Using the Area Model to Teach Multiplying, Factoring, and ......For example, 4x2 – 3x1 + 6x0 is a...

Using the Area Model to Teach Multiplying, Factoring, and

Dividing Polynomials

For more information about the materials you find in this packet, contact:

Darrell Trussell [email protected]

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© 2006, 2013 CPM Educational Program. All rights reserved. 2

1-15. Finding and using a pattern is an important problem-solving skill you will use in algebra. The patterns in Diamond Problems will be used later in the course to solve other types of algebraic problems.

Look for a pattern in the first three diamonds below. For the fourth diamond, explain how you could find the missing numbers (?) if you know the two numbers (#).

Copy the Diamond Problems below onto your paper. Then use the pattern you

discovered to complete each one.

a. b. c. d. e.

3-25. Your teacher will distribute a set of algebra tiles for your team to use during this

course.

a. The tiles have a positive side and a negative side. In this text the positive side will be the shaded side. Flip the tiles so that the positive side of each tile is facing up. Trace one of each of the six tiles provided by your teacher on your paper. Leave plenty of space between each tracing.

b. The dimensions of some of the tiles are shown at right. Label the dimensions of all the tiles next to the tracings you made.

c. The algebra tiles will be named according to each of their areas. Write the name of each tile in the center of your tracing with a colored pen or pencil. Make the name of the tile stand out.

d. Below each tile write “P =” and then find the perimeter of each tile.

2 5 –2 –1 8

6

4

12

–7

10 5 2

7

6 2 3

5

4 –1 – 4

–5

? # #

?

x 1

x x

y

y


3.2.2 What can I do with rectangles? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Exploring an Area Model In the last lesson, you used tiles to represent algebraic equations. Today you will use algebra tiles again, but this time to represent expressions using multiplication. 3-45. Your teacher will put this group of tiles on the overhead:

a. Using your own tiles, arrange the same group of tiles into one large rectangle, with the x2-tile in the lower left corner. On your paper, sketch what your rectangle looks like.

b. What are the dimensions (length and width) of the rectangle you made? Label your sketch with its dimensions, then write the area of the rectangle as a product, that is, length · width.

c. The area of a rectangle can also be written as the sum of the areas of all its parts. Write the area of the rectangle as the sum of its parts. Simplify your expression for the sum of the rectangle’s parts.

d. Write an equation that shows that the area written as a product is equivalent to the area written as a sum.

x2

x

x

x

x

x

x


3-46. Your teacher will assign several of the expressions below. For each expression, build a rectangle using all of the tiles, if possible. Sketch each rectangle, find its dimensions, and write an expression showing the equivalence of the area as a sum (like x2 + 5x + 6) and as a product (like (x + 3)(x + 2)). If it is not possible to build a rectangle, explain why not.

a. x2 + 3x + 2 b. 6x + 15

c. 2x2 + 7x + 6 d. xy + x + y + 1

e. 2x2 + 10x + 12 f. 2y2 + 6y

g. y2 + xy + 2x + 2y h. 3x2 + 4x + 1

i. x2 + 2xy + y2 + 3x + 3y + 2 j. 2xy + 4y + x + 2 3-47. LEARNING LOG Make a rectangle from any number of tiles. Your rectangle

must contain at least one of each of the following tiles: x2, y2, xy, x, y, and 1. Sketch your rectangle in your Learning Log and write its area as a product and as a sum. Explain how you know that the product and sum are equivalent. Title this entry “Area as a Product and as a Sum” and label it with today’s date.

The area of a rectangle can be written two different

ways. It can be written as a product of its width and length or as a sum of its parts. For example, the area of the shaded rectangle at right can be written two ways:

ETHODS AND MEANINGS

MA

TH N

OTE

S

Multiplying Algebraic Expressions with Tiles

(x + 4)length (x + 2)

width = x

2 + 6x + 8area

area as a product = area as a sum

x x x x x2

x x


3-48. For the entire rectangle at right, find the area of each part

and then find the area of the whole. 3-49. Write the area of the rectangle at right as a product and as a sum.

