February 27, 2009 Distributive property and its connection to areas page 1 Distributive property and its connection to areas Recap: distributive property The distributive property says that when you multiply two sums, every term has to be multiplied with every term. We arranged all the multiplications in a multiplication table. Examples: (x + 3) (x + 2) = x 2 + 2x + 3x + 6 x 2 x x 2 2x 3 3x 6 (6 + 4) (7 + 3) = 42 + 18 + 28 + 12 7 3 6 42 18 4 28 12 Using area to represent distributive property equations Goal: To understand why the distributive property works, by seeing distributive property equations as statements about how various rectangle areas are related. The area of an a-by-b rectangle is ab. So, this rectangle provides a good way of visualizing the multiplication ab. a b ab For example, to visualize the multiplication 6 7 = 42, think of a 6-by-7 rectangle having an area of 42. If you drew this rectangle on graph paper, it would be made up of 42 1-by-1 squares. Using several of these rectangles, here is the kind of picture that you can draw to represent the distributive property equation (6 + 4) (7 + 3) = 42 + 18 + 28 + 12 Notice that the areas of the rectangles are exactly the same numbers that appear in the multiplication table. Also notice that the area of the entire picture is 10 10 = 100, and the four smaller rectangle areas add up to the same number: 42 + 18 + 28 + 12 = 100. This kind of picture is the reason that the multiplication table process works! (6 + 4) (7 + 3) = 42 + 18 + 28 + 12 = 100
February 27, 2009 Distributive property and its connection to areas page 2 You try it: drawing area pictures of the distributive property Directions: On graph paper, draw a rectangle diagram representing each of the following distributive property equations. Each diagram should be labeled in the same way as the picture at the bottom of page 1. For problems 2 5 and 8, you are only given one side of the equation, so you ll need to figure out the other side using a multiplication table. 1. (2 + 4) (5 + 3) = 10 + 6 + 20 + 12. 2. (8 + 2) (4 + 6) = 3. (9 + 1) (3 + 2) = 4. = 15 + 6 + 35 + 14 5. = 6 + 10 + 18 + 30 6. (x + 3) (x + 2) = x 2 + 3x + 2x + 6 Hint for 6 8: Since the value of x isn t a specific number, you ll need to use a length of your choice 7. (x + 5) x = x 2 + 5x to represent x on the graph paper. 8. (x + 4) (x + 5) =
February 27, 2009 Distributive property and its connection to areas page 3 Using algebra tiles to represent area Watch the presentation in class on using algebra tiles to represent areas and distributive property equations. Take notes by filling in these sentences: Each yellow tile has dimensions by, so its area is. Each green tile has dimensions by, so its area is. Each blue tile has dimensions by, so its area is. 9. On the overhead projector you will see an algebra tile arrangement that represents the equation (x + 3) (x + 2) = x 2 + 2x + 3x + 6. Draw a picture of it in the space below. 10. Write the distributive property equation represented by this algebra tile picture. Make sure you give both sides of the equation (with an = between them).
February 27, 2009 Distributive property and its connection to areas page 4 You try it: algebra tiles You will be given a set of algebra tiles to use during this class. IMPORTANT: You are responsible for putting all of your tiles back in the bag at the end of the period. Directions: Find an algebra tile arrangement that represents each of the following distributive property equations. Then, sketch a picture of your tile arrangement on paper. 11. (x + 4) (x + 2) = x 2 + 2x + 4x + 8 12. (2x + 2) (x + 3) = 2x 2 + 6x + 2x + 6 13. 2x (x + 3 + 1) = 2x 2 + 6x + 2x Put away the algebra tiles now, but we will work with them more at our next class.
February 27, 2009 Distributive property and its connection to areas page 5 Homework: more distributive property practice Here are more distributive property problems like yesterday s problems. Be especially careful to correctly follow exponent rules and methods for combining like terms. 14. Do each of these multiplications using a distributive property multiplication table. Combine like terms to simplify your answers. a. (x 5)(4 x) b. (x 2 6x + 5) ( x) c. (x 4 + 3x 7 ) (x 2 2x 5 ) d. (x 2 + 3x 4) (2x 2 5x + 6)
February 27, 2009 Distributive property and its connection to areas page 6 15. Do each of these multiplications using a distributive property multiplication table. Combine like terms to simplify your answers. a. (x 4)(x + 4) b. (x k)(x + k) c. (x 4) 2 Hint: (x 4) 2 means (x 4) (x 4) d. (x + 4) 2 e. (x + k) 2