Lesson 26-30 6.0 p.indd

Algebra Success 

T675 

LESSON 29: Polynomials 

[OBJECTIVE] 

The student will learn how to add and subtract polynomials. 

[MATERIALS] 

Student pages S262–S268 

Transparencies T684, T686, T688, T690, T692 

Overhead algebra tiles 

Algebra tiles for students 

Optional: Red and yellow overhead pens and colored pencils 

[ESSENTIAL QUESTIONS] 

1. What are some other words that we know that could help us understand what the 

words monomial, binomial, trinomial, and polynomial 

mean 

2. Is subtracting polynomials the same as adding polynomials 

[GROUPING] 

Cooperative Pairs, Whole Group, Individual 

[LEVELS OF TEACHER SUPPORT] 

Modeling (M), Guided Practice (GP), Independent Practice (IP) 

[MULTIPLE REPRESENTATIONS] 

SOLVE, Algebraic Formula, Verbal Description, Concrete Representation, Pictorial 

Representation 

[WARM-UP] (5 minutes – IP) S262 (Answers on T683.) 

• Have students turn to S262 in their books to begin the Warm-Up. Students will 

practice identifying like terms. Remind students that like terms have the same 

variable, or the same “last name,” and can be combined in addition or subtraction. 

Give students 3 minutes to complete the problems and then spend 2 minutes 

reviewing the answers as a class. {Algebraic Formula} 

[HOMEWORK]: (5 minutes) 

Take time to go over the homework from the previous night. 

[LESSON]: (45–55 minutes – M, GP, IP)

T676 



SOLVE Problem 

(5 minutes – GP) T684, S263 (Answers on T685.) 

Have students turn to S263 in their books, and place T684 on the overhead. The 

first problem is a SOLVE problem. You are only going to complete the S step with 

students at this point. Tell students that during the lesson they will learn how 

to add and subtract polynomials. They will use this knowledge to complete this 

SOLVE problem at the end of the lesson. {SOLVE} 

Algebra Tiles (7 minutes – M) 

Pass out the algebra tiles to each student. Use the following activity to introduce 

the algebra tiles to students. {Algebraic Formula, Verbal Description, Concrete 

Representation, Pictorial Representation} 

MODELING 

Introduce Algebra Tiles 

Step 1: Review the definition of monomial with students. A monomial is one term 

which is a number, a variable, or the product of the two. 

Examples of monomials: 5x, xy 2 , 8, 9mn 3 . 

Step 2: Write binomial, trinomial, and polynomial on the board or overhead. Ask 

your students if the prefixes of the words give them any clues to their 

meaning. (Students may know that poly means “many,” 

bi means “two,” 

and tri 

means “three.” It is possible to lead them to the meanings of bi 

and tri by asking about the number of wheels on a bicycle or tricycle.) 

A binomial is the sum of two monomials. A trinomial is the sum of three 

monomials. A polynomial is the sum of two or more monomials. 

Examples of binomials: x + 2, 

y + 

x, 2x x + 4 

y 

Examples of trinomials: x 3 + 3x x + 6, 

y 2 + 2y y + 1, 

x 2 + xy + 

y 2 

Examples of polynomials: x 3 + 4x 2 + 5, x + 5, 

x 3 + x 2 y + 

y 2 , x + y 

Step 3: Pass out the algebra tiles. Introduce the ones tiles first. Tell students to 

put the small yellow square on their desks. Put one small yellow square 

on the overhead. Explain that the small yellow square represents the 

number 1. Ask students how they would represent the number 4. (with 4 

small yellow squares) Put 4 small yellow squares on the overhead. 

Repeat this process with the small red squares, each of which represents 

- 1. Show 

- 4, 

- 6, and 

- 2 using the small red squares.


T677 


Step 4: Introduce the x 

tiles. Tell students to put one of the long yellow pieces 

on their desks. Ask them how long they think this piece is. Let students 

guess the length of the long piece. Place one long yellow piece (one yellow 

x tile) on the overhead. Tell students that the length of the long piece 

is not known. Ask students what they use in mathematics to represent 

something that they do not know. They should say that they use a variable. 

Explain that the long piece’s length is x 

units, so students should refer to 

the long pieces as x tiles. 

Have students compare the width of the x 

tile to that of a small square. 