3.2.3 How can I rewrite a product? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Multiplying Binomials and the Distributive Property In Lesson 3.2.2, you made rectangles with algebra tiles and found the dimensions of the rectangles. Starting with the area of a rectangle as a sum, you wrote the area as a product. Today you will reverse the process, starting with the product and finding its area as a sum. 3-54. For each of the following rectangles, find the dimensions (length and width) and write

the area as the product of the dimensions and as the sum of the tiles. Remember to combine like terms whenever possible.

a. b.

7

3

11 8

x x x2

x2

x x x

x x x

y

x xy x2


3-55. Your teacher will assign your team some of the expressions below. Use your algebra tiles to build rectangles with the given dimensions. Sketch each rectangle on your paper, label its dimensions, and write an equivalence statement for its area as a product and as a sum. Be prepared to share your solutions with the class.

a. (x + 3)(2x + 1) b. 2x(x + 5)

c. x(2x + y) d. (2x + 5)(x + y + 2)

e. (2x + 1)(2x + 1) f. (2x)(4x)

g. 2(3x + 5) h. y(2x + y + 3) 3-56. With your team, examine the solutions you found for

parts (b), (c), (g), and (h) of problem 3-55. This pattern is called the Distributive Property. Multiply the following expressions without using your tiles and simplify. Be ready to share your process with the class.

a. 2x(6x + 5) b. 6(4x + 1)

c. 3y(4x + 3) d. 7y(10x + 11y) 3-57. CLOSED SETS Whole numbers (positive integers and zero) are said to be a closed set under addition:

if you add two whole numbers, you always get a whole number. Whole numbers are not a closed set under subtraction: if you subtract two whole numbers, you do not always get a whole number: 2 – 5 = –3 (–3 is not a whole number).

a. Investigate with your team whether the integers are a closed set under addition, and whether the integers are a closed set under subtraction. Give examples. If you find that integers are closed under either of the operations, can you explain how you know they are closed for all integers?

b. Read the Math Notes box that follows. Are polynomials a closed set under addition? Are polynomials a closed set under subtraction? That is, if you add or subtract two polynomials, will you always get a polynomial as your answer? Give examples and explain how you know your answer is always true.


A mathematical expression is a combination of numbers, variables, and operation symbols. Addition and subtraction separate expressions into parts called terms. For example, 4x2 – 3x + 6 is an expression. It has three terms: 4x2, 3x, and 6. The coefficients are 4 and 3. 6 is called a constant term.

A one-variable polynomial is an expression which only has terms of the form:

For example, 4x2 – 3x1 + 6x0 is a polynomial, so the simplified form, 4x2 – 3x + 6 is a polynomial.

The function f (x) = 7x5 + 2.5x3 – 12 x + 7 is a polynomial function.

The following are not polynomials: 2x – 3, 1x2−2

, and x − 2 .

A binomial is a polynomial with only two terms, for example, x3 – 0.5x and 2x + 5.

3-58. Examine the rectangles formed with tiles below. For each figure, write its area as a

product of the width and length and as a sum of its parts.

a. b.

x x

ETHODS AND MEANINGS M

ATH

NO

TES Vocabulary for Expressions


3-59. Find the total area of each rectangle below. Each number inside the rectangle represents the area of that smaller rectangle, while each number along the side represents the length of that portion of the side.

a. b.

3.2.4 How can I generalize the process? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Using Generic Rectangles to Multiply You have been using algebra tiles and the concept of area to multiply polynomial expressions. Today you will be introduced to a tool that will help you find the product of the dimensions of a rectangle. This will allow you to multiply expressions without tiles. 3-64. Use the Distributive Property to find each product below.

a. 6(–3x + 2) b. x2(4x – 2y)

c. 5t(10 – 3t) d. –4w(8 – 6k2 + y) 3-65. Write the area as a product and as a sum for the

rectangle shown at right.

