1. Mathematics is often described as the study of patterns, so it is not surprising that area models have many patterns. You saw one important pattern in Lesson 4.1.1 (Casey’s pattern from problem 4-4). Today you will continue to use patterns while you develop a method to factor trinomial expressions.  4-12. Examine the area model shown at right. a. Review what you learned in Lesson 4.1.1 by writing the area of the rectangle at right as a sum and as a product. (5x – 2)(2x – 7) = 10x 2 − 39x + 14 b. Does this area model fit Casey’s pattern for diagonals? State the pattern and justify your answer. The products of the diagonals are equal; − 35x · − 4x = 10x 2 · 14 = 140x 2

2. 4.1.2 Factoring with Area Models November 16, 2015

3. Objectives  CO: SWBAT develop an algorithm to factor quadratic expressions without using algebra tiles.  LO: SWBAT explain to a partner how to factor quadratic expressions.

4. 4-13. FACTORING QUADRATIC EXPRESSIONS a. Using algebra tiles, factor 2x 2 + 5x + 3; that is, use the tiles to build a rectangle, and then write its area as a product. (2x + 3)(x + 1) b. When you factor using tiles, you need to determine how to arrange the tiles to form a rectangle. Using an area model to factor requires a slightly different process. Miguel wants to use an area model to factor 3x 2 + 10x + 8. He knows that 3x 2 and 8 go into the rectangle along one of the diagonals, as shown at right. Finish the rectangle by deciding how to split and place the remaining term. Then write the area as a product. (3x + 4)(x + 2)  A polynomial in the form ax 2 + bx + c, with a ≠ 0, is called a quadratic expression in standard form. To develop a method for factoring quadratic expressions without using algebra tiles, you will first model how to factor with algebra tiles, and then look for connections within an area model. Algebra tiles

5. 4-13. FACTORING QUADRATIC EXPRESSIONS c. Kelly wants to determine a method for factoring 2x 2 + 7x + 6. She remembers Casey’s pattern for diagonals. She places 2x 2 and 6 into the rectangle in the locations shown at right. Without actually factoring yet, what do you know about the missing two parts of the area model? Their sum is 7x, and their product is 12x 2 d. To complete Kelly’s area model, create and solve a Diamond Problem using the information from part (c). The product 12x 2 should be placed at the top of the diamond problem, 7x at the bottom, and the terms 3x and 4x should be in the middle. e. Use your results from the Diamond Problem to complete the area model for 2x 2 + 7x + 6, and then write the area as a product of factors. (2x + 3)(x + 2)

6. So, then…? 1. Place the x 2 -term and the constant term of the quadratic expression in opposite corners of the area model. 2. Determine the product of the x 2 -term and the constant term. Place this product in the top of a Diamond Problem. 3. Place the x-term from the expression in the bottom (sum) of a Diamond Problem. 4. Solve the Diamond Problem by identifying two terms whose product and sum match those in the diamond. 5. Place the solutions from the Diamond Problem into the two remaining corners of the area model. 6. Determine the dimensions of the area model. 7. Write the area as the product of the dimensions.

7. 4-14. Factoring with an area model is especially convenient when algebra tiles are not available or when the number of necessary tiles becomes too large to manage. Using a Diamond Problem helps avoid guessing and checking, which can at times be challenging. Use the process from problem 4 ‑ 13 to factor 6 x 2 + 17 x + 12.  (2x + 3)(3x + 4) 1. Place x 2 & # in box 2. Set up diamond 3. Solve diamond 4. Find dimensions 5. Write product

8. 4-16. Use the process you developed in problems 4-13 and 4-14 to factor the following quadratic expressions, if possible. If a quadratic cannot be factored, justify your conclusion. a. x 2 + 9x + 18 b. 4x 2 + 17x – 15 c. 4x 2 – 8x + 3 d. 3x 2 + 5x – 3 a. (x + 3)(x + 6) b. (4x − 3)(x + 5) c. (2x − 3)(2x − 1) d. not factorable