They will see that the x 

tile has the same width as a small square. Explain 

that this means that the x 

tile has a width of 1, so the area of the 

x 

tile 

is 1 • x, or x. Place 4 yellow x tiles together on the overhead and have 

students do the same on their desks. Explain that because 1 yellow x 

tile represents 1x, or x, 4 yellow x 

tiles represent 4x. Do a few more 

examples with students. 

Repeat this process with the long red pieces (red x tiles), each of which 

represents - 1x, or - x. 

Step 5: Introduce the x 2 tiles. Have students put one of the large yellow squares 

on their desks. Put one on the overhead. Tell students that they have to 

come up with a name for this piece. Tell them to use a yellow x 

tile to 

measure the length and width of the large square. Help students see that 

each side of the large square is x units long. Thus, the area of the large 

square is x • 

x, or x 2 . Tell students that they should refer to the large 

yellow squares as x 2 tiles. 

Repeat this process with the large red squares (red x 2 tiles), each of 

which represents - x 2 . 

Add Polynomials 

(8 minutes – (M, GP) T684, S263 (Answers on T685.) 

Use the following activity to model for students how to add polynomials at the 

concrete level using the overhead algebra tiles. Students should model the 

problems at their desks using their red and yellow algebra tiles. {Algebraic Formula, 

Verbal Description, Concrete Representation, Pictorial Representation}

T678 



MODELING 

Add Polynomials 

Step 1: Discuss the process of addition before you begin to model the first problem. 

If you were going to teach addition to first graders, you might put 3 

apples on one side of your desk and 4 apples on the other side. Then, to 

add 3 apples and 4 apples, you would push together, or combine, the two 

groups to get 7 apples. You will show students how to add polynomials in 

the exact same way—by representing the first polynomial and the second 

polynomial as two separate groups and then pushing the two groups 

together. 

Step 2: For Problem 1, have students first represent x 2 + 3x x + 1 using 1 yellow 

x 2 tile, 3 yellow x 

tiles, and 1 small yellow square. Then have students 

represent 2x 2 + 5x x 

+ 3 using 2 yellow x 2 tiles, 5 x 

tiles, and 3 small yellow 

squares. 

Step 3: Have students push the two groups together. Explain that students should 

put all the x 2 tiles together and all the x 

tiles together. Tell students that 

doing this is combining like terms. 

Step 4: Once the like terms are combined, students should see that the answer 

is 3 yellow x 2 tiles, 8 yellow x 

tiles, and 4 small yellow squares, or 

3x 2 + 8x x + 4. 

Step 5: For Problem 2, have students represent 2x 2 + 5x x – 2 using 2 yellow 

x 2 tiles, 5 yellow x tiles, and 2 small red squares. Then have students 

represent x 2 – 7x x 

+ 5 using 1 yellow x 2 tile, 7 red x 

tiles, and 5 small 

yellow squares.


T679 


Step 6: Have students push the two groups together and combine like terms. 

Once the like terms are combined, explain to students that they need to 

have the x 

tiles all in one color and the small squares all in one color. To 

do this, students should form zero pairs. 

Have students form zero pairs from 5 yellow x tiles and 5 red 

x tiles. Have 

students cancel the zero pairs one at a time. Then have students form 

zero pairs from 2 small yellow squares and 2 small red squares and then 

cancel the zero pairs one at a time. Once the canceling of zero pairs is 

complete, students should see that the answer is 3 yellow x 2 tiles, 2 red 

x tiles, and 3 small yellow squares, or 3 

x 2 – 2x x + 3. 

Step 7: After students have modeled the problems with the algebra tiles, have 

them go back and model each problem pictorially as you do the same on 

T684. Answers are shown on T685. 

Add and Subtract Polynomials 

(10 minutes – M, GP) T686, S264 

(Answers on T687.) 

Have students turn to S264 in their books, and place T686 on the overhead. Use 

the following activity to model for students how to add and subtract polynomials 

at the concrete level using the overhead algebra tiles. Students should model 

the problems at their desks using their red and yellow algebra tiles. {Algebraic 

Formula, Verbal Description, Concrete Representation, Pictorial Representation}

T680 



MODELING 

Add and Subtract Polynomials 

Step 1: For Problem 3, have students represent 3x 2 – 7x x + 5 using 3 yellow 

x 2 tiles, 7 red x tiles, and 5 small yellow squares. Then have students 

represent x 2 – x 

– 6 using 1 yellow x 2 tile, 1 red x 

tile, and 6 small red 

squares. 

