Lecture Notes for
Math 135:
Algebraic Reasoning
for Teaching Mathematics
Steffen Lempp
Department of Mathematics, University of Wisconsin,
Madison, WI 53706-1388, USA
URL:http://www.math.wisc.edu/~lempp
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Lecture Notes for Algebraic Reasoning for Teaching Math | MATH 135, Study notes of Mathematics
Material Type: Notes; Professor: Lempp; Class: Algebraic Reasoning for Teaching Math; Subject: MATHEMATICS; University: University of Wisconsin - Madison; Term: Spring 2008;
Typology: Study notes
Pre 2010
1 / 163
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Lecture Notes for Algebraic Reasoning for Teaching Math | MATH 135 and more Study notes Mathematics in PDF only on Docsity!
Lecture Notes for
Math 135:
Algebraic Reasoning
for Teaching Mathematics
Steffen Lempp
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA E-mail address: [email protected] URL: http://www.math.wisc.edu/~lempp
Key words and phrases. algebra, equation, function, inequality, linear function, quadratic function, exponential function, student misconception, middle school mathematics
These are the lecture notes for the course Math 135: “Algebraic Reasoning for Teaching Mathematics” taught at UW-Madison in spring 2008.
The preparation of these lecture notes was partially supported by a faculty development grant of the College of Letters and Science and by summer support by the School of Education, both of the University of Wisconsin-Madison.
Feedback, comments, and corrections are welcome!
This text is currently in a constant state of change, so please excuse what I am sure will be numerous errors and typos!
Thanks to Dan McGinn and my Math 135 students in spring 2008 (Rachel Burgan, Erika Calhoon, Lindsay Feest, Amanda Greisch, Celia Hagar, Brittany Hughes, and Shannon Osgar) for helpful comments and corrections.
© 2008 Steffen Lempp, University of Wisconsin-Madison
This material may only be reproduced and distributed for cost of reproduction and distribution for educational/non-profit purposes. This book is currently available for free download from http://www.math.wisc.edu/~lempp/teach/135.pdf.
Any dissemination of this material in modified form requires the author’s written approval.
CONTENTS v
- Chapter 1. Numbers and Equations: Review and Basics Introduction to the Student x - A Review 1.1. Arithmetic in the Whole Numbers and Beyond:
- 1.1.1. Addition and Multiplication
- 1.1.2. Subtraction and Division
- 1.2. Letters and Algebraic Expressions
- 1.2.1. Letters in Algebra
- 1.2.1.1. Letters as Variables
- 1.2.1.2. Letters as Unknowns
- 1.2.1.3. Letters as Parameters
- 1.2.1.4. Letters as Constants:
- 1.2.2. Algebraic Expressions
- 1.3. Equations and Identities
- 1.4. Manipulating Algebraic Expressions
- 1.4.1. Using the definition of subtraction
- 1.4.2. “Combining Like Terms”
- 1.4.3. Removing Parentheses
- 1.5. Conventions in Algebra
- 1.5.1. Notation for Sets, and Some Special Sets of Numbers
- 1.5.2. Order of Operations and Use of Parentheses
- 1.6. Algebra as “Undoing Expressions”
- Chapter 2. Linear Equations
- 2.1. Ratio and Proportional Reasoning
- 2.2. Velocity and Rate of Change
- 2.3. Linear Expressions
- 2.4. Line Equations
- 2.4.1. Graphing a Linear Equation
- 2.4.2. Different Forms of Equations for Lines
- 2.4.3. Varying the Parameters in a Line Equation iv CONTENTS
- 2.5. Solving a Linear Equation in One Variable
- Variables 2.6. Solving Simultaneous Linear Equations in Two or More
- Chapter 3. Order and Linear Inequalities
- 3.1. Ordering the Numbers
- 3.2. Order and the Arithmetical Operations
- 3.3. Linear Inequalities in One Unknown
- 3.4. Estimation and Approximation - Related to Linear Functions Chapter 4. The Concept of a Function, and Functions Closely
- 4.1. The Concept of a Function
- and Inverse of a Function 4.2. Range of a Function, Onto and One-to-One Functions,
- 4.2.1. Range of a Function, and Onto Functions
