seen only in isolation before;
give a graphical interpretation of the behaviour of a function; make an
inference, from a graph or a table of data, about a function or a physical
relationship.
When you begin working on these problems, you may feel that you do not
know how to get started on a problem or where you should end up.
That's only natural. In fact, some of the problems can be
approached in a variety of ways and have no single answer. Since the
purpose of all the problems in this volume is to help you develop a better
understanding of calculus, a good way to get started is to see if you
understand the question. Talk it over with a classmate and see if the
two of you have the same interpretation. If you don't check in the
textbook to see if you have the right meanings for the crucial words in the
problem. Draw a picture, if possible, to illustrate the problem.
If you encounter a function that is hard to graph, use a computer or a
graphing calculator to draw the graph. In fact, all uses of
computers and calculators are legitimate in working on these problems.
If you are still stuck, talk it over some more with a classmate or ask
for a discussion in class, but be prepared to offer the thoughts you have
developed about the problem.
The keys to getting the most out of these problems are thinking, discussing
and writing. When you recognize a concept or technique that is likely
to be involved in a problem, ask yourself what you know about it and how it
might be applied, and be prepared to reread your textbook or lecture notes
to refresh your understanding Then test your ideas by discussing them
with a classmate or in class. Finally, write up your conclusions in complete
English sentences that convey your understanding as clearly as you know how.
With practice, you will discover that discussing and writing promote
clear thinking and thus help you develop a better understanding of the
material that you are studying. | 677.169 | 1 |
At the end of the period, remain in your seat until you are dismissed.
How to be prepared for class:
All students are required to have a 3-ring binder with paper for class notes.Pencils are preferred, pen is fine.I can provide calculators and will give you a packet of class notes for each chapter.
Report Card Grade:
Your grade has four components:
Tests:Tests count as 40% of your grade.At the beginning of each chapter, you will be given a chapter packet of notes and practice that we will work through.We will also build a calendar together which will include the date of the chapter test so you can be prepared.
Quizzes:All quizzes are pop quizzes, but they are open notes.They also make up a large portion of your grade.If you perform better on the chapter test than you did on the quiz, I will replace your quiz grade with the chapter test grade.
Homework:HOMEWORK IS PRACTICE.IN MATH, JUST LIKE ABOUT ANYTHING ELSE IN LIFE, YOU DON'T REALLY KNOW WHAT YOU CAN DO UNTIL YOU TRY IT ON YOUR OWN.
oHomework is checked and graded based on effort each day as a 0, 1, 2, 3, or 4.
oTo receive credit, your paper must be legible and your work must be shown.
oNo late papers accepted.If you are absent, the homework must be turned in to me on the following day.
Class participation
Two great resources to help ensure your success:
Our building has a universal learning lab built into the schedule each day 3rd period.Students are required to get a presigned pass from any teacher they would like to work with for that period.All teachers in the building are available this period for extra help.
We also offer a Math Lab in room 106B.The math lab is run by certified math teachers periods 2 through 9, with the exception of 3rd period.The offer assistance in homework, studying for tests, or reteaching the material.Any student who has a free period is invited to the math lab.Please bring your math supplies when you attend and be sure to sign in so I can give you credit for attending. | 677.169 | 1 |
e... read more
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Bonus Editorial Feature:
Morris Kline (1908–1992) had a strong and forceful personality which he brought both to his position as Professor at New York University from 1952 until his retirement in 1975, and to his role as the driving force behind Dover's mathematics reprint program for even longer, from the 1950s until just a few years before his death. Professor Kline was the main reviewer of books in mathematics during those years, filling many file drawers with incisive, perceptive, and always handwritten comments and recommendations, pro or con. It was inevitable that he would imbue the Dover math program ― which he did so much to launch ― with his personal point of view that what mattered most was the quality of the books that were selected for reprinting and the point of view that stressed the importance of applications and the usefulness of mathematics. He urged that books should concentrate on demonstrating how mathematics could be used to solve problems in the real world, not solely for the creation of intellectual structures of theoretical interest to mathematicians only.
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Master Math: Probability is a comprehensive reference guide that explains and clarifies the principles of probability in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced, the book helps clarify probability using step-by-step procedures and solutions, along with examples and applications. A complete table of contents and a comprehensive index enable readers to quickly find specific topics, and the approachable style and format facilitate an understanding of what can be intimidating and tricky skills. Perfect for students studying probability and those who want to brush up on their probability skills.
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Catherine A. Gorini, Ph.D., received her A.B. in mathematics from Cornell University, M.S. and Ph.D. in mathematics from the University of Virginia, and D.W.P. from Maharishi European University. She is Dean of Faculty and Professor of Mathematics at Maharishi University of Management. She is the editor of Geometry at Work and author of Facts On File Geometry Handbook. Her numerous awards for teaching include the Award for Distinguished College or University Teaching of Mathematics from the | 677.169 | 1 |
This book is full of clear revision notes, worked examples and practice questions for GCSE Maths. It covers all the major Year 10 and 11 topics for the OCR, Edexcel, AQA and WJEC exam boards. It's packed with useful tips for doing well in your exams and every few pages there's a quick warm-up test and some exam-style questions (answers are at the back). It's easy to read and revise from - everything's explained simply, in CGP's chatty, straightforward style. | 677.169 | 1 |
Trigonometry, Hybrid Edition - 2nd edition
Summary: Reflecting Cengage Learning's commitment to offering flexible teaching solutions and value for students and instructors, these new hybrid versions feature the instructional presentation found in the printed text while delivering end-of-section exercises online in Enhanced WebAssign. The result: a briefer printed text that engages students online! TRIGONOMETRY is designed to help you learn to ''think mathematically.'' With this text, you can stop merely memorizing facts and mimicking ...show moreexamples--and instead develop true, lasting problem-solving skills. Clear and easy to read, TRIGONOMETRY illustrates how trigonometry is used and applied in the real world, and helps you understand how it can apply to your own life. ...show less
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Used - Good Textbook Only. Hybrid Edition. 2nd Edition | 677.169 | 1 |
"The best Algebra tutorial program I have seen... in a class by itself." Macworld
We start with a word problem one would never have to solve in life. Super. The answer is 2n so I choose n/2 to see what happens. The software says "Incorrect" in red, does not offer the correct answer, and simply encourages me to go on to the next slide. Very helpful. I hit Next.
Eventually we get to play with two helicopters to solve x/2 - 3 = 1. I found myself wondering where the two expressions came from and what a helicopter would be doing hovering at x/2 - 3, but I was always a troublemaker in school. Nowhere does the software talk about needing to keep the two helicoptesr at the same altitude, let alone why we have to. Me, I like see-saws which are level when the two sides are the same, just as we want to preserve an equation's truth as we transform it. Anyway...
Clicking +3 on the first guy moves it up but leaves the expression as x/2 - 3. It should have changed to x/2, the way the other guy changed from 1 to 4 when I clicked +3. Clicking x2 (meaning multiply, not the variable x) finally changes the first guy to x. The other guy continues to work and becomes 8.
Now the material simply goes wrong, saying we have to add before we multiply. No, that just makes it easier. And it gets worse: the text says that if we multiply by 2 first we will end up with the wrong answer, x=5. Nonsense, as the graphic shows: we end up with x - 3 = 5, what it calls "an incomplete solution". Thought one: well then it is not a solution! Add 3 to both sides!! And how on Earth did we get to x-3=5? By going x/2-3=1 to 2(x/2-3)=2*1 to x-6=2 to x-6+3=2+3. Right, they accepted as inevitable the two operations of adding at most 3 and multiplying by at most 3, with nothing else permitted. Hunh?
Just this little bit of material is wrong in one place, inconsistent with itself, confusing, unmotivating, and plain leaves out the fundamental concept of preserving the truth of the equation as necessary. On-line and interactive is only as good as the underlying fundamental material, and in that regard Monterey comes up short.
A blow-by-blow replay of a disappointing on-line Algebra experience at the Monterey Institute. | 677.169 | 1 |
John Bird?s approach, based on numerous worked examples and interactive problems, is ideal for students from a wide range of academic backgrounds, and can be worked through at the student?s own pace. Basic mathematical theories are explained in the simplest of terms, supported by practical engineering examples and applications from a wide variety of... more...
John Bird?s approach, based on numerous worked examples and interactive problems, is ideal for students from a wide range of academic backgrounds, and can be worked through at the student?s own pace. Basic mathematical theories are explained in a straightforward manner, being supported by practical engineering examples and applications in order to... more... and breakfasts. Bird grew civilized through his pursuits, moving on to found an international movement that works with troubled people: ex... more...
Unlike most engineering maths texts, this book does not assume a firm grasp of GCSE maths, and unlike low-level general maths texts, the content is tailored specifically for the needs of engineers. The result is a unique book written for engineering students, which takes a starting point below GCSE level. Basic Engineering Mathematics is therefore... more...
In this book John Bird introduces engineering science through examples rather than theory - enabling students to develop a sound understanding of engineering systems in terms of the basic scientific laws and principles. The book includes 575 worked examples, 1200 problems, 440 multiple choice questions (answers provided), and the maths that students... more...... more...
Newnes Engineering Science Pocket Book is a uniquely versatile and practical tool for a wide range of engineers and students. All the fundamentals of electrical and mechanical engineering science and physics are covered, with an emphasis on concise descriptions, key methods, clear diagrams, formulae and how to use them. John Bird's presentations of... more...
Newnes Engineering Mathematics Pocket Book is a uniquely versatile and practical tool for a wide range of engineers and students. All the essentials of engineering mathematics are covered, with clear explanations of key methods, and worked examples to illustrate them. Numerous tables and diagrams are provided, along with all the formulae you could... more...
Engineering Mathematics is a comprehensive textbook for vocational courses and foundation modules at degree level. John Bird's approach, based on numerous worked examples supported by problems, is ideal for students of a wide range of abilities, and can be worked through at the student's own pace. Theory is kept to a minimum, placing a firm emphasis... more...
This textbook for courses in electrical principles, circuit theory, and electrical technology takes students from the fundamentals of the subject up to and including first degree level. The coverage is ideal for those studying engineering for the first time as part of BTEC National and other pre-degree vocational courses, especially where progression... more... | 677.169 | 1 |
Algebra ½ covers all topics normally taught in prealgebra, as well as additional topics from geometry and discrete mathematics (used in engineering and computer sciences). With Algebra ½ , students can deepen their understanding of prealgebraic topics. Algebra ½ includes: instruction and enrichment on such topics as compressions, approximating roots, polynomials, advanced graphing, basic trigonometry, and more. [via] | 677.169 | 1 |
1 : Basic Operations on whole and rational numbers 4.00.005 2 : Developments & applications 4.00.005
Mathematics program intended for High School pupils (age 15-17) This Title comprises 20 3 : Analysis, vectors, trigonometry, probabilities… 2.01.002
PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a ... mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of transcendental functions. PARI .... Free download of PARI/GP 2.3.4
... is an easy to use, general purpose Computer Algebra System, a program for symbolic manipulation of mathematical ... of scripts that implement many of the symbolic algebra operations; new algorithms can be easily added to the library. YACAS comes with extensive documentation (hundreds of pages) covering the scripting language, the functionality .... Free download of Yacas 1.3.3
A Program for Statistical Analysis and Matrix Algebra MacAnova is a free, open source, interactive ... are analysis of variance and related models, matrix algebra, time series analysis (time and frequency domain), and (to a lesser extent) uni- and multi-variate exploratory statistics. The current version is 5.05 release 1. .... Free download of MacAnova 5.05.3 Windows 12.09.0 Mac OS X 12.09.0 4.2.30 for Mac 4.2.30 4.0 64bit 4.0
Algebra1 : Basic Operations on whole and rational numbers 4.00.005
... needs to solve problems ranging from simple elementary algebra to complex equations. Its underling implementation encompasses high precision, sturdiness and multi-functionality. MultiplexCalc also has the unlimited ability to extend itself by using user-defined variables. You can add your own variables to MultiplexCalc in order to convenience your work. Any instance .... Free download of Multipurpose Calculator - MultiplexCalc 5.4.8
... needs to solve problems ranging from simple elementary algebra to complex equations. In .... Free download of Innovative Calculator - InnoCalculator 1.1.8 | 677.169 | 1 |
Cary, IL GeIntegral calculus is the branch of calculus focusing on accumulations; for example, areas under curves and volumes enclosed by surfaces. The two branches are connected by the Fundamental Theorem of Calculus discovered independently by Isaac Newton and Gottfried Leibnitz. My first exposure to calculus was in high school | 677.169 | 1 |
Math & Science Classes
Math Classes
Algebra I
This course is the foundation for all math that you will learn in high school. From real numbers and radicals to proportions and polynomials, the skills you learn in this class will serve you in all future math and many science courses.
Geometry
This course is designed to develop students' deductive reasoning skills through the study of spatial relationships. An emphasis is placed on proof. Topics include introductory terminology, segments and angles, triangle congruency, parallel lines, quadrilaterals, similarity, circles, area and volume. Occasionally we will take advantage of the campus and farm for "geometry in real life" activities.
Algebra II
Advanced Algebra continues and expands on concepts taught in Algebra I and Geometry. Advanced Algebra covers the following topics: functions and graphs, systems of linear equations, polynomials, quadratic equations, inequalities, exponents and radicals, exponential and logarithmic functions, rational expressions, sequences and series, and trigonometric functions. Students will become proficient in the use of algebraic expressions and sentences to model real-world situations.