12 39 11

4

18

40

8 6

x2


3-66. Now examine the following diagram. How is it similar to the set of tiles in problem 3-65? How is it different? Talk with your teammates and write down all of your observations.

3-67. Diagrams like the one in problem 3-66 are referred to as generic rectangles. Generic

rectangles allow you to use an area model to multiply expressions without using the algebra tiles. Using this model, you can multiply with values that are difficult to represent with tiles.

Draw each of the following generic rectangles on your paper. Then find the area of

each part and write the area of the whole rectangle as a product and as a sum.

a. b.

c. d.

e. How did you find the area of the individual parts of each generic rectangle?

3-68. Multiply and simplify the following expressions using either a generic rectangle or

the Distributive Property. For part (a), verify that your solution is correct by building a rectangle with algebra tiles.

a. (x + 5)(3x + 2) b. (2y – 5)(5y + 7)

c. 3x(6x2 – 11y) d. (5w – 2p)(3w + p – 4)

3 y +5

x

2x

+5

x

2x –3

–7 4y

6y –1

12x

8x2

15

10x

4x + 5

+ 3

2x


3-69. THE GENERIC RECTANGLE CHALLENGE Copy each of the generic rectangles below and fill in the missing dimensions and areas.

Then write the entire area as a product and as a sum. Be prepared to share your reasoning with the class.

a. b.

c. d.

3-83. Find the dimensions of the generic rectangle at right. Then

write an equivalency statement (length · width = area) of the area as a product and as a sum.

xy 3y

x

+ 5 12x

x2

–2 –10 –4x

–3y

–3xy

6

x x2

x2

–5x

3x –15


8.1.1 How can I find the product? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Introduction to Factoring Quadratic Expressions In Chapter 3 you learned how to multiply algebraic expressions using algebra tiles and generic rectangles. This section will focus on reversing this process: How can you find a product when given a sum? 8-1. Review what you know about products and sums below.

a. Write the area of the rectangle at right as a product and as a sum. Remember that the product represents the area found by multiplying the length by the width, while the sum is the result of adding the areas inside the rectangle.

b. Use a generic rectangle to multiply (6x – 1)(3x + 2). Write your solution as a sum.

8-2. The process of changing a sum to a product is called factoring. Can every expression

be factored? That is, does every sum have a product that can be represented with tiles?

Investigate this question by building rectangles with algebra tiles for the following

expressions. For each one, write the area as a sum and as a product. If you cannot build a rectangle, be prepared to convince the class that no rectangle exists (and thus the expression cannot be factored).

a. 2x2 + 7x + 6 b. 6x2 + 7x + 2

c. x2 + 4x + 1 d. 2xy + 6x + y2 + 3y

x

y

xy x2


8-3. Work with your team to find the sum and the product for the following generic rectangles. Are there any special strategies you discovered that can help you determine the dimensions of the rectangle? Be sure to share these strategies with your teammates.

a. b. c.

8-4. While working on problem 8-3, Casey noticed a

pattern with the diagonals of each generic rectangle. However, just before she shared her pattern with the rest of her team, she was called out of class! The drawing on her paper looked like the diagram below. Can you figure out what the two diagonals have in common?

8.1.2 Is there a shortcut? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Factoring with Generic Rectangles Since mathematics is often described as the study of patterns, it is not surprising that generic rectangles have many patterns. You saw one important pattern in Lesson 8.1.1 (Casey’s pattern from problem 8-4). Today you will continue to use patterns while you develop a method to factor trinomial expressions. 8-13. Examine the generic rectangle shown at right.

a. Review what you learned in Lesson 8.1.1 by writing the area of the rectangle at right as a sum and as a product.

b. Does this generic rectangle fit Casey’s pattern for diagonals? Demonstrate that the product of each diagonal is equal.