- 4.2.2. 1–1 Functions
- 4.2.3. Inverse of a Function
- 4.3. Some Functions Closely Related to Linear Functions
- 4.3.1. Step Functions
- 4.3.2. Piecewise Linear Functions
- 4.3.3. The Absolute Value Function
- 4.3.4. Direct and Inverse Variation
- Chapter 5. Quadratic Functions, Equations and Inequalities
- 5.1. Introduction to Quadratic Functions
- 5.2. Solving Quadratic Equations
- 5.2.1. Reviewing the Binomial Laws
- 5.2.2. Solving Quadratic Equations by Factoring
- 5.2.3. Solving Quadratic Equations by Taking the Square Root - Square 5.2.4. Solving Quadratic Equations by Completing the
- 5.2.5. Solving Quadratic Equations by the Quadratic Formula
- 5.2.6. The number of real solutions of a quadratic equation
- and the Complex Numbers 5.3. A Digression into Square Roots
- 5.3.1. Square Roots
- 5.3.2. The Number i and the Complex Numbers
- Solving Quadratic Equations Graphically 5.4. Graphing Quadratic Functions and
- 5.5. Quadratic Inequalities - and Inequalities Chapter 6. Exponential and Logarithmic Functions, Equations
- 6.1. Exponentiation: A Review of the Definition
- 6.2. Motivation for Exponentiation
- 6.3. The Exponential Functions
- 6.3.1. Properties of Exponential Functions
- 6.3.2. Another Application of Exponential Functions
- Equations 6.4. The Logarithmic Functions and Solving Exponential
- 6.4.1. Definition and Properties of Logarithmic Functions
- 6.4.2. Applications of Logarithmic Functions
- 6.4.2.1. The slide rule
- 6.4.2.2. Radioactive Decay
- 6.4.2.3. Earthquakes and the Richter Scale
- 6.4.2.4. Sound Volume and Decibels
- Bibliography
- Index
PREFACE vii
century). Until the Renaissance, most of the attention in algebra cen- tered around solving equations in one variable. How to solve quadratic equations by completing the square was already known to the Babyloni- ans; the Italian mathematicians Scipione del Ferro and Niccol`o Fontana Tartaglia independently, and the Italian mathematician Lodovico Fer- rari, gave the first general solution to the cubic and quartic equation, respectively.^1 (The Italian mathematician Gerolamo Cardano was the first to publish the general solutions to cubic and quartic equations, and the formula for the cubic equation bears his name to this day.)
The French mathematician Fran¸cois Vi`ete (1540 - 1603, more com- monly known by his Latinized name Vieta) is credited with the first attempt at giving the modern notation for algebra we use today; before him, very cumbersome notation was used. The 18th and 19th century then saw the birth of modern algebra in the other sense mentioned above, which also led to much more general techniques for solving equa- tions, including the proof by the Norwegian mathematician Niels Hen- rik Abel (in 1824) that there is no general “formula” to solve a quintic equation. (For a brief history of algebra, see the Wikipedia entry on algebra at http://en.wikipedia.org/wiki/Algebra#History.)
(^1) Here, a quadratic equation is an equation involving numbers, x and x (^2) ; a cubic
equation is one also involving x^3 ; a quartic equation is one additionally involving x^4 ; and a quintic equation (mentioned in the following paragraph) also involves x^5.
Introduction to the Instructor
These lecture notes first review the laws of arithmetic and discuss the role of letters in algebra, and then focus on linear, quadratic and ex- ponential equations, inequalities and functions. But rather than simply reviewing the algebra your students will have already learned in high school, these notes go beyond and study in depth the concepts un- derlying algebra, emphasizing the fact that there are very few basic underlying idea in algebra, which “explain” everything else there is to know about algebra: These ideas center around
- the rules of arithmetic (more precisely, the ordered field ax- ioms), which carry over from numbers to general algebraic expressions, and
- the rules for manipulating equations and inequalities.