Pre-Calculus
Students entering Pre-Calculus should have a thorough grounding in high school algebra, geometry and right-triangle trigonometry. However, this background is not sufficient to begin studying Calculus or other higher-level mathematics courses. In particular, students have likely had little or no exposure to logarithms, advanced trigonometry, polar coordinates, parametric equations, conic sections, probability, and some advanced topics like the Fundamental Theorem of Algebra.
Statistics
Students develop skills that will allow them to gather, organize, display, and summarize data. They should be able to draw conclusions or make predictions from the data and assess the relative chances for certain events happening. Topics include: descriptive statistics, basic probability and distribution of random variables, estimation and hypothesis tests for means and proportions, regression and correlation, analysis of count data.
AP Calculus
AP Calculus is a college-level mathematics course and the expectations for students are extremely high. Students will explore the following concepts: limits and continuity; differentiation and its application, including extrema and related rates; differential equations and slope fields.
Science Classes
Biology
The study of life is a study of how millions of different types of living things survive on this planet. In many ways the strategies are the same for all organisms and so we will study what unifies all life forms — the workings of cells and the mechanics of heredity. But it's also true that different species have very different approaches to survival and so we will study how such diversity of life has arisen and how different species interact with units about evolution and ecology. Principles we study in the classroom will be illustrated in the fields, streams, woods, and farmland surrounding Olney.
Conceptual Physics
Physics is the study of matter and energy, and how they interact. Like other sciences, physics is based on the assumption that rules govern the way the universe behaves. Testing the world with repeatable methods will yield predictable results, which allow us to describe those rules that govern our universe. Conceptual Physics is a non-mathematical approach to the topic, intended to make connections in your mind between what you study and your everyday world. Conceptual Physics requires basic algebra skills, but most often you will learn by describing physical phenomena with words, as well as observing both real-world and digital demonstrations. Hands-on and electronic labs will allow active exploration of the topics you learn.
Chemistry
Chemistry deals with the composition of matter and the changes that matter undergoes. The goal of this course is to provide students with a core foundation in understanding the principles of chemistry, to help develop critical thinking skills, and to relate learning to your lives. This is a laboratory-based and project-based course. My role will be as a guide for the journey, a facilitator, an events planner, and occasionally a source of information. Other sources of information will be the textbook, charts and diagrams, videos, and internet sources selected by the guide and by you.
Environmental Science
Environmental Science is a culminating science course at Olney. Following Conceptual Physics, Chemistry, and Biology, it draws on and expands the students' knowledge in these disciplines. Yet, our study of the environment is even more interdisciplinary than that. Students learn of the interplay among economics, politics, ethics, and the sciences. They learn that solutions to environmental problems are simple and straightforward – until the human element is taken into consideration.
A chief goal of this course is that the student gains greater awareness of how fragile is the "human niche" in the web of life and of how their daily actions affect the environment. We will take advantage of our own local environment to illustrate ecological principles and to research solutions to environmental problems. In early May, selected students will have the opportunity to compete in Ohio's Envirothon.
AP Physics: Mechanics
The AP Physics C course covers topics typically found in a first-year introductory college physics course and advances the student's understanding of concepts normally covered in high school physics. AP Physics C Part I covers Newtonian Mechanics in depth. It provides a solid preparation for the AP Physics C Mechanics exam. Topics of study in mechanics include: kinematics, Newton's laws of motion, work and energy and power, systems of particles and linear momentum, circular motion and rotation, and oscillations and gravitation. Laboratory work is an integral component of this course. Students will learn the applications for the material being studied in Calculus. Students must have completed or be enrolled in AP Calculus to enroll in this course. This course will prepare students to take the Advanced Placement Exam, by which students may earn up to one year of college credit.
AP Physics: Electricity and Magnetism
AP Physics C Part II includes topics of study in electricity and magnetism. Topic areas include: electrostatics, conductors and capacitors and dielectrics, electric circuits, magnetic fields, and electromagnetism. Laboratory work is an integral component of this course. To enroll in AP Physics C Part II, students must have successfully completed AP Calculus. | 677.169 | 1 |
Elementary Number Theory:
Primes, Congruences, and Secrets
November 2008
This is a textbook about classical elementary number theory and
elliptic curves. The first part discusses elementary topics such as
primes, factorization, continued fractions, and quadratic forms, in
the context of cryptography, computation, and deep open research
problems. The second part is about elliptic curves, their applications
to algorithmic problems, and their connections with problems in number
theory such as Fermats Last Theorem, the Congruent Number Problem, and
the Conjecture of Birch and Swinnerton-Dyer. The intended audience of
this book is an undergraduate with some familiarity with basic
abstract algebra, e.g. rings, fields, and finite abelian groups.
On November 2008, this book was published
by Springer-Verlag and can be purchased from Amazon.com. Springer-Verlag has also very generously agreed to let me
make this book completely free online. So please feel free to download it:
Of course, I would greatly appreciate it if you support the book by buying it.
This book is based upon work supported by the National Science Foundation under Grant No. 0653968. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. | 677.169 | 1 |
Overview - ACADEMIC SUPPORT PROGRAM FOR MATHEMATICS GR 6
Review, Practice, and Assess Math Skills in Grades 6–12! Provide support for students in Grades 6–8 math and algebra. Program includes lessons in problem solving, reasoning and proof, communication, and connections while addressing content strands from the NCTM. Units in each of the more than 450-page texts combine direct instruction, guided practice, and hands-on activities.
Binder includes 120 hours of lessons with reproducible activity sheets, test-taking strategies and practice test items, station-based activities for small group work, and a teacher's guide.
Grade 6, Grade 7, Grade 8: Provides review, remediation, and hands-on application opportunities for four strands of mathematics: number and operations, algebra, geometry and measurement, and data analysis and probability. | 677.169 | 1 |
Math BSI: This full-year course serves to reinforce the foundational knowledge from other mathematics courses. Students study basic skill clusters (1. Number and Numerical Operations, 2. Geometry and Measurement, 3. Patterns and Algebra, 4. Data Analysis, Probability, and Discrete Mathematics) of material which will help them succeed on standardized tests. Students will utilize workbooks, computer lab and online assignments, and various software.
Pre-Algebra: This course is designed for those students who are in preparation for Algebra 1. Topics include graphing, writing algebraic expressions, solving equations and inequalities, operations with signed numbers, and applications.
Algebra 1CP: The course includes the study of real number properties, solving equations and inequalities, finding solutions to word problems, solving systems of equations, and solving quadratic equations. Real world application and problem-solving techniques are stressed.
Algebra 2H: This course focuses on and enhances subjects discussed in Algebra in grade 8. Topics include the study of linear, quadratic, polynomial, exponential and logarithmic functions, each integrating technology and real world applications.
Geometry CP: This course includes the study of plane geometry. It is a structured course building upon concepts which develop logical thinking through deductive as well as inductive reasoning. Topics include the geometry of points, lines, and planes, properties of congruence and similarity, circles and spheres, coordinate geometry, area, and volume.
Geometry H: The advanced level of geometry encompasses in greater depth all of the topics in Geometry. The course includes challenging problem-solving.
Algebra 2CP: This course expands the study of algebra to include complex numbers, quadratics, conic sections and logarithms. These concepts are implemented through the use of cooperative learning with an emphasis on technology and real world applications.
Precalculus CP: This course includes a semester of elementary functions, composite functions, logarithmic functions, and exponentials as well as a semester of trigonometry. Emphasis in the trigonometry portion of the course includes an analysis and graphic interpretation of the six trigonometric functions. Throughout the entire course, relevance to practical applications in the real world is stressed.
Precalculus H: This rigorous course approaches the study of polynomial, exponential, logarithmic, rational and trigonometric functions: numerically, graphically, algebraically and analytically. Series, sequences, conic sections and their applications are developed and applied. Limits of continuous functions are defined and applied as a foundation for the calculus course.
Calculus H: This honors level calculus course consists of a full year of development and application of derivatives and integrals. Several projects are introduced to enhance the understanding of the material. The course is designed to help students master their college calculus classes.
AP Calculus (AB): In this course topics include elementary functions and limits with an emphasis on differential and integral calculus and their applications. Students must take the Advanced Placement CalculusAB examination for college credit.
AP Calculus (BC): (7 periods per week includes 2 lab periods). This course includes all of the topics taught in AP Calculus (AB), but is more extensive and includes an emphasis on theory. Additional topics are complex integration, infinite series, vectors, and polar coordinates. Students must take the Advanced Placement Calculus BC examination for college credit.
AP Statistics: This course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. The four broad themes include: explaining data observing patterns and departures from patterns, planning a study deciding what and how to measure, anticipating patterns producing models using probability and simulating, and statistical inference guiding selection of appropriate models. Students must take the Advanced Placement AP examination for college credit.
Statistics H: This course will cover all the topics of AP Statistics without the rigor and depth required in AP Statistics.
Discrete Mathematics: Students in this course will apply the concepts and methods of discrete mathematics to model and explore a variety of practical situations. The course has five major themes, including systematic counting, using discrete mathematical models, applying literative patterns and processes, organizing information, and finding the best solutions using algorithms. Discrete topics include: graph theory, matrix models, planning and scheduling, map coloring, social decision making, and election theory. Students will also study descriptive and inferential statistics, which includes representing data visually, calculating measures of central tendency, and computing standard deviation and z-scores. During probability, laboratory experiments are used to explore how often particular events are expected to occur. Overall, the course incorporates individual and small group problem solving.
Visual Computer Programming 1: In this course the student is instructed in principles of computer science. Using our computer labs in the high school, the student studies the structure, capabilities and limitations of computers. The student learns to program the computer using a high-level computer language and to use computers to assist problem-solving.
Computer Programming H: The major topics for this course include programming methodology, features of programming languages, data types, and algorithms.
AP Computer ScienceAB: This course continues the study of programming and includes additional features of programming languages, data structures, algorithms, and applications. Students must take the Advanced Placement Computer ScienceAB examination for college credit.
Robotics: This course is designed to enhance computer programming skills through the study of robotics. Topics include mechanics, electronics, software, and sensory systems associated with the robot. Students also have the opportunity to do research, analyze, and implement independent projects throughout the school year. Presentation skills are developed throughout the course.
Multivariable Calculus: This course is the final course in the accelerated course sequence. Topics included in this course are: vectors and the Geometry of Space, Vector-Valued Functions, Functions of Several Variables, Multiple Integration, and Vector Analysis. Vectors have many applications in geometry, physics, engineering, and economics. The student builds on many of the ideas of calculus of a single variable to calculus of several variables | 677.169 | 1 |
Mount Wilson CalculusIt is essential to understand the students? thought processes because each student is uniquely suited to a particular method. In order to lead students in a right direction, I ask them what problems they are really having difficulty with and take the necessary time to explain the concepts to them. However, understanding the theories is fundamental but is not enough | 677.169 | 1 |
A. Beck, M. Bleicher and D. Crowe,Excursion into Mathematics,
Worth Publishers, 1969 (ISBN 0-87901-004-5). The first chapter (about 80 pages)
introduces graph theory and many of its most interesting topics. This
book is written for those with two or three years of high school mathematics.
K. H. Rosen, Discrete Mathematics and its Applications, Random
House, NY, 1988. (ISBN 0-394-36768-5, QA39.2 R654) This college text,
written for students that have completed college algebra, present graph
theory in chapters seven and eight, and does so from an algorithmic viewpoint.
L. Steen editor, For All Practical Purposes : Introduction to
Contemporary Mathematics 3ed. W. H. Freeman and Company, New York 1994.
(ISBN 0-7167-2378-6, QA7.F68 1994) This excellent text by the Consortium
for Mathematics and Its Applications aims to show what mathematics is
good for and what mathematicians do. This freshman college level book
starts with two very practical chapters on graph theory. | 677.169 | 1 |
Screenshots
Description
* You can use functions: all trigonometry, all hyperbolic, logarithm, arithmetic and statistics. And logically the operators "+,-,/,*,^,!,_".
* You can enter 4 different numerical systems in the same expression: binary, decimal, octal, hexadecimal.
* You may have the answer in 4 different number systems (binary, decimal, octal, hexadecimal)
* You can create variables of type "variable name=expression" and use them in the following expressions.
* You can standardize the form of calculations (degrees, radians, gradians) or/and if you want you can force a function to be calculated in radians, degrees, and gradians, using "rad", "deg", "gon". Example "radsin(pi/6)" ="degsin(30)".
* You can use three types of parentheses "{}" "[,]" "(,)"
* You can solve expressions with exponents like "2^_2^3^_4" and using parentheses you can calculate with even more complexity.
* You can get the answer in the form of SI prefixes (milli, micro, everyone!) Example "1E-9" = "1n".
* And more... | 677.169 | 1 |
I have recently completed my second semester of calculus, and this textbook was used for each class. I was quite impressed by it. It starts out simply enough with a unit on functions and gradually leads the student deeper into more advanced concepts, such as differentiation, integration, integration techniques, numerical series, and vector analysis. The book also includes a chapter giving an introduction to differential equations, and it contains several appendices dealing with trigonometry, logarithms, complex numbers, and more; making this book excellent as a math reference as well as a class textbook. That's why I have no intention of selling it back after the class!
Each new concept is illustrated with several examples, and numerous exercises accompany each section. The author strikes a good balance between being overly abstract and overly concrete, so you can make this book work for you no matter what your style of learning math may be. It contains interesting side notes on the history of mathematics, and the pages are laid out in a way that's pleasing to the eye. All in all, a very well-constructed book.