2x 5

6x2 15x

–2y –6

5xy 15x

–9x –12

12x2 16x

2x 5

6x2 15x

14 –35x

–4x 10x2


8-14. FACTORING QUADRATIC EXPRESSIONS To develop a method for factoring without algebra tiles, first model how to factor

with algebra tiles, and then look for connections within a generic rectangle.

a. Using algebra tiles, factor 2x2 + 5x + 3; that is, use the tiles to build a rectangle, and then write its area as a product.

b. To factor with tiles (like you did in part (a)), you need to determine how to arrange the tiles to form a rectangle. Using a generic rectangle to factor requires a different process.

Miguel wants to use a generic rectangle to factor 3x2 + 10x + 8. He knows that 3x2 and 8 go into the rectangle in the locations shown at right. Finish the rectangle by deciding how to place the ten x-terms. Then write the area as a product.

c. Kelly wants to find a shortcut to factor 2x2 + 7x + 6. She knows that 2x2 and 6 go into the rectangle in the locations shown at right. She also remembers Casey’s pattern for diagonals. Without actually factoring yet, what do you know about the missing two parts of the generic rectangle?

d. To complete Kelly’s generic rectangle, you need two x-terms that have a sum of 7x and a product of 12x2. Create and solve a Diamond Problem that represents this situation.

e. Use your results from the Diamond Problem to complete the generic rectangle for 2x2 + 7x + 6, and then write the area as a product of factors.

product

sum

8

3x2

?

?

6

2x2


product

sum

8-15. Factoring with a generic rectangle is especially convenient when algebra tiles are not available or when the number of necessary tiles becomes too large to manage. Using a Diamond Problem helps avoid guessing and checking, which can at times be challenging. Use the process from problem 8-14 to factor 6x2 + 17x + 12. The questions below will guide your process.

a. When given a trinomial, such as 6x2 + 17x + 12, what two parts of a generic rectangle can you quickly complete?

b. How can you set up a Diamond Problem to help factor a trinomial such as 6x2 + 17x + 12? What goes on the top? What goes on the bottom?

c. Solve the Diamond Problem for 6x2 + 17x + 12 and complete its generic rectangle.

d. Write the area of the rectangle as a product.

8-16. Use the process you developed in problem 8-14 to factor the following quadratics, if possible. If a quadratic cannot be factored, justify your conclusion.

a. x2 + 9x + 18 b. 4x2 + 17x – 15

c. 4x2 – 8x + 3 d. 3x2 + 5x – 3

Why does Casey’s pattern from problem 8-4 work? That is, why does the product of the terms in one diagonal of a 2-by-2 generic rectangle always equal the product of the terms in the other diagonal?

Examine the generic rectangle at right for (a + b)(c + d). Notice that each of the resulting diagonals have a product of abcd. Thus, the product of the terms in the diagonals are equal.

d

c

a b

bc ac

bd ad

Product = abcd Product = abcd

ETHODS AND MEANINGS

MA

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S Diagonals of Generic Rectangles


8.2.5 How can I write it in graphing form? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Completing the Square In Lesson 8.2.3 you found that writing the equation of a parabola in graphing form, f (x) = a(x – h)2 + k, made it easier to find the vertex and the x-intercept. But how can you change standard form, f (x) = ax2 + bx + c, into graphing form? In this lesson, you will learn a new method called “completing the square.” 8-98. Use the Zero Product Property to find the exact vertex of y = x2 + 8x + 10. What

went wrong? How could you work around this problem? 8-99. The function from problem 8-98 can be rewritten in graphing form as

y = (x + 4)2 – 6. Find the exact x-intercepts and vertex. Make a sketch, then check your sketch with your graphing calculator.

8-100. COMPLETING THE SQUARE Jessica was at home struggling with her Algebra homework. She had missed class

and did not understand the new method called completing the square. She was supposed to use it to change y = x2 + 8x + 10 to graphing form. Then her precocious younger sister, who was playing with algebra tiles, said, “Hey, I bet I know what they mean.” Anita’s Algebra class had been using tiles to multiply and factor binomials.

