copyright © cengage learning. all rights reserved. chapter 2 fundamental concepts

Copyright © Cengage Learning. All rights reserved.

CHAPTER 2

Fundamental Concepts

Copyright © Cengage Learning. All rights reserved.

SECTION 2.2

Algebraic Thinking

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What Do You Think?

• What does it mean to say that two sets are functionally related?

• What are some examples of functions in everyday life?

• Can we have functions without numbers?

• What is the reason for developing algebraic thinking in elementary school?

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Algebraic Thinking

Standard 2: Algebra

Instructional programs from pre-kindergarten through grade 12 should enable all students to:

• understand patterns, relations, and functions, sort, classify, and order objects in different ways, recognize, describe, and extend geometric and numerical patterns using words, tables, and graphs;

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Algebraic Thinking

• represent and analyze mathematical situations and structures using algebraic symbols, recognize and apply principles and properties of operations, like commutativity, use concrete, pictorial, and verbal representations to understand different symbolic notations, express mathematical relationships using equations;

• use mathematical models to represent and understand quantitative relationships, model situations with operations on whole numbers using manipulatives, pictures, and symbols;

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Algebraic Thinking

• analyze change in various contexts, describe change both qualitatively and quantitatively using graphs, tables, and equations, identify and describe situations with constant and varying rates of change.

Algebra is not a discrete subject but rather a strand that should be integrated with other branches of mathematics, including geometry, statistics, and discrete mathematics.

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Variables and Symbols

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Variables and Symbols

For many people, algebra is like going into a foreign country with different rules, procedures, and language.

Yet many of the rules and procedures in algebra are actually extensions of rules and procedures in arithmetic.

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Investigation A – A Variable by Any Other Name is Still a Variable

A crucial idea in algebraic thinking is that of the variable. In each of the following examples, the variables have aslightly different meaning. On a separate piece of paper, write down what each of the variables means.

C = d

5x = 30

sin x = cos x tan x

1 = n

y = kx

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Investigation A – Discussion

The first example is a formula. C and d stand for circumference and diameter, whose values vary according to the circle; however, the value of (3.14) does not vary.

The second example is generally called an equation. Although x is the variable, in this case its value is 6.

The third example is an example of an identity; it is true no matter what the value of x.

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The fourth example represents a property. In this case, n is used as a symbol to represent a property that is true for all numbers (except 0); the formal name of this propertyis the multiplicative inverse property.

The fifth example represents a family of functions in which the independent variable (x) and the dependent variable (y)are related in a certain way—in this case, a linear

relationship.

cont’d

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Table 2.3 illustrates some of the important differences in these examples.

Table 2.3

cont’d

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The Equals Sign and Equivalence

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The equals sign is universally associated with algebra. However, many elementary school children think of theequals sign as a call for action as opposed to a relationship between quantities.

That is, they think of the equals sign as “what comes next,” or “solve this.” For example, if you present 8 + 4 = 12to children and ask them to verify whether it is true or not, they will do so.

However, if you present 12 = 8 + 4, many children in the early grades will balk at this representation and say that it’s “not right.”

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In learning to work with equals signs in more powerful ways, the larger idea is equivalence, which is one of the big ideas of mathematics.

When you work with young children, your curriculum will provide various explorations to help students develop aricher understanding of the equals sign.

With respect to the equals sign signifying that the quantities on each side have the same value. When we say that sin x = cos x tan x, we are saying that cos x tan x has the same value as sin x.

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Structures

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Structures

All mathematical structures can be described usingalgebraic notation. Many of these structures are at the heart of the elementary school curriculum:

• Properties of operations—closure, commutative, associative, identity, inverse, distributive, etc.

• Properties of numbers—for example, an “even” number can be divided into two equal halves, whereas an odd number cannot. The algebraic representation of an even number is 2n, and the algebraic representation of an odd number is 2n + 1 or 2n – 1.

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Structures

• Connections between operations, such as a – b = a + –b

A very important structure in the system of whole numbers is the commutative property of addition.

It is stated in its most concise form as a + b = b + a for any whole numbers a and b.

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Relationships Among Quantities and Functions

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In elementary school, children will not investigate functions and other algebraic ideas at a formal level.

However, through their investigations, they do need to realize that relationships often exist between differentvariables (for example, the area of a rectangle varies according to the length and the width of the rectangle); that patterns and relationships can be represented in various ways (with words, symbols, graphs, and pictures); and that being able to communicate clearly the patterns they see is important.

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A simple example of a function is the hourly wage. If you know that you are being paid at the rate of $8.00 per hour, you expect to receive $320 for working 40 hours, or $160for working 20 hours, or $40 for working 5 hours.