I'm NOT saying that this book makes learning calculus easy-such a book does not exist, unless you're a math prodigy! Stewart's Calculus will, however, give you thorough guidance as you learn this difficult yet fascinating subject. Calculus is very hard to learn with any approach, but with confidence and plenty of effort, it can be mastered and I've found it a very fulfilling area of study. Hopefully you will also.
The bookseller (dreamboat books) was great to deal with, would gladly work with them again. The book is o.k., for a government book. | 677.169 | 1 |
This course emphasizes the extension of mathematical knowledge and skills to prepare students for success in their everyday lives, in the workplace, strengthening and extending key foundational mathematical concepts and skills by solving authentic, everyday problems. Students have opportunities to extend their mathematical literacy and problem-solving skills and to continue developing their skills in reading, writing, and oral language through relevant and practical math activities.
MFM2P - Mathematics: Foundations (Applied):
This course enables students to consolidate their understanding of relationships relationships. Students will investigate similar triangles, the trigonometry of right-angled triangles, and the measurement of three-dimensional objects. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
MPM2D - Mathematics: Principles (Academic): multistep problems and communicate their thinking. | 677.169 | 1 |
Mathrealm
Algebra 1 Explorer software will turn struggling students into successful math learners and average students into accelerated math learners!
Spark your students' interest in algebra with state-of-the-art graphing tools and interactive lessons that encourage students to explore and learn algebraic concepts.
The Algebra 1 Explorer features a comprehensive algebra 1 curriculum that is packed with resources for your class, including student self-paced tutorials, interactive class presentation material, as well as student learning activities and worksheets.
Algebra World software will turn struggling students into successful math learners and average students into accelerated math learners!
Algebra World teaches and reinforces introductory algebra concepts and meets NCTM standards. Mathematics topics have series of lessons and real world examples that accompany them. Equations and their relationship to word problems are emphasized throughout the program.
Pre-Algebra World creatively develops key concepts with images that students will remember for a lifetime. Learning activities in this highly visual and interactive program actively engage students and reflect NCTM standards. This entire imaginary world is designed to motivate students to learn and use math.
Thousands of students, teachers and schools are using Pre-Algebra World to get great results. It provides a proven and dynamic learning environment that motivates students to succeed.
This program provides comprehensive overview of trigonometry fundamentals and helps students make connections from abstract concepts to the world around them. This engaging and easy-to-follow program builds many bridges from math to science, history, and everyday occurrences.
The Trigonometry Explorer offers something for every student of Trigonometry. For those who need to review the basic building blocks of Trigonometry, the creative animated and engaging lessons on degrees, angles and triangles provide a great starting place. The functions, plotting and inverse functions lessons are for those who have a grasp of the basics. For the student who is bored and uninspired, with Trigonometry, the applications on radio waves, sound, surveying and many more show how Trigonometry is used to explain everyday phenomena and solve real word problems. | 677.169 | 1 |
Websites that are usefull
by
interactmath.org This is a tutorial website for many high school topics. When you get to the website, just click "enter". The new screen will ask you for a textbook. The Pearson 2011 edition is very similar to the one we use. Select it and choose a level to start. It checks your answer for correctness and will walk you through a problem if you wish.
phschool.com/atschool/txtbk_res_math.html This is the website of the textbook manufacturer. Choose the Alg II 2004 edition. You will be able to pick a particular section from any chapter for a tutorial on the major topics from the section.
Geogebra.org This is an interactive graphing program that is rated very high in quality. You can download it for free. It is used anywhere from simple geometry to high level calculus. The tutorial walks you through many examples. I highly recommend this for every high school student.
Purplemath.com This is a web site that has excellent concept development for students to understand an area of mathematics. I use many of their explanations in the Daily Lessons section.
KhanAcademy.com This is a web site contain short instructional videos developed by a MIT professor. It coverts the basic procedures on a very wide range of topics covered in the high school curriculum. This is highly recommended for the student to view before they even cover the material in the class. | 677.169 | 1 |
Understandable Statistics (Hardcover)
9780618949922
ISBN:
0618949925
Edition: 9 Publisher: Houghton Mifflin Company
Summary: This algebra based text is a thorough yet approachable statistics guide for students. The new edition addresses the growing importance of developing students' critical thinking and statistical literacy skills with the introduction of new features and exercises.
Ships From:Alpharetta, GAShipping:Standard, Expedited, Second Day, Next DayComments: 0618949925 MULTIPLE COPIES AVAILABLE-Very Good Condition-May have writing or highlighting-May ha... [more] 06189499 | 677.169 | 1 |
College textbooks may be in the top-ten of the worst things for sale, ever. It's not bad enough that the universe makes you feel worthless if you don't get a degree, and then laughs at you when you want a job. No, along the four- or five-year journey to your worthless diploma, they make you buy dozens of textbooks.
The future has brought slight reprieve to the textbook problem - you can buy them online for cheap, get free shipping, and resell them for more than the snotty guy at the campus bookstore wants to give you when the class is over. But the fundamental issue remains that introductory calculus, or chemistry, or whatever, has not changed in at least twenty years. The only difference is the word problems have changed.
Skrillex buys an ice cream cone whose height is h and radius r, topped with a sphere of ice cream with radius 1.1r. His friend Deadmau5 texts him on an iPhone 4, and while he texts back, the ice cream melts and runs into the cone. The cone has a leak which allows the melted ice cream to run out the bottom at rate 0.031r3 per minute (t). Express the surface area of the cone filled with melted ice cream as a function of time. Do not use rage faces in your solution. | 677.169 | 1 |
Pre-Algebra. Nice choice! During our long and celebrated (OK, so maybe we're exaggerating
a little) years in various math classes, we've found that a solid foundation
is extremely important. So we're glad you came here, and we hope
it helps you out!
In this section
of the site, we'll try to clear up some common problems encountered in
pre-algebra. We'll cover everything from the basics of equations
and graphing to everyone's favorite - fractions.
After each section,
there is an optional (though highly recommended) quiz that you can take
to see if you've fully mastered the concepts. Don't forget to visit the
message board
and the formula database. | 677.169 | 1 |
More About
This Textbook
Overview
An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.
Updates include a chapter by J. H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader.
The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.
Related Subjects
Meet the Author
Roger Heath-Brown F.R.S. was born in 1952, and is currently Professor of Pure Mathematics at Oxford University. He works in analytic number theory, and in particular on its applications to prime numbers and to Diophantine equations.
Table of Contents
Preface to the sixth editionAndrew Wiles Preface to the fifth edition
1. The Series of Primes (1)
2. The Series of Primes (2)
3. Farey Series and a Theorem of Minkowski
4. Irrational Numbers
5. Congruences and Residues
6. Fermat's Theorem and its Consequences
7. General Properties of Congruences
8. Congruences to Composite Moduli
9. The Representation of Numbers by Decimals
10. Continued Fractions
11. Approximation of Irrationals by Rationals
12. The Fundamental Theorem of Arithmetic in k(l), k(i), and k(p)
13. Some Diophantine Equations
14. Quadratic Fields (1)
15. Quadratic Fields (2)
16. The Arithmetical Functions ø(n), µ(n), *d(n), σ(n), r(n)
17. Generating Functions of Arithmetical Functions
18. The Order of Magnitude of Arithmetical Functions
19. Partitions
20. The Representation of a Number by Two or Four Squares
21. Representation by Cubes and Higher Powers
22. The Series of Primes (3)
23. Kronecker's Theorem
24. Geometry of Numbers
25. Elliptic Curves, Joseph H. Silverman Appendix List of Books Index of Special Symbols and Words Index of Names General | 677.169 | 1 |
Formulas for functions of one variable This is a chart of functions with one variableLicense information
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Basic Math - Number Patterns Studying number patterns is important for two reasons. First, they help one better understand the concepts of arithmetic and provide a basis for understanding the concepts of more complex mathematics (algebra, trigonometry, calculus). Second, pattern recognition is a useful problem-solving skill, both in mathematics and in real-world situations. Patterns involving odd and even numbers are investigated. Patterns in multiples of certain numbers lead to an understanding of divisibility rules. SequeStatistical Reasoning II Statistical Reasoning in Public Health II provides an introduction to selected important topics in biostatistical concepts and reasoning through lectures, exercises, and bulletin board discussions. Author(s): John McGreadyMethods in Biostatistics II Presents fundamental concepts in applied probability, exploratory data analysis, and statistical inference, focusing on probability and analysis of one and two samples. Author(s): Brian Caffo
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Content within individual OCW courses is (c) by the Johns Hopkins University and individual authors unless otherwise noted. JHSPH OpenCourseWare materials are licensed under a Creative Commons License
Analyzing Statistics S.S. Europe and Russia Students will gather statistical information on countries in Europe and Russia from almanacs. The information will be recorded in a chart. Students will then take the information and make line or bar graphs. Students will analyze the information by answering higher level thinking questions.Supporting Teachers Intervention in Collaborative Knowledge Building In the context of distributed collaborative learning, the teacher's role is different from traditional teacher-centered environments, they are coordinators/facilitators, guides, and co-learners. They monitor the collaboration activities within a group, detect problems and intervene in the collaboration to give advice and learn alongside students at the same time. We have designed an Assistant to support teachers intervention in collaborative knowledge building. The Assistant monitors the collabo Author(s): Chen Weiqin
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The Effective Provision of Pre-School Education (EPPE) Project: Technical Paper 8a - Measuring the IA changing climate for educational research? The role of research capability-building As part of the Teaching and Learning Research Programme, the ESRC have funded a totally new kind of project, which is likely to be watched with interest by others in social science more generally. This Research Capacity-Building (RCB) project (grant number L139251106) is an innovative attempt to invigorate an entire research field. Among its aims are to support and encourage: the management of complex projects, a widening of methodological approaches, the further combination of different approac Author(s): Creator not set
Artificial Intelligence: Natural Language Processing This course is designed to introduce students to the fundamental concepts and ideas in natural language processing (NLP), and to get them up to speed with current research in the area. It develops an in-depth understanding of both the algorithms available for the processing of linguistic information and the underlying computational properties of natural languages. Wordlevel, syntactic, and semantic processing from both a linguistic and an algorithmic perspective are considered. The focus is on m Author(s): No creator set
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Examining the Burdens of Gendered Racism: Implications for Pregnancy Outcomes Among College-Educated Objectives: As investigators increasingly identify racism as a risk factor for poor health outcomes (with implications for adverse birth outcomes), research efforts must explore individual experiences with and responses to racism. In this study, our aim was to determine how African American college-educated women experience racism that is linked to their identities and roles as African American women (gendered racism).
Methods: Four hundred seventy-four (474) African American women collaborate Author(s): Jackson, Fleda Mask,Phillips, Mona Taylor,Hogue, C
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Formulas for functions of two variables This website features a chart of functions with two variables | 677.169 | 1 |
Globalshiksha has come up with LearnNext Uttarakhand Board Class 10 CDs for Maths and Science. This CD contains the entire syllabus for Uttarakhand Board Class 10 Mathematics and Science for the current year. Included lessons are in audio and visual format. Solved examples, practice workout, experiments, tests and many more tests related to Maths and Science. It also includes various set of visual tools and activities on each Lesson with Examples, Experiments, Summary and workout, which is benefit for the students. Students can understand the concepts well; clear all doubts with ease through this Educational compact disk and get score in the exams.
This multimedia comes with a useful Exam Preparation like Lesson tests usually 20-30 minutes in duration, which will help you to evaluate the understanding of each lesson and Model tests usually 150-180 minutes in duration, that cover the whole subject on the lines of final exam pattern and package that can help you sharpen your preparation for exams, identify your strengths and weaknesses and know answer to all tests with a thorough explanation, overcome exam fright and get scores in exams. | 677.169 | 1 |
Mathematical Challenges VI Problems and solutions from Years 2003 to 2006 In the years of Mathematical Challenges covered by this book, questions were set at four levels: Primary, Junior, Middle and Senior...
Can You Prove It? Mathematics which is proof-orientated develops pupils' ability to reason logically and deepens their understanding of mathematical concepts... | 677.169 | 1 |
Mathematics Faculty: Curriculum Overview 2012-13
The grid below gives an overview of the curriculum for this academic year.
Year 7
Students cover work from all strands of Mathematics each term. They consolidate and improve both their mental and written methods for addition, subtraction, multiplication and division of whole numbers, decimals, integers and fractions. They begin to develop both algebraic and geometric reasoning skills and statistical inquiry; studying probability, sequences, functions, graphs, transformations, ratio, percentages, measurement and aspects of discrete Mathematics.
Year 8
Students further develop their skills gained in Year 7 with an increased focus on algebraic manipulation and application to real-life contexts.
Year 9
Each half term has a broad theme to enable students to perceive the links between different topics. In order, these themes are: Fractions, percentages and applications; decimals and approximation; formulae manipulation and measurement; sequences and graphs; geometry and equations and finally angles with trigonometry.
Year 10
All students study the Linear GCSE (Edexcel 1MA0) course; a few students are also entered in Entry Level Mathematics (OCR). Students review and develop their non-calculator numerical skills, measurement, algebraic manipulation, graphical representation and angle properties, statistics and probability. Elements of functional skills are developed, together with the quality of students' written communication.
Year 11
Students continue with the course they began in Year 10. There is greater emphasis on the topics at the higher end of the grades applicable for their tier of entry (either foundation or higher). Topics include trigonometry, circle theorems, vectors, congruence proofs and transformations of functions at Higher level and quadratics, Pythagoras' theorem, transformations, construction and loci, inequalities, measurement and scatter graphs at Foundation level.
