1. Where We’ve Been……. Properties, Part I…….. If I say “order,” you say……… If I say “grouping,” you say……… If I say “identity,” you say……… commutative associative Value stays the same

2. Where We Are Going……. Today, we are going to investigate one of the most important properties you will use this year and in future classes. Distributive Property Algebra Just listen. Associative PropertiesCommutative Properties Identity Properties Order of Operations Translating Expressions

3. The Distributive Property 6.6 p. 485

4. What Is Our Objective? p.485 Use the distributive property to rewrite and simplify equivalent multiplication problems Three friends went to a baseball game. Each ticket cost $20 and all three friends bought a baseball hat for $15 each. Notes. Let’s look at this expression: 3(20 + 15). The 3 represents: The 20 represents: The 15 represents: Three friends Price of the ticket Price of the hat

5. Three friends went to a baseball game. Each ticket cost $20 and all three friends bought a baseball hat for $15 each. 2. Let’s evaluate the expression…write the 3 in front of the parentheses…… 3(20 + 15) = 3 x 35 = 105 The price of three tickets The price of three hats 60 + 45 = 105 3. What does the expression 3 x 20 + 3 x 15 represent? 3 x 20 represents 3 x 15 represents 4. Evaluating this expression: 3 x 20 + 3 x 15 3 x 20 = 3 x 15 = 45 60 What do you notice about the answers of these two problems?

6. The Distributive Property Words: To multiply a sum by a number, multiply each addend by the number outside the parentheses. 2(7 + 4) = 2 x 7 +2 x 4 a(b + c) = ab + ac 9(4) + Ask yourself, “What is one-third of 9”? 36 + 3 = 39

7. WAIT…..Let’s practice this with just whole numbers first! Insert this practice page with your other notes for this lesson. Let’s talk through the first one. 5 x 27 Let’s take the larger factor and write it in expanded form. 5 (20 + 7) Remember, no sign means to multiply! Let’s “ distribute “ the 5. It is the factor used on both the 20 and the 7. (5 x 20)+(5 x 7) = (100) + (35) = 135

8. 4 x 26 Take the larger factor and write it in expanded form. 4 (20 + 6) Let’s “ distribute “ the 4. It is the value used on both the 20 and the 8. (4 x 20) + (4 x 6) = (80) + (24) = 104 7 x 32 =7 (30 + 2 ) (7 x 30) + (7 x 2) =210 + 14 = 224

9. 6(56) = 6 ( ) 8(73) = ___ ( ) 50 + 6 6(50) + 6(6) 300 + 36 =336 8 70 + 3 8(70) + 8(3)= 560 + 24 = 584

10. Back to the text. Example 1. 9(4) 36 + 3 =39 p. 486

11. ( + ) 9 4 1/3 9(4) + 9(1/3) Use your imagination! 36 + 3= 39

12. SHOW YOUR STEPS as you evaluate each of these. 5(2) 10 + Think of 15 ÷ 5 3 =13 12(2) 24 + 3 =27 2 x 3.6 = 2(3) 6 + 1.2 = 2 ( 3 +.6) = 7.2 p. 486

13. Let’s stop here for today. What was the goal of our lesson? Tonight you will practice only the process of evaluating with the distributive property.

14. Adding Variables to the Process… xx 1 1 11 1 1 We will use the distributive property to rewrite the expression that describes this model: 2(x + 3) 2(x + 3) = 2x + 2(3) = 2x + 6 Can we do anything else????? NO!! We do not know the value of x, so we have to stop here. 8(x + 3) 5(9 + x) 2(x + 3) 8x + 8(3) 45 + 5x 5(9) + 5x 8x + 24 2x + 2(3) 2x + 6

15. Distributive Property II In our previous lesson, we rewrote expressions in a simpler way using the distributive property. 7 x 35 was rewritten as……… 7(30 + 5) 7(30) + 7(5) 210 + 35 = 245 We also used variables. 3 ( b + 5) 3b + 15

