3.1+Walking+Together+0910

15/4/10 SM Big Idea: Equations can be used to model real things. Essential Question: **What ** **are the distributive and commutative properties useful for? **  Using distributive and the commutative properties
 * Notes **

Distributive property: A mathematical property used to rewrite expressions involving addition and multiplication. The distributive property states that for any three quantities //a, b,// and //c, a(b+c)=ab+ac//. If the expression is written as a factor multiplied by a sum, you can use the distributive property to //multiply// the factor by each term in the sum.  4(5+//x//) = 4(5)+4(//x//)=20+4//x//  If an expression is written as a sum of terms and the terms have a common factor, you can use the distributive property to rewrite the expression as the common factor multiplied by a sum. This processing is called //factoring//.  20+4//x// = 4(5) +4(//x//) = 4(5+//x//)  Commutative property of addition: A mathematical property that staes that the order in which quantities are added does not matter. For example, 5+7 = 7+5 and 2//x//+4 = 4+2//x//. The commutative property of addition is sometimes called the //rearrangement property of addition.//

Commutative property of multiplication: A mathematical property that states that the order in which quantities are multiplied does not matter. For example 5x7 = 7x5 and 2//x// (4) = (4)2//x//. The commutative property of multiplication is sometimes called the //rearrangement property of multiplication.//

<span style="font-family: "Arial","sans-serif";">A Leanne: //16x// A Gilberto: //7(2x)// A Alana: 11(0.5x+5) A Total: 16x+7(2x)+11(0.5x+5) <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="font-family: "Arial","sans-serif";">$33.50 per mile <span style="font-family: "Times New Roman","serif";">
 * Problem 3.1** <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">A. For each student, write an equation for the amount of money the student will raise if he or she walks //x// miles. Then write an equation for the total amount the three-person team will raise if they walk //x// miles. **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">B. Alana asked each of her 11 sponsors to pledge $5 in addition to an amount per mile, so the team will raise $55 regardless of how far they walk. **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * 1) **<span style="font-family: "Arial","sans-serif";">Excluding $55, how much will their team raise per mile? **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">A total//: 33.5x+55//
 * <span style="font-family: "Arial","sans-serif";">2. Use your answer from part 1 to help you write a different equation for the total amount the team will raise if they walk //x// miles. **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";">A Total: 16x+7(2x) +11(0.5x+5) 16x+14x+11(0.5x+5) <span style="font-family: "Arial","sans-serif";">16x+14x+5.5x+55 <span style="font-family: "Arial","sans-serif";"> 30x+5x+55 <span style="font-family: "Arial","sans-serif";"> **35.5x+55** <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">A Total: **35.5x+55** <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">Both these equations are equivalent <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">If x=1 <span style="font-family: "Arial","sans-serif";">16x+7(2x) +11(0.5x+5) <span style="font-family: "Arial","sans-serif";">No. of Sponsors <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Amount per mile <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Total per person <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Leanna <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">16 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$16 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Gilberto <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">7 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$2 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$14 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Alana <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">11 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$5.50 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$60.50 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Total <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> ||  ||   || <span style="font-family: "Arial","sans-serif";">$90.50 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";"> 35.5x+55
 * <span style="font-family: "Arial","sans-serif";">C. 1. Use the distributive property and commutative properties to show that the two expressions you wrote for the total amount the team will raise are equivalent. **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * 1) **<span style="font-family: "Arial","sans-serif";">Verify that the expressions are equivalent by making and comparing tables or graphs.  **

<span style="font-family: "Arial","sans-serif";">No. of Sponsors <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Amount per mile <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Total per person <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Leanna <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">16 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$1 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$16 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Gilberto <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">7 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$2 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$14 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Alana <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">11 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$5.50 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">$60.50 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";">Total <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> ||  ||   || <span style="font-family: "Arial","sans-serif";">$90.50 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These equations are equivalent because they have the same tables.

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> **<span style="font-family: "Arial","sans-serif";">In questions 1-12, tell whether the expressions are equivalent, and explain your reasoning. **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">Follow-Up 3.1 **<span style="font-family: "Arial","sans-serif";">

**<span style="font-family: "Arial","sans-serif";">1. 3x + 5x and 8x. **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="font-family: "Arial","sans-serif";">These are equivalent because the commutative property of addition applies. 8x is the simplified version of 3x + 5x. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are not equivalent because 3x + 5 cannot be simplified further. <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are not equivalent because the simplified version of the distributive property would be 4 (x) + 4 (7), not 4x + 7 so the actual equivalent equation is 4x+28 <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">2. 3//x// + 5 and 8//x// **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">3. 4(x + 7) and 4x + 7 **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are equivalent because 5 times x equals 5x, and 5 times 2 equals 10. Therefore 5x + 10 and 5 (x + 2) are equivalent. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">4. 5(x + 2) and 5x + 10 **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are equivalent because 4 (3 + 2x) simplified becomes 8x + 12. When 4 is multiplied by 3 and 4 is multiplied by 2x the expressions remain equivalent. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">5. 12 + 8x and 4 (3 + 2x) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">No, these are not equivalent because 4x + x = 5x. 5x + 2x = 7x. 7x is not equivalent to 8x. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">6. 4x + x + 2x and 8x **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">Yes, these are equivalent because these two are simply reversed by the commutative property of addition. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">7. 7 + 5x and 5x + 7 **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are not equivalent because the commutative properties cannot flip the amount of x’s from one number to the other. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">8. 3x + 8 and 8x + 3 **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are equivalent because the commutative property of addition is being used. Both of these expressions add up to 12x. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">9. 5x + 3x + 4x and 4x + 5x + 3x **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are equivalent because using the distributive property we can find out that 3 times 2 equals 6 and 2 times t equals 2t. If you put them together you get 6 + 2t. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">10. 6 + 2t and 2 (t + 3) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">These are equivalent because of the distributive property. 2 times L equals 2L and 2 times 2 becomes 4. So, 2L + 4 + 2W is the same as 2L + 2W + 4. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">11. 2(L+2) + 2W and 2L + 2W +4 **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">Yes, it is equivalent because of the distributive property. 2 times L is 2L. 2 times is 2W and 2 times 2 is 4. Therefore 2L + 2W + 4 = 2L + 2W + 4. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">
 * <span style="font-family: "Arial","sans-serif";">12. 2L + 2W + 4 and 2 (L + W + 2) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">