Statistics (Year 10 and 11)
Students may study GCSE Statistics as an option at Year 10 and 11. Statistics GCSE extends the handling data and probability elements of GCSE Mathematics. It is a project based course. Each project begins with some kind of data collection task, and this data is then used to illustrate and practice the techniques learnt. Students collect data in a variety of ways; experiments, from the internet and by using a questionnaire. This data is then displayed using statistical graphs and summarised by statistical calculations. ICT programs such as Excel and Fathom are used to analyse data, and pencil and paper techniques are also learnt. | 677.169 | 1 |
This module introduces you to the mathematical notation and techniques relevant to studying engineering at undergraduate level. The emphasis is on developing the skills that will enable you to analyse and solve engineering problems. You cover algebraic manipulation and equations, the solution of triangles, an introduction to vectors and an introduction to probability and descriptive statistics. | 677.169 | 1 |
Product Description
From the Back Cover
Master MATLAB®!
If you want a clear, easy-to-use introduction to MATLAB®, this book is for you! The Third Edition of Amos Gilat's popular MATLAB®, An Introduction with Applications requires no previous knowledge of MATLAB and computer programming as it helps you understand and apply this incredibly useful and powerful mathematical tool.
Thoroughly updated to match MATLAB®'s newest release, MATLAB® 7.3 (R2007b), the text takes you step by step through MATLAB®'s basic features—from simple arithmetic operations with scalars, to creating and using arrays, to three-dimensional plots and solving differential equations. You'll appreciate the many features that make it easy to grasp the material and become proficient in using MATLAB®, including:
Sample and homework problems that help you hands-on practice solving the kinds of problems you'll encounter in future science and engineering courses
New coverage of the Workspace Window and the save and load commands, Anonymous Functions, Function Functions, Function Handles, Subfunctions and Nested Functions
By showing you not just how MATLAB® works but how to use it with real-world applications in mathematics, science, and engineering, MATLAB®, An Introduction with Applications, Third Edition will turn you into a MATLAB® master faster than you imagined.
About the Author
Amos Gilat, Ph.D., is a Mechanical Engineering Professor at the Ohio State University. Dr. Gilat's research has been supported by the National Science Foundation, NASA, FAA, Department of Energy, Department of Defense, and various industries.
If you do not know anything about MATLAB, this is the book you should have at the first step. It teaches you every basic steps and how to apply them to real engineering or mathematical problems in an interactive environment. It has very good screen shots and real world problems to show how to use MATLAB. It reinforces the concepts with quality exercise questions. It is very easy to read and understand. It is absolutely a beginner book not for an advance user.
20 of 21 people found the following review helpful
5.0 out of 5 starsThe perfect introductory text for MATLABDec 7 2005
By shuttledude - Published on Amazon.com
Format:Paperback
If you are completely new to MATLAB then you will find no better book to guide you through the basics. It is perfectly suited for teaching yourself several basic but still very interesting and useful programming techniques. Topics are presented to the reader in an order carefully determined to produce maximum benefit and knowledge. The book is short and very readable, with many example programs.
In short: if you want a FIRST introductory textbook for MATLAB, you can't beat this book. And it covers the latest version (Release 14).
21 of 23 people found the following review helpful
2.0 out of 5 starsNot all bad, but it does fall shortJan 20 2009
By Neurofox - Published on Amazon.com
Format:Paperback
Not a bad book per se - if you are an absolute beginner with Matlab.
However, for the asking price, there are some rather glaring deficiencies.
Here are the major ones:
1. It doesn't go far enough. After working through the examples, it leaves one not entirely self-sufficient and confident.
2. There are quite a few copy-editing mistakes, particularly in the current edition.
3. Feels too generic. In a nutshell, one can't shake the feeling that this represents a somewhat annotated help file. As the in-built Matlab help is getting better and better all the time, the need for such a generic book becomes questionable.
4. Outdated. This book is 3 releases behind the current version of Matlab - and it shows. Some basic features that were introduced in the more recent releases of Matlab are - naturally - not discussed here.
Recommendation: Release a new edition that catches up with the evolution of Matlab, fixes the copyediting mistakes, goes farther and includes more distinctive examples. | 677.169 | 1 |
Many of the problems students experience with A-Level Physics are associated with the mathematics involved. This title deals with this problem offering support for mathematics in physics. 'Maths boxes' present the mathematics needed to grasp a concept. It includes: objectives stated; color illustrations; and graduated questions and practice. | 677.169 | 1 |
Trigonometry - 6th edition
Summary: This easy-to-understand trigonometry text makes learning trigonometry an engaging, simple process. The book contains many examples that parallel most problems in the problem sets. There are many application problems that show how the concepts can be applied to the world around you, and review problems in every problem set after Chapter 1, which make review part of your daily schedule. If you have been away from mathematics for awhile, study skills listed at the beginning of the first...show more six chapters give you a path to success in the course. Finally, the authors have included some historical notes in case you are interested in the story behind the mathematics you are learning. This text will leave you with a well-rounded understanding of the subject and help you feel better prepared for future mathematics courses | 677.169 | 1 |
Summary: Provides completely worked-out solutions to all odd-numbered exercises within the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.2.00 | 677.169 | 1 |
The Mathematics Division is designed to help all students succeed. Almost every student attending Clark College will have to take some math classes to fulfill the requirements for their degrees or certificates. The Mathematics Division provides a variety of instructors, class meeting times, formats and extra help to accommodate our students' needs.
Everyone can be successful in math!
Everyone struggles with mathematics at some point, but with patience, practice and persistence we learn the principles and skills that will help us move to the next level. Advances in science, technology, social science, business, industry and government are all dependent upon precise analysis of data. A basic understanding of math is essential to all people who will be entering the job market.
Benefits
Students gain a better understanding of math concepts and processes, giving them the skills and experience necessary to succeed in college and in careers.
Approximate Costs
Costs to the student can widely vary. General tuition fees and textbook costs apply to every student taking a math course. A graphing calculator is generally recommended for student use. Costs for graphing calculators start at $115.00. Alternatively, students may borrow a graphing calculator form the Mathematics Division for one quarter at no cost. | 677.169 | 1 |
RELATED LINKS
Math 180 - Elementary Functions – Fall 2006
MWRF 8:00 - 8:50 MC 346
Instructor: Diane Overturf, M.S.
Office: MRC 571 Telephone: 796-3654 Email: [email protected]
Office Hours: R 9:00-9:50. Other times can be arranged by appointment.
Additional help: Individual help is also available in the Learning Center located in MC 332. You can sign up for individual tutoring at any time or drop in for homework help.
Text: Required: Precalculus: Functions and Graphs, tenth edition by Swokowski and Cole
Optional: Student's Solutions Manual for the text listed above. Although this is not required, I strongly recommend purchasing it.
Course Catalog Description: Topics include polynomial, exponential, logarithmic, and trigonometric functions and an introduction to vectors and analytic geometry.
Core Skill Objectives:
Communication Skills
Writes competently within the major and for a variety of purposes and audiences.Applies the skills of planning, monitoring and evaluating.
Life Values
Analyzes, evaluates and responds to ethical issues from an informed personal value system.
Aesthetic Skills
Develops an aesthetic sensitivity.
Cultural Skills
Participates in activities that broaden the student's customary way of thinking.
Course Objectives:
Communication Skills
Use graphs to represent mathematical behavior.
Model problems from geometry and other disciplines using function concepts.
Attendance and Academic Honesty: Attendance is essential. You are adults and mature enough to realize that in order to succeed in this class it is vital that you be here. If you cannot make it to class and have any questions, contact someone in the class or myself. To make up a missed quiz or exam you must contact me before the start of class.
Tardiness is a disruptive influence on the class and will affect your grade as follows. After 4 times of being tardy, one point will be deducted from your quiz total points for each additional day you are tardy.
You are responsible for all information given during class. Missed quizzes and exams may be made up if and only if you contact me before the quiz or exam and have a legitimate excuse.
Cheating will not be tolerated. First offense will be a zero for the particular work; a second offense will result in an F for the course.
Responsibility: My responsibility is to help you learn the material in this class through presenting new concepts, modeling the process of solving problems, and challenging you do your best. I will do this to the best of my ability.
Your responsibility is to be actively engaged in the process of learning through attending class, reading the text, listening attentively, taking notes, practicing the concepts through doing daily assigned homework, asking questions when you need clarification, and seeking outside help when you need it. You will not succeed in this class if you are unwilling to put time into practicing the concepts outside of class. I encourage you to study with others and to seek a tutor if you find the material difficult. You are responsible for all information and assignments given during class, even if absent.
Americans with Disabilities Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see Wayne Wojciechowski in Murphy Center, Room 335 (796 - 3085) within ten days to discuss your accommodation needs.
Chapter Exams: (30%) Assignments: (30%)
Individual and group assignments, for grade, will be given throughout the semester. Group assignments are to be completed as a group. Every member of the group will receive the same grade. If a member of your group is not pulling his/her weight contact me. Any student who does not actively participate in completing group assignments may be asked to complete them alone. Writing Assignments: I will assign a number of related writing projects during the semester. I will collect and read them twice - at mid-semester and at the end of the semester. Writing Assignments will be graded on accuracy, completeness, thought put into your responses, and writing skills. Quizzes: (10%)
Quizzes will be given at least once a week, with the possible exception of exam weeks. You will have a quiz the last day of every week (usually Friday). A pop quiz can occur at any time. Final Exam: (30%)
Your final exam grade will consist of a take home group, open book exam and a comprehensive individual, closed book exam given on Wednesday Dec 13 from 7:40 to 9:40 PM. The two will be combined to form one Final Exam grade as follows: group exam (1/3), individual exam (2/3).
Late Assignments will be accepted up to three days late. For each day late your grade will be deducted 10 percentage points (one grade level). After three days, a zero will be given for that assignment. Any extra credit assignments will not be accepted late. Extra Credit assignments may be offered during the semester. Extra credit assignments are not accepted late. Extra Credit is graded on a point for each problem correctly done and the points are added to your quiz scores. Extra Credit will not raise your grade more than one half grade level. I.e. it can raise your grade from a BC to a B but not from a C to a B. Missed Quizzes and Exams: Missed quizzes and exams may be made up if and only if you contact me before the quiz or exam and have a legitimate excuse.
Schedule:
This schedule may change as we progress through the course. You will be notified of any changes. You are responsible for knowing these dates. Graded assignment due dates will be announced as they are assigned. | 677.169 | 1 |
books.google.com - Description: As technology continues to move ahead, modern engineers and scientists are frequently faced with difficult mathematical problems that require an ever-greater understanding of advanced concepts. This professional reference book is designed as a self-study text for practicing engineers and... techniques for engineers and scientists | 677.169 | 1 |
MATH ESSENTIALS BOOKS and DVDs
New from Math Essentials!!!!
America's Math Teacher DVD Series
Take learning to a whole new level. Now, each of the following Math Essentials books have a complete tutorial DVD to go along with it.
Each DVD guides students through each and every topic. It's like having your very own personal tutor. Only Mr. Richard W. Fisher, America's math teacher, is available anytime, 24/7.
These DVD sets will ensure master of concepts necessary for success in algebra and beyond.
DVD's are sold separately, but it is highly recommended that they are used along with the Mastering Essential Math Skills books. Each DVD can be ordered with the companion book from the Mastering Essential Math Skills Book Series or without the book.
AS AN INTRODUCTORY OFFER, EACH SET WILL COME WITH A FREE A+ MATH KIT. (A $4.99 VALUE)
Mastering Essential Math Skills Book Series
The key to this program's success is that every lesson is fun and exciting. Each daily 20 minute session is short, concise and self contained. Students don't have time to get bored or discouraged. Consistent review is built into lessons so students are able to master and reinforce their math skills. Students can see their progress and this helps increase their confidence and build self-esteem.
This book covers the four decimal operations and shows the close relationship that exists between decimals, fractions, and percents. Students will learn much more than computational skills. Practical, real-life problem solving will equip students with all the necessary tools for success in future math classes as well as real-life situations.
This book covers all four operations for fractions. Specific and easy-to-follow instructions ensure that students master the "dreaded" fraction. With consistent, built-in review included in each lesson, students will conquer this topic that gives many students so much difficulty. Each book also contains plenty of practical, real-life problem solving.
This book will provide students with al the essential geometry skills. Vocabulary, points, lines, planes, perimeter, area, volume, and the Pythagorean theorem are just some of the topics that are covered. There is plenty of practical, real-life problem solving that shows students the importance of geometry in the real world.
This book is a must for students who are about to start their first algebra class. Exponents, scientific notation, probability and statistics, equations, algebraic word problems, and coordinate systems are just some of the topics covered. Learn and master the essential topics that will ensure success in algebra and beyond.
This book will ensure that students master the "much-feared" math word problem. Learn to apply the math operations to real-life situations. Included is a short review for whole number, fraction, and decimal operations. The book begins with simple one-step problems and progresses slowly to multi-step problems, with plenty of built-in reviews to ensure mastery and success. | 677.169 | 1 |
Category Archives: NewsWe are excited to announce that The Text and Academic Authors Association (TAA) awarded the first edition of Big Ideas Math: A Common Core Curriculum Algebra 1, by Ron Larson and Laurie Boswell, the TAA 2013 Most Promising New Textbook award.
The Most Promising New Textbook award was created in 2012, to recognize current textbooks and learning materials, still in their first editions. Judges are published textbook authors.
As part of our continual effort to improve your experience on our website, we are making an update to the Teachers tab that will be implemented tomorrow on the Big Ideas Math website.
