Algebra Success. [OBJECTIVE] The student will learn how to multiply monomials and polynomials.

Transcription

1 Algebra Success T697 [OBJECTIVE] The student will learn how to multiply monomials and polynomials. [MATERIALS] Student pages S269 S278 Transparencies T704, T705, T707, T709, T711, T713, T715 Red and yellow algebra tiles [ESSENTIAL QUESTIONS] 1. Does the distributive property change when variables are involved? 2. Do you follow the same rules for multiplying monomials when distributing a monomial to a polynomial? [GROUPING] Cooperative Pairs, Whole Group, Individual [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP), Cooperative Pairs (CP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Concrete Representation, Pictorial Representation [WARM-UP] (5 minutes IP) S269 (Answers on T703.) Have students turn to S269 in their books to begin the Warm-Up. Students will review multiplying monomials. 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]: (47 55 minutes M, GP, IP)

2 T698 Algebra Success SOLVE Problem (2 minutes GP) T705, S271 (Answers on T706.) Have students turn to S271 in their books, and place T705 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 multiply polynomials. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE} Monomial Binomial (7 minutes M) T704, T705, S270, S271 (Answers on T706.) Have students turn to S270 in their books, and place T704 on the overhead. Pass out the algebra tiles to each student. Students used the algebra tiles to add and subtract polynomials in Lesson 29. Remind students that the algebra tiles still represent the same values and variables. Use the following activity to model for students how to multiply monomials by binomials at the concrete level using the overhead algebra tiles. Students will model the problems at their desks using their red and yellow algebra tiles. Students can work independently or in pairs. {Algebraic Formula, Verbal Description, Concrete Representation, Pictorial Representation} MODELING Multiply a Monomial by a Binomial Step 1: Review with students how to use arrays to represent multiplication. Model with students how to use the small squares from the algebra tiles to make a concrete representation of 2 3 (2 groups with 3 items in each group) on T704, as shown below. The area created by the small squares is 2 3 = 6. Explain to students that they will use the same method to multiply polynomials. They will represent the first factor vertically and the second factor horizontally. The area created by the factors will represent the answer.

3 Algebra Success T699 Step 2: For Problem 1 on S271, have students represent the first factor, x, vertically with 1 yellow x tile. Then have students represent the second factor, x + 3, horizontally with 1 yellow x tile and 3 small yellow squares. The correct representation is shown below. Step 3: Model for students how to find the product of a monomial and a polynomial by finding the area they create with a length of x and a width of x + 3. Multiply x by x to create a product of x 2. Multiply x by 3 to create a product of 3x. The correct product is shown below. Step 4: Explain to students that the product is 1 yellow x 2 tile and 3 yellow x tiles, or x 2 + 3x. Step 5: Model Problem 1 pictorially on T705 while students do the same in their books on S271. The answer is shown on T706. Step 6: For Problem 2, have students represent the first factor, x + 3, vertically with 1 yellow x tile and 3 small yellow squares. Then have students represent the second factor, x, horizontally with 1 yellow x tile. The correct representation is shown below.

4 T700 Algebra Success Step 7: Model how to find the product of a monomial and a polynomial by finding the area they create with a length of x + 3 and a width of x. Multiply x by x to create a product of x 2. Multiply 3 by x to create a product of 3x. The correct product is shown below. Step 8: Explain to students that the product is 1 yellow x 2 tile and 3 yellow x tiles, or x 2 + 3x. Step 9: Model Problem 2 pictorially on T705 while students do the same in their books on S271. The answer is shown on T706. More Multiplication (20 minutes M, GP) T704, T707, T709, S270, S272, S273 (Answers on T708, T710.) Repeat the steps above to model Problems 1 6 on S272 and S273 (T707 and T709) with students at the concrete level using the overhead algebra tiles. Students will model the problems at their desks using their red and yellow algebra tiles. Students can work independently or in pairs. Remind students that when they multiply negative tiles by positive tiles, the answers will be negative, and when they multiply negative tiles by negative tiles, the answers will be positive. {Algebraic Formula, Verbal Description, Concrete Representation, Pictorial Representation} More Practice (7 minutes IP) T711, S274 (Answers on T712.) Give students 5 minutes to complete the three problems on S274 (T711), using their algebra tiles. If your students are struggling, complete the problems as a class. Use 2 minutes to review the answers. {Algebraic Formula, Verbal Description, Concrete Representation, Pictorial Representation} Without Algebra Tiles (10 minutes M, CP, GP) T713, S275 (Answers on T714.) Have students turn to S275 in their books, and place T713 on the overhead. Use the following activity to model for students how to use the distributive property to multiply polynomials. Model the odd-numbered problems only. {Algebraic Formula, Verbal Description}

