Multiply Polynomials The Area Model Focus 9 Learning

Multiply Polynomials – The Area Model

Focus 9 Learning Goal – (HS. A-SSE. A. 1, HS. A-SSE. A. 2, HS. A-SEE. B. , HS. A-APR. A. 1, HS. A-APR. B. 3, HS. A-REI. B. 4) = Students will factor polynomials using multiple methods, perform operations (excluding division) on polynomials and sketch rough graphs using key features. 4 In addition to level 3, students make connections to other content areas and/or contextual situations outside of math. 3 Students will factor polynomials using multiple methods, perform operations (excluding division) on polynomials and sketch rough graphs using key features. - Factor using methods including common factors, grouping, difference of two squares, sum and difference of two cubes, and combination of methods. - Add, subtract, and multiply polynomials, - Explain how the multiplicity of the zeros provides clues as to how the graph will behave. - Sketch a rough graph using the zeros and other easily identifiable points. 2 Students will factor polynomials using limited methods, perform operations (excluding division) on polynomials, and identify key features on a graph. - Add and subtract polynomials. - Multiply polynomials using an area model. - Factor polynomials using an area model. - Identify the zeros when suitable factorizations are available. - Identify key features of a graph. 1 Students will have partial success at a 2 or 3, with help. 0 Even with help, the student is not successful at the learning goal.

An area model is a useful tool in performing mental multiplication. If we want to multiply 23 and 46, assign each number to a side of a rectangle. Our goal is to calculate the area of the full rectangle by splitting the side lengths into more manageable numbers, like multiples of 10 and single digits. 23 is the sum of 20 and 3. Subdivide the side of the rectangle. 46 is the sum of 40 and 6. Subdivide the top of the rectangle. 46 23 40 20 3 6

An area model is a useful tool in performing mental multiplication. Now we will fill in the area of each internal rectangle. 20(40) = 800 40 20(6) = 120 3(40) = 120 20 6 800 120 18 3(6) = 18 Next add up all of the areas: 800 + 120 + 18 = 1058 3

Use the area model to multiply binomials. If we want to multiply (x + 1)(3 x – 2), assign each binomial to a side of the rectangle. Our goal is to calculate the area of the full rectangle by splitting the side lengths into more manageable parts. Use “x + 1” and subdivide the side of the rectangle. Use “ 3 x – 2” and subdivide the top of the rectangle. 3 x - 2 x + 1 3 x x 1 -2

Use the area model to multiply binomials. Now we will fill in the area of each internal rectangle. x(3 x) = 3 x 2 3 x -2 x 3 x 2 -2 x 1 3 x -2 x(-2) = -2 x 1(3 x) = 3 x 1(-2) = -2 Next add up all of the areas: 3 x 2 – 2 x + 3 x - 2 = 3 x 2 + x - 2

Use the area model to multiply binomials. Multiply (3 x + 1)(2 x – 9) 3 x(2 x) = 6 x 2 2 x 3 x(-9) = -27 x 1(2 x) = 2 x 3 x -9 6 x 2 -27 x 2 x -9 1(-9) = -9 Next add up all of the areas: 6 x 2 – 27 x + 2 x - 9 = 6 x 2 - 25 x - 9 1