Below is a preview of the new Teachers tab. Here you will notice that we have added a light blue box with three separate divisions that contain the new features.
On the left-hand side, you will see Steps 1 and 2. Step 1 prompts you to enter your login information and Step 2 prompts you to select a Big Ideas Math book from the drop-down menu.
Once you have successfully logged in, the Step 1 area will become gray. You will see two options under your username including My Account and Logout.
Next on Step 2, you will select your textbook from the drop-down menu. Once you have selected a book, the Teacher Resources below the blue box will be unlocked. You can select a different textbook at any time.
You will also see the Technical Support information on the right-hand side of the blue box.
We hope that these updates will make logging in and selecting a textbook on the Teachers tab more efficient for you!
If you have any questions regarding the changes made to the website, please contact us at (877) 552 – 7766.
The Big Ideas Math online textbooks have been updated from PDF format to a new interactive format that displays in your internet browser. Designed with multiple devices in mind, the new format is compatible with iPads and interactive whiteboards in addition to computers.
You can now easily navigate the textbook in your internet browser through the menu on the left-hand side of the screen after you select a book. You can also access the numerous student resources and tools from the menu. Everything you need is right at your fingertips!
As the Common Core State Standards are sweeping the nation, more and more teachers and parents are wondering about the assessment of the new curriculum.
PARCC, The Partnership for Assessment for Readiness for College and Careers, is a consortium of states working together to develop a common set of K-12 assessments. The goal is to dramatically increase the rates at which students graduate from high school prepared for success in college and the workplace.
It will provide students, educators, policymakers and the public tools needed to identify whether students are on track for postsecondary success and where gaps need to be addressed before students enter college or the workforce. These new assessments will assess the full range of the Common Core Standards, both content and practices.
The assessment system will be comprised of four components.
Two summative, required assessment components designed to:
- Make "college- and career-readiness" and "on-track" determinations,
- Measure the full range of standards and full performance continuum, and
- Provide data for accountability uses, including measures of growth.
Two non-summative, optional assessment components designed to:
- Generate timely information for informing instruction, interventions, and professional development during the school year.
- An additional third non-summative component will assess students' speaking and listening skills.
Assessments will be computer based and will be graded via computer scoring and human scoring. PARCC assessments will begin during the 2014-15 school year. | 677.169 | 1 |
Summer 2012 quarter
In this course, we'll study standard topics in discrete mathematics including logic and proof; sets, relations, and functions; combinatorics; basic probability; and graph theory. Along the way, we'll focus on skills and techniques for problem-solving. This is an excellent course for teachers and future teachers, people wanting to broaden their mathematical experience beyond algebra, and students considering advanced study in mathematics and/or computer science. | 677.169 | 1 |
Signature Math
Signature Math is a blended-learning model that combines one-to-one computing and teacher-led, small-group, hands-on learning activities. Students begin by taking an assessment to determine their level of knowledge of Algebra Readiness and Algebra I concepts and are then assigned Individualized Prescriptive Lessons™ (IPLs™) designed to build mastery of the math concepts. Learning is reinforced and the concepts are applied in hands-on, culminating group activities, or CGAs, which are teacher led in a whole-class learning environment. As students master each concept, they advance to the next lesson series. | 677.169 | 1 |
ACT Math
ACT Math is a collection of pre-algebra, elementary algebra, intermediate algebra, geometry, and trigonometry; basically all the courses that should have been taking by the end of the eleventh grade year.
Algebra 1
Expanding basic algebra concepts in understanding linear equations. This course also focuses on simplifying expressions, solving equations and inequalities, using numerical representations, along with graphical representations and algebraic notation.
Algebra 2
Algebra 2/3 develops advanced algebra skills such as systems of equations, advanced polynomials, imaginery and complex numbers, quadratics, and concepts and includes the study of trigonometric functions,logarithmic and exponential equations, and introduces matrices and their properties.
Elementary Math
Elementary math is basic computation skills such as adding, subtracting, multiplying and dividing. Also focus on working with fractions, greatest common factors, least common multiples, and primes. There is some introduction of geometry and algebra as well.
GED
The GED Mathematics Test assesses an understanding of mathematical concepts such as problem-solving, analytical, and reasoning skills; focuses on Numbers Operations and Number Sense, Measurement and Geometry, Data Analysis, probability, and algebra.
Eric C.
Erica
has consented to a background check to be run by Lexis Nexis upon request for
$7.99. You will be able to run a background check after you email
Erica
with your tutoring inquiry. | 677.169 | 1 |
Here are some words of explanation, advice, and motivation from past students:
I thought it was strange, and impossible, on the first day of class, when Professor Chris told us that we would learn to see basically any life situation in a mathematical way. Well, he was right.
This course has been very challenging for me. The homeworks are very long; it usually took me many hours to complete them. I had to learn a new language, Mathematica, on the computer. I remember thinking that I would not survive the course.
Well, I did survive it, and I'm very glad that I did. Professor Chris's course has opened up my eyes to a whole world of math. Math really is everywhere, though I wouldn't have known that if it weren't for Math 245 (math modeling). I worked in a study group throughout the semester with my two friends, and that helped a lot. I didn't realize it at the time, but as the semester went on I became more used to the course, and it became a little easier. Now, I see how the course really helps in every day life. As a math student, we are encouraged to see math and its applications in our everyday lives. I now cannot leave math alone--I sit in the waiting room at the doctor's office and write out Mathematica loops in my head, which can model the situation in the waiting room. It sounds insane, but it really is awesome.
My advice to you, a future student: realize all that this course has to offer and do not give up. Work hard because it is worth it. Pick a project topic you are sincerely interested in. Complete the Mathematica tutorials. Go to Prof. Chris early on if you need help. Do not give up. You can do it, and you should!
Good luck!
At the beginning of the semester, you might be confused about how one topic relates to the next, and this is understandable. This is because you are learning topics which are the
tools for mathematical modeling.
The goal of this class is to teach you how to build a model on your own, so you will be assigned a group project to complete. In completing the group project is when you will use all the modeling knowledge you learned during the semester. It is as if you have wood, nails, a hammer, and paint without knowing that you will be building a table. The only thing
you need to do is to learn how to use them. The materials in this example are the
different topics in the lessons, and the wooden table is your mathematical model.
Before you start to build your table, you need to learn how to use the tools. The same is true in mathematical modeling.
I hope that you will learn a lot in professor Hanusa's class. Have a great semester.
This class is not a hard class but requires a lot of thinking. Most math classes you can jump right in and start doing a problem, while this class is different. Before you start doing any problem you will first need to think about what the question is asking you and how you are going to answer the question. Then you can start actually doing the question and once you are done and have completed the question Professor Hanusa asks you to write out in words what you have done. That the reason why the homework can sometimes take so long.
While doing the homework make sure you are thorough. At the time writing out everything in words can be very annoying but once it comes on the test, it will be very helpful. Sometimes in other classes you look back at your notes and homework assignments and you do not remember why you did something. But when you have the words written out it makes it a lot easier to study from.
When it comes the notes in class, Professor Hanusa will give you power point slides. But the slides are not necessarily sufficient; you should also be paying attention in class and taking notes onto the slides. There are things that Professor Hanusa will say but aren't on the slides. You should also write down the examples that he does in class because they will be helpful when are trying to study.
This class was different from any other math class you have ever taken
because it teaches you the math you already know and REALLY connects
them to the real world. It breaks the mold of the typical real world
connection (such as "hair grows at an exponential rate.") It helps to
explain the other factors that our teachers usually brushed of with the
excuse "that is beyond the scope of this course." This gives you the
power to include the things you really want to study and ignore the
things you don't as long as it is backed by logical reasoning.
Though this class will require a lot of time devoted to homework,
studying, and the main mathematical modeling project, looking back will
reveal a picture that over the course of time the class has painted. As
long as you do your homework, come prepared to class, study for tests
and prepare the mathematical modeling project well, this class will pay
off in the end. This class is like a blender, during the class your mind
is constantly turning but when the blender is off, you realize that
something else was created.
Dear Future Mathematical Modeling Student,
Hi. You're probably sitting in your chair and wondering what to expect from this class. If you're like me, you probably signed up for this class because you need math credits and it was one of the only courses with seats still open. In that case you just know you need to do well no matter what the course entails. Or maybe you picked the class because the two sentences describing it on the Queens College registration website piqued your interest. Maybe it just fit in well with the rest of your schedule. In any case, here's what you need to know about succeeding in Professor Hanusa's Mathematical Modeling course.
The first thing you need to know is that there is a lot of writing involved, much more so than you've probably ever had to do for another math class. And I'm not talking about writing numbers, equations, and formulas, though that is part of it. In addition to those, for your homework assignments you will be writing very detailed explanations about concepts learned in class and explanations about how you used them. You need to brush up on your sentence structuring and grammar to make sure you can accurately convey the points you'll be trying to make. If a homework problem does involve numbers and formulas, in addition to actually solving the equation you'll need to explain your methodology and why you set things up the way you did to solve the problem.
All of the notes will be posted online on the course website. Actually, pretty much everything you need to know about the course, from the general syllabus to homework assignments to due dates, will be on the site. It's an extremely helpful resource. Use it well. Professor Hanusa posts the notes online before the class he uses them in but I recommend bringing a notebook and writing the notes every day, using the online notes as a backup only. From a personal standpoint, I find it easier to understand things and remember them later when I write them myself. Sometimes there are typos too, so it's easier to just write things correctly instead of trying to remember what in the online notes needed correcting.
You will also use Mathematica, a very powerful mathematical software program. This will be essential to your studies, as the professor uses it A LOT in the notes and homework assignments. If you have ever taken a computer programming course you will know what to expect from Mathematica, as it's very similar to Java and C++ in certain respects. If you've never taken a programming course I will tell you now, prepare to feel in over your head. When you're asked to modify code or write your own it gets very complicated very quickly, especially if you have a full course load and don't have time to devote to mastering the nuances of the programming language.
There's also a very large project due towards the end of the year. The first time you see it, it will appear daunting, and it does not get talked about in class until after the first midterm. DO NOT PANIC! You will learn about the concepts that you will apply for the project during the semester, so talking about it any earlier will only confuse you. My advice would be that as the first half of the semester progresses, start giving some thought as to what experiments you could focus your project on that would use concepts that get introduced. Also, I wouldn't recommend doing the project with more than one other person. You're allowed to team up with up to four people, but I have always found that it's easier to make sure everyone contributes evenly the fewer people you have. Do not go this one alone.
In summary, prepare to spend a lot more time on this class than you thought you would. It's a very interesting class, but you need to make a significant investment to get the most out of it. Time management will be key, especially if you have other writing intensive courses and no freedom to shift your schedule around, like I did. If I can get through this class, then so can you. Good luck! | 677.169 | 1 |
Love it! My students and my own children can't wait to get on the computer, and I then have a hard time stopping them from using the site!"
Teacher, Illinois, U.S.A.
We are pleased to announce the arrival of Algebra on IXL!
Our new algebra content is all that you'd expect from the Web's most comprehensive math site: no fewer than 240 practice skills, infinite computer-generated questions, and a vibrant, game-like format that can engage students of any age.
We have everything you'll need for your algebra class this year, and we mean EVERYTHING. IXL covers skills from basic operations and one-step linear equations to advanced material like matrices and quadratic functions. Your students will learn each concept from every angle as they encounter fill-in questions, word problems, and our ever-popular interactive graphing skills. IXL features question-specific, step-by-step explanations for even the toughest algebra skills, and immediate feedback helps students solidify understanding at every turn.
Our 30-day free trial is the perfect way to see what's new at IXL and get your students excited about algebra at the same time! Click here to sign up.
If you have any questions about our new algebra content, or IXL in general, we'd love to hear from you! E-mail us anytime at [email protected]. | 677.169 | 1 |
How to Navigate
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A: This is the menu bar. It appears in every page of The Net Equation, letting you move about our site easily and quickly. Click on a link to go to the main page of that section.
B: The title banner for each section appears here. This is a convenient reminder of which area you are currently in.
C: This side table is the Section Navigator. It acts as a specialized menu for the section you are in, as yet another method of making navigation easier. Eash section will have a complete listing of topics in the Section Navigator, and it appears in the same place on every page, immediately below the main menu. In sub-sections, you can also navigate via "next", "back", and "index" links located at the bottom of the page.
D: This area contains the text of the page. All lessons, example problems, diagrams, and other information will appear in this area. The text section can be quite long, so scroll bars will appear if needed.
E: This link table appears at the bottom of each page. It, like the menu bar on the left, contains links to every major section of our web site. A "back-to-top" link is also provided, allowing you to jump to the top of the page without scrolling. Finally, "Back," "Index," and "Next" links are included to let you move back a page in the lesson areas, return to the section's main page, or continue to the next lesson. | 677.169 | 1 |
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Description
Pearson Mathematics homework program for Year 9 provides tear-out sheets which correspond with student book sections, providing systematic and cumulative skills revision of basic skills and current class topics in the form of take-home exercises. With over 120 double-sided worksheets, Pearson Mathematics provides a complete homework program. Worksheets contain basic skills and revision questions, and leave enough room for student working.