These basic underlying facts are contained in the few “Propositions” sprinkled throughout the lecture notes. These notes, however, attempt to not just “cover” the material of algebra, but to put it into the right context for teaching algebra, by focusing on how real-life problems lead to algebraic problems, multi- ple abstract representations of the same mathematical problem, and typical student misconceptions and errors and their likely underlying causes. These notes just provide a bare-bones guide through an algebra course for future middle school teachers. It is important to combine them with a lot of in-class discussion of the topics, and especially with actual algebra problems from school books in order to generate these discussions. For these, we recommend using the following Singapore math schoolbooks:
- Primary Math Textbook 5A (U.S. Edition),
- Primary Math Textbook 6A (U.S. Edition),
- New Elementary Math Textbook 1 (Syllabus D),
- New Elementary Math Textbook 2 (Syllabus D),
- New Elementary Math Textbook 3A (Syllabus D), and
- New Syllabus Additional Mathematics Textbook. viii
Introduction to the Student
These notes are designed to make you a better teacher of mathe- matics in middle school. The crucial role of algebra as a stepping stone to higher mathe- matics has been documented numerous times, and the mathematics curricula of our nation’s schools have come to reflect this by placing more and more emphasis on algebra. At the same time, it has be- come clear that many of our nation’s middle school teachers lack much of the content knowledge to be effective teachers of mathematics and especially of algebra. (See the recent report of the National Mathemat- ics Advisory Panel available at http://www.ed.gov/about/bdscomm/ list/mathpanel/report/final-report.pdf, especially its chapter 4 on algebra.) After some review of the laws of arithmetic and the order of oper- ations, and looking more closely at role of letters in algebra, we will mainly study linear, quadratic and exponential equations, inequalities and functions. But rather than simply reviewing the algebra you have already learned in high school, we will go beyond and study in depth the concepts underlying algebra, emphasizing the following topics:
- “problem-solving”: modeling real-life problems (“word prob- lems”) as mathematical problems and then interpreting the mathematical solution back into the real-world context;
- “proofs”: making mathematically grounded arguments about mathematical statements and solutions;
- “analyzing student solutions”: examining the rationale behind middle school and high school students’ mathematical work and how it connects to prior mathematical understanding and future mathematical concepts; analyzing the strengths and weaknesses of a range of solution strategies (including standard techniques); and recognizing and identifying common student misconceptions;
- “modeling”: flexible use of multiple representations such as graphs, tables, and equations (including different forms), and x
INTRODUCTION TO THE STUDENT xi
how to transition back and forth between them; and using functions to model real-world phenomena;
- “symbolic proficiency”: solving equations and inequalities, sim- plifying expressions, factoring, etc. These lecture notes try to make the point that school algebra centers around a few basic underlying concepts which “explain” all (or at least most) of school algebra. These are
- the rules of arithmetic for numbers (including those for the ordering of numbers), and how they naturally carry over from numbers to general algebraic expressions; and
- the rules for manipulating equations and inequalities, and how they allow one to solve equations and inequalities.
We will generally label these basic facts “Propositions” to highlight their central importance. Many, if not most, of the student misconcep- tions and errors you will encounter in your teaching career will come from a lack of understanding of these basic rules, and how they natu- rally “explain” algebra.
CHAPTER 1
Numbers and Equations: Review and Basics
1.1. Arithmetic in the Whole Numbers and Beyond: A Review 1.1.1. Addition and Multiplication. Very early on in learning arithmetic, we realize that there are certain rules which make com- puting simpler and vastly decrease the need for memorizing “number facts”. For example, we see that
(1.1) 2 + 7 = 7 + 2
(1.2) 6 + 0 = 6
(1.3) 3 × 5 = 5 × 3
(1.4) 6 × 0 = 0
(1.5) 3 × 1 = 3
For a more elaborate example, we also see that
(1.6) 2 × (3 + 9) = 2 × 3 + 2 × 9
Is there something more general going on? Of course, you know the answer is yes, and we could write down many more such examples. However, this would be rather tiring, and we certainly cannot write down all such examples since there are infinitely many; so we’d rather find a better way to state these as general “rules”. In ancient and medieval times, before modern algebraic notation was invented, such rules were stated in words; e.g., the general rule for (1.1) would have been stated as
“The sum of any two numbers is equal to the sum of the two numbers in the reverse order.” Obviously, this is a rather clumsy way to state such a simple rule, and furthermore, when writing down rules this way, it is not always easy to write out the rule unambiguously. (Pause for a moment and think how you would write down the general rule for (1.6)!) And again, you know, of course, how to write down these rules in general: We use “letters” instead of numbers and think of the letters as “arbitrary”
1
2 1. NUMBERS AND EQUATIONS: REVIEW AND BASICS
numbers. So the above examples turn into the following general rules:
(1.7) a + b = b + a
(1.8) a + 0 = a
(1.9) a × b = b × a
(1.10) a × 0 = 0
(1.11) a × 1 = a
(1.12) a × (b + c) = a × b + a × c
Two related issues arise now: (1) For which numbers a, b, and c do these rules hold? (2) How do we know these rules are true for all numbers a, b, and c? In elementary school, these rules can be “proved” visually for whole numbers by coming up with models for addition and multiplication. A common model for the addition of whole numbers a and b is to draw a many “objects” (e.g., apples) in a row, followed by b many objects in the same row, and then counting how many objects you have in total. Similarly, a common model for the multiplication of whole numbers a and b is to draw a rectangular “array” of a many rows of b many objects (e.g., apples), and then counting how many objects you have in total. All the above rules are now easily visualized. (E.g., see Figure 1.1 for a visual proof of (1.1) and Figure 1.2 for a visual proof of (1.3).)
