Factoring Trinomials using Algebra Tiles Student Activity Materials: Algebra Tiles (student set) Worksheet: Factoring Trinomials using Algebra Tiles Algebra Tiles: Each algebra tile kits should contain three different sizes of tiles: tile tiles x 1 by 1 tiles 1 The red tiles represent negative quantities and all other colors represent positive quantities. The length of an x tile is unknown since x is a variable. Therefore, one cannot use 1x1 tiles to match the length of an x tile. Terminology: Term: A number, a variable, or a product or a quotient of numbers and / or variables. Terms are separated by plus signs. For example, in 2 + 3x + 1, the terms are 2, 3x, and 1. Polynomial: An expression with more than one term. Examples: 2x + 3, + 2x + 1 Binomial: A polynomial with two terms. Example: 2x + 1. Trinomial: A polynomial with three terms. Example: 2 + 3x + 1.
Product: Two or more numbers or terms that are multiplied. Example: 3(x + 2). Factor: To factor means to rewrite an expression as a product. Example: 3x + 6 = 3(x + 2) The parts of a product are called factors. Example: 3 and (x + 2) are factors of 3x + 6. Prime: If an expression cannot be factored, we say that it is prime. Lesson: Demonstration 1. a. In order to factor a trinomial such as + 3x + 2, you first begin by gathering your tiles: 1 - tile 3 - tiles x 2-1 by 1 tiles 1 b. You next arrange the tiles to form a perfect rectangle. For example, + 3x + 2 forms the following rectangle. x x x 1 1 d. The length of the above rectangle is (x + 2) and the width is (x + 1). The area of a rectangle is length x width. Therefore, + 3x + 2 = (x + 2) (x + 1).
Guided Practice 2. Now in groups of three, try factoring + 5x + 6. What tiles do you need to gather? Now form a rectangle using these tiles. What is the length of the rectangle? What is the width of the rectangle? What is the factored form for the area of the rectangle? How do the numbers 5 and 6 relate to the factored form? To check an answer, we multiply the two binomials. To multiply two binomials, we distribute one term at a time. For example: (x + 3) (x + 2) = x(x + 2) + 3(x + 2) = + 2x + 3x + 6 We now add like terms 2x + 3x to get 5x. Therefore, our final answer is + 5x + 6, which matches what we started with in the beginning. Practice 3. Now try factoring the following trinomials on your own using algebra tiles and fill in the factored answer. Check your answers. Remember that some trinomials do not factor. If it does not factor, we say that it is prime. Trinomial Factored Answer + 2x + 3 + 4x + 4 + 8x +12 + 12x + 36 + 11x + 24 + 17x + 42 2 + 11x + 12 3 + 14x +15 6 + 19x +15
Reflection: Pattern Recognition List at least three patterns that you noticed when doing the check? Pattern 1: Pattern 2: Pattern 3: Work in groups of three to determine at least three patterns you noticed when forming rectangles with the algebra tiles. Pattern 1: Pattern 2: Pattern 3: Suppose + bx + c = (x + m)(x + n). How are m and n related to b? How are m and n related to c? Lecture/Demonstration: Using red or negative Algebra Tiles Remember that red tiles represent negative quantities. Also remember that 1 + (-1) = 0. Adding zero to something is referred to as the identity property of addition since it doesn t change the value of the original quantity. Therefore, we can always add equal quantities of black and red tile of the same size since they add to zero. Sometimes when factoring trinomials using algebra tiles, we need to add equal quantities of black and red tiles. Example: When factoring 3x 4 we need the following tiles: 1 - tile
3 - tiles(red) x 4-1 by 1 tiles (red) 1 However, we cannot construct a rectangle using these tiles. But if we add one more red tile and one more black tile then we can construct a rectangle as shown below. Black x x x x Therefore, 3x 4 = (x 4)(x + 1). Signed Numbers: Remember the following rules when dealing with signed numbers. Addition: 1. If the signs are the same, add the numbers and keep the sign. 2. If the signs are different, subtract the numbers and keep the sign of the larger number. Subtraction: 1. Change to addition and use the rules for addition. Multiplication and Division: 1. If the signs are the same, the answer is positive. 2. If the signs are different, the answer is negative.
Practice: Try factoring the following in groups of three using algebra tiles. Check your answers by multiplying. Remember to add equal quantities of black and red tiles if needed. Trinomial - 5x - 6-7x + 6 + 5x - 6-7x - 8 - x - 20 4x - 21 11x + 18 2x - 15 + 2x - 15-7x +12 3 + x - 4 2 + x - 6 2-9x - 5 3 + x - 2 6 - x - 2 Factored Answer When you are done, please fill out your evaluation form.