Table of contents
Financial mathematics
Pythagoras's Theorem
Algebra
Measurement
Linear relationships
Geometric reasoning
Trigonometry
Statistics and probability
Non-linear graphs
Features & benefits
A complete homework program
Specifically developed and written for the Australian Curriculum
Over 120 double-sided worksheets which can be torn out, so they are easy to take home
Basic skills revision and current classroom learning reinforcement
Plenty of space for student working
Target audience
Suitable for Year 9 | 677.169 | 1 |
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HINARI requires you to log in before giving you full access to articles from Algebra of Polynomials | North-Holland Mathematical Library, 5 Algebra of Polynomials | North-Holland Mathematical Library, 5 as a member of the public. You will not have full access, unless your institution participates in other arrangements. | 677.169 | 1 |
independent activities, covering aspects of this area of statistics for those studying the S1 Module of AS Level Statistics. The tasks are designed to encourage students to apply their knowledge of binomial distributions to solve challenging problems, giving them the opportunity toCarom Maths provides this resource for teachers and students of A Level mathematics.
This presentation looks at the theorem of cross-ratio, of four complex numbers, which is of great interest in a field of mathematics known as projective geometry and has an ancient history.
The activity is designed to explore aspects of the…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation investigates the value of π in different types of geometry and provides a link for students to experiment with Hyperbolic geometry. The classic definition of a distance function, or metric, is given and a version of the Manhattan…
Carom Maths provides this resource for teachers and students of A Level mathematics.
Cyclotomic polynomials are explained in this presentation, which uses a regular polygon to illustrate the complex roots of unity and establish which of those are primitive, before demonstrating some intriguing algebra.
The activity is designed…
Carom Maths provides this resource for teachers and students of A Level mathematics.
The Fibonacci sequence is an example of a linear recurrence relation (LRS). A matrix is used to calculate future terms, as well as running the sequence backwards to see how many zeroes appear. Algebra is used to prove the maximum number of zeroes…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation shows how, when placing triominoes onto a chessboard, there is always one empty square. An algebraic proof is developed to show that the empty square will always appear in the same location, or in one of its rotations.
The…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation uses the fact that 1729 is the smallest number that can be expressed as two cubes in two different ways to introduce the topic of Elliptic curves, which are used more and more in the field of number theory.
The activity is…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation begins by introducing students to the technique used to find the radical of an integer before progressing to the ABC conjecture, which is recognised by mathematicians as being an important unsolved problem in number theory.
The…
Carom Maths provides this resource for teachers and students of A Level mathematics.
A triangle can have more than one centre and this presentation demonstrates the application of vectors, in three different situations, to show that the circumcentre, the centroid and the orthocentre of a triangle are indeed positioned at the centre…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation demonstrates how the interesting idea of Hikorski triples was developed from writing a GCSE Equations worksheet in 2002. The triples are identified as (p, q, pq+1/p+q).
The activity is designed to explore aspects of the subject…
Carom Maths provides this resource for teachers and students of A Level mathematics to explore aspects of the subject which may not normally be encountered, to encourage new ways to approach a problem mathematically and to broaden the range of tools that an A Level mathematician can call upon.
This presentation on Quadratic reciprocity…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation begins with a reminder of some transformations of the plane before introducing the less well known transformation of Inversion. Students are able to experiment with an Autograph file to explore this concept and the Steiner Chain…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation provides a spreadsheet, which models a population of mice, and allows students to vary the input and thus change the behaviour of the model. There are some surprising discoveries to be made when the results are analysed.
The…
Carom Maths provides this resource for teachers and students of A Level mathematics.
There are six ways to write the total edge length, total surface area and volume of a cube, or cuboid, in order of size. This presentation challenges students to find a cube for each order. Autograph files are provided to help with this problem.
Thw…
Carom Maths provides this resource for teachers and students of A Level mathematics.
Colouring maps, so that no two countries sharing a border are shaded with the same colour, is the focus of this presentation, which includes both the history and mathematical proofs to this problem.
This activity is designed to explore aspects…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation introduces the Conics and provides a Geogebra file for students to explore how, by changing the eccentricity of a curve, the locus of a point is altered. They are also challenged to vary the constants of an equation to see how…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation provides students with an opportunity to explore their own grasp of logic and introduces a logical tautology called Modus Tollens.
This activity is designed to explore aspects of the subject which may not normally be encountered,…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation describes the concept of Tangles, which were the idea of John Conway. Students are given some rules from which they create the tangle representing 2/5, before creating their own tangle numbers for others to untangle. The activity…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation explores the effect of placing a single tile into an existing pattern which tessellates, referred to as a pearl tiling. New descriptions and notation are introduced and questions regarding regular polygons, polyominoes, and quadrominoes…
Carom Maths provides this resource for teachers and students of A Level mathematics.
In this presentation students are given a function and explore the mapping of natural numbers, as well as being challenged to find the inverse function. A link to the online encyclopaedia of integer sequences is provided for students to check…
Carom Maths provides this resource for teachers and students of A Level mathematics.
The Game of Life is a simulation of how a population might grow, if subject to a few simple rules. Initially students try the rules out for themselves on squared paper, before following a link to a computer program that shows the patterns which…
Carom Maths provides this resource for teachers and students of A Level mathematics.
This presentation explores the Mandelbrot Set, a topic from Chaos Theory, which involves complex numbers and the Argand diagram. Following a set of rules, students use a spreadsheet to establish if complex numbers, on the Argand diagram, are within… | 677.169 | 1 |
Cost = number of items
x price per item
Income = hours
worked x wage per
hour
Value = number of
items x value per item
Money problems
involve the use of
several
transformations.
Money problems
involve the use of the
5-step plan when they
are solved.
One and two
step equations
Using several
transformations
to solve
equations
Investment Formula:
Interest = Amount
invested x Interest
Rate
Investment problems
use charts to help in
organization of
information.
Investment problems
use algebraic fractions
in finding their
solutions.
Investment problems
involve the use of the
5-step plan when they
are solved.
Rate of Work
Formula: Work done
= Work rate x Time
worked
Rate of Work
problems use charts to
help in organization of
information.
Rate of Work
problems use algebraic fractions in finding
their solutions .
Rate of Work
problems involve the
use of the 5-step plan
when they are solved.
Catalog Description: This course focuses on solution methods for
quadratic equations and inequalities, graphs of
quadratic equations , quadratic models, and the use of these methods in problem
solving. A student who is required by the
college to take this course must pass it with a C or better before being allowed
to take a higher-level course in the mathematics sequence .
Course Outcomes: After the successful completion of this course, you will
be able to solve application problems using
the following skills:
ASK Outcome: This course will provide an opportunity to develop your
skills, not only as a mathematician, but also as
an independent learner . Ask outcome: to become competent in math and
statistical methods
Attendance: Regular and punctual class and laboratory attendance is
required for success. Students who are not fully
succeeding and are not regularly attending may be withdrawn from this course.
Tardiness is a form of absenteeism and
may be regarded as grounds for withdrawal if significant progress is not shown.
Assessment: Course grade determination will include at least three,
one-hour exams and a comprehensive final exam.
Laboratory assignments, homework, quizzes, projects, and class participation may
also be considered. Your instructor
will provide you with specific methods of assessment and evaluation plus a
tentative schedule of topics that will include
test dates.
IP: An IP is an earned grade, awarded only to
students in development, who demonstrate significant progress in a
course without achieving a level of skill sufficient to be successful in their
next level. A person who has been awarded an
IP twice before in this course is not eligible for a third.
The Value of Integrity: Northwest Vista College values integrity;
therefore cheating will not be tolerated. Please
read the complete set of new policies and procedures regarding
academic integrity.
ADA Disability Statement: As per section 504 of the Vocational
Rehabilitation Act of 1973 and the Americans with
Disabilities Act (ADA) of 1990, if a student needs an accommodation, contact
Sharon Dresser at 348-2020.
Phones/Texting/Laptops/Etc. : You are not allowed to have your
phone/laptop on your desk or
in your hands during class. If you are on the phone or texting during class you
will be asked
to leave and 10 points will be taken off your next exam. If you are expecting an
important
call just let me know before class begins.
Sleeping/Being Disruptive: If you are too tired to stay awake in this
course then don't come to
class. If you are sleeping in class I will ask you to leave class and 5 points
will be taken off
your next exam. If you are being disruptive in class I will ask you to leave
class and 5 points
will be taken off your next exam.
Strand Trace Algebra
Performance Indicators Organized by Grade Level and Band under Major
Understandings
Multiply a binomial by a monomial or a binomial (integer
coefficients).
8.A.9 Var. & Express
Divide a polynomial by a monomial (integer coefficients).
8.A.10 Var. & Express
Factor algebraic expressions using the GCF.
8.A.11 Var. & Express
Factor a trinomial in the form ax2 + bx + c; a=1 and c having no
more than three sets of factors.
8.A.12 Eqns. & Ineqs.
Apply algebra to determine the measure of angles formed by or
contained in parallel lines cut by a transversal and by intersecting
lines.
8.A.13 Eqns. & Ineqs.
Solve multi-step inequalities and graph the solution set on a number
line.
8.A.14 Eqns. & Ineqs.
Solve linear inequalities by combining like terms, using the
distributive property, or moving variables to one side of the inequality
(include multiplication or division of inequalities by a negative
number).
Performance Indicators Organized by Grade Level and Band under Major
Understandings
Students will perform algebraic procedures accurately.
A.A.12 Var. & Express
Multiply and divide monomial expressions with a common base, using
the properties of exponents.
A.A.13 Var. & Express
Add, subtract, and multiply monomials and polynomials.
A.A.14 Var. & Express
Divide a polynomial by a monomial or binomial, where the quotient
has no remainder.
A.A.15 Var. & Express
Find values of a variable for which an algebraic fraction is
undefined.
A.A.16 Var. & Express.
Simplify fractions with polynomials in the numerator and denominator
by factoring both and renaming them to lowest terms.
A.A.17 Var. & Express
Add or subtract fractional expressions with monomial or like
binomial denominators.
A.A.18 Var. & Express.
Multiply and divide algebraic fractions and express the product or
quotient in simplest form.
A.A.19 Var. & Express.
Identify and factor the difference of two perfect squares.
A.A.20 Var. & Express.
Factor algebraic expressions completely, including trinomials with a
lead coefficient of one (after factoring a GCF).
A.A.21 Eqns. & Ineqs.
Determine whether a given value is a solution to a given linear
equation in one variable or linear inequality in one variable.
A.A.22 Eqns. & Ineqs.
Solve all types of linear equations in one variable.
A.A.23 Eqns. & Ineqs.
Solve literal equations for a given variable.
A.A.24 Eqns. & Ineqs.
Solve linear inequalities in one variable.
A.A.25 Eqns. & Ineqs.
Solve equations involving fractional expressions.
A.A.26 Eqns. & Ineqs.
Solve algebraic proportions in one variable which result in linear
or quadratic equations.
A.A.27 Eqns. & Ineqs.
Understand and apply the multiplication property of zero to solve
quadratic equations with integral coefficients and integral roots.
A.A.28 Eqns. & Ineqs.
Understand the difference and connection between roots of a
quadratic equation and factors of a quadratic expression.
Determine the center-radius form for the equation of a circle in
standard form.
A2.A.48 Coordinate
Write the equation of a circle, given its center and a point on the
circle.
A2.A.49 Coordinate
Write the equation of a circle from its graph.
A2.A.50 Coordinate
Approximate the solution to polynomial equations of higher degree by
inspecting the graph.
A2.A.51 Coordinate
Determine the domain and range of a function from its graph.
A2.A.52 Coordinate
Identify relations and functions, using graphs.
A2.A.53 Coordinate
Graph exponential functions of the form y = bx for positive values
of b, including b = e.
A2.A.54 Coordinate
Graph logarithmic functions, using the inverse of the related
exponential function.
A2.A.55 Trig Fcns
Express and apply the six trigonometric functions as ratios of the
sides of a right triangle.
A2.A.56 Trig Fcns
Know the exact and approximate values of the sine, cosine, and
tangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles.
A2.A.57 Trig Fcns
Sketch and use the reference angle for angles in standard position.
A2.A.58 Trig Fcns
Know and apply the co-function and reciprocal relationships between
trigonometric ratios.
A2.A.59 Trig Fcns
Use the reciprocal and co-function relationships to find the value
of the secant, cosecant, and cotangent of 0º, 30º, 45º, 60º, 90º, 180º,
and 270º angles.
A2.A.60 Trig Fcns
Sketch the unit circle and represent angles in standard position.
A2.A.61 Trig Fcns
Determine the length of an arc of a circle, given its radius and the
measure of its central angle.
A2.A.62 Trig Fcns
Find the value of trigonometric functions, if given a point on the
terminal side of angle θ.
A2.A.63 Trig Fcns
Restrict the domain of the sine, cosine, and tangent functions to
ensure the existence of an inverse function.
A2.A.64 Trig Fcns
Use inverse functions to find the measure of an angle, given its
sine, cosine, or tangent.
A2.A.65 Trig Fcns
Sketch the graph of the inverses of the sine, cosine, and tangent
functions.
A2.A.66 Trig Fcns
Determine the trigonometric functions of any angle, using
technology. | 677.169 | 1 |
Course Requires a Media Kit to be Purchased by Course Sponsor
(see additional details below):
No
Description:
The course emphasizes the relationships among geometric figures and concepts and applies them to real world applications.
The concepts of points, lines, planes, parallel lines, congruence, similarity, polygons, coordinate geometry, area, volume, circles, and right triangle trigonometry will be explored. Students will apply logical thinking throughout the course, without an emphasis on formal proofs. Students are expected to have, and be able to use, solid algebra skills to solve problems in each topic area. Internet resources are used throughout the course to explore and instruct the topics presented.
Media:
There is no text for this course. All instruction materials are either provided in the course or located on content related websites. Geogebra, a free geometry software application, will be downloaded and used for constructions and investigations | 677.169 | 1 |
How do I go about studying Further Mathematics?