17. Think Critically Fran is making a pair of earrings and a bracelet for four friends. Each pair of earrings uses 4.5 cm of wire and each bracelet uses 13 cm of wire. Write two equivalent expressions and then find out how much wire is used. 4(4.5 + 13) and 4(4.5) + 4(13) are equivalent. 18 + 52 = 70 cm 4(17.5) = 70 cm Sort all of the information you have: What do you need for one set of stuff???? Earrings: 4.5 cm bracelet: 13 cm How many are being made? 4 p. 487

18. Think Critically Each day, Martin lifts weights for 10 minutes and runs on the treadmill for 25 minutes. Write two equivalent and find the total minutes that Martin exercises in 7 days. 7(10 + 25) and 7(10) + 7 ( 25) 7(35) 245 minutes 70 + 175 245 minutes 7(10) + 7(20) + 7(5) ??????? 70 + 140 + 35 105 + 140 = 245 p. 487

19. Take a moment to look at the operations we used with the Distributive property. Did we leave something out?????? We rewrote all problems as addition. Let’s look at two problems and change them to subtraction.

20. Our first problem in the Guided Practice was 5 x 27 Could we use a subtraction problem to create a value of 27???? 30 – 3 = 27 5 x 27 = 5 ( )30 - 3 5(30)- 5(3) 150 - 15 = 135

21. Could we use a subtraction problem to create a value of 8(78)???? 80 – 2 = 78 8 x 78 = 8 ( )80 - 2 8(80)- 8(2) 640 - 16 = 624

22. Factoring Expressions I am going to take a slightly different approach to this skill than the one shown in your notes. We will cover prime factorization later in the year. Factoring the Expression: We will take a simple addition problem and rewrite it using the distributive property. Let’s factor the numerical expression 12 + 8 12 = 2 ·2·3 8 = 2 ·2·2 Ask yourself, “What the largest number 12 and 8 share?” We know it is 4. OK…..if we divide each of the numbers by 4 – factor the 4 out of each one – what’s left??? Think of the values as multiplication problems with 4’s. 4 is the shared factor…..What do we have left if we just write it once???

23. Let’s look at that again……this is not in your notes. Okay. Let’s do the practice problems at the bottom of p. 487. 9 + 21 14 + 28 80 + 56 3 x 3 + 3 x 7 3 ( 3 + 7) 7 x 2 + 7 x 4 ????? 14 x 1 + 14 x 2 14 ( 1 + 2) 8 x 10 + 8 x 7 8( 10 + 7) 3 14 8

24. Let’s add some variables! 16 + 4x 7x + 42 36x + 30 4 · 4 + 4x 4(4 + x) 7x + 7 · 6 7(x + 6) 6 · 6 x + 6 · 5 6(6x + 5)

25. Let’s try one more of those……….

26. Guided Practice p. 488 3x + 3(1) 3x + 3 5x + 5(8) 5x + 40 4x + 4(6) 4x + 24 5(5 + 12) 4( x + 10) Six friends: Admission is $9.50. One ride is $1.50 6(9.5 + 1.5) = 6(9.5) + 6(1.5) =6(11) = $66

27. Independent Practice p. 489 Talk to each other. Talk to me. 9(40 + 4) 9(40) + 9(4) 360 + 36 396 7(3 +.8) 7(3) + 7(.8) 21 + 5.6) 26.6 4. 8(x + 7) 5. 6(11 + x) 6. 8(x + 1) 8x + 8(7) 8x + 56 6(11) + 6x 66 + 6x 8x + 8(1) 8x + 8

28. Reasoning…..Subtraction??? A coyote can run up to 43 mph while a rabbit can run up to 35 mph. We will write two equivalent expressions to Show the difference in how far they would run after 6 hours. Choose one to solve. 6( 43 – 35) or 6(43) – 6(35) 6( 8) = 48 or 258 – 210 = 48

29. Factor Each Expression 8.8 + 16 9. 54 + 24 10. 63 + 81 11. 11x + 55 12. 32 + 16x 13. 77x + 21 8(1) + 8(2) 8(1 + 2) 6(9) + 6(4) 6(9 + 4) 9(7) + 9(9) 9(7 + 9) 11x + 11(5) 11(x + 5) 16(2) + 16x 16(2 + x) 7(11)x + 7(3) 7(11x + 3)

30. What have we done? We have used the distributive property to solve multiplication problems. We have written problems with variables using the distributive property.