Multiplying Polynomial Expressions Student Probe Find the product: x x 5 Answer: x x 10. Students who struggle with this concept might answer x 10. Lesson Description This lesson makes use of the area model for multiplication to teach students to multiply monomials and binomials using the distributive property. The lesson begins with students thinking about how numbers are multiplied before transitioning into algebraic expressions. The area model is more robust than the frequently used FOIL model and helps students develop a strategy that is applicable to more types of multiplication. Rationale In order for students to be successful in algebra courses at all levels, they must be able to multiply monomials and binomials with fluency. The inability to do this efficiently can hinder students ability to understand factoring, finding zeros of functions, and simplification of expressions. This fluency is further tested when students must reverse the process when factoring algebraic polynomials. By enabling students to develop a strategy that can applied in multiple situations and is reversible, they will be able to concentrate on the conceptual understanding of these topics rather than only on the symbolic manipulation. Preparation Prepare a display for the problems and their solutions. At a Glance What: Use the area model of multiplication to multiply polynomials (distributive property) Common Core State Standard: CC.9-1.A.APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Matched Arkansas Standard: AR.9-1.LA.AI.1.5 (LA.1.AI.5) Perform polynomial operations (addition, subtraction, multiplication) with and without manipulatives. Mathematical Practices: Look for and make use of structure. Look for and express regularity in repeated reasoning. Who: Students who do not use distributive property correctly when multiplying binomials Grade Level: Algebra 1 Prerequisite Vocabulary: terms, monomial, binomial, distributive property, simplify, polynomial Prerequisite Skills: multiply monomials, add and subtract like terms, integer operations Delivery Format: individual, small group Lesson Length: 15 45 minutes Materials, Resources, Technology: none Student Worksheets: Multiplying a Monomial and Binomial Multiplying Binomials (.pdf)
Lesson The Expect students to say or do If students do not, then the 1. We are going to use a diagram to help us multiply polynomials. Let s begin by looking at a multiplication problem using numbers: 8 7. We can think of 7 as 0 7. So, 8 0 7 8 0 8 7. By using the Distributive Property, 8 0 7 8 0 8 7 160 56 16 We can make a diagram of this called an area model. This is what the diagram looks like: 0 7 8160 56 Can you explain where the numbers in the diagram come from? What are the dimensions of our model? Why do you think this is so? 7 0 7 8 0 160 8 7 56 1 We are multiplying a one digit number by a two digit number. Relate the number sentences to the diagram.
The Expect students to say or do If students do not, then the. Now you try 5 14. 14 10 4 10 4 5 50 0. Explain how your area model matches the numbers in the multiplication problem. How do you know? 4. Let s use the area model to find this product: x. In some ways this is even easier than using numbers! Draw the model, but use the x in place of the number. What are the dimensions of our model? Why? 5. What is the product? How does your area model show you? 6. Let s check our answer by letting x 7. What is 7? What is 7 6? Since they are both equal to 7, our model is correct. (Make sure that students see that these are the same.) 50 0 70 5 14 =70 5 10 4 5 10 5 4 50 0 70 The distributive property tells us. x 1, because we are multiplying a monomial by a binomial. x x 6 x x 6 9 7 1 6 7 x 6 Relate the number sentences to the diagram. If students struggle with the variable, ask what if the number is 1 instead of x. Let x 5 or some other number to verify.
The Expect students to say or do If students do not, then the 7. Now let s try x 4x 7. This looks more difficult, 4x 7 but it works exactly the x 8x 14x same way. 8. Model several more examples until students appear comfortable using the area model. (See Multiplying a Monomial and Binomial worksheet for additional exercises.) 9. What if both numbers have two digits such as 45? How can we rewrite? How can we rewrite 45? What dimensions do we need for our area model? How do you know? 10. This means 45 0 40 5 Let s create the area model for this problem. x 4x 7 x 4x x 7 0 45 40 5 8x 14x We are multiplying a two digit number by a two digit number. 0 40 5 800 100 10 15 45 0 40 5 0 40 0 5 40 5 800 100 10 15 105 multiplying monomials, refer to Multiply Monomial Expressions. adding like terms, refer to Addition and Subtraction of Polynomials. Twenty-three has how many tens? How many ones? Forty-five has how many tens? How many ones?
