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What are different ways that you can teach students multiplication of 2 binomials? 1 Why do students struggle with factoring? 2 Using Algebra Tiles for Student Understanding Examining the Tiles Colors and shapes = 1, = x, = -1, = -x, = x2 Model the following expressions +3 -2x x–4 x2 + 3x - 2 = -x2 Zero Pairs Called zero pairs because they are additive inverses of each other. When put together, they model zero. 5 Model Simplify Initial illustration: before combining terms Final illustration Modeling Polynomials Algebra tiles can be used to model expressions. Model the simplification of expressions. Add, subtract, multiply, divide, or factor polynomials. 7 Modeling Polynomials 2x2 -4x 3 or +3 8 Polynomials Represent each of the given expressions with algebra tiles. Draw a pictorial diagram of the process. Model the symbolic expression -2x + 4 9 Modeling Polynomials 2x2 + 3 4x – 2 10 Add/Subtract Polynomials This process can be used with problems containing x2. Model each of the following using tiles. (2x2 + 5x – 3) + (-x2 + 2x + 5) = x2 + 7x + 2 (2x2 – 2x + 3) – (3x2 + 3x – 2) = -x2 - 5x + 5 Verify your solutions graphically. 11 Multiplying Polynomials Algebra tiles can be used to multiply polynomials. Use tiles and frame to represent the problem. The factors will form the dimensions of the frame. (vertical) and (horizontal) The product will form a rectangular array inside frame. “Area Model” 12 Multiplication using “Area Model” (2)(3) = Place 2 sm. squares on the vertical and 3 sm. squares on the horizontal Fill in the interior of the area model with appropriate algebra tiles to form a rectangular array. 2x3=6 13 Multiplying Polynomials What are all the different ways that you can demonstrate multiplication of two binomials? Tiles Box method Distributive Property (aka rainbow or foil) 14 Multiplying Polynomials (x )(x + 3) Fill in each section of the area model Algebraically 15 x2 + x + x + x = x2 + 3x Multiplying Polynomials (x + 2)(x + 3) Fill in each section of the area model Algebraically 16 x2+ 2x+ 3x + 6 = x2+ 5x + 6 Multiplying Polynomials (x – 1)(x + 4) Fill in each section of the area model Make zero pairs or combine like terms and simplify x2 + 4x – 1x – 4 = x2 + 3x – 4 17 Multiplying Polynomials Use the tile frame and tile pieces to model the product of each problem below. Think about how you will connect the algebraic procedure to the model. Verify your solution using the box method and distributive property. 18 (2x + 3)(x – 2) (x – 2)(x – 3) Virtual Algebra Tiles http:..media.mivu.org/mvu_pd/a4a/homework/applets_applet_home.html 19 Dividing Polynomials Algebra tiles can be used to divide polynomials. Use tiles and frame to represent the problem. Dividend should form the rectangular array inside the frame. Divisor will form one of the dimensions (one side) of the frame (vertical side). Be prepared to use zero pairs in the dividend. 20 Dividing Polynomials = (x + 3) 2x2 + 6x 2x Now, determine the tiles that would represent the horizontal axis: quotient (answer) 21 Dividing Polynomials x2 + 7x + 6 x+1 = (x + 6) Now, determine the tiles that would represent the horizontal axis: quotient (answer) 22 Dividing Polynomials Practice dividing polynomials using tiles. 23 2x2 + 5x – 3 x+3 x2 – x – 2 x–2 x2 + x – 6 x+3 Factoring Polynomials Algebra tiles can be used to factor polynomials. Use tiles and the frame to represent the problem. Use the tiles to form a rectangular array inside the frame. (area model) Be prepared to use zero pairs to fill in the array. Draw pictures. 24 Factoring When using tiles….. Big squares can't touch little squares. Little squares must all be together. Only equal length sides may touch. You may not lay two equally sized tiles of different colors next to each other. Use all of the pieces to make a rectangle. Once you have correctly arranged the tiles into a rectangle, the factors of the quadratic are the length and width of the rectangle. 25 Factoring Polynomials 3x + 3 = 3 · (x + 1) 2x – 6 = 2 · (x – 3) Note the two are positive, this needs to be developed 26 Factoring Polynomials x2 + 6x + 8 = (x + 2)(x +4) x2 + 4x + 2x + 8 27 Factoring Polynomials x2 – 5x + 6 = (x – 2)(x – 3) x2 - 3x - 2x + 6 28 Factoring Polynomials x2 + 5x + 8 = prime 29 Factoring Polynomials x2 – x – 6 = (x + 2)(x – 3) x2 - 3x + 2x - 6 30 Factoring Polynomials x2 – 9 = (x + 3)(x – 3) x2 - 3x + 3x - 9 31 Factoring Polynomials 2x2 + x – 6 = (2x - 3)(x + 2) 32 2x2 - 3x + 4x - 6 Factoring Polynomials 2x2 + 3x – 4 = prime 33 Virtual Algebra Tiles http:..media.mivu.org/mvu_pd/a4a/homework/applets_applet_home.html 34 Factoring Polynomials Practice factoring x2 + x – 6 x2 – 4 4x2 – 9 2x2 – 3x – 2 3x2 + x – 2 -2x2 + x + 6 35