5 Algebra Success T701 MODELING Multiply Polynomials without Algebra Tiles Step 1: For Problem 1, have students draw arrows to show how to distribute the monomial to all terms in the polynomial. Problem 1: 2x(4x x + 5) Step 2: Have students multiply the monomial by the first term in the polynomial: 2x x 4 x = 8x 2. Then have students multiply the monomial by the second term in the polynomial: 2x x 5 = 10x. The final answer is 8x x. Step 3: For Problem 3, have students first draw arrows to show how to distribute the monomial to all terms in the polynomial. Problem 3: 3x(x 2 x 1) Step 4: Have students multiply the monomial by the first term in the polynomial: 3x x x 2 = 3x 3. Then have students multiply the monomial by the second term in the polynomial: 3x x - x = - 3x2. Finally, have students multiply the monomial by the third term in the polynomial: 3x x - 1 = - 3x. The final answer is 3x 3 3x 2 3x. Repeat the steps above to model Problems 5, 7, and 9 with students. Then have students complete the even-numbered problems in pairs.

6 T702 Algebra Success SOLVE Problem (7 minutes GP) T715, S276 (Answers on T716.) 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} If time permits... (8 minutes IP) S277 (Answers on T717.) Have students complete Problems 1 10 on S277. Students may use algebra tiles or draw pictures to help them solve the problems. Review the answers as a class. {Algebraic Formula} [CLOSURE]: (5 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. Does the distributive property change when variables are involved? (No, you are still multiplying the outside term by the inside terms.) Do you follow the same rules for multiplying monomials when distributing a monomial to a polynomial? (Yes, you multiply coefficients and add the exponents of like variables.) [HOMEWORK]: Assign S278 for homework. (Answers on T718.) [QUIZ ANSWERS] T719 T D 2. D 3. A 4. C 5. D 6. C 7. A 8. A 9. B 10. C The quiz can be used at any time as extra homework or to see how students did on understanding multiplying monomials by polynomials.

7 Algebra Success T703 Here is the key to S269. Warm Up Directions: Find each product (x 2 ) - 3x ( - 4x) - 8x 3. x( - x) - x (2x 3 ) - 10x x (x 2 ) - x 2

8 T704 Algebra Success TRANSPARENCY MASTER

9 Algebra Success T705 TRANSPARENCY MASTER Directions: Complete the following SOLVE problem with your teacher. You will only complete the S step. The length of a rectangle is represented by the polynomial 2x 2 + 4x x + 5. The width of the rectangle is represented by the monomial 3x. What is the area of the rectangle? S Underline the question. This problem is asking me to find. Directions: Complete the rest of the page with your teacher. 1. x(x x + 3) 2. (x x + 3) x

10 T706 Algebra Success Here is the key to S271. Directions: Complete the following SOLVE problem with your teacher. You will only complete the S step. The length of a rectangle is represented by the polynomial 2x 2 + 4x x + 5. The width of the rectangle is represented by the monomial 3x. What is the area of the rectangle? S Underline the question. This problem is asking me to find the area of the rectangle. Directions: Complete the rest of the page with your teacher. 1. x(x x + 3) = x 2 + 3x 2. (x x + 3) x = x 2 + 3x