The Expect students to say or do If students do not, then the 11. We can use the same strategy to multiply binomials. We are multiplying a binomial by a binomial. Let s create an area x model for x x. What dimensions do we need for our area model? Why? x x x x 6 Relate the number sentences to the diagram. How did you complete each square of the model? 1. Now let s combine like terms where we can, and determine our answer. 1. Let s try another one similar to this. Find the product y 4 y. x x x x 6 x x x x x x 6 x 5x 6 adding like terms, refer to Addition and Subtraction of Polynomials. y 4 y y 4y y 1 y y y y y 4 4 1 y 7y 1 Relate the number sentences to the diagram adding like terms, refer to Addition and Subtraction of Polynomials. 14. Now find the product x 4 x. What do you notice that is the same as before? What do you notice that is different? Answers may vary, but listen for, We are multiplying two binomials. Answers may vary, but listen for, One binomial is uses subtraction instead of addition. And One of the terms is x instead of x.
The Expect students to say or do If students do not, then the 15. Let s set up our area model for this problem. x 4 x x -4x x -1 What do you notice is different in your area model than from before? 16. Now let s combine like terms, where we can, and get our answer. 17. Model several more examples until students appear comfortable using the area model. (See Multiplying Binomials worksheet for additional exercises.) We had to use negative numbers. x 4 x x x 4x 1 There aren t any like terms, so this is our answer. with integer operations, refer to Integer Multiplication. adding and subtracting like terms, refer to Addition and Subtraction of Polynomials. Teacher Notes 1. Be aware that the mnemonic device FOIL has two major drawbacks. Students may forget how to apply FOIL or forget what the letters stand for. If students are multiplying polynomials other than two binomials, this device is ineffective.. The method demonstrated in this lesson can be extended to polynomials multiplied by any monomial or binomial.. Stress that the dimensions for the area model diagram are related to the number of terms in each factor. 4. It is extremely important for students to relate the number or algebraic sentences to the diagram. 5. While it is inconsequential which factor is listed vertically and which is horizontal in the area model, in this lesson the first factor is always horizontal and the second factor is always vertical. 6. The dimension of the models is always spoken of as rows columns. Since rows columns is the convention for stating the dimensions of matrices, it is done in this lesson for consistency.
Variations 1. Use this method to multiply any polynomial by any monomial or binomial. For example, x x 4x would require a 1 model and x 1 x x 1 would require a model.. The strategy presented in this lesson could be used to teach factoring. Present a completed matrix and ask students to determine the factors. For example, the model below illustrates the product x x. x x Formative Assessment Find the product x x 5. Answer: x x 15. References Paulsen, K., & the IRIS Center. (n.d.). Algebra (part ): Applying learning strategies to intermediate algebra. Retrieved on April 5, 011. Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved 5, 011, from rti4sucess.
Multiplying a Monomial and Binomial (x + ) x + Correct response: (x + 1) x + 1 (x + 4) x + 8 4(x 6) 4x - 4 (y ) y - 6 5(a 1) 5a - 5 x (x + 5) x + 5x x (4x + 8) 8x + 16x x(4x + x) 8x + 6x 4y(y + 5y) 1y + 0y 8y(y y) 16y 8y
Multiplying Binomials (x + 5)(x + 7) x + 1x + 5 (x + )(x + 8) x + 10x + 16 (x + 4)(x + 1) x + 5x + 4 (x - 5)(x + 7) x + x 5 (x + 4)(x - 5) x - 1x - 0 (x - )(x - 9) x - 1x + 7