2 7
7 2
Figure 1.1. A visual proof that 2 + 7 = 7 + 2
An easy extension of these arguments works for fractions (and in- deed for positive real numbers) if we go from counting objects to work- ing with points on the positive number line: On the positive number line (starting at a point identified with the number 0), mark off a “unit interval” from 0 to a new point called 1. Repeatedly marking off more unit intervals gives us points we identify with the whole numbers 2, 3, 4, etc. For a whole number n > 0, we can subdivide the unit interval into n many subintervals of equal length (i.e., of length (^) n^1 ). Now again,
4 1. NUMBERS AND EQUATIONS: REVIEW AND BASICS
for each to convince yourself that they are true. For (1.14), you will have to go to a three-dimensional array.)
Exercise 1.1. Give pictures for each of (1.8) and (1.10)–(1.14) in the whole numbers.
1.1.2. Subtraction and Division. At first, students are given problems of the type 2 + 5 = or 3 × 4 = , asking them to fill in the answer. A natural extension of this is to then assign problems of the form 2 + = 7 or 3 × = 12, asking them to fill in the missing summand or factor, respectively. Let’s look at the problem of the “missing summand” first: Solving 2 + = 7 corresponds, e.g., to the following word problem:
“Ann has two apples. Her mother gives her some more apples, and now she has seven apples. How many apples did her mother give her?” Students will quickly learn that this requires the operation of sub- traction: Ann’s mother gave her 7−2, that is 5, apples. More generally, the solution to a + = c is c − a. But then a problem arises: What if a > c?^1 Obviously, you now have to go from the “objects” model to a more “abstract” model, e.g., the money model with debt: If you have $2 and spend $5, then you are $3 “in debt”, i.e., you have −$ dollars. (It is a good idea to emphasize the distinction in the use of the symbol − in these two cases by reading 2 − 5 as “two minus five”, but −3 as “negative three”.) So allowing subtraction for arbitrary numbers forces us to allow a “new” kind of numbers, the negative numbers. So we proceed from the whole numbers 0 , 1 , 2 ,... to the integers 0 , 1 , − 1 , 2 , − 2 ,.... You will need to allow your students to adjust to this change, since now your answer to a question like “Can I subtract a larger number from a smaller number?” will change, possibly confusing your students. Of course, one should never answer “no” to this question without some qualification; some students will know about the negative numbers at an early age (especially in Wisconsin in the winter!). But now the correct answer should be “it depends”, namely, it depends on which number system we work in. At this point, you will want to extend the number line to the left also (see Figure 1.3, it can also be used to include fractions and negative fractions). But this transition from the whole numbers to the integers imme- diately raises three new questions about the arithmetical operations:
(^1) I can’t resist at this point to tell you my favorite German math teacher joke:
“What does a math teacher think if two students are in the class room and five students leave? – If three students come back, the room is empty!”
1.1. ARITHMETIC IN THE WHOLE NUMBERS AND BEYOND: A REVIEW 5
-3 -2 -1 0 ½ 1 2 3
Figure 1.3. The number line
(1) How should we define a + b and a − b when one or both of a and b are negative? What about multiplication? (2) How do we know that the rules (1.7)–(1.14) hold for addition and multiplication of arbitrary (positive or negative) numbers? (3) Do these rules also hold for subtraction and multiplication of arbitrary (positive or negative) numbers? Of course, you already know the answer to item (1). But how would you justify it to your students? Maybe there is another way to define addition with negative numbers? The answer is twofold: First of all, one can justify the common definition of addition of integers with a “real-life” model (such as the money-and-debt model) or with the number line (which is really just an abstraction from a real-life model). The justification for subtraction and multiplication is a bit more delicate and may not appeal to your students as much. But one can also justify the common definition of addition and multiplication of integers by the fact that we want the rules (1.7)–(1.14) to be true for addition and multiplication of integers also. So, in a sense, the rules in item (2) above really are “rules” we impose rather than rules we can “prove”, unless you are satisfied with the justification for item (1) by the models for addition and multiplication. So what about subtraction? We first define (−b) as the integer which when added to b gives 0:
(1.15) b + (−b) = 0
This is now a new “rule” about addition, and (−b) is called the (ad- ditive) inverse of b. Note that there is lots of room for confusion here: (−b) (or simply −b for short) is sometimes called the “negative” of b, a term I would avoid, since −b need not be negative. In fact, −b is positive if b is negative; −b is 0 if b is 0; and −b is negative if b is positive! (This will confuse quite a few of your students through high school and even college, so be sure to emphasize this early on.) We can then “define” subtraction via addition: a − b is defined as a + (−b) for any integers a and b. Note that this works well with real- life models of subtraction: Adding a capital of -$5 (i.e., a debt of $5) to a capital of $2 is the same as subtracting $5 from $2.
