Honors Math 2 Date: ______________ Name: ____________________________ Block: ____________ Investigating Squares and Rectangles Directions: 1) In your group, use the paper “xy” ruler to measure the sides of each of the squares and rectangles. Write the measurements on each shape so that you know the dimensions of each shape. a. Use those measurements to find the area of each shape in terms of x and y. b. Write the area on each shape. Note: for this activity, we are modeling products of expressions as area (just like with a rectangle where l x w = area) 2) Cut out the shapes (if its not already done for you) 3) Use the shapes to model an area that is 𝑥 ! − 𝑦 ! . Sketch the shape below. 4) Using the shapes, find at least 2 ways that you can represent the same area from (3). Sketch your results below. 5) For each of the models that you found in part 4: a. Write expressions for each area model you made in (4) b. Write each expression as a product of two factors. What do you notice? 6) Imagine a square that has side length 𝑥 + 𝑦. Find a group of shapes that you can use to model that square. Sketch your results below. 7) Using the shapes, find at least 2 ways that you can represent the same area from (6). Sketch your results below. 8) For each of the models that you found in part 7: a. Write expression for the area. b. Expand and combine like terms. What do you notice? 2 9) In part 5, you should have have found the expression (𝑥 + 𝑦)(𝑥 − 𝑦) as a representation of the area of 𝑥 ! − 𝑦 ! . Prove that 𝑥 ! − 𝑦 ! = (𝑥 + 𝑦)(𝑥 − 𝑦) is an identity. (This identity is called the Difference of Squares.) 10)In part 8, you should have found (𝑥 + 𝑦)! and 𝑥 ! + 2𝑥𝑦 + 𝑦 ! as representations of the area. Prove that (𝑥 + 𝑦)! = 𝑥 ! + 2𝑥𝑦 + 𝑦 ! is an identity. (This identity is called the Perfect Square Trinomial.) 11) Use the identity that you proved in parts 9 and 10 to rewrite each of the expressions as a product of two binomials. (These are two different forms of factoring.) a. 𝑥 ! − 8𝑥 + 16 b. 16 − 𝑟 ! c. 4𝑗 ! + 4𝑗 + 1 d. 𝑚 ! 𝑘 ! − 81 e. 25𝑝 ! − 30𝑝𝑞 + 9𝑞 ! f. 25𝑥 ! − 9𝑦 ! g. 50𝑥 ! − 18 h. 81𝑟 ! − 18𝑟𝑡 + 𝑡 ! i. 𝑚 ! − 𝑛! Homework: Read “Minds in Action,” page 119-­‐120, then do page 121 #7, 8, 9, 11, 13, 17 3