T755 [OBJECTIVE] The student will learn how to multiply polynomials. [MATERIALS] Student pages S289 S297 Transparencies T765, T767, T769, T771, T773, T775 [ESSENTIAL QUESTIONS] 1. When multiplying polynomials, is it possible to determine the number of terms you should have before combining like terms? 2. Is it possible to have more than two like terms when multiplying polynomials? [GROUPING] Cooperative Pairs, Whole Group, Individual [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Algebraic Formula, Concrete Representation, Pictorial Representation [WARM-UP] (5 minutes IP) S289 (Answers on T764.) Have students turn to S289 in their books to begin the Warm-Up. Students will review multiplying binomials. 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)
T756 Algebra Success SOLVE Problem (2 minutes GP) T765, S290 (Answers on T766.) Have students turn to S290 in their books, and place T765 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} Binomial Trinomial (7 minutes M) T765, S290 (Answers on T766.) Use the following modeling activities to model for students how to multiply binomials by trinomials. {Algebraic Formula, Verbal Description} Problem 1 Step 1: Explain to students that, in Problem 1, they will model how to multiply the binomial x + 2 by the trinomial x 2 + 3x x + 1. Explain that the first factor is represented vertically, and, because the first factor is a binomial, students will need two rows, one for each term in the binomial. Explain that the second factor is represented horizontally, and, because the second factor is a trinomial, students will need three columns, one for each term in the trinomial. Step 2: Have students label the box as shown below. Label the rows x and Label the columns x 2, + 3x, and + 1. + 2. x 3 + 3x 2 + x + 2x 2 + 6x x + 2 = x 3 + 5x 2 + 7x + 2.
T757 Problem 2 Step 1: Explain to students that, in Problem 2, they will model how to multiply the binomial 2x x + 1 by the trinomial x 2 x + 4. Remind students that the first factor is represented vertically, and, because the first factor is a binomial, students will need two rows. The second factor is represented horizontally, and, because the second factor is a trinomial, students will need three columns. Step 2: Have students label the box as shown below. Label the rows 2x x and Label the columns x 2, - x, and + 4. + 1. 2x 3 2x 2 + 8x x + x 2 x + 4 = 2x 3 x 2 + 7x x + 4.
T758 Algebra Success Trinomial Trinomial (8 minutes M, GP) T767, S291 (Answers on T768.) Have students turn to S291 in their books, and place T767 on the overhead. Use the following activities to model for students how to multiply trinomials. {Algebraic Formula, Verbal Description} Problem 1 Step 1: Explain to students that, in Problem 1, they will model how to multiply the trinomial x 2 + 2x x + 1 by the trinomial x 2 + 3x x + 2. Remind students that the first factor is represented vertically, and, because the first factor is a trinomial, students will need three rows. The second factor is represented horizontally, and, because the second factor is a trinomial, students will need three columns. Step 2: Have students label the box as shown below. Label the rows x 2, + 2x, and + 1. Label the columns x 2, + 3x, and + 2. x 4 + 3x 3 + 2x 2 + 2x 3 + 6x 2 + 4x x + x 2 + 3x x + 2 = x 4 + 5x 3 + 9x 2 + 7x x + 2.
T759 Problem 2 Step 1: Explain to students that, in Problem 2, they will model how to multiply the trinomial 3x 2 x + 4 by the trinomial x 2 + 2x x 1. Remind students that the first factor is represented vertically, and, because the first factor is a trinomial, students will need three rows. The second factor is represented horizontally, and, because the second factor is a trinomial, students will need three columns. Step 2: Have students label the box as shown below. Label the rows 3x 2, - x, and + 4. Label the columns x 2, + 2x, and - 1. 3x 4 + 6x 3 3x 2 x 3 2x 2 + x + 4x 2 + 8x x 4 = 3x 4 + 5x 3 x 2 + 9x x 4.
T760 Algebra Success Trinomial Binomial (8 minutes M, GP) T769, S292 (Answers on T764.) Have students turn to S292 in their books, and place T769 on the overhead. Use the following activities to continue to model for students how to multiply polynomials. {Algebraic Formula, Verbal Description} Problem 3 Step 1: Explain to students that, in Problem 3, they will model how to multiply the trinomial x 2 + 5x x + 4 by the binomial 2 x 3. Remind students that the first factor is represented vertically, and, because the first factor is a trinomial, students will need three rows. The second factor is represented horizontally, and, because the second factor is a binomial, students will need two columns. Step 2: Have students label the box as shown below. Label the rows x 2, + 5x, and + 4. Label the columns 2x x and - 3. 2x 3 3x 2 + 10x 2 15x x + 8 x 12 = 2x 3 + 7x 2 7x x 12.
T761 Problem 4 Step 1: Explain to students that, in Problem 4, they will first represent the product 2x(x x + 1) in the leftmost box. Remind students that the first factor is represented vertically, and because the first factor is a monomial, students will need one row. The second factor is represented horizontally, and, because the second factor is a binomial, students will need two columns. Next, explain that students will represent the product 3x(x 2 5x x + 1) in the rightmost box. In this product, the first factor is a monomial, so students will need one row. The second factor is a trinomial, so students will need three columns. Point out to students that the problem is asking them to add these two products. Step 2: Have students label the boxes as shown below and write a plus sign between them. Step 3: For each box, have students multiply each row by each column and find the products in the boxes, as shown below. 2x 2 + 2x x + 3x 3 15x 2 + 3x x = 3x 3 13x 2 + 5x.
T762 Algebra Success More Multiplication (7 minutes IP) T771, S293 (Answers on T772.) Have students turn to S293 in their books. Give students 5 minutes to complete the three problems using the box method. If your students are struggling, complete the problems with your students. Use 2 minutes to review the answers. {Algebraic Formula} Use the Distributive Property (8 minutes M, GP) T773, S294 (Answers on T774.) Have students turn to S294 in their books, and place T773 on the overhead. Use the following activity to model for students how to use the distributive property to multiply polynomials. You may need to review the rules for multiplying monomials. {Algebraic Formula, Verbal Description} Use the Distributive Property to Multiply Polynomials Model each problem for your students. Draw lines to show how to distribute each term in the first polynomial to all terms in the second polynomial. The model for Problem 1 is shown below: (x x + 1)( x 2 x 2) (x x 2 = x 3 ); (x x - x = - x 2 ); (x x - 2 = - 2x ); (1 x 2 = x 2 ); (1 - x = x 3 x 2 2x x + x 2 x 2 = x 3 3x x 2 - x); ) (1-2 = - 2) Continue to model the rest of the problems, drawing lines and multiplying the polynomials using the distributive property. SOLVE Problem (7 minutes GP) T775, S295 (Answers on T776.) 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) S296 (Answers on T777.) Have students complete Problems 1 6 on S296, using the distributive property or the box method. Students can work in cooperative pairs or independently. Give your students 6 minutes to complete the problems. Use 2 minutes to review the answers. {Algebraic Formula}
T763 [CLOSURE]: (5 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. When multiplying polynomials, is it possible to determine the number of terms you should have before combining like terms? (Yes, multiply the number of terms in each factor to determine the total number of terms in the product before combining like terms.) Is it possible to have more than two like terms when multiplying polynomials? (Yes, depending on the types of polynomials you are multiplying.) [HOMEWORK]: Assign S297 for homework. (Answers on T778.) [QUIZ ANSWERS] T779 T780 1. B 2. C 3. D 4. A 5. A 6. D 7. A 8. C 9. A 10. C The quiz can be used at any time as extra homework or to see how the students did on understanding multiplying polynomials.