If possible you should study Further Mathematics through timetabled classes at your school or college. Where this is not possible, with the cooperation of your school or college, the Further Mathematics Support Programme can help you, either by providing all of your Further Mathematics tuition, or by sharing the teaching with your school or college.
Taking AS Further Mathematics in year 13 is an excellent option if you have decided during year 12 that you are going to apply to for a mathematics-rich degree course at university. It will really help you on your university degree course and will look very impressive on your university application form.
If you are able to enrol in a Further Mathematics class at your school or college, but cannot access all of the options you would like to study, you could share your tuition between your school/college class and tuition through the FMSP.
The flowchart below will help you to decide how best to access Further Mathematics tuition.
Updated by CS 02/08/09
Quotes
"It's been an excellent experience, introducing me to some more interesting mathematics, and making my regular Maths stronger." | 677.169 | 1 |
Seat Lookup for math exams
Using this
form, you may look up seating assignement for upcoming exams from math
classes such as Calculus 131 and 132. Please input your last name, the
course number (e.g. 131,132), and the exam number (f for final). | 677.169 | 1 |
Mathematica
Mathematica is a program and computer language for use in mathematical applications.
More information
Mathematica can be used as a calculator with a much higher degree of precision than traditional calculators. Mathematica can perform operations on functions, manipulate algebraic formulas, and do calculus. Mathematica is also able to produce both two- and three-dimensional graphs. | 677.169 | 1 |
Complete Idiot's Guide to Calculus (Complete Idiot's Guides)
Synopses & Reviews
Publisher Comments:
According to figures released by ACT Inc., many more U.S. high school students are taking courses in mathematics than was the case a decade ago. In fact, the portion of college-bound students taking calculus increased from 16 percent in 1987 to 27 percent in 2000. Let's face it, most students and adults who take calculus do so not for the fun of it, but rather to advance within a job or fulfill a degree requirement. The Complete Idiot's Guide to Calculus will take the sting out of this complex math by putting its uses, functions and limitations in perspective of what is already familiar to readers-algebra. Once readers have brushed up on their algebra and trigonometry skills, they'll be eased into the fundamentals of calculus.
Synopsis:
Cast off the curse of calculus!About the Author
W. Michael Kelley is a former award-winning calculus teacher and the author of The Complete Idiot's Guide to Calculus, The Complete Idiot's Guide to Precalculus, and The Complete Idiot's Guide to Algebra. | 677.169 | 1 |
Graphing Calculators Replace PCs For Mathematics Instruction
04/01/97
Looking at the Texas Instruments TI-92 mathematics instructional tool, itís easy to see why both students and teachers like it so much. For students who grew up handling video game controllers , the unit fits comfortably in both hands, with a horizontal orientation that suits its 240 x 128-pixel display. It even has an eight-direction thumbpad like many video game controllers, complementing its full QWERTY keyboard, separate function keys and numeric keypad.
For teachers, the TI-92ís functionality and versatility are just some of the reasons for its popularity. It handles a broad range of math from algebra through calculus including interactive geometry, symbolic manipulation, statistics and even 3D graphing.
Best of all, the TI-92 has been priced with the notion of giving each student their own, powerful mathematics tool. Sounds great, right? But d'es it all come together in the classroom? Can this hand-held unit replace, or even supplement, traditional PC-based mathematics instruction? In answer to these very important questions, Kathy Longhart, mathematics teacher at Flathead High School in Kalispell, Montana, would give a resounding ìYes!î
PCs vs. Calculators
In 1992, while teaching a new curriculum that had been specifically written to take advantage of technology, Longhart and the teachers at Flathead High School tried using traditional computers in a lab-based setting to teach mathematics. However, this initial effort did not prove to be a very successful experiment.
With four kids to a computer, each student had to share time and space, making sure the technologically adept or mathematically inclined didnít ìhogî the machine. The more reticent students would sit back and, while not participating, let those so inclined make good use of the lab time. And when class was over, the computers, with their math software, stayed put, while the students who needed them went home.
Enter the TI-92. Now, Flathead High School has three classroom sets of these mathematics tools, which works out to about 80 or 90 calculators. Longhart uses them in all three of the mathematics classes she teaches, starting with her Math 2 class.
ìThe TI-92 has a symbolic manipulator, data package, geometry package and graphing package in one unit, and I like to pick problems where we do a lot of moving between the packages,î she says. ìBy incorporating all of the packages, we can look at more powerful problems.î The calculator even has a two-graph mode for creating separate graphing environments, letting one compare different functions and graphs.
Longhart mentions that the unit acts as a kind of ìelectronic chalkboard, where students can manipulate problems by hand.î They type problems into the calculator using its standard-style keyboard and keypad, and the calculator lets them do the same manipulations that they would do on paper, with one important difference: it lets them instantly see if they are correct.
Instead of shuffling through a time-consuming calculation, marking up an entire page and finally realizing that a minor arithmetic mistake screwed up what was otherwise a correct procedure, the calculator simply d'esnít make these kinds of errors, letting students concentrate on learning procedures and formulas rather than worrying about every arithmetic problem that needs to be solved.
Students Vote with Own Money
After getting used to the TI-92s, students actually ask Longhart to go get them, she says. "At first, my Math 2 class was a bit frustrated because I hadnít given them enough guidance," she admits. But after showing them how helpful the calculators could be, her students had a change of heart. About 40 or 50 students have even purchased their own TI-92, which speaks spades about the students' opinions of the calculator.
For those who donít have their own, the school lets them check out calculators to take home for homework. "I love [the TI-92], I think it's great. It's easy to learn, very user-friendly," mentions Longhart. "You don't get bogged down teaching a machine, you can focus on teaching math." | 677.169 | 1 |
Designed to help Advanced Placement students succeed in their studies and achieve a '5' on the AP Exam, AP Achiever for European History provides: A thorough explanation of course expectations, exam parameters, preparation suggestions, as well as comprehensive tips on writing essays for the document-based and free-response section of the Exam, all to help your students maximize studies and time. Each chapter includes a thorough ...
Glencoe Pre-Algebra is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique language of mathematics, so that they are prepared for every type of problem-solving and assessment situation.
We learn often in life, but only once as a child. This popular book will help future teachers make the most of this special time. Here is complete coverage of how children learn, what they can learn, and how to teach them. The focus is on creating a child-centered curriculum that addresses children's needs in all developmental areas—physical, social, emotional, creative, and cognitive. The authors provide a wealth of meaningful teaching ...
Six-Way Paragraphs , a three-level series, teaches the basic skills necessary for reading factual material through the use of the following six types of questions: subject matter, main idea, supporting details, conclusions, clarifying devices, and vocabulary in context.
THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! Glencoe Pre-Algebra is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments.
Chemistry: Matter and Change is a comprehensive chemistry course of study, designed to for a first year high school chemistry curriculum. The program incorporates features for strong math-skill development. The Princeton Review has review and authenticated all in-text assessment items to validate them to be unbiased.
Applying AutoCAD 2008 introduces new features and enhancements to existing capabilities. What's New in 2008? * A new default workspace called 2D Drafting & Annotation has been added. * The new 2D Drafting and Annotation workspace employs a new Dashboard containing panels and toolbars for drawing and dimensioning in two dimensions, while new panels and features improve the Dashboard for the 3D Modeling workspace. * New dimensioning ... | 677.169 | 1 |
Almost all adults suffer a little math anxiety, especially when it comes to everyday problems they think they should be able to figure out in their heads. Want to figure the six percent sales tax on a $34.50 item? A 15 percent tip for a $13.75 check? The carpeting needed for a 12½-by-17-foot room? No one learns how to do these mental calculations in school, where the emphasis is on paper-and-pencil techniques. With no math background required and no long list of rules to memorize, this book teaches average adults how to simplify their math problems, provides ample real-life practice problems and solutions, and gives grown-ups the necessary background in basic arithmetic to handle everyday problems quickly.
This comprehensive volume covers a wide range of duality topics ranging from simple ideas in network flows to complex issues in non-convex optimization and multicriteria problems. In addition, it examines duality in the context of variational inequalities and vector variational inequalities, as generalizations to optimization. Duality in Optimization and Variational Inequalities is intended for researchers and practitioners of optimization with the aim of enhancing their understanding of duality. It provides a wider appreciation of optimality conditions in various scenarios and under different assumptions. It will enable the reader to use duality to devise more effective computational methods, and to aid more meaningful interpretation of optimization and variational inequality problems.
"An understanding of the relationship between the product and the process in election polling is often lost. This edited volume unites ideas and researchers, with quality playing the central role." --J. Michael Brick, PhD, Director of the Survey Methods Unit, Westat, Inc. Elections and Exit Polling is a truly unique examination of the specialized surveys that are currently used to track and collect data on elections and voter preferences. Employing modern research from the past decade and a series of interviews with famed American pollster Warren Mitofsky (1934-2006), this volume provides a relevant and groundbreaking look at the key statistical techniques and survey methods for measuring voter preferences worldwide. Drawing on the most current studies on pre-election and exit polling, this book outlines improvements that have developed in recent years and the results of their implementation. Coverage begins with an introduction to exit polling and a basic overview of its history, structure, limitations, and applications. Subsequent chapters focus on the use of exit polling in the United States election cycles from 2000-2006 and the problems that were encountered by both pollsters and the everyday voter, such as how to validate official vote count, confidentiality, new voting methods, and continuing data quality concerns. The text goes on to explore the presence of these issues in international politics, with examples and case studies of elections from Europe, Asia, and the Middle East. Finally, looking to the upcoming 2008 U.S. presidential election, the discussion concludes with predictions and recommendations on how to gather more accurate and timely polling data. Research papers from over fifty eminent practitioners in the fields of political science and survey methods are presented alongside excerpts from the editors' own interviews with Mitofsky. The editors also incorporate their own reflections throughout and conclude eaNew probabilistic model, new results in probability theoryOriginal applications in computer scienceApplications in mathematical physicsApplications in finance
use back cover copy A 'down-to-earth' introduction to the growing field of modern mathematical biology Also includes appendices which provide background material that goes beyond advanced calculus and linear algebra
The 11th International Workshop on Dynamics and Control brought together scientists and engineers from diverse fields and gave them a venue to develop a greater understanding of this discipline and how it relates to many areas in science, engineering, economics, and biology. The event gave researchers an opportunity to investigate ideas and techniques from outside their own fields of expertise, enabling a cross-pollination of dynamics and control perspectives. Now there is a book that documents the major presentations of the workshop, providing a foundation for further research.
The range and diversity of papers in Dynamical Systems and Control demonstrate the remarkable reach of the subject. All of these contributed papers shed light on a multiplicity of physical, biological, and economic phenomena through lines of reasoning that originate and grow from this discipline.
The editors divide the book into three parts. The first covers fundamental advances in dynamics, dynamical systems, and control. These papers represent ideas that can be applied to several areas of interest. The second part deals with new and innovative techniques and their application to a variety of interesting problems, from the control of cars and robots, to the dynamics of ships and suspension bridges, and the determination of optimal spacecraft trajectories. The third section relates to social, economic, and biological issues. It reveals the wealth of understanding that can be obtained through a dynamics and control approach to issues such as epidemics, economic games, neo-cortical synchronization, and human posture control.
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the Écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated.Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.
This self-contained monograph provides systematic, instructive analysis of second-order rational difference equations. After classifying the various types of these equations and introducing some preliminary results, the authors systematically investigate each equation for semicycles, invariant intervals, boundedness, periodicity, and global stability. Of paramount importance in their own right, the results presented also offer prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one. The techniques and results in this monograph are also extremely useful in analyzing the equations in the mathematical models of various biological systems and other applications.
Each chapter contains a section of open problems and conjectures that will stimulate further research interest in working towards a complete understanding of the dynamics of the equation and its functional generalizations-many of them ideal for research projects or Ph.D. theses. Clear, simple, and direct exposition combined with thoughtful uniformity in the presentation make Dynamics of Second Order Rational Difference Equations valuable as an advanced undergraduate or a graduate-level text, a reference for researchers, and as a supplement to every textbook on difference equations at all levels of instruction.
Extending and generalizing the results of rational equations, Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures focuses on the boundedness nature of solutions, the global stability of equilibrium points, the periodic character of solutions, and the convergence to periodic solutions, including their periodic trichotomies. The book also provides numerous thought-provoking open problems and conjectures on the boundedness character, global stability, and periodic behavior of solutions of rational difference equations. After introducing several basic definitions and general results, the authors examine 135 special cases of rational difference equations that have only bounded solutions and the equations that have unbounded solutions in some range of their parameters. They then explore the seven known nonlinear periodic trichotomies of third order rational difference equations. The main part of the book presents the known results of each of the 225 special cases of third order rational difference equations. In addition, the appendices supply tables that feature important information on these cases as well as on the boundedness character of all fourth order rational difference equations. A Framework for Future Research The theory and techniques developed in this book to understand the dynamics of rational difference equations will be useful in analyzing the equations in any mathematical model that involves difference equations. Moreover, the stimulating conjectures will promote future investigations in this fascinating, yet surprisingly little known area of research.
Engage students in grades K–1 and build their confidence using Early Graphing: Hidden Pictures. This 64-page resource teaches essential early graphing skills through hands-on activities using popular kindergarten and first-grade themes. Students work to reveal hidden pictures while practicing reading order, fine-motor skills, attention to detail, concentration, and color words. They take pride in the finished product and look forward to the next one lesson. The lessons are ideal for independent practice, centers, and homework. This book aligns with state, national, and Canadian provincial standards.