In other words, there is a clear, consistent relationship between the variable “hours worked”and the variable “dollars earned.”

We can create a table of values for this function (Table 2.4).

Table 2.4

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Formally, we define a function as a relationship between two sets in which each element of the first set, called the domain of the function, is matched with exactly oneelement of the second set, called the range of the function.

In the wage example, the first column (hours worked) represents the domain, and the second column (dollars earned) represents the range.

Technically, what is in these columns is a subset of the domain and a subset of the range, respectively.

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Not all relationships between two sets are functions. For example, consider the relationship between the set of positive whole numbers and their square roots.

In this case, if we ask, “What is the square root of 9?” the answer is +3 and – 3.That is, each member of the set of positive whole numbers is matched with two square roots.

Inputs, outputs, and function notation:

Many elementary textbooks use a game called “What’s my rule?” to introduce children to the idea of functions.

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The students give a number (the input), then the teacher performs an operation and gives them the numberassociated with that input (the output).

The students then try to guess what the teacheris doing to the inputs. A game of “What’s my rule?” is given below (see Figure 2.17).

Figure 2.17

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TEACHER: What’s my rule?

STUDENT: 3

TEACHER: 6

STUDENT: 4

TEACHER: 7

STUDENT: I think I know the rule.

TEACHER: What is your rule?

STUDENT: You are adding 3. If I give you 10, you will say 13.

TEACHER: That’s right!

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The teacher is adding 3. Algebraically, we would sayy = x + 3.

We can also use function notation to write the rule for adding 3 to any number: f (x) = x + 3.

This mathematical sentence is read “f of x equals x plus 3.” That is, f (x) means “function of x.”

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Function notation is more precise because it emphasizes the relationship between the input, which is another name for a member of a function’s domain, and the output, whichis the corresponding member of its range.

In this particular case of “What’s my rule?” the output is f (x) determined by adding 3 to the input x, and so we have

f (3) = 3 + 3 = 6

f (4) = 4 + 3 = 7

f (10) = 10 + 3 = 13

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Let’s play another round. Look at the inputs and outputs in Figure 2.18.

Using everyday English, we could say that the rule (or function) is to triple the input and add 1.

Figure 2.18

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Using function notation, we could say that f (x) = 3x + 1.

Elementary teachers often make a “function machine” to illustrate these situations. Function machines for the two situations just presented can be seen in Figures 2.17 and 2.18.

In this game, the domain and range do not have to be sets of numbers. Here is one example from Margie Hoey, a second-grade teacher with whom I worked.

One student placed some students in one group and some students in another group.

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The determining factor was whether the student was wearing blue or not. That is, the domain set consisted of the set of students in the class.

The range set had two elements:

“is wearing blue” and “is not wearing blue.” For example,

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Investigation B – Baby-sitting

Let us extend our understanding of functions by examining another relationship.

Let’s say Ellen baby-sits and charges $8 per hour. Describe how you would determine how much to pay Ellen.Determine whether the relationship between the twovariables—hours and dollars—is a function. Whether yousay it is or is not a function, can you justify your response?

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Investigation B – Discussion

A common response to describing how to pay Ellen is to multiply the number of hours sat by 8.

Let us use this situation to introduce several different ways to represent functions. Then we will examine more closely the relationship between time sat and dollars paid.

For purposes of simplicity, let us first look at different representations of this function using only whole-hour amounts.

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Tables:We can represent this function with a table in which h represents hours and d represents dollars (Table 2.5).

We can then use the table to determine how much money Ellen will make from baby-sitting.

Equations:We can represent this function with an equationin which h represents hours and d representsdollars:

d = 8h

Table 2.5

cont’d

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Graphs:We can represent this function with a graph (Figure 2.19).

We can refer to the input as the independent variable and to the output as the dependent variable.

In this case, we say that the independent variable is the hours babysat and the dependent variable is the dollars earned.

That is, the number of dollars a baby-sitter earns is dependent on the number of hours that he or she baby-sits. Figure 2.19

cont’d

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Ordered pairs: We can represent this function as a set of ordered pairs in which the first element of each ordered pair represents hours and the second element represents dollars:

B = {(1, 8),(2, 16), (3, 24), (4, 32), . . .}

Mappings:Finally, we can representfunctions with arrow diagrams. Mathematicians often refer to thisrepresentation as a mapping of one set onto another set.

h d1 → 82 → 163 → 244 → 32

cont’d

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A closer look at paying the baby-sitter:At first glance, the question of how much to pay the baby-sitter is simple: Multiply the hours sat by 8.

However, let us use the problem-solving strategy “act it out” to examine this problem more closely.

cont’d

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For example, what if Ellen baby-sat from 7 to 11:15?