11 Algebra Success T707 TRANSPARENCY MASTER Directions: Complete this page with your teacher. 1. x(x x + 2) 2. (x x + 4) x 3. 2x(x x 3)

12 T708 Algebra Success Directions: Complete this page with your teacher. Here is the key to S x(x x + 2) = x 2 + 2x 2. (x x + 4) x = x 2 + 4x 3. 2x(x x 3) = 2x 2 6x

13 Algebra Success T709 TRANSPARENCY MASTER Directions: Complete this page with your teacher. 4. (x x + 2)3x 5. - x(x x + 3) 6. (x x 2) - 4x

14 T710 Algebra Success Directions: Complete this page with your teacher. 4. (x x + 2)3x = 3x 2 + 6x Here is the key to S x(x + 3) = - x 2 3x 6. (x 2) - 4x = - 4x 2 + 8x

15 Algebra Success T711 TRANSPARENCY MASTER Directions: Complete the following problems with your partner. 1. 2x(x x + 2) 2. (x x 1)2x 3. - x(x x 2)

16 T712 Algebra Success Directions: Complete the following problems with your partner. 1. 2x(x x + 2) = 2x 2 + 4x Here is the key to S (x x 1)2x = 2x 2 2x 3. - x(x x 2) = - x 2 + 2x

17 Algebra Success T713 TRANSPARENCY MASTER Directions: Complete the following odd-numbered problems with your teacher and the even-numbered problems with your partner. 1. 2x(4x x + 5) 2. (x x 1)4x 3. 3x(x 2 x 1) 4. (2x 2 + 3x x 4)2x x(3x x + 1) 6. ( - 5x x + 2) - 3x 7. 2x(3x 2 2x) 8. (2x 2 2)x x(3x 2 + 5x x 1) 10. (x 2 4x x + 1)6x

18 T714 Algebra Success Here is the key to S275. Directions: Complete the following odd-numbered problems with your teacher and the even-numbered problems with your partner. 1. 2x(4x x + 5) = 8x x 2. (x x 1)4x = 4x 2 4x 3. 3x(x 2 x 1) = 3x x 3 3x 2 3x 4. (2x 2 + 3x x 4)2x = 4x x 3 + 6x 2 8x x(3x x + 1) = - 6x 2 2x x 6. ( - 5x x + 2) - 3x = 15x 2 6x 7. 2x(3x 2 2x) = 6x x 3 4x 2 8. (2x 2 2)x = 2x x 3 2x x(3x 2 + 5x x 1) 10. (x 2 4x x + 1)6x = - 6x x 3 10x 2 + 2x = 6x x 3 24x 2 + 6x

19 Algebra Success T715 TRANSPARENCY MASTER Directions: Complete the following SOLVE problem with your teacher. The length of a rectangle is represented by the polynomial 2x 2 + 4x x + 5. The width of the rectangle is represented by the monomial 3x. What is the area of the rectangle? S Underline the question. This problem is asking me to find. O Identify the facts. Eliminate the unnecessary facts. List the necessary facts. L Choose an operation or operations. Write in words what your plan of action will be. V Estimate your answer. Carry out your plan. E Does your answer make sense? (Compare your answer to the question.) Is your answer reasonable? (Compare your answer to the estimate.) Is your answer accurate? (Check your work.) Write your answer in a complete sentence.

20 T716 Algebra Success Directions: Complete the following SOLVE problem with your teacher. Here is the key to S276. The length of a rectangle is represented by the polynomial 2x 2 + 4x x + 5. The width of the rectangle is represented by the monomial 3x. What is the area of the rectangle? S Underline the question. This problem is asking me to find the area of the rectangle. O Identify the facts. Eliminate the unnecessary facts. List the necessary facts. Length = 2x 2 + 4x x + 5 Width = 3x L Choose an operation or operations. Multiplication Write in words what your plan of action will be. Multiply the length by the width V Estimate your answer. A polynomial Carry out your plan. (2x 2 + 4x x + 5)(3x) = 6x x x x E Does your answer make sense? (Compare your answer to the question.) Yes. Is your answer reasonable? (Compare your answer to the estimate.) Yes. Is your answer accurate? (Check your work.) Yes. Write your answer in a complete sentence. The area of the rectangle is 6x x x x.

