Why Teach the Area Models of Multiplication? Can area models of multiplication be linked to factoring polynomials? MC2 has the wonderful opportunity to partner with classroom teachers to reflect on mathematics teaching and learning to enhance student understanding. When we work with elementary teachers, we are often asked, Why do the Common Core Standards emphasize teaching students the area model of multiplication? or How soon can we begin to teach the standard algorithm? This presentation will explore some possible answers to these questions. mc2.nmsu.edu

One-Digit by One-Digit Multiplication 8 x 7 8 (5 + 2) 5 2 8 40 16 Area models can play an important role in a student’s ability to work flexibly with numbers and can bridge to Algebraic understanding. Decomposing one factor to make simpler problems helps students think about multiplication as a series of easier problems. For instance, let’s begin with an example of one-digit by one-digit multiplication. A student may consider 8 X 7 as two simpler problems by decomposing the 7 into 5 and 2 then multiplying the 8 by 5 to get 40 and 8 times 2 to arrive at 16. The 40 and 16 can then be added together to equal 56. This area model connects to the distributive property (8 x 5, 8 x 2) 40 + 16 = 56

Two-Digit by Two-Digit Multiplication 18 x 12 10 2 (10 + 8) (10 + 2) 10 100 20 The same model also works for multi-digit multiplication. Take 18 x 12, for example. A student may decompose the 12 into one ten and two ones and the 18 into one ten and eight ones, then multiply 10 X 10 to get 100….10 X 2 to arrive at 20…8 x 10 to get 80 and 8 x 2 to make 16. This can also be illustrated using the distributive property where 10 x 10 = 100…10 x 2 = 20…8 x 10 = 80 and 8 x 2 = 16. Adding the four products together equals 216. (check language of 216) 100 + 16 8 80 16 + 20 + 80 = 216

Multiplication of Algebraic Expressions (X+3)(X+2) (X + 3) (X + 2) X 3 X 2 X2 3X So how does this link to Algebra? The same area model lends itself to multiplication of algebraic expressions. Let’s consider (X+3) times (X+2). Place X + 3 along one side of the model and X + 2 along the other. As before, this model highlights that 4 sets of factors need to be multiplied together resulting in 4 terms. X times X is X2…X times 2 is 2X…3 times X is 3X….and 3 times 2 is 6. Linking the distributive property to the model highlights the four resulting factors. (pause) Once like terms are added together the simplified expression becomes X2 + 5X + 6. 2X 6 X2 + 2X + 3X + 6 = X2 + 5X + 6

Try using the area model to multiply this problem. (3X + 4) (X + 2) You may want to try this problem on your own.

Factoring Polynomials X2 + 6X + 8 1 x 8 -1 x -8 X 2 2 x 4 -2 x -4 X2 2X X 1 + 5 Let’s examine if the same process will work for factoring a polynomial. How can we factor the polynomial X2 + 6X + 8 using the area model? From previous experiences with numbers and variables, we know that one way to get X2 is to multiply X times X. Next place 8 in the lower right-hand box. In order to find the other two terms, consider the factors of 8 (1 x 8 , 2 x 4, and the negative factors, -1 x -8 and -2 x -4) and the addends of 6 (1 + 5, 2 + 4, and 3 + 3). In this case, 2 and 4 will work because they meet both constraints. Place the 2 along the top edge of the box and the 4 along the side or vice versa. Complete the multiplication to get 4X in one box and 2X in the remaining box. The factored form becomes (X+2) times (X+4). Keep in mind that providing your students with multiple opportunities to work with area models can develop their flexibility with numbers and create a foundation for factoring polynomials. 4 4X 8 2 + 4 3 + 3 (X + 2) (X + 4)

Try using the area model to factor this problem. X2 + 11X + 24 You may want to try this problem on your own.

For other interesting uses of flexibility with numbers view Jo Boaler’s video, What is Number Sense? https://www.youcubed.org/what-is-number-sense/ For other interesting uses of flexibility with numbers view Jo Boaler’s video, What is Number Sense? at the link shown here. Thank you for your interest!

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