Anita explained: “x2 + 8x + 10 would look like this;”

“Yes,” said Jessica, “I’m taking Algebra too, remember?” Anita continued, “And you need to make it into a square!”

Problem continues on next page. →

x2 x x x x x

x

x

x 1

1 1

1 1

1

1

1 1

1


8-100. Problem continued from previous page. “OK,” said Jessica, and she

arranged her tiles on an equation mat as shown at right.

“Oh,” said Jessica. “I need

16 small unit tiles to fill in the corner!”

“But you only have 10,”

Anita reminded her. “Right, I only have ten,”

Jessica replied. She drew the outline of the whole square and said: “Oh, I get it! To complete the square, I need to add six tiles to each side of the equation:”

“Oh, I see,” said Anita. “You started with y = x2 + 8x + 10, but now you can rewrite

it as y + 6 = (x + 4)2.” “Thank you so much, Anita! Now I can easily write the function in graphing form,

y = (x + 4)2 – 6.” How can you use your graphing calculator to verify that y = x2 + 8x + 10 and

y = (x + 4)2 – 6 are equivalent functions?

+ +

_ _

x

x2

x x x

x x x x

x + 4

x + 4

1 1 1 1 1 1 1 1 1 1

y

+ +

_ _

y 1 1 1 1 1 1 x

x2

x x x

x x x x

x + 4

x + 4

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1


8-101. Write each function in graphing form, then state the vertex and y-intercept of each parabola.

a. f (x) = x2 + 6x + 7 b. f (x) = x2 + 4x + 11 8-102. Help Jessica with a new problem. She needs to complete the square to write

y = x2 + 4x + 9 in graphing form.

a. Draw tiles to help her figure out how to make this expression into a square. Does she have too few or too many unit squares this time? Write her function in graphing form.

b. Find the vertex and the x-intercepts. What happened? What does that mean?

c. Algebraically find the y-intercept. Sketch the graph. 8-103. How could you complete the square to change f (x) = x2 + 5x + 2 into graphing form?

How would you split the five x-tiles into two equal parts? Jessica decided to use force! She cut one tile in half, as shown below. Then she

added her two small unit tiles.

a. How many small unit tiles are missing from Jessica’s square?

b. Write the graphing form of the function, name the vertex and y-intercept, and sketch the graph.

x2 x x

x x

x + 2.5

x

+

2.5

x2 x x

x x

x + 2.5

x

+

2.5

1 1

Figure A Figure B


8.3.1 How can I divide polynomials? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Polynomial Division When you graphed polynomial functions in the first section of this chapter, you learned that the factored form of a polynomial is very useful for finding the roots of the function or the x-intercepts of the graph. But what happens when you do not have the factored form and you need to find all of the roots? You will investigate the answer to this question in this lesson. 8-113. Andre needs to find the exact roots of the function f (x) = x3 + 2x2 – 7x – 2. When he

uses his graphing calculator, he can see that one of the x-intercepts is 2, but there are two other intercepts that he cannot identify exactly.

Andre remembers that he learned how to multiply binomials and other polynomials

using area models. He figures that since division is the inverse (or undo) operation for multiplication, he should be able to reverse the multiplication process to divide. As he thinks about that idea, he comes across the following news article.

What do you think Andre needs to be able to do to find the other roots?

Polydoku Craze Sweeping Nation! (CPM) – Math enthusiasts around the nation have entered a new puzzle craze involving the multiplication of polynomials. The goal of the game, which enthusiasts have named Polydoku, is to fill in squares so that the multiplication of two polynomials will be completed.

The game shown at right, for example, represents the multiplication of (3x – 2)(2x3 – x2 + 3x – 1) = 6x4 – 7x3 + 11x2 – 9x + 2.