Some people say $32. Some people say $36—they round up to the nearest half-hour. In actuality, different people have different ways of determining how much to pay a babysitter.

Let us examine the case of a couple, who rounds up the time to the nearest half-hour. We could now represent their process for paying the baby-sitter in each of the ways we have just examined.

cont’d

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Mathematical modeling:As you have discovered, a problem is often more complicated than it first appears. This is what makes “real-life” problems so challenging.

The NCTM has stated that mathematical modeling should be an important part of mathematics education in school. A model of an object is not the object and is often a scaled-down version of the object.

We speak of model airplanes, and architects and engineers build a model of a building or a bridge before constructing the actual object.

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In one sense, a model is a representation of the original object. There are two important features of many models.

First, the model contains many of the properties of theoriginal object. Second, the model can be manipulated andstudied to help us better understand the (usually larger andmore complex) object.

A mathematical model is a mathematical structure that approximates the features of a situation. A mathematical model can be an equation (or a set of equations), a graph, or some other kind of diagram.

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In this case, we have seen that the original straight-line graph turns out not to be a useful or accurate model of the baby-sitting situation.

The step graph is a more accurate representation. This process of examining a situation and then developing a model that accurately represents that situation is called mathematical modeling.

Constructing and interpreting mathematical models is one of the more important uses of mathematics in the real world.

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Investigation C – Choosing Between Functions

Jackie has been promoted to chief salesperson of Southside Computers. She has two choices as to how she will be paid.

She can receive $50 for every computer system she sells, or she can receive $250 a week plus $25 for every computer system she sells. Looking at the past year, she finds that she has averaged 9.3 sales per week, and she figures that she can do a little better this year. What would you recommend?

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Investigation C – Discussion

You may want to start by plugging in a few numbers—that is, using guess–check–revise. Do you see that if Jackie sells only a few computers, she will be better off with the second plan but that the first plan is better if she sells a lot of computers?

This realization leads to a specific mathematical question: At what point will the two plans give her the same money?

Once she can answer that question, she can base her decision on whether she believes she will sell more or less than that amount.

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Below we see three very different strategies for solving this subproblem: guess–check–revise, use algebra, and make a graph.

Strategy 1: Guess–Check–Revise

This strategy is illustrated in Table 2.6.

Table 2.6

cont’d

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It means that Jackie makes the same amount under both plans when she sells 10 computers a month. If she thinks she will sell more than 10, then plan A is better for her.

Strategy 2: Use Algebra

Let y = Jackie’s weekly salary.

Let x = the number of computer systems she sells.

The equation that represents plan A is y = 50x

cont’d

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The equation that represents plan B is

y = 250 + 25x

Do you understand how both of these equations were constructed?

We have two equations in two unknowns that we can solve:

50x = 250 + 25x

25x = 250

x = 10

cont’d

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Strategy 3: Make a Graph

Where the two lines cross will show the point at which her earnings under both plans will be equal. The two lines cross at the point (10, 500) (Figure 2.20).

In other words, the ordered pair (10, 500) is an element of both graphs.

cont’d

Figure 2.20

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Algebra as a Tool to Understand Change:

Many of the investigations and explorations deal with the question of answering questions about change. Understanding change is an important way in which mathematics is used.

For example, manufacturers test different shapes and materials for cups and then measure the temperature of the coffee to see how fast it cools;

scientists measure the effectiveness of a new medicine by measuring how fast it kills the harmful bacteria.

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Investigation D – Matching Graphs to Situations

Below are descriptions of three runners in a race and three graphs. Match each description to the correct graph and explain your choice.

The independent variable is time, and the dependent variable is distance from the start.

Alex started slowly, then ran a bit faster, and then ran even faster at the end of the race.

Manuel started quickly but then tired and slowed down a bit and then slowed down even more at the end.

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Investigation D – Matching Graphs to Situations

Ragib started quickly, stopped to tie his shoe, and then ran even faster than before.

Discussion:A slower speed has a smaller slope than a faster speed. Stopping means that the runner gets no closer to the end and so is represented by zero slope (horizontal line).

cont’d

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Investigation D – Discussion

Thus the second graph matches Alex—less steep than the others at the beginning and then steeper and steeper.

The third graph matches Manuel—steeper than Alex at the beginning and then less and less steep.

The first graph matches Ragib—steep at the beginning, horizontal in the middle to show that he stopped, and then even steeper than before.

cont’d

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Investigation E – Developing “Graph Sense”

Which of the graphs below best represents the following scenario? A runner is running at a steady rate and then comes to a hill, which causes her to run at a slower rate.

Once she reaches the top of the hill, she runs down the hill very fast. Upon reaching the bottom of the hill, she resumes her original pace.