Because elementary mathematics is vital to be able to properly design biological experiments and interpret their results. As a student of the life sciences you will only make your life harder by ignoring mathematics entirely. Equally, you do not want to spend your time struggling with complex mathematics that you will never use. This book is the perfect answer to your problems. Inside, it explains the necessary mathematics in easy-to-follow steps, introducing the basics and showing you how to apply these to biological situations. Easy Mathematics for Biologists covers the basic mathematical ideas of fractions, decimals and percentages, through ratio and proportion, exponents and logarithms, to straight line graphs, graphs that are not straight lines, and their transformation. Direct application of each of these leads to a clear understanding of biological calculations such as those involving concentrations and dilutions, changing units, pH, and linear and non-linear rates of reaction. Each chapter contains worked examples, and is followed by numerous problems, both pure and applied, that can be worked through in your own time. Answers to these can be found at the back.
Though Economics as a discipline arose in Great Britain and France at the end of the eighteenth century, it has taken two centuries to reach the threshold of scientific rationality. Previously, intuition, opinions, and conviction enjoyed equal status in economic thought; theories were vague, often unverifiable. It is no wonder, then, that bad economic policies ravaged entire nations during the twentieth century. In Economics Does Not Lie, noted French journalist Guy Sorman examines the state of economic affairs today. Virtually everywhere, the public sector has given ground to privatization and market capitalism. The results have been breathtaking. Opening economies and promoting trade have helped reconstruct Eastern Europe after 1990 and lifted 800 million people out of poverty across the globe. Economics Does Not Lie reveals that behind all this unprecedented growth is not only the collapse of state socialism but also a scientific revolution in economics–one that is as of yet dimly understood by the public but increasingly embraced by policymakers around the globe. No longer does economics lie; no longer would Baudelaire be able to write that "economics is a horror." For the mass of mankind, on the contrary, economics has become a source of hope.—whether it is a Fortune 500 company, a small accounting firm or a vast government agency—This books holds the keys to success for systems administrators, information security and other IT department personnel who are charged with aiding the e-discovery process.Comprehensive resource for corporate technologists, records managers, consultants, and legal team members to the e-discovery process, with information unavailable anywhere elseOffers a detailed understanding of key industry trends, especially the Federal Rules of Civil Procedure, that are driving the adoption of e-discovery programsIncludes vital project management metrics to help monitor workflow, gauge costs and speed the processCompanion Website offers e-discovery tools, checklists, forms, workflow examples, and other tools to be used when
Due to the increase in computational power and new discoveries in propagation phenomena for linear and nonlinear waves, the area of computational wave propagation has become more significant in recent years. Exploring the latest developments in the field, Effective Computational Methods for Wave Propagation presents several modern, valuable computational methods used to describe wave propagation phenomena in selected areas of physics and technology. Featuring contributions from internationally known experts, the book is divided into four parts. It begins with the simulation of nonlinear dispersive waves from nonlinear optics and the theory and numerical analysis of Boussinesq systems. The next section focuses on computational approaches, including a finite element method and parabolic equation techniques, for mathematical models of underwater sound propagation and scattering. The book then offers a comprehensive introduction to modern numerical methods for time-dependent elastic wave propagation. The final part supplies an overview of high-order, low diffusion numerical methods for complex, compressible flows of aerodynamics. Concentrating on physics and technology, this volume provides the necessary computational methods to effectively tackle the sources of problems that involve some type of wave motion.
Effective Experimentation is a practical book on how to design and analyse experiments. Each of the methods are introduced and illustrated through real world scenario drawn from industry or research. Formulae are kept to a minimum to enable the reader to concentrate on how to apply and understand the different methods presented.The book has been developed from courses run by Statistics for Industry Limited during which time more than 10,000 scientists and technologists have gained the knowledge and confidence to plan experiments successfully and to analyse their data. Each chapter starts with an example of a design obtained from the authors' experience. Statistical methods for analysing data are introduced, followed, where appropriate, by a discussion of the assumptions of the method and effectiveness and limitations of the design.The examples have been chosen from many industries including chemicals, oils, building materials, textiles, food, drink, lighting, water, pharmaceuticals, electronics, paint, toiletries and petfoods.This book is a valuable resource for researchers and industrial statisticians involved in designing experiments. Postgraduates studying statistics, engineering and mathematics will also find this book of interest.The EPUB format of this title may not be compatible for use on all handheld devices. | 677.169 | 1 |
Linear algebra and abstract algebra simultaneously?
Linear algebra and abstract algebra simultaneously?
Is this a good idea (provided the university will allow it)? I'll be going into my sophomore year at my university. But I'm unfamiliar with exactly how much linear algebra an intro course in abstract algebra would require. In hindsight I probably should have taken linear last semester, but scheduling issues meant that I would have had to sacrifice a lot to do it. I'm interested in doing a major in pure math and maybe going to grad school, and thus I want to get involved with research as early as possible. For this reason I want to start taking advanced courses soon, and abstract felt like a good place to start.
There are other math courses I could take instead, but most of them seem to require linear algebra, and the ones that don't either interfere with my schedule or are only offered spring semester. Since abstract algebra seems to build from fundamentals, it felt like a good option. Thoughts?
I think you hit a key point with "provided the university will allow it". Why would you take the recommendations of a bunch of folks on the internet over those of the university?
Courses in Linear vary from "a bag of computational tricks" all the way to "abstract lite". We can't tell how much rigor your particular university's version has, and we can't tell how much prior knowledge the abstract professor expects the students to walk in with. That's what prereqs and co-reqs are for. Taking them concurrently might work out, or you might get squashed like a bug. | 677.169 | 1 |
Based on a series of lectures for adult students, this lively and entertaining book proves that, far from being a dusty, dull subject, geometry is in fact full of beauty and fascination. The author's infectious enthusiasm is put to use in explaining many of the key concepts in the field, starting with the Golden Number and taking the reader on...
The Yang-Mills theory of gauge interactions is a prime example of interdisciplinary mathematics and advanced physics. Its historical development is a fascinating window into the ongoing struggle of mankind to understand nature. The discovery of gauge fields and their properties is the most formidable landmark of modern physics. The expression of the... more...
A practical, accessible introduction to advanced geometry Exceptionally well-written and filled with historical and bibliographic notes, Methods of Geometry presents a practical and proof-oriented approach. The author develops a wide range of subject areas at an intermediate level and explains how theories that underlie many fields of advanced mathematics... more... | 677.169 | 1 |
A First Course in Discrete Mathematics
Discrete mathematics has now established its place in most undergraduate mathematics courses. This textbook provides a concise, readable and accessible introduction to a number of topics in this area, such as enumeration, graph theory, Latin squares and designs. It is aimed at second-year undergraduate mathematics students, and provides them with many of the basic techniques, ideas and results. It contains many worked examples, and each chapter ends with a large number of exercises, with hints or solutions provided for most of them. As well as including standard topics such as binomial coefficients, recurrence, the inclusion-exclusion principle, trees, Hamiltonian and Eulerian graphs, Latin squares and finite projective planes, the text also includes material on the ménage problem, magic squares, Catalan and Stirling numbers, and tournament schedules | 677.169 | 1 |
4-color hardback text w/complete text-specific instructor and student print/media supplement package AMATYC/NCTM Standards of Content and Pedagogy integrated in Exercise Sets, Sourced-Data Applications (students are also asked to generate and interpret data), Scientific and Graphing Calculator Explorations Boxes, Mental Math exercises, Conceptual and Writing exercises, geometric concepts, Group Activities, Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews 6 step Problem-Solving Approach introduced in Chapter 2 and reinforced throughout the text in applications and exercises helps students tackle a wide range of problems Early and intuitive introduction to the concept of graphing reinforced with bar charts, line graphs, calculator screens, application illustrations and exercise sets. Emphasis on the notion of paired data in Chapters 1 and 2 leads naturally to the concepts of ordered pair and the rectangular coordinate system introduced in Chapter 3. Graphing and concepts of graphing linear equations such as slope and intercepts reinforced through exercise sets in subsequent chapters, preparing students for equations of lines in Chapter 7 | 677.169 | 1 |
by Patricia W. Hammer, Department
of Mathematics and Statistics
and Jessica A. King, Department of Computer Science
Hollins University
and
Steve Hammer, Department of Mathematics
Virginia Western Community College
In this project, students will complete a series
of modules that require the use of polynomial and trigonometric functions to
model the paths of straight stretch roller coasters. These modules
involve the mathematical definition of thrill and calculation of thrill for
several real coasters (Module A), design and thrill analysis of single drop
coaster hills (Modules B and C) and design and thrill analysis of several drop
coasters (Modules D and E). The ultimate goal of this interactive project is
successful completion of an optimization problem (Module F) in which students must design a straight stretch roller
coaster that satisfies the following coaster restrictions regarding height,
length, slope and differentiability of coaster path and that has the maximum
thrill (as defined below.)
Roller Coaster Restrictions
The total horizontal length of the straight
stretch must be less than 200 feet.
The track must start 75 feet above the
ground and end at ground level.
At no time can the track be more than 75
feet above the ground or go below ground level.
No ascent or descent can be steeper than 80
degrees from the horizontal.
The roller coaster must start and end with
a zero degree incline.
The thrill of a drop is defined to be the
angle of steepest descent in the drop (in radians) multiplied by the total
vertical distance in the drop. The thrill of the coaster is defined
as the sum of the thrills of each drop.
The path of the coaster must be modeled
using differentiable functions.
Students must use Maple8 (or any later version) to complete the project.
Students must already be familiar with derivatives and their
use in determining maximum and minimum function values. These ideas play a crucial role in the design and analysis
of the coasters.
To complete this project, student should work
through each of the modules given below.
A.Introduction to Roller Coaster Design - In this
module, students use an
interactive coaster window to mark peaks and valleys of real-life coasters and
then calculate the thrill of each drop using the above definition. Be
sure to record the x and y coordinates of the peak and valley points and the
slope at the steepest point. You will need this information to complete
parts B - E below.
B. Design
and Thrill of One Coaster Drop Using a Trig Function- In this module,
students model one drop of a coaster by marking the peak and valley of the drop
and then by fitting (in height and slope) a trig function of the form f(x) =
Acos(Bx+C)+D to the marked points. Once the function has been determined,
students then calculate the thrill of the single drop. A downloadable
Maple worksheet with commands and explanation is provided.
C.
Design and Thrill
of One Coaster Drop Using a Polynomial Function - In this module, students
model one drop of a coaster by marking the peak and valley of the drop and then
by fitting (in height and slope) a cubic polynomial to the marked points.
Once the function has been determined, students then calculate the thrill of the
single drop. A downloadable Maple worksheet with commands and explanation
is provided.
D. Design and Thrill
of a Straight Stretch Coaster Using Trig Functions - In this module,
students model a straight stretch coaster (several hills) by marking peak and
valley points and then by fitting (in height and slope) a trig function to each
consecutive pair of marked points. Once the functions have been
determined, students then calculate the thrill of the coaster. A
downloadable Maple worksheet with commands and explanation is provided.
E. Design and Thrill
of a Straight Stretch Coaster Using Polynomial Functions - In this
module, students model a straight stretch coaster (several hills) by marking
peak and valley points and then by fitting (in height and slope) a cubic
polynomial function to each consecutive pair of marked points. Once the
functions have been determined, students then calculate the thrill of the
coaster. A downloadable Maple worksheet with commands and explanation is
provided.
F. Project Assignment
- Design the Most Thrilling Straight Stretch Coaster - Students use
the ideas from modules A- E above to design a coaster (that satisfies all
restrictions) with the maximum possible thrill. Completion of this project
requires ingenuity, creativity and extension/modification of many of the ideas
and Maple commands presented in modules A-E. | 677.169 | 1 |
**The statistics class invites you to take a quick on-line survey about your music listening habits. Thank you for participating!
Statistics 2011-12 Instructor: Eric Rhomberg
How can we understand and communicate about numbers and data that describe real, practical situations?
How can we best prepare ourselves in terms of mathematics to be successful in our post-high school lives?
How can we increase our comfort and confidence level in mathematics?
These will be the essential questions that guide our work throughout the year. This course is for students who want an alternative to pre-calculus and calculus – students who would benefit more from solidifying basic skills, preparing to be "college ready" in terms of computational skills, and developing their practical math fluency and confidence.
In this course, we will:
Use Statisitics as a playing field for developing our overall math fluency.
Review math skills and concepts from basic computations through algebra and geometry.
Play math games, solve puzzles, engage in problem solving challenges and "number talks" in order to develop our "math minds" and our fluency with numbers.
Support your peers. Do your part to create a safe and effective collaborative environment.
Do 20 minutes of focused math homework each night. If you get stuck on a problem, always write it out as far as you can take it (even if that just means writing out the initial problem). Bring your homework to class everyday, and be ready to go over it and ask questions.
Complete projects by the due date. Communicate with the instructor in advance to negotiate extensions.
Try to have fun with all of this!
The flow of the class:
Our class periods will vary depending on what we are working on. We will often start class with a "Warm-up" problem or challenge to activate our math thinking.
We will then divide the period up between two or three of the following:
Direct skill instruction to the whole class.
Individualized work and practice, with instructor(s) doing 1-on-1 coaching.
Games and Puzzles that liven things up and develop our math fluency.
Group problems that emphasize collaboration.
"Number Talks" in which we really break down and articulate how we think about and solve math problems. | 677.169 | 1 |
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