Activities

1. Use the distributive property to rewrite the following expression:  -4(7x³ – 11x² + 8x – 2) .

2. Most mathematicians like to write polynomials in standard form, that is, from highest exponent to lowest. What property would allow a mathematician like yourself to rewrite
   8x + 3x² – 4 + 5x⁴ as 5x⁴ + 3x² + 8x – 4 ?

3. According to the associative property, and without doing any calculations, if (9 + 4) + x = 27, then what must 9 + (4 + x) equal?

4. Using the distributive property, what value for a in the expression a(3y + 12) would make it equivalent to the expression  3y + 3y + 3y + 3y + 12 + 12 + 12 + 12.

5. Katee was given the following list of numbers to add: 19 + 25 + 81 + 17 + 25 + 3. She thought that if she rearranged the numbers, it could help her to add the numbers more quickly. Her new order was 19 + 81 + 25 + 25 + 17 + 3. Next, she grouped the numbers like this to make the problem easier: (19 + 81) + (25 + 25) + (17 + 3). She used two properties in her mental math, what were they, and in what order did she use them?

6. When x = y we know that x + 2 = y + 2 by the addition property of equality. Write the names of the properties described here, assuming that x = y.

1. x – 7 = y – 7
2. -5x = -5y
3. y/10 = x/10

7. Crissy correctly solved the following problem below. In the space beside each step, write the property that justifies the equation determined in that step.

-3(-8 + 9x) = 5x + 2

24 – 27x = 5x + 2              Property: _____________________

-27x + 24 = 5x + 2            Property: _____________________

-32x + 24 = 2                    Property: _____________________

-32x = -22                         Property: _____________________

x = 11/16                           Property: _____________________

8. Tasha was asked to solve the equation -2.75 – 3(2 – m) = 0. You are given the task of explaining how to find the solution for x. Create a list of the properties used to solve the equation, in the correct order, so that you will be able to explain the steps using the correct terminology to Tasha.

9. What is the solution to the linear equation ?

10. What are the values of a, b, and c using the following series of linear equations:

1. When 6a = -7a
2. Then 8(a – b) = -7(1 + b)
3. Then

11. To factor the quadratic equation x² + 7x + 12 = 0, Belinda rewrites the equation as the following.  (x + 3)(x + 4) = 0. Belinda then discovers that she could plug the values x= -3 or x= -4 in for both x values and get a true statement. Which mathematical property does Belinda use to find the solutions to this equation?

12. There are seven basic properties used when solving linear equations. They are: associative, distributive, commutative, addition property of equality, subtraction property of equality, multiplication property of equality, and division property of equality. Write a one-sentence description of each property and give an example of how it could be used.

13. Simplify the following polynomial using the distributive property:

2(5x⁵ + 3(4x⁴ + 4(3x³ + 5(2x² + 6(x)))))

14. The following equation determines how much Jessica is paid in a month, P, based on how many hours she works, h. If she is paid $2,138.75 in the month of October, how many hours did she work during that month?

P = 14.50(h – 1) – 3.625h

15. The following equation was solved incorrectly. Describe the two mistakes that were made and explain why those are common mistakes.

5b + 6 – 3b = -1(b – 2)
8b + 6 = -b – 2
9b + 6 = -2
9b = -8
b = -(8/9)

16. Which of the following equations shows the associative property?

a. 2(3) – 4(5) = 2(3 – 2(5))                  b. 4(-1) = -4                 c.

d. (1 + 2) + 6 = (1 + (2 + 6))               e. 5(0) = 0                    f. (4 – 2) – 3 = 4 – (2 – 3)

17. Match the three properties (associative, commutative, and distributive) with the three situations below:

1. At a party, Josh and Bryan were first standing next to each other while Luke was alone, but later Josh was alone while Luke was talking with Bryan.
2. Samantha, Brielle, and Tina were all at the same party. The host, Nikki, brought them all a piece of pizza.
3. Meanwhile, Dave and Lisa were taking a selfie. Originally Dave was on the left, but because of poor lighting, they decided to switch sides.

18. Create an equation that utilizes the addition, subtraction, multiplication, and division properties of equality and has a final solution of x = -4.

Answer Key/Rubric

1. -28x³ + 44x² – 32x + 8

2. Commutative Property

3. 27

4. a = 4

5. Commutative then Associative Properties.

6. a.  Subtraction Property of Equality
   b.  Multiplication Property of Equality
   c.  Division Property of Equality
   d.  -3(-8 + 9x) = 5x + 2
        24 – 27x = 5x + 2                    Property: Distributive
        -27x + 24 = 5x + 2                  Property: Commutative
        -32x + 24 = 2                          Property: Subtraction Property of Equality
        -32x = -22                               Property: Subtraction Property of Equality
        x = 11/16                                 Property: Division Property of Equality
   e.  Distribution, Combine Like Terms, Addition Property of Equality, Division Property of Equality
   f.  b = 25
   g.  a = 0, b = 7, c = 18

7. Multiplicative Property of Zero

8. Various Answers
   1. Associative: The grouping symbols can be rearranged when performing addition or multiplication.  (ab)c = a(bc)
   2. Distributive: When multiplying the sum of two numbers you can multiply by each number then add or add the numbers then multiply. a(b + c) = ab + ac
   3. Commutative: Order does not matter when adding or multiplying numbers abc = bca
   4. Addition property of equality: When starting with an equation you can add equal amounts to both sides and the equation will remain equal. If a = b, then a + x = b + x
   5. Subtraction property of equality: When starting with an equation you can subtract equal amounts to both sides and the equation will remain equal. If a = b, then a – x  = b – x
   6. Multiplication property of equality: When starting with an equation you can multiply both sides of the equation by the same number and the equation will remain equal.
   7. Division property of equality: When starting with an equation you can divide by equal amounts on both sides and the equation will remain equal.

9. 10x⁵ + 24x⁴ + 72x³ + 240x² + 720x

10. h = 198

11. First, when combining like terms on the left side of the equals sign, 5b and -3b should equal 2b, not 8b. This is a common mistake because the student tried to add 3b to both sides of the equations although there is not an equals sign in between the 5b and -3b terms. Second, when distributing on the right side of the equation -1 times (b – 2) should equal –b + 2. This is a common mistake because students may forget to include the negative sign when distributing and recall the multiplying two negatives should become a positive.

12. b, c, and d

13. 
    
    1. Associative Property
    2. Distributive Property
    3. Commutative Property

14. Various answers including: