MATH UEST 2 - Texas State University

MATH QUEST 2TABLE OF CONTENTS

i

CHAPTER 1: PATTERNS, GRAPHS, AND TABLES

1.1 Graphing on a Coordinate Plane 1

1.2 Lining Up 10

CHAPTER 2: FUNCTIONS AND GRAPHS

2.1 Adding, Subtracting, and Multiplying Integers 29

2.2 Functions 61

CHAPTER 3: EXPLORING FRACTIONS

3.1 Modeling Fractions 85

3.2 Adding and Subtracting Fractions 102

3.3 Multiplying Fractions 118

APPENDIX

Additional Resources 139

MATH QUEST 2ACKNOWLEDGMENTS

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AUTHORS

Hiroko Warshauer

Max Warshauer

Terry McCabe

TECHNICAL EDITOR

Genesis Dibrell

TECHNICAL AND CONTENT INPUT

Sammi Yarto

Claudia Hernández

STAFF SUPPORT

Patty Amende

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MATH QUEST 2PREFACE

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Welcome to Math Quest Level 2. As you work through the book, you will be introduced to functions and

fractions. In Chapter 1, you will begin by looking at patterns and graphing on a coordinate plane. In Chapter 2,

you will learn about the basic operations – addition, subtraction, and multiplication, as well as functions and

their graphs. Finally, in Chapter 3, you will explore fractions and operations using fractions.

This 6th edition of Level 2 revises earlier editions with the contents updated and revised. A special thanks to

Genesis Dibrell and Sammi Yarto for their help in editing, and to all of our Math Camp teachers for their help

and suggestions. Particularly we would like to thank Everett Mungia and Maureen Bakenhus for piloting earlier

editions, and suggestions for changes.

1

CHAPTER 1PATTERNS, GRAPHS, SEQUENCES

SECTION 1.1 GRAPHING ON A COORDINATE PLANE

OBJECTIVES

• Graph ordered pairs

• Set up equations with two variables

• Graph a straight line

• Cost and sales applications using straight lines

Numbers are often used to represent location on number lines. If we draw our number line horizontally, then positive numbers are located to the right of 0 and negative numbers are located to its left. Suppose we draw our number line vertically like a thermometer. In this case, positive numbers are above 0, and negative numbers are below 0.

When we plot points on a coordinate plane, we use two numbers instead of one number to locate the points. We call these numbers coordinates. A coordinate plane is constructed as follows:

We begin by drawing a horizontal number line and call it the horizontal axis or the x–axis.

01234 1 2 3 4origin

- - - -

Chapter 1 Pat terns, Graphs, Sequences

2

Locate the zero point, which is called the origin. Next, draw a vertical number line through the origin, so that the two zero points overlap at the origin. On graph paper, our picture looks like this.

Quadrant II Quadrant I

Quadrant III Quadrant IV

P

y-axis

x-axis

The vertical number line is called the vertical axis or the y–axis.

These axes divide the coordinate plane into four regions. We call these regions Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. The coordinates for the points in the plane are pairs of numbers.

For example, the point P has coordinates (4,3), where 4 represents the x-coordinate and 3 represents the y-coordinate. The points in the plane are written in the form of (x, y) and are called ordered pairs. Points (x, y) with x and y integers are called lattice points.

Sect ion 1 .1 Graphing on a Coord inate P lane

3

EXPLORATION: PLANE POINTS

1. On the graph paper below, construct and label a coordinate plane.

2. Plot and label the following points on the coordinate plane. State the quadrant or axis that each point is located.

N (−5, 2) ______________________

B (3, 5) ______________________

M (−4, −3) ______________________

S (4, −1) ______________________

C (0, −3) ______________________

R (−2, 0) ______________________


4

3. Make a list of characteristics of the points in each of the quadrants. Make a list of the characteristics of points found on the x-axis, and a list for the y-axis.

4. On the graph paper below, construct and label a coordinate plane.

Name a point in each of the quadrants. Explain your choice.

In Quadrant I: _______________

In Quadrant II: _______________

In Quadrant III: _______________

In Quadrant IV: _______________

On the x-axis: _______________

On the y-axis: _______________


5

EXERCISE A

1. Write the coordinates for each point shown on the coordinate plane below.

1

2

3

4

5

67

-1

-2

-3

-4

-5

-6

-7

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

D

L

G

H

E

F

A

B

K

C

J

I

x-axis

y-axis

A

B

C

D

E

F

G

H

I

J

K

L

2. Use the coordinate plane from Problem 1 to plot the following points.

P (0, 3)

Q (−6,6)

R (2, 4)

S (−5, −3)

T (−3, 5)

U (3, 0)

W (1, 1)

Z (4, −5)


6

3. For each condition below make a list (or T-Chart) of 6 points. Plot these points on the coordinate plane below.

1

2

3

4

5

67

-1

-2

-3

-4

-5

-6

-7

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x-axis

y-axis

a. Each point must have an x-coordinate equal to zero and a positive y-coordinate.


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b. Each point must have a y-coordinate that is equal to the x-coordinate.

c. Each point must have a y-coordinate that is double the x-coordinate

d. Each point must have a y-coordinate which is the negative of the x-coordinate. Which quadrants do these points lie in?


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GAME: HIDDEN DIAMOND

Materials:

One sheet with a coordinate system for each student with x ranging from ¯5 to 5 and y ranging from 5 to¯5. Another sheet to keep a record of guessed ordered pairs.

Goal of the game:

Each student will have a partner and each student tries to locate his or her partner’s diamond in the least number of guesses. To locate the diamond, you must name all four of the vertices of the diamond and its center point for a total of 5 points identified.

1. Each student draws a diamond on his or her coordinate system without letting others see it. The diamond should have its vertices on lattice points one unit from the center. Also locate the center of your diamond.

2. Decide who will guess first and begin. The first player, A, guesses a point by naming an ordered pair. The other player, B, tells A if that ordered pair is a vertex or center of his/her diamond. If the guessed point is a vertex or center of B’s diamond then A gets to guess another point until A guesses a point not on the diamond or center.

3. B now gets to guess an ordered pair that is on A’s diamond. If B guesses a point on A’s diamond then (s)he continues to guess until B guesses a point not on A’s diamond.

4. Continue taking turns until one of the players gets all four vertices and the center of the other player’s diamond. This player has found the hidden diamond!


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SUMMARY:

Write about what you learned in this section. Include some of the new vocabulary and what they mean using pictures and words.


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SECTION 1.2 LINING UP

In this section, you will explore lines and the points on those lines. See if you discover a pattern or rule for the points on the line, especially between the coordinates on the points.

LINE 1:

Plot the points (1, 2) and (3, 4) on a coordinate plane. Carefully draw a straight line connecting these two points. Write a list of four more points that you think are on this line. Explain why you think the points are on this line.

Sect ion 1 .2 L in ing Up

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1. Make a list of at least 8 points on the line.

2. How did you organize this list of points?

3. Make a chart of these points by filling in the second coordinates that are missing. We can say that these points “lie” on this line.

First Coordinate

−2 −1 0 1 2 3 4 5 10

Second Coordinate

4. Name 2 or 3 other points on this line that are not on the chart.

5. Identify other points that would be on this line that do not fit on the coordinate plane on the previous page.


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6. What point will have 30 as its first coordinate? ( 30 , ____________ ) Explain how you found the second coordinate.

7. What point will have 30 as its second coordinate? ( ____________ , 30 ) Explain how you found the first coordinate.

8. Try to write the second coordinate in terms of x, where x stands for the first coordinate? What point will have x as its first coordinate? ( x , ____________) Explain.

9. If (x, y) is a point on this line , then write y in terms of x in an equation form, y = ____________. Explain how you arrived at your equation.


13

We take the coordinate chart in part 3) on the previous page and write the coordinates as ordered pairs. We can also take the horizontal chart and view it vertically as you can see below.

( ¯2 , ¯1 )

( ¯1 , 0 )

( 0 , 1 )

( 1 , 2 )

( 2 , 3 )

( 3 , 4 )

( 4 , 5 )

( 5 , 6 )

( 6 , )

( 7 , )

( 15 , )

( 24 , )

Do you see the pattern? Explain. Use the pattern that you observe to find the y coordinate missing in the chart above.

If p is the first coordinate of a point on this line, what is the second coordinate of the point in terms of p? ( p , )

From above, we have an equation of the line y = x + 1.

If x is the first coordinate of a point on this line, the second coordinate would be x + 1. If (x, y) is a typical point in this line, then we have discovered that y is the same as x + 1. We write this as y = x + 1.

This statement is called an equation for the line because it describes the relationship between the first coordinate, x, and the second coordinate, y, of a typical point, (x, y), on the line.

x y−2 −1

−1 0

0 1

1 2

2 3

3 4

4 5

5 6

6

7

15

24


14

LINE 2:

Plot the points (2, 6) and (4, 8) on the coordinate plane below. Draw a straight line connecting these two points.

Write down as many points on this line as you can.

Fill in the chart below.

First Coordinate −2 −1 0 1 2 3 4 5 10

Second Coordinate 6 8


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1. What point(s) will have 5 as its first coordinate? Explain. (5, ____________ )

2. What point(s) will have 24 as its second coordinate? Explain. (____________ , 24)

3. Name any point(s) where this line intersects the x-axis. Explain.

4. Name any point(s) where the line intersects the y-axis. Explain.

5. What equation can be used to describe the y-coordinate for the points that lie on this line? Explain. ( x , ____________ )

6. What equation can be used to describe the x-coordinate for the points that lie on this line? Explain. (____________ , y )

7. If (x, y) is a point on this line, y = ____________ is the equation that describes this line. Explain.


16

LINE 3:

Let us start with an equation of a line given by y = x − 3.

1. Make a list of possible points on this line. Write the points as ordered pairs (x, y).

2. Create a table of these points.

3. Plot the points on your coordinate system and draw a line that passes through these points.


17

LINE 4:

Plot the points (1, 5) and (4, 2) on a coordinate plane. Draw a straight line connecting these two points.

Write a list of 4 more points on this line. We can say that these points belong to this line or lie on this line.

Make a list of all these points by filling in the chart below.

First Coordinate –2 –1 0 1 2 3 4 5 6 7 8

Second Coordinate 5 2

You may wish to list the points in a vertical column or table.


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1. What pattern do you notice in these ordered pairs of numbers?

2. Use this pattern to write an equation for this line involving a typical point (x, y).

3. Use this equation to find the point that has 12 as its first coordinate.

4. Use this equation to find the point that has 10 as its second.


19

LINE 5:

Madeline opens a lemonade stand. Suppose it costs her $5 to set up the stand and $1 for the ingredients to make each glass of lemonade. Complete the following chart.

Number of glasses

0 1 2 3 4 5 6 7 8 9 10

Cost in $

How many glasses of lemonade could she make for $7?

1. Plot this data as points on the coordinate system below with the first coordinate representing the number of glasses and the second coordinate representing the cost in dollars. Label the x–axis as the “number of glasses” and the y–axis as the “cost in $.” Can you have negative coordinates for this line? Why or why not?


20

2. Describe any patterns you notice in the points plotted.

3. How would you describe the way they appear to someone who cannot see the graph?

4. If Madeline made 10 glasses of lemonade, what would be the cost? How about 20 glasses? 30 glasses?

( 10 , )

( 20 , )

( 30 , )


21

5. What is a relationship between the first and second coordinates of these points?

6. If Madeline made x number of glasses, what would be the cost? Write an equation for this line that relates x to the cost y.

7. If Madeline has $30 to spend on making lemonade, how many glasses could she make?


22

8. If Madeline has $46 to spend on making lemonade, how many glasses could she make?

9. If Madeline has $29 to spend on making lemonade, how many glasses can she make? Solve this problem algebraically.

10. Make up a question about this problem and see if you can answer it algebraically.


23

LINE 6:

Claudia wants to sell her lemonade for $2 per glass. Complete the chart below. How much money will she collect selling x number of glasses of lemonade?

Number of glasses

0 1 2 3 4 5 6 7 8 9 10

Sales (revenue) in dollars ($)

1. Plot this data as points on the coordinate plane below. The first coordinate represents the number of glasses and the second coordinate represents the amount of sales in dollars.

Amou

nt o

f Sal

e

Number of Glasses

2. What relationship can you describe between the first and second coordinates of these points?


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3. Do any of your points lie outside the first quadrant? Explain your answer.

4. If Claudia sold 10 glasses of lemonade, what would be the amount of sales (revenue)? How about 20 glasses? 30 glasses?

( 10 , )

( 20 , )

( 30 , )

5. If Claudia wanted to take in $32 in sales, how many glasses would she need to sell?


25

LINE 7:

Use the equation y = 2x + 4 to produce a table of points. Plot 6 of those points.


26

a. Do they all lie on a straight line? Explain why you do or do not think so.

b. At what point does the line intersect the x-axis? Explain how you determined this point.

c. At what point does the line intersect the y-axis? Explain how you determine this point.


27

LINE 8:

Consider the points (1,6) and (4,3). Draw a straight line that contains those two points. Name two other points, P and Q, that lie on line 8.

a. Name two points S and T that do not belong to the line L.

b. What properties do the points P and Q have that the points S and T do not have? Explain


28

SUMMARY:

Write what you learned in this section. Include the key ideas for the day in your journal.

29

CHAPTER 2FUNCTIONS AND GRAPHS

SECTION 2.1 ADDING, SUBTRACTING, AND MULTIPLYING INTEGERS

OBJECTIVES:

• Review addition and subtracting of integers using the number line

• Model multiplication of integers using number lines and patterns

Adding and Subtracting Integers on the Number Line

Addition is a mathematical operation for combining integers. Pictorially, using the “set model,” when we add two integers we are combining the sets. To add 4 and 3 we draw the picture below:

+ =

We also use our number line model to describe addition and subtraction. For example, we model the addition problem 7 + 2 on the number line, with cars or with frog leaps.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Let us record exactly what happened to our hopping frog when we add 7 + 2. First, notice that our frog begins at 0 facing in the positive direction because our the first number, 7, is positive. When the frog lands on 7, it notices that the next number is two (again, a positive number). The frog then faces in the positive direction and hops 2 more. It lands on 9, so we can say that 9 is the sum of 7 and 2.

Chapter 2 Funct ions and Graphs

30

Try writing the steps to find the sum of this new problem, 7 + (–2). Discuss this with your classmates first, and then we will present a model that is similar to our Car Model for Addition.

We again start at 0 and hop to 7, facing the positive direction. Be careful with the next step! The frog sees that the next number is a negative 2 and not a positive 2. The frog now faces in the negative direction, and then hops forward 2 places. Do you see where it landed? Draw the process of adding these two numbers on the number line below.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Did your frog land on 5?

Sect ion 2 .1 Adding, Subtract ing, and Mult ip ly ing Integers

31

Try helping the frog in this problem: 7 + –2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Let's summarize what we would do for any addition problem using the Frog Jump Model of two integers A and B (notice A and B could be positive, negative, or zero):

FROG JUMP MODEL OF A + B

Step 1: Start at 0 and face in the direction of the sign of A.

Step 2: Hop as many places as the absolute value of A.

Step 3: Face in the direction of the sign of B.

Step 4: Hop as many places forward as the absolute value of B. This locates the sum of A + B.


32

PRACTICE PROBLEMS

1. Find the following sums, and use the Frog Jump Model for addition on a number line to show the process.

a. –7 + –2 =

b. –7 + 2 =


33

c. 4 + – 6 =

d. –4 + – 6 =


34

How might 7 − 2 look on the number line?

Notice that the numbers 7 and 2 in this problem are the same as the first addition problem that we did with the frog, 7 + 2. The only difference is that the operation is subtraction and not addition.

Does your model look like this?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Let's record what happened here in the Frog Jump Model for subtraction.

First, notice that our frog begins at 0 facing in the positive direction because the first number 7 is positive. When the frog lands on 7, it notices that the next number is 2, again a positive number. The frog faces in the positive direction, but this time, because the operation is subtraction, it hops backward 2 places. The frog lands on 5, so we say 5 is the difference of 7 and 2.

The only difference between this problem and the first addition problem is that in addition, the frog goes forward facing in the direction of the second number, while in subtraction the frog goes backward while facing in the direction of the second number.


35

Try writing the steps to find the different of the number line for 7 – (–2). Discuss this with your classmates first, and use the number line below to justify your steps.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

First, start at 0 and face in the positive direction. Hop forward to 7. Next, examine the second number. Because it is negative 2, face in the negative direction, then, noting that we are subtracting and not adding, hop backward 2 places. We should end at 9.

Let's summarize what we would do for any subtraction problem using the Frog Jump Model of two integers A and B (notice A and B could be positive, negative, or zero):

FROG JUMP MODEL OF A – B

Step 1: Start at 0 and face in the direction of the sign of A.

Step 2: Hop as many places as the absolute value of A.

Step 3: Face in the direction of the sign of B.

Step 4: Hop as many places backward as the absolute value of B. This locates the difference of A – B.


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PRACTICE PROBLEMS

2. Find the following differences, and use the Frog Jump Model for subtraction on a number line to show the process.

a. –7 – (–2) =

b. –7 – 2 =


37

c. 4 – (–6) =

d. –4 – (–6) =


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MULTIPLYING INTEGERS

Just as we modeled addition and subtraction on the number line, we can use the number line to model multiplication.

For example, skip counting by 3’s generates the list 3, 6, 9, 12, 15, 18, 21, and so on, continuing indefinitely. The number line can use the idea of "skip counting" to model multiplication.

On the number line, the frog’s jumps correspond to the numbers we are adding. In order to multiply, we can think of a frog that jumps along the number line. For example, when you multiply 4 · 3,

• the result of the multiplication is called the product

• the first factor indicates which direction the frog should face and the length of each jump

• the second factor indicates the number of jumps

The picture below models the multiplication 4 · 3 = 12. Notice the frog is facing in the positive direction because the first factor, 4, is positive. The frog takes 3 jumps, and each jump is 4 units long. The final location is the product 12.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14


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EXPLORATION 1: FROG JUMP MULTIPLICATION

Complete Table 1, in which each jump is 4 units long.

TABLE 1

Length of Jump(factor)

Number of Jumps(factor)

Frog’s Location(product)

4 0 0

4 1 4

4 2 8

4 3

4 4

4 5

4 6

4 10

4 20

4 n

Does this table look familiar? You might recognize these numbers from a multiplication table of 4’s where the pattern is 4 · 1 = 4; 4 · 2 = 8; 4 · 3 = 12; 4 · 4 = 16.

You can think of (4) (3) as (4 units per jump) (3 jumps) = 12 units.


40

Complete Table 2 as you did for Table 1, but this time use jumps of directed length 7.

TABLE 2

Length of Jump Number of Jumps Frog’s Location

7 0 0

7 1 7

7 2

7 3

7 4

7 5

7 6

7 10

7 20

7 n

Multiplication of 7 and 12, often written as 7 × 12, can also be written as 7 · 12, 7 *12, or (7) (12).


41

PROBLEM 1

Compute the following products. Explain how you arrive at your answer using the number line.

a. (8) (6) =

b. (8) (5) =


42

c. (8) (3) =

d. (8) (7) =

Let’s summarize the frog model:

The first factor tells us the length of each jump, and the second factor tells us the number of jumps.


43

Using the frog model, complete the following table using the length 15.

TABLE 3

Directed Length of Jump Number of Jumps Frog’s Location

15 0 0

15 1 15

15 2

15 3

15 4

15 6

15 10

15 20

15 100

15 n

Use this table to compute the following products:

a. (15) (7) =

b. (15) (30) =

c. (15) (12) =

d. (15) (1000) =


44

You learned how to add positive and negative integers in Level 1. Is there a way to think about multiplying a negative integer times a positive integer? You can use the frog model to multiply –4 · 3, as shown below.

-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Copy and fill the skip counting Table 4 as you did in Table 1, but this time use jumps of directed length ¯4.

TABLE 4


–4 0 0

–4 1 –4

–4 2

–4 3

–4 4

–4 5

–4 6

–4 10

–4 20

–4 n

Using the pattern demonstrated in this table, compute the product –3 · 4.


45

The picture below models the product –3 · 4. The first factor tells us which direction the frog should face and the length of each jump; the second factor tells us the number of jumps.

-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

The frog is facing left because we are modeling a jump of −3 units per jump.

Use the number line to compute the following products:

a. (–3) (1) =

b. (–3) (3) =


46

c. (–3) (5)

d. (–3) (6)


47

How can we make sense of the product (3) (–4)? This is the first example where the second factor is negative.

The first number, 3 or +3, gives the length of each jump, and the direction the frog is facing. Because the number is positive, the frog faces right.

The second factor gives the number of jumps. What do we mean by the number –4 as the number of jumps? If we think of the jumps taking place at equal time intervals, we can imagine the frog jumping along a line.

We pick one location, call it 0, and name the time as the “0 jump.” When the frog takes its first jump, jump 1, the frog lands at location 3. When the frog takes its second jump, jump 2, the frog lands at location 6.

Let’s go back to the 0 location and ask where the frog was on the jump before it arrived at 0. We call this jump –1. Because the frog jumps 3 units to the right every jump, the frog must have been at location –3, which is 3 units to the left of 0. Two jumps before reaching 0, the frog was at location –6. We can now fill out the table below.

TABLE 5


3 –6

3 –5

3 –4

3 –3

3 –2

3 –1

3 0 0

3 1 3

3 2 6

3 3 9


48

It is now possible to answer the earlier question. What do we mean by –4 jumps? This means we jump backward in time, or simply jump backward.

Use the number line to compute the following products. Verify that your answers agree with the table.

a. (3) (–1)

b. (3) (–3)


49

c. (3) (–5)

d. (3) (–6)


50

Let’s summarize the frog model:

The first factor tells us which direction the frog should face and the length of each jump.

The second factor tells us the number of jumps and the direction of the jump. When the second factor is positive, the frog jumps forward; when the second factor is negative, the frog jumps backward.

Using the frog model, compute the product (–3) (–4). The directed length of each jump is –3. Determine what happens when the frog jumps backward in time.

0 1 2 3 4 5 6 7 8 9 10 11 12


51

Complete the following table, starting at the bottom and working up.

TABLE 6

Directed Length of Jump

Number of Jumps Frog’s Location

–3 –6

–3 –5

–3 –4

–3 –3

–3 –2

–3 –1

–3 0 0

–3 1 ¯3

–3 2 ¯6

–3 3


52

EXERCISES

1. Suppose we want to make a table for the line given by the equation y = 3x before you plot points.

a. Fill in the table below for this line. Notice that we start with negative values for the input x.

Input x Output y = 3x

–5 y = (3) (–5) =

–4

–3

–2

–1

0

1

2

3

4

5


53

b. Plot a few of these points on a coordinate system.


54

2. Suppose we want to make a table for the line given by the equation y = –2x before you plot any points.

a. Fill in the table below for this line. Notice that we start with negative values for the next line input x.

Input x Output y = –2x

–5 y = (–2) (–5) =

–4

–3

–2

–1

0

1

2

3

4

5


55



56

3. Suppose we want to make a table for the line given by the equation y = –4x before you plot any points.

a. Fill in the table below for this line. Notice that we start with negative values for the next line input x.

Input x Output y = –4x

–5 y = (–4) (–5) =

–4

–3

–2

–1

0

1

2

3

4

5


57



58

EXPLORATION 2

In San Marcos, Texas, the temperature rises an average of 3 °F per hour over a twelve hour period from 7 a.m.

to 7 p.m. The temperature at 7 a.m. is 72 °F. Let x be the number of hours after 7 a.m.

a. Make a table that shows a relationship between the time and temperature over the twelve

hours. Make a column for the number of hours since 7 a.m. and a column for the temperature.

Time After 7 a.m. Temperature at the Time

0 72 °

1

2

3

4

5

6

7

8

9

10

11

12

b. What was the temperature at 3 p.m.?


59

c. What x-value corresponds to 12 p.m.? What is the temperature at 12 p.m.? How many hours will

it take for the temperature to rise 12 degrees? What time will that be?

d. When is the temperature 63 °F?

e. Is it possible to use multiplication to help determine the temperature in parts b and c? If so, explain

how.


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SUMMARY:

Write what you learned in this section.

Sect ion 2 .2 Funct ions

61

SECTION 2.2 FUNCTIONS

KEY OBJECTIVES:

• Review variables and introduce functions.

• Build tables to represent functions.

• Solve equations.

We saw in the last section how lines in the plane are made up of points that have coordinates (x, y) with a special relationship between the first coordinate x and the second coordinate y. This relationship is often written as an equation, such as y = 2x, where the second coordinate is twice the first coordinate. In this section, we explore a more general setting in which to think about relating two quantities. We investigate the idea of a function. Before we begin, let us review the idea of a variable such as the x and y that we used above.

A variable is a symbol, often a letter, used to represent a quantity. For example, suppose Chris has a collection of books. We can use the letter B to represent the number of books Chris has to start. B is called a variable; its value may change depending on how many books Chris has. For example, he may receive some more books for his birthday or give some away to a friend. Variables are like the alphabet of algebra and are useful for writing expressions and equations which are like the phrases and sentences used in algebra.

For example, if Chris had B books but receives 2 more books today, then algebraically, we would write the expression, B + 2 to represent the number of books Chris has now. And if instead Chris had received 3 books, we would write B + 3 to represent the number of books Chris has. If Chris starts with B books but decides to give away 1 book to a book sale, what algebraic expression would represent the number of books Chris has now?

We begin with an exploration to learn about what is a function.


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EXPLORATION 1: USING TABLES TO REVEAL A PATTERN

Sarah builds model airplanes. She makes two airplanes each day. How many airplanes will she make in 4 days? 10 days? Organize the information to reveal a pattern in the number of airplanes she makes in a given number of days. How did you organize the information?

Do you see a pattern in the number of airplanes she can make in a given number of days?

One way to organize such information is to build a table such as the following. Notice that the first column is the number of days, and the second column is the total number of airplanes that Sarah can make in that many days.

DaysTotal Number of

PlanesOrdered Pairs

0

1

2

3

5

10

x


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What do you notice about this table?

Why is this a good way to organize the information?

This is an example of a function. There is a rule, or function, to determine how many planes Sarah has produced based on the number of days she has worked. You can also think of a function as a machine with inputs and outputs. The input is the number of days Sarah worked. The output is the number of planes she produced.

In mathematics, we usually name a function with a letter such as F for this function. It can be a little confusing because we also said that letters can represent variables. We will try to keep the use of the letters straight by always saying what the letter is being used for.

Let’s think a little more about the function machine just mentioned by first viewing an example of what it looks like.

Input OutputFunction

RuleDomain Range


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ACTIVITY: FUNCTION MACHINE:

Now that we have an idea of what a function does, let’s go ahead and make a more formal definition.

DEFINITION: FUNCTIONA function is a rule which assigns to each element of one set (called

the domain) one and only one element of a second set (called the

range).

The domain is the set of inputs to the function.

The range is the set of outputs.

We can draw a picture that shows the pairing of this function as follows:

Domain Range

2

6

4

1

2

3

Notice that a function produces pairs of numbers. From the table and the picture above, we can see that the following pairs belong to the function F: (0,0), (1,2), (2,4), (3,6), (5,10) and so on.


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ACTIVITY: GUESS MY NUMBER

There are 2 card readers, 2 recorders, and the rest of the class as participants. A deck of function cards are provided for the card readers. The objective is to guess the function rule.

1. The card readers select a card from the stack and look at the card without revealing the function rule to the rest of the class. As information and hints about the function, they ask the recorders to write two input/output pairs that satisfy the function rule.

2. The readers then ask a class participant to guess an input/output pair.

3. The reader gives thumbs up if the input/output pair satisfies the function or thumbs down if the input/output pair does not satisfy the function. If it is thumbs down, one of the recorders keeps a record of the input/output pair that did not work in a certain portion of the blackboard. If it is thumbs up, then the other recorder keeps a record of the input/output pair that did work in another portion of the blackboard.

4. Another class participant makes another input/output pair guess.

5. The reader gives thumbs up or thumbs down and the recorders organize the information.

6. The recorder asks the students to examine the information on the board and asks if anyone wants to guess the rule of the function.

7. Continue this process until the correct rule of the function is determined.


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Let’s say the Function Rule is 5x − 1. Think of a number, or value of x, to input into the machine. We’ll again organize our information on a table.

Input (Domain)

Function Rule 5x–1 Output (Range) Ordered Pair

1 5(1) – 1 4 (1, 4)

2 5(2) – 1

3

4

To continue, substitute the value of 1 in for x, and find what is 5(1) –1. Write your answer in the output column. We could also write this as an ordered pair, (1, 4), meaning that when the x value is 1, the y value is 4.

If the input is x = 1, could the y value be any other number? ___________

Explain your answer.

Continue inputting the values of x to find the corresponding output and record it on the table.


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ACTIVITY: THE FUNCTION CHALLENGE

1. Each group has two index cards with the name of their group written at the top. On one of the index cards which we call the function card, the group names a function (e, f, g, h, r, or s) and writes a letter to represent the name of the function along with the function rule. This card will be kept face down for later reference. On the other card, the group writes the function letter and then identifies 5 input/output pairs that satisfy the function rule.

2. The groups trade their input/output cards and each group tries to determine the function rule that corresponds to the given input/output pairs.

3. After a set time, the groups can check their guess with the function cards. A team gets a point if they correctly guess the function rule. The input/output cards are traded and the new group tries to determine their function. Continue the trading until each team has had a chance to try to guess all the other function rules except theirs. Let each team keep their own record and tell them not to reveal the rules to the other teams so that each team can get a fresh start in guessing the rules.

We introduce a mathematical way of writing a relationship between numbers in the domain with numbers in the range using function notation. Just as in the Function Challenge Activity, we begin by giving our function a letter name. Let’s use F for the name of our function and refer our first Exploration with Sarah’s airplane. She built 2 airplanes each day. If we start with day 1, we see that this function F pairs the number 1 with 2. We can write this symbolically as

F(1)=2

to indicate that 2 is the output corresponding to the input 1.

Similarly, because the function F pairs the number 2 in our domain with the number 4 in the range, we write,

F(2) = 4, read “F of 2 equals 4”.

We also have F(3) = 6. We can continue this pattern but we can also express this rule in general as follows:

F(x) = 2x

So F(x) = number of planes that can be produced in x days.


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Using this rule, what is the value of F(4)?

F(4) = _____________

What is the value of F(5)?

F(5) = ______________

We can also represent F by a table of values. Fill in the table below.

x 0 1 2 3 5 10 x

F(x) = 2xF(0) = F(1) = F(2) = F(3) = F(5) = F(10) = F(x) =

Each pair of numbers in the table, such as (2, 4), can be thought of as a point on the coordinate system and as a point on the graph of F. The ordered pairs on the table for the function F have the special property that the first and the second coordinates are related by the function rule for F. In other words, for the function F, every first coordinate, x, must have second coordinate 2x. We call the second coordinate F(x), the value of the function F with input x. Notice that the points on the graph of F have coordinates (x, F(x)).

In our example, the first coordinate x = the number of days that Sarah has worked.


69

Using the table, write the ordered pairs and plot their corresponding points on the following coordinate system. Can you produce three more pairs that belong to this list? Plot them.

F(x)

x


70

SMALL GROUP ACTIVITY:

1. Let the variable x = the number of days it takes to produce 12 airplanes. Write an equation that can be used to answer this question. Try to set up your equation before reading on. Using Exploration 1, determine how many days it takes to make 12 airplanes.

We have the rule F(x) = 2x and we also know that F(x) must equal 12 for this particular x. Therefore 2x = 12. Solve for x.

2. How many days will it take to make 20 airplanes?

Do you see a pattern?


71

3. How would you figure out the number of days to make 5 airplanes? or 9 airplanes?


72

Let’s practice building functions to represent different problems.

EXERCISE A

1. Sonora sells pencils for 6 cents each. For example, if Sonora sells 3 pencils, then Sonora earns 18 cents. We say Sonora’s revenue for selling 3 pencils is 18 cents.

If we let R(x) = Sonora’s revenue for selling x pencils, make a chart showing R(0),R(1), R(2), R(5), R(x).

x = # of pencils 0 1 2 3 5 10 x

R(x) = Revenue


73

Make a list of the ordered pairs, and plot these points on the following coordinate system.

R(x)

x


74

2. In order to sell lemonade, Terry paid $20 for a sign and a table. It then cost him $4 for each gallon of lemonade he made. What was the total cost for making 2 gallons (including the $20 set-up cost)?

Fill out the cost table below, where C(x) = the cost to produce x gallons of lemonade

x = # of gallons

0 1 2 3 5 10 x

C(x) = Cost of

x gallons

The second coordinate numbers are relatively large and difficult to fit on the coordinate system below. You can consider scaling the y-axis by 4, for example, while the x-axis can remain scaled by 1.

Plot the ordered pairs on the coordinate system.

C(x)

x


75

C(4) = _____________________________

Write a rule for the function C in terms of x.

C(x) =_____________________________

Suppose we now know the cost. Then we can determine the number of gallons that can be produced at that cost. Complete the table below.

x = # of gallons x

C(x) = Cost of x gallons in dollars 32 44 52 60 26 31

How many gallons of lemonade can be made for $30?


76

FUNCTION F(x) = 2x + 3

Consider the function F given by the rule F(x) = 2x + 3. Find the following.

a. F(0) =

b. F(1) =

c. F(2) =

d. F(−1) =

e. F(−2) =

f. F(3) =

g. F(−3) =


77

FUNCTION F(x) = 2x + 3 (cont.)

Make a table of five other ordered pairs using the above formula for F.

x F(x) = 2x + 3


78


Graph these ordered pairs on the following coordinate system.

You can use your graphing calculator to enter the function y = 2x + 3. First, check your answers above using the Table key. Next graph the function using your calculator. How does it compare to your graph?


79


For the function F(x) = 2x + 3 above, fill out the table below.

x

F(x) = 2x + 3 3 5 7 9 10 23 4

How can you find the answers using your graphing calculator?

Try to set up an equation to find each answer value of x with the F(x) given. Remember to show each step for solving the equation.

For example:

What is the value of x if F(x) = 9?

Since the rule for F(x) = 2x + 3, then 2x + 3 = 9 for our particular x.

Solving for x we get x = 3.

What value of x will give us the output of

a) F(x) = 5

b) F(x) = 7

c) F(x) = 23

d) F(x) = 4


80

FUNCTION H(x) = 3x + 1

Consider the function H given by the rule H(x) = 3x + 1. Compute the following:

a. H(0) =

b. H(1) =

c. H(2) =

d. H(–1) =

e. H(–2) =

f. H(–3) =

Use the values you found to build the table for H:

x

H(x) = 3x + 1


81

FUNCTION H(x) = 3x + 1 (cont.)

Graph these ordered pairs, and then connect the points to graph the function H(x).

H(x)

x

For the function H(x) = 3x +1 above, fill out the table below. Try to set up an equation to find each answer.

x

H(x) = 3x +1 13 16 22 –5 –8 –11 31


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7. Oliver has 9 M&M’s. He eats them very slowly; in fact, he takes 1 minute to eat each one.

Make a table for the number of M&M’s Oliver has left.

Time (min.)

0 1 2

# M&M’s left

Plot the points on the graph below.

Find a formula for M(x), the number of M&M’s Oliver has left after x minutes.

When will Oliver run out of M&M’s?


83

SUMMARY:

Think about a situation that you can model as a function. Describe your function using words and mathematical language. How did what you learned help you to think of this?

Important vocabulary:

• variable

• function

• input

• output

• domain

• range

• coordinate system

• graph

• first coordinate

• second coordinate


84

85

CHAPTER 3EXPLORING FRACTIONS

SECTION 3.1 MODELING FRACTIONS

OBJECTIVES:

• Understand what fractions are.

• Find equivalent fractions.

Have you heard expressions like “I can only eat half that apple” or “You can have two thirds of my marbles”? Each of these statement contains a number called a fraction. What is a fraction?

Picture an apple divided into two equal parts.

"Half an apple" could be drawn by shading one of the two parts.

Imagine the pile of marbles divided into three equal groups. Two-thirds of the marbles would be two of these three equal groups.

One-half and two-thirds are fractions.

Chapter 3 Exp lor ing Fract ions

86

A picture representing one-half can be drawn using a Fudge Model where the one shaded part out of two equal pieces represents one-half of the whole pan of fudge.

Similarly, two-thirds would be shaded as:

We can write these mathematically as follows:

One-half = 21 and Two-thirds = 3

2

There are two parts of a fraction, the numerator and the denominator. The denominator is on the bottom; it represents the number of parts in the whole. The numerator is on the top; it indicates how many parts are used.

Use the Fudge Model to draw the fractions below. Write the fractions using a numerator and denominator. Identify the numerator and denominator of each fraction.

1. Three-fourths =

Numerator: ________

Denominator:________

Sect ion 3 .1 Model ing Fract ions

87

2. Two-fifths =

Numerator: ________

Denominator: ________

3. One-fifth =

Numerator: ________


4. One-tenth =

Numerator: ________


5. Five-sixths =

Numerator: ________



88

PIZZA BREAD STICK GAME:

1. Make a line of students (12 is best). They become a pizza bread stick. It needs to be divided into pieces.

2. Suppose there are 2 people who want to split the bread stick. Have someone not in line divide the giant bread stick into 2 equal pieces.

3. Suppose instead, 3 people want to divide the bread stick equally. Have someone else do this.

4. Now divide the bread stick for 4 people, then 6 people.

5. What would be the problem with trying to divide the stick into 5 equal parts?

6. Record the results in each of the steps 1 through 5.

Let’s look at how the fractions are used with the concept of time. Remember that 60 minutes is equal to one hour. Look at our clock.

For example, what part of an hour is 30 minutes?

Did you notice that 30 minutes is 30 parts out of 60, or 6030 of an hour? We often say 30 minutes

is half an hour. That means 6030 = 2

1 .


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What fraction of an hour is 15 minutes?

6015 is correct. This fraction represents 15 minutes out of an hour or 60 minutes total. There are four 15-minute intervals in one hour, so 15 minutes is 4

1 of an hour. 6015 = 4

1 . What fraction is 20 minutes? Shade in a 20 minute representation on the clock below.

Can you find two ways to write the fraction?

Can you write 10 minutes as a fraction of an hour without drawing a clock? Are there two or more ways to express this fraction?


90

EXERCISE A

1. Jeremy practiced juggling for 40 minutes. For what fraction of an hour did Jeremy practice?

2. Daniel practiced trombone for 45 minutes. For what fraction of an hour did Daniel practice?


91

3. Mike played the piano for 1 21 hours. For how many minutes did Mike play?

4. Margaret spent 65 of an hour making dinner. After dinner, Juliana spent 6

5 of an hour cleaning up. How much total time did the two sisters take in making dinner and cleaning up afterwards? Write your answer in minutes as well as in hours (you may need to use fractions).


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5. Use the following measuring conversions to answer the questions:

2 cups = 1 pint

2 pints = 1 quart

4 quarts = 1 gallon

a. How many pints are in 1 gallon?

b. How many pints are in 21 gallon?

c. How many cups are in 3 pints?


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6. Use the following measuring conversions to answer the questions:

1 cup = 16 tablespoons

1 tablespoon = 3 teaspoons

a. How many teaspoons are in 1 cup?

b. How many tablespoons are in 41 cup?

c. How many teaspoons are in 31 cup?


94

It is possible for two fractions with different numerators and denominators to represent the same amount? Consider the following example:

21 =

Shading 2 equal parts out of 4 is equivalent to shading 1 part out of 2. Thus, we say that the fractions 2

1 and 42 are equivalent. Another way to show that the fraction 2

1 is equivalent to the fraction 4

2 is to take the picture representing 21 and draw a horizontal slice as shown below.

21 =

EXPLORATION

Suppose we make two horizontal cuts as shown below. To what fraction is this equivalent?

21 =

As you probably observed, the picture represents both 21 and 6

3 . These fractions are equivalent. Now try writing two more fractions that are equivalent to one-half.

21 = _______________ = _______________


95

ACTIVITY: LARGE HERSHEY’S CHOCOLATE BAR

Form two groups: Group A and Group B. Each group has a large chocolate bar.

1. Instruct Group A to eat 41 of the candy bar. The Group B is to eat 12

3 of the candy bar. Each group records how much remains. Is it the same? What does it say about 4

1 and 123 ?

2. Have Group A eat 31 of the remaining candy bar. The Group B is to eat 9

3 of the remaining candy bar. Each group records how much remains. What can you conclude about the fractions

31 and 9

3 ?

3. Group A now eats 21 of the uneaten portion and Group B eats 6

3 . Each group records how much remains.

4. As a final step, Group A eats 33 of the remaining candy bar. The Group B eats what remains.

5. Write all the equivalent fractions that occur in every step above.


96

GROUP ACTIVITY

Get into groups of four. Have each group draw 3 equivalent fractions for a given fraction on butcher paper with crayons or markers. Every group should be able to explain their drawings. Do you notice anything about the numerator and denominator of the equivalent fractions that each group chose?

Can each group find one more equivalent fraction for each of the other groups without having to draw a picture?


97

EXERCISE B

Write an equivalent fraction for each problem. Include a visual representation (for example, an area model) to justify your answers.

1. 31 =

2. 41 =

3. 51 =

4. 52 =

5. 101 =

6. 103 =

7. 125 =

8. 61 =


98

In general, we can find equivalent fractions by multiplying the numerator and denominator by the same number. For example,

41 =

( )( )( )( )2 42 1 = 8

2

Pictorially,

41 =

Multiplying the numerator and denominator by 2 has the effect of doubling the number of slices, as shown below:

41 = 8

2 =

Multiplying the numerator and denominator by the same number will double, triple, etc. the number of parts, but yields an equivalent fraction. This is an important result and we state it as the

EQUIVALENT FRACTION PROPERTYFor all non-zero numbers k, we have

ba = k b

k a::


99

PRACTICE PROBLEMS

Write fractions which are equivalent to each of the following fractions. Draw a picture of each.

1. 71

2. 72

3. 121

4. 31


100

TEAM GAME 1: LOTS OF EQUIVALENT FRACTIONS

This game is a competition to see which team can find the most equivalent fractions (to a given fraction) in one minute. The fractions must be written down. You can use the board or a lap board so everyone can see how many equivalent fractions were correctly found.

Time: Every team plays at the same time and has one minute to make its list.

Scoring: A team gets +1 point for each correct equivalent fraction. An incorrect answer counts as ¯1 point. There will be 5 rounds. At the end of the 5th round, the team that has the most points wins the game.

TEAM GAME 2: A SPECIAL FRACTION

Objective: Find the equivalent fraction with the smallest possible denominator.

Procedure:

1. Every team is given a fraction and one minute to find this equivalent fraction.

2. After one minute is up each team must explain how it got the particular fraction by showing its work.

3. After 5 rounds the team with the most correct answers wins.


101

SUMMARY:


VOCABULARY LIST:

• fractions

• numerator

• denominator

• equivalent fractions


102

SECTION 3.2 ADDING AND SUBTRACTING FRACTIONS

OBJECTIVES:

• Add fractions with like denominators.

• Subtract fractions with like denominators.

• Find equivalent fractions to mixed fractions.

• Solve word problems using mixed fractions.

When we add 1 foot and 2 feet together our total is 3 feet. 1 apple plus 2 apples equals 3 apples. Notice that we keep track of both the numbers and the units. So when we add the fractions 1 fifth and 2 fifths together the sum is 3 fifths.

Let’s look at a picture model. Suppose that you had 51 of a candy bar, and a friend had 5

2 of the same candy bar.

Then together, both people would have 53 of the candy bar. We write

51 + 5

2 = 53

Notice that we added the numerators 1 and 2 because they tell us how many pieces we are combining. The denominator tells us the size of each piece, which in our case is one-fifth, written

51 .

When we added, the size of the pieces remained the same. So the denominator remains 5.

Sect ion 3 .2 Adding and Subtract ing Fract ions

103

SMALL GROUP ACTIVITY: EGG CARTON ADDITION

Work in groups of 3.

1. One person puts objects into some of the egg holders and writes the fraction that represents the filled portion of the egg carton.

2. Second person puts more objects into some of the remaining empty egg holders and writes only the fraction he/she filled on the same sheet.

3. Third person adds the two fractions on the paper and checks it with the egg carton. Is there more than one way to write the fractions?

4. Switch roles.

Did you discover that to combine fractions with the same denominators, we add the numerators, and the resulting sum has the same denominator as the fractions being added? We can record this rule algebraically.

ADDING FRACTIONS WITH LIKE DENOMINATORS

da + d

b = d(a b)+

where da and d

b represent two fractions.

It is important to remember that in order to use the above rule, the fractions must have the same denominator. If the fractions have the same denominator then we combine the fractions by adding the numerators, and the result has the same denominator.

The same principle applies when we subtract fractions. If we begin with 1 candy bar and then take away 3

2 of the candy bar, how much of the candy bar is left?


104

As mathematical fractions, the problem looks like the following.

In order to perform this calculation, we can begin by drawing a picture of a candy bar and dividing it into three pieces.

We first convert 1 into the fraction 33 .

1 =

What happens when we subtract 32 of the candy bar? Shade the portions that are to be taken

away.

We can write this as:

1 − 32 = 3

3 − 32 = 3

1 with the corresponding picture.


105

SMALL GROUP ACTIVITY: EGG CARTON SUBTRACTION

Work in groups of 3.

1. One person puts objects into some of the egg holders and writes the fraction that represents the filled portion of the egg carton.

2. Second person takes out some of the objects from the egg holders and writes the fraction that he/she takes out on the same sheet.

3. Third person performs the subtraction of the two fraction on paper and checks it with the egg carton. Is there more than one way to write the fractions?

4. Switch roles.

Again, when we combine fractions, we first make sure that the fractions have the same denominators. If we are adding the fractions, we then add the numerators; if subtracting the fractions we subtract the numerators. The denominator in our difference remains the same. Like the addition rule, we have.

SUBTRACTING FRACTIONS WITH LIKE DENOMINATORS

da − d

b = ( )

da b+

where da and d

b represent two fractions.


106

PRACTICE PROBLEMS

1. Add or subtract the following fractions. Draw a picture of each problem using the number line or rectangles as above.

a. 52 + 5

1 =

b. 43 + 4

1 =

c. 53 + 5

3 =

d. 2 + 51 =

e. 53 − 5

1 =

f. 76 − 7

2 =

g. 1 − 52 =

h. 2 − 43 =


107

2. Combining fractions

a. 63 + 6

2 =

b. 105 + 10

2 =

c. 74 + 7

3 =

d. x2 + x

1 =

e. y3 + y

2 =

f. t3 + t

4 =

g. x1 + x

1 =

h. x2 + x

3 =


108

CLASS DISCUSSION:

How can we represent a person’s height? What units can we use? What devices do we use to measure a person’s height?

Discuss when you might use just one unit or perhaps two units. For example, you might express a measurement in feet and inches.

Person A is 5 feet 6 inches tall.

Person B is 5 21 feet tall.

Person C is 66 inches tall.

Who is the tallest?


109

CLASS ACTIVITY:

1. Use the tape measure to find the heights of each student. Record all the heights, expressing the answers in feet and inches, only feet, and only inches.

2. Order the heights from tallest to shortest.


110

Let’s look at measuring time. Consider expressing amounts of time using all minutes, both hours and minutes, and then just hours. For example, if you are in school for 6 hours and 15 minutes, we can also say this time in all minutes as 375 minutes or as 6 4

1 hours.

Use the hours and minutes you are in school and express this in minutes only. Then in hours only.

Another example can involve seeing how old you are using years and months, using only months, or using only years.

Can you find other such paired units that we often use to measure quantities?


111

A number such as 2 31 is called a mixed fraction or mixed number. This is actually shorthand

for the number 2 + 31 . Notice that we have two whole parts and the fractional amount of 3

1 .

This can also be written as 37 .

When the numerator is greater than or equal to the denominator, we call this type of fraction an improper fraction.

Notice that in our mixed fraction, 2 31 , we really have 3

3 + 33 + 3

1 = 37 .

Write 4 31 as an improper fraction.

Did you get 33 + 3

3 + 33 + 3

3 + 31 = 3

13 ?

Now write 4 32 as an improper fraction.


112

ACTIVITY: FRACTION ART I

1. Draw the fraction 1 61 . Mark the subdivisions clearly so that we know what is the whole part

and what fractional part you are specifying.

2. Use your fractional drawing to show the equivalent improper fraction.

3. What does 3 61 look like in your drawing and what is the equivalent improper fraction?

4. Without drawing 5 61 , try to write this as an improper fraction and explain how you arrived at

your answer.

5. Look at some other examples and see if you observe a pattern for writing a mixed fraction as an improper fraction. Write in your own words the process you use.


113

ACTIVITY: FRACTION ART II

1. Draw the fraction 58 . Are your pieces the same size?

2. Show in your drawing that there is a whole part and a fractional part. Can you express this as a mixed fraction? Record this on your picture.

3. Draw the fraction 518 .

4. Show the whole parts and the leftover fractional part. How can we write 518 as a mixed

fraction?

5. Look at some other examples and see if you observe a pattern or process for writing an improper fraction as a mixed fraction. Write in your own words what this process is.


114

PRACTICE PROBLEMS

1. Rewrite the mixed fraction as an improper fraction and as a sum.

a. 3 21

b. 2 43

c. 5 52

d. 4 75

2. Rewrite the improper fraction as a mixed fraction. Draw a fudge model for each and determine the “whole part” and the “fractional part.”

a. 23

b. 35

c. 46

d. 512


115

3. Susie has 4 41 gallons of honey. Her friend Peter has 2 4

1 gallons of honey. How much honey do they have altogether?

4. Diann is making a cake that requires 2 31 cups of milk. If she pours her milk out of a container

that has 4 cups of milk, how much will she have left?


116

ACTIVITY: BASKETBALL

The teacher chooses two teams: Team Archimedes and Team Einstein.

1. A member of each team goes up to the board and is given a problem of a type: mixed to improper fraction, improper to mixed fraction, or adding or subtracting fractions with like denominators.

2. The students show their work on the board.

3. The first person to solve the problem gets a chance to shoot the “basketball” into the trash can. There is a 3-point line. If the basket is made, the team is awarded 3 points. If the ball does not go in, the team is awarded 2 points. If the other player also solved the problem, then the student can shoot from the 2-point line. If the basket is made, the team is awarded 2 points. If the ball does not go in, the team is awarded 1 point.


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SUMMARY:


IMPORTANT VOCABULARY:

• mixed fractions

• improper fractions


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SECTION 3.3 MULTIPLYING FRACTIONS

KEY OBJECTIVES:

• Multiplication with fractions

• Model fraction multiplication

When we add or subtract fractions, we can visualize the operation using a Fudge Model. Now let’s continue our study of operations with fractions by considering what it means to multiply fractions.

EXAMPLE 1:

What is 21 of 6?

We can look at some different ways to model this problem; 21 of 6 can be modeled as:

or:

In either case, 21 of 6 would be equal to 3. Explain how each of these would give us the answer 3.


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EXERCISES:

1. a. 21 of 8

b. 21 of 9

c. 21 of 10

d. 21 of 11

2. a. 13 of 8

b. 31 of 9

c. 13 of 10

d. 31 of 11

3. a. 54 of 6

b. 43 of 5

c. 65 of 8

d. 72 of 9


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EXAMPLE 2:

Now consider the following problem:

Susan has 21 of a pan of fudge. She wants to divide this evenly among three friends. This means

that each friend would receive one-third of half of a pan of fudge. Pictorially we have:

Dividing this 21 of a pan of fudge into three equal pieces, each friend’s portion would look like this:

a. What part of a pan of fudge does each friend get?

We can translate the phrase “ 31 of 2

1 ” into the multiplication problem 31

21

61=a ak k . The

word “of” in this case translates into “times” or “multiplied by” when using fractions.

b. What if Susan wants to share an equal part of this half pan of fudge with her 3 friends? How much would each of these 4 persons get?


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EXERCISES:

1. Translate " 21 of 5

1 " into a multiplication problem and draw the corresponding picture to find the product.

2. Translate " 41 of 3

1 " into a multiplication problem and draw the corresponding picture to find the product.


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3. Describe any patterns that you see.

4. What would 31 of 5

1 be?


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CLASS ACTIVITY: FRACTION ART MULTIPLIES

1. Each person selects a different pair of fractions k1 and d

1 , with k and d being a counting number. Use butcher paper to draw the multiplication of k

1 · d1 , using the Fudge Model. Include

the multiplication problem and the product with your drawing. How many pieces did your fudge end up being divided into?

2. Now take the same two fractions, but this time draw the multiplication problem with the factors in reverse order. Remember the first drawing is of k

1 · d1 . The second is now of d

1 · k1 . How

many pieces is your second fudge divided into now? Is there any relationship to the first?

Thus, we see that when we multiply a given fraction d1 by a fraction k

1 , this has the effect of multiplying the denominator d of the given fraction by k. This yields the rule:

Rule: Multiplying Fractions

k d k d1 1 1$

$=a ak k

Now, what do you suppose happens if we multiply the general fraction ba by k

1 ?


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CLASS ACTIVITY: SMART ART

1. What is 21 of 3

2 ? Draw a Fudge Model of the problem.

2. What is 31 of 5

4 ? Draw a Fudge Model and find the answer.


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EXAMPLE:

Lisa has 32 of a bar of fudge. If she divides this evenly among her five children, how much does

each child get?

Pictorially, we can model this situations as:

Each child gets 51

32

152=a ak k of a bar of fudge as shown.

We could also think of this as 32 ÷ 5 = 3

251a ak k because 5

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32

51=a a a ak k k k. Draw a Fudge Model

below to check that this works.

We know from our Fudge Model that 51

32

152=a ak k , so 3

2 ÷ 5 = 32

51a ak k = 15

2

Dividing a fraction by 5 is the same as multiplying the fraction by 51 . The effect is to multiply the

denominator by 5.


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EXAMPLE:

We see now how much fudge each of Lisa’s children will get. How much fudge will two of the children get?

We know from the work we just did that each child gets 152 of the fudge.

Two children should get 2 · 152 or 1

2152

154

$ = since 152 + 15

2 = 154 .

Notice when we multiply a whole number like 2 by a fraction ba , we have b

a2 ; the whole number multiplies the numerator.

We can also think of Lisa’s 2 children as being part of 5 of her total brood. So putting all our work together,

52 of 3

2 = 52

32a ak k = 15

4 .

Observe that the numerator 4 comes from 2 · 2, the product of the numerators of the two fractions, and the denominator 15 comes from 5 · 3, the product of the denominators of the two fractions.

SMALL GROUP ACTIVITY:

Draw a picture that models 32 of 5

4 and compute two fractions, ba

dc` `j j?

We can now see the basic pattern for multiplying fractions:

Rule: Multiplying Fractions

ba

dc

b da c$$=` `j j


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EXERCISES:

Translate each problem below into a multiplication problem and compute the product. Use the Fudge Model anytime you would like or need to. Simplify your fractions.

1. a. 51 of 5 b. 5

3 of 25

2. a. 32 of 5

4 b. 52 of 4

3

3. a. 43 of 5

2 b. 21 of 5

4


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4. a. 41 of 3

2 b. 31 of 4

2

5. a. 53 of 3

5 b. 32 of 2

3

6. a. 32 of 4 2

1 b. 31 of 2 2

1


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EXAMPLE:

Let’s consider the following problem: what number should be multiplied by 32 , to equal 1?

We want to find a number N which, when we multiply it by 32 , gives us 1. We can write this as an

equation to read 1N32$ = .

Because 32

23 1$ = , we would say that N = 2

3 is the number we seek.

We call 23 the reciprocal of 3

2 . The reciprocal of a number ba is simply a

b .

Notice the property about reciprocals is that

ba

ab

b aa b 1$$= =` aj k , for any number a, b not equal to 0.


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EXERCISES:

1. Find the reciprocal of each of the numbers below, and multiply to check your answer.

a. 31

b. 94

c. 53

d. 32

e. 3

f. 2

g. 49

h. 34


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2. Does every number have a reciprocal? What about parts e and f from Exercise 1? Does 6 have a reciprocal? Does 5 have a reciprocal? Does 1 have a reciprocal? Does 0 have a reciprocal?


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GAME: CONCENTRATION

Materials: one pack of 3 by 5 index cards or 20 sheets of paper.

1. Form groups of 2 or 3.

2. Take 20 index cards. Write a different fraction on each of 10 index cards. On the other 10 index cards write the corresponding reciprocal of the fractions.

3. Turn the cards face down.

4. A player turns two cards up to try and match a fraction with its reciprocal. If successful, pick up the pair and turn up two more cards to try and make a match. If unsuccessful, the turn goes to the next player.

EXAMPLE:

On our summer car trip, we notice that the gas gauge is at the half full mark. Because the next gas station is far away, we decide to stop and fill the car up with gas. It takes 9 gallons to fill the car. How much gasoline does the gas tank hold? Can you use an equation to model the situation?

Here is one way to model the problem where we are trying to find the capacity of the gas tank. Let’s call the total capacity of the gas tank C. Then we know that half of C is 9.

We can write this in equation form C21 9: = and try to find the value of C.

To solve an equation we can add, subtract or multiply an equation by the same number and get an equivalent equation. In this case, we would want to multiply our equation by 2. Can you see why?

Notice 2 is the reciprocal of 21 . Lets see what happens when we multiply both sides of the equation

by 2.

2 · 21 · C = 2 · 9

Because the left side really looks like

12

21: · C = 2 · 9

we have

C = 18


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EXERCISES:

1. Rewrite the following sentences as equations and solve each equation.

a. 21 of what number equals 10? (Use x for the unknown number.)

b. 31 of what number equals 15? (Use x for the unknown number.)

c. 53 of what number equals 25? (Set up this statement as an equation and solve.)


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2. Nathan’s car gas tank was 31 full with gas. Nathan is sure that he has 5 gallons in his tank. How

much gas does his gas tank hold when it is completely full? If he were to fill his present tank to full, how much gas would he need to purchase? Set up an equation that uses G = gas tank capacity in gallons and solve for G.

3. Ms. Garcia's classroom is 21 girls. If there are 12 girls in the class, how many students are in

Ms. Garcia's class?


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5. Joe wants to make a cake that calls for 2 31 cups of milk. If he wants to make a cake 2

1 the size of the recipe, how much milk should he use?


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6. A recipe for banana pudding calls for 3 cups of milk. If Jeremy wants to make the pudding recipe 1 2

1 times as big, how much milk will Jeremy need?


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7. Simplify the following expressions

a. 31 of 6x

b. 21 of 8x

c. 32 of 6x

d. 32 of 5x

e. 43 of x

2

f. 53 of x

9


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MATH UEST 2 - Texas State University