Name:_________________________ Class: IM8 Block:___________

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Investigation 1: Mathematical Properties and Order of Operations Mathematical Properties

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Practice Level 1: Write the name of the