1.1. ARITHMETIC IN THE WHOLE NUMBERS AND BEYOND: A REVIEW 7
makes no sense since it would amount to solving the original problem 0 × = c; and there will either be no solution if c 6 = 0, or any number is a solution if c = 0; so c ÷ 0 never makes sense.) We proceed from the whole numbers to fractions (which will always be positive or 0), or more generally from the integers to the rational numbers (which may also be negative). The number line from Figure 1.3 still serves as a good visual model. But this transition again raises a number of questions: (1) How should we define the operations of addition, subtraction, multiplication, and division for fractions, or, more generally, for rational numbers? (2) How do we know that the rules (1.7)–(1.15) still hold? (3) Do these rules also hold for division?
In answering item (1), we need to be able to come up with real- life examples to justify the “correct” way to perform the arithmetical operations on fractions and more generally rational numbers. (See Exercise 1.3 below.) But again one can also justify this by wanting the rules (1.7)–(1.15) to still hold for these numbers, addressing item (2). An elegant way to think about division is to first define the (multi- plicative) inverse of a nonzero number a as that number, denoted by (^) a^1 , which when multiplied by a gives 1:
(1.16) a ×
a
This is now a new rule for multiplication; and it works not only when a is a positive whole number but indeed for any nonzero rational number. And again, this removes the need for new rules for division since we can reduce division to multiplication by defining a ÷ b as a × (^1) b. (See Exercise 1.4.) We summarize all the rules about the arithmetical operations of number which we have discovered in the following
Proposition 1.2. The following rules hold for all real numbers a, b and c:
(commutative law of addition) a + b = b + a
(associative law of addition) (a + b) + c = a + (b + c)
(law of the additive identity) a + 0 = a
(law of the additive inverse) a + (−a) = 0
8 1. NUMBERS AND EQUATIONS: REVIEW AND BASICS
(commutative law of multiplication) a × b = b × a
(associative law of multiplication) (a × b) × c = a × (b × c)
(law of the multiplicative identity) a × 1 = a
a ×
a
(law of the multiplicative inverse) = 1 (when a 6 = 0)
(distributive law) a × (b + c) = a × b + a × c
Exercise 1.3. Give several word problems for the division 12 ÷ 13 , both for partitive division (as in the apple example in the main text above), and for measurement division (as in the footnote above).
Exercise 1.4. Explain which of the rules (1.9)–(1.12) and (1.14) carry over to division; if they don’t carry over, are there corresponding rules for division?
1.2. Letters and Algebraic Expressions 1.2.1. Letters in Algebra. Arithmetic only deals with numbers and numerical expressions, i.e., expressions involving numbers and the arithmetical operations (addition, subtraction, multiplication, division, later on also roots, exponentiation, and logarithms). Each such ex- pression has a definite value (or is undefined for some reason, such as division by 0 or taking the square root of a negative number). Word problems often can be reduced to evaluating numerical expressions, e.g.,
“Ann has four apples. Her friend Mary gives her three more apples each day for a (five-day) week. How many apples does she have at the end of the week? ” Clearly, this gives rise to the numerical expression 4 + (5 × 3), and evaluating this numerical expression then gives the solution
“Ann has 19 apples at the end of the week.” One extension of this problems is to leave some numbers, e.g., the number of apples Ann has at the beginning, and the number of apples she receives each day, unspecified. In this case, we typically use letters to denote these unspecified quantities, and it is crucial that these letters are clearly and unambiguously specified. So let’s set
a = number of apples Ann has at the beginning of the week b = number of apples Ann receives from Mary each day
Now the word problem becomes