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Investigation E – Discussion

Many students pick the first graph because that is what the race looks like—level, then a hill, and then level again. But the graph is a picture not of the layout of the race but, rather, of the person’s speed.

A steady speed means that the speed is not changing and is thus represented by a horizontal line.

When the runner slows down the slope of the graph decreases (that is, becomes negative); when she speeds up, the slope of the graph increases. Thus the second graph is correct.

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Algebra as Generalized Arithmetic

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Algebra as Generalized Arithmetic

The value of being able to recognize and use patterns insolving problems. Exploring patterns in elementary school is an important part of developing algebraic reasoning.

However, it is important to state that patterns alone do notnecessarily develop algebraic thinking. Algebraic thinking develops as the students learn to analyze various patterns(numerical and visual) in order to make and test predictions

and finally to make generalizations.

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Investigation F – Looking for Generalizations

The figure below shows an example of what are called growing patterns in elementary school. How many little squares will it take to make the nth figure in this pattern?

Discussion:There are often different strategies that can be used to solve a problem and that embody different representations of the problem.

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Investigation F – Discussion

Strategy 1: Make a Table and Look For Patterns

Looking at the relationship between the corresponding numbers in the first and second sets (input, output), many people will quickly see that the output number is always1 less than double the input number.

Thus the number of squares in the nth L number is simply 2n – 1.

cont’d

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The third row in the table below illustrates this relationship very clearly.

Strategy 2: Break It Apart

Each of the L shapes can be broken down (decomposed) into two “arms” and a base (or “arm connector”).

cont’d

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The base is always a single square. In each case, the arms are the same length, and the length is 1 less than the figure number. That is, the length of each of the arms of the5th figure is 4 (1 less than 5).

Thus the nth figure will have two arms, each of whoselength is (n – 1),plus one square that represents the base.

Thus the number of squares of the nth figure will be equal to (n – 1) + (n – 1) + 1, which is mathematically equivalent to 2n – 1.

cont’d

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Strategy 3: Break it apart another way

We could also have broken the shape apart this way: a bottom and a top.

The bottom begins with one square and grows by one each time. The top begins with the second figure and also grows by one each time. The number of squares in the bottom always matches the figure number.

cont’d

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That is, the fourth figure has a base of four, the fifth figure has a base of five, and so the nth figure will have a base of n. The number of squares in the top is always one lessthan the number of squares in the bottom.

Thus, the nth figure will have a bottom of n and a top of (n – 1). Adding the two together, we find that thetotal number of squares is n + (n – 1) , or 2n – 1.

cont’d

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Investigation G – How Many Dots?

Predict how many dots it will take to make the nth figure.

1st 2nd 3rd 4th

Discussion: I was fascinated by the discussion of this problem at a conference I attended because of the many different ways it was seen by children.

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Investigation G – Discussion

Let us examine three of those ways.

Strategy 1: Seeing Two Lines

One group of children saw two lines. Each line consists of an odd number of dots and grows by 2 each time.

cont’d

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The biggest conceptual problem here is how to connect the figure number to the number of dots in each line.

The connection is that the number of dots in each line is always one less than double the figure number: 2n – 1.

Thus the total number of dots is 2 (2n – 1) – 1 = 4n – 3.

cont’d

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Strategy 2: How Many Pairs Of Dots?

Another group of children saw the problem in terms of a center and then pairs of dots. The second figure is the center plus 2 pairs of dots. The third figure is the center plus 4 pairs of dots.

cont’d

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As in the previous strategy, the challenge now is how to connect the figure number to the number of pairs.

In this case, we see that the number of pairs is always double the previous figure number.

That is, in the third figure, the number of pairs is 4, which is 2 times 2.

In the fourth figure, the number of pairs of dots is 6, which is 2 times 3.

cont’d

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Strategy 3: How Much Does It Grow By?

Another group of children saw that the number of dots increases by 4 each time.

From this observation, there are still multiple ways to determine the nth case. One is by trial and error.

cont’d

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Another is by using the observation that the pattern grows by 4 each time and representing the amount this way, such as

4 + 1, 2 4 + 1, 3 4 + 1,etc.

From this point, we can see that the multiplier of 4 is simply one less than the figure number.

Thus the nth case will have (n – 1)4 + 1 dots, which simplifies to 4n – 4 + 1 = 4n – 3.

cont’d

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Yet another way to use the observation of increasing by 4 each time is to remember that in a straight line, the slope is simply the rate of change.

In this case, the rate of change is 4, and so the slope is 4. Thus, the equation is y = 4x + b.

Now it is a matter of trial and error to determine that b = –3. That is, y = 4x – 3 is the number of dots in any figure.

cont’d

copyright © cengage learning. all rights reserved. chapter 2 fundamental concepts

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