Most of the squares are blank at the start of the game. While the beginner level provides the factors (in the gray squares), some of the factors are missing in the more advanced levels.

1 2 3 4 5

A × 2x3 – x2 + 3x – 1

B 3x 6x4 –3x3 9x2 –3x

C –2 –4x3 2x2 –6x 2

6x4 –7x3 +11x2 –9x +2


8-114. Andre decided to join the craze and try some Polydoku puzzles, but he is not sure how to fill in some of the squares. Help him by answering parts (a) and (b) below about the Polydoku puzzle in the news article he read (found in problem 8-113), then complete part (c).

a. Explain how the term 2x2 in cell C3 of the news article was generated.

b. What values were combined to get –7x3 in the news article answer?

c. Copy and complete the Polydoku puzzle at right.

8-115. POLYDOKU TEAM CHALLENGE Work with your team to complete

the puzzle at right. Find the factors and the product for the puzzle. If you get stuck, you can consult parts (a) through (c) below for ideas.

a. How is cell B2 related to the answer?

b. How did you find the third term in the answer?

c. What cells did you use to get the value in cell B5?

1 2 3 4 5

A × 4x3 + 6x2 – 2x – 5

B 2x

C – 3

1 2 3 4 5

A × –2x

B x 2x4

C – 4 12x2

12x


8-116. Jessica is about to start the intermediate-level Polydoku puzzle shown at right. Show Jessica how to complete the puzzle. Make sure you can justify your solution.

Use your results to complete the statements

below.

6x3 + 7x2 −16x +102x + 5

= __________ and (2x + 5) ⋅ _____________ = ____________

8-117. Unfortunately, Jessica made a mistake when she copied the problem. The constant

term of the original polynomial was supposed to have the value + 18 (not + 10). She does not want to start all over again to solve the puzzle.

a. Jessica realizes that she would now have 8 remaining from the original expression. What is the significance of this 8?

b. Jessica writes her work as shown below:

6x3 + 7x2 −16x +182x + 5

= (6x3 + 7x2 −16x +10) + 8

2x + 5= 3x2 − 4x + 2 , remainder 8

Gina thinks that there is a way to write the answer without using the word “remainder.” Discuss this with your team and find another way to write the result. Be prepared to share your results and your reasoning with the class.

c. Use Jessica and Gina’s method to divide (6x3 + 11x2 – 12x – 1) ÷ (3x + 1).

8-118. Create your own Polydoku puzzles that can be used to solve each of the polynomial-division problems below. Express any remainders as fractions and use your results to write a multiplication and a division statement like those in problem 8-117.

a. 6x4 − 5x3 +10x2 −18x + 53x −1

b. (x4 − 6x3 +18x − 4) ÷ (x − 2)

c. x − 3 x3 + x2 −14x + 3 d. x5 −1x −1

8-119. Now work with your team to help Andre solve his original problem (problem 8-113).

Find all of the roots (exact zeros) of the polynomial.

1 2 3 4

A ×

B 2x

C + 5

6x3 +7x2 –16x +10


Therefore, (x4 − 6x3 +18x −1) ÷ (x − 2) = x3 − 4x2 − 8x + 2 + 3x−2 and

(x − 2)(x3 − 4x2 − 8x + 2 + 3x−2 ) = x

4 − 6x3 +18x −1 .

The examples below show two methods for dividing x4 – 6x3 + 18x – 1 by x – 2. In both cases, the remainder is written as a fraction.

ETHODS AND MEANINGS M

ATH

NO

TES Polynomial Division

Using long division:

Answer:

Using area models: x4 – 6x3 + 0x2 + 18x – 1 Answer: x3 – 4x2 – 8x + 2 +

x3 –4x2 – 8x + 2

x x4 –4x3 – 8x2 + 2x 3

–2 –2x3 + 8x2 + 16x – 4

Remainder

Using the Area Model to Teach Multiplying, Factoring, and ......For example, 4x2 – 3x1 + 6x0 is a...

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