Step 2: Have students push the two groups together and combine like terms. 

Once the like terms are combined, explain to students that they need to 

have the small squares all in one color. To do this, students should form 

zero pairs. 

Have students form zero pairs from 5 small yellow squares and 5 small red 

squares and then cancel the zero pairs one at a time. Once the canceling 

of zero pairs is complete, students should see that the answer is 4 yellow 

x 2 tiles, 8 red x tiles, and 1 small red square, or 

4x 2 – 8x x – 1. 

Step 3: Before modeling Problem 4, remind students about the rules for subtracting 

integers. Explain that students will represent the first polynomial and take 

away the second polynomial. If they need to take away something that is 

not there, they will create the possibility by adding zero pairs. 

Step 4: For Problem 4, have students represent 2x 2 + 5x x – 4 using 2 yellow 

x 2 

tiles, 5 yellow x tiles, and 4 small red squares. Explain to students that 

the problem asks students to take away x 2 + 3x x + 1, which is 1 yellow x 2 

tile, 3 yellow x tiles, and 1 small yellow square. 

Step 5: Model for students how to take away the yellow x 2 tile and the 3 yellow 

x 

tiles. Explain that students do not have a small yellow square to take 

away, so they must create the possibility. Model for students how to add 

a zero pair (1 small yellow square and 1 small red square) and then 

take away 1 small yellow square. Students should see that the answer is 

x 2 + 2x x – 5.


T681 


Step 6: Another way to solve the same problem is to use your rules for subtracting 

integers. Remind students that subtracting is the same as adding the 

opposite. For Problem 4, model for students how to distribute the 

subtraction sign to all terms in the second polynomial and rewrite the 

problem as an addition problem. 

(2x 2 + 5x x – 4) – ( 

x 2 + 3x x + 1) = (2 

x 2 + 5x x – 4) + 

(2x 2 + 5x x – 4) + ( 

- x 

2 

– 3x x – 1) 

- 1(x 

2 

+ 3x x + 1) = 

Step 7: Have students model the two polynomials, push them togethers and 

combine like terms, and then cancel zero pairs. Students should see that 

the answer is the same as the one they found when they subtracted, 

x 2 + 2x x – 5. 

Repeat Steps 3–7 above to model Problem 5 with students. After students have 

modeled Problems 3–5 on S264 (T686) with the algebra tiles, have them go back 

and model each problem pictorially as you do the same on T686. Answers are 

shown on T687. 

If time permits... 

(10 minutes – IP) T688, S265 (Answers on T689.) 

Have students complete Problems 1–4 on S265 (T688). Have students model 

each problem with their algebra tiles and then pictorially. Students can work in 

cooperative pairs or independently. Give students 8 minutes to complete the 

problems and then use 2 minutes to review the answers. {Algebraic Formula, 

Verbal Description, Concrete Representation, Pictorial Representation} 

No Algebra Tiles 

(10 minutes – M, GP) T690, S266 (Answers on T691.) 

Have students turn to S266 in their books, and place T690 on the overhead. Model 

Problems 5–10 with students without using algebra tiles. Have students change 

each subtraction problem to an addition problem before finding the difference. 

{Algebraic Formula, Verbal Description} 

SOLVE Problem 

(5 minutes – GP) T692, S267 (Answers on T693.) 

Remind students that the SOLVE problem is the same one from the beginning of 

the lesson. Complete the SOLVE problem with your students. {SOLVE, Algebraic 

Formula}

T682 



[CLOSURE]: (5 minutes) 

• To wrap up the lesson, go back to the essential questions and discuss them with 

students. 

• What are some other words that we know that could help us understand what 

the words monomial, binomial, trinomial, and polynomial 

mean (polygon, 

bicycle, tricycle, triangle) 

• Is subtracting polynomials the same as adding polynomials (Subtraction is 

not exactly the same as adding. Subtraction is adding the opposite.) 

[HOMEWORK]: Assign S268 for homework. (Answers on T694.) 

[QUIZ ANSWERS] T695–T696 

1. D 2. B 3. A 4. C 5. A 6. B 7. A 8. D 9. B 10. D 

The quiz can be used at any time as extra homework or to see how students did on 

understanding adding and subtracting polynomials.

Lesson 26-30 6.0 p.indd

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