21 Algebra Success T717 Directions: Find the products. Here is the key to S x(3x x + 6) = 3x 2 + 6x 2. (2x x 5)3x = 6x 2 15x 3. 2x( - x 2 2x x + 4) = - 2x x 3 4x 2 + 8x 4. (2x 2 + x + 6) x = 2x x 3 + x 2 + 6x x(5x x 2) = - 20x 2 + 8x x 6. ( - 7x x + 9) - 2x = 14x 2 18x 7. 5x(x 2 x) = 5x x 3 5x 2 8. ( - 3x 2 + 1) - x = 3x x 3 x 9. 2x( - x 2 4x x 5) = - 2x x 3 8x 2 10x 10. (x 2 x + 6)2x = 2x x 3 2x x

22 T718 Algebra Success Here is the key to S278. Homework Directions: Find the products. 1. 6x(4x 2 3x) 2. 3y 3 (5y 2 y) 3. 5x(3x 2 x) 24x x 3 18x 15y y 5 3y y 4 15x x 3 5x 2 4. m 3 n(6m 2 6m + n) x 3 y(4x 3 3y y 1) 6m 5 n 6m 4 n + m 3 n 2-12x x 6 y + 9x x 3 y 2 + 3x x 3 y 6. Find the area of a rectangle that has a width of 2x x and a length of 2x 2 7x x x (2x 2 7x x + 5) = 4x x 3 14x x 7. Find the area of a rectangle that has a length of x 4 and a width of 3x x 9. x 4 (3x x 9) = 3 x 5 9x x m 3 (6m 4 9m) 9. 6x 2 (x 3 + 2x) xy 3 (4 x) - 24m m 4 6x x x x 3-4x xy 3 + x 2 y 3

23 Algebra Success T719 Name Quiz Date 1. Multiply: x(2x x + 1) A. 2x x + 1 B. 2x x + x C. 2x D. 2x 2 + x 2. Multiply: 3(3x x 2) A. 6x x 2 B. 6x x 6 C. 9x x 5 D. 9x x 6 3. Multiply: 2x(x x + 4) A. 2x 2 + 8x B. 2x C. 2x x + 4 D. 2x x Multiply: - x(2x x 5) A. 2x 2 5x B. 2x 2 + 5x C. - 2x 2 + 5x D. - 2x 2 5x 5. Multiply: (2x x + 3)4x A. 6x 2 + 7x B. 6x x C. 8x 2 + 7x D. 8x x

24 T720 Algebra Success 6. Multiply: 4x(x x 3) A. 4x x 12x B. 4x 2 12 C. 4x 2 12x D. 4x x Simplify: 6x(2x 2 4x x + 5) A. 12x 3 24x x B. 16x 2 + 7x x + 2 C. 16x 3 24x x x + 2 D. 2x 2 + 2x x Simplify: (2x 2 + 3x x + 5)( - 3x3 ) A. - 6x 5 9x x 4 15x 3 B. - 6x 6 9x 3 15x 3 C. - 3x 3 + 2x 2 + 3x x + 5 D. - x 2 x 2 + 2x 3 9. Simplify: 7c(5 2c 2 + c) A. 12c c 14c 3 + 7c 2 B. 35c c 14c 3 + 7c 2 C. 12c c + 5c 3 + 7c 2 D. 35c c 14c 2 + 7c 10. To calculate the area of a rectangle, you multiply the length by the width. If the length of a rectangle is 2x, and the width is (3x x 2), what is the area of the rectangle? A. 5x x 4 B. 5x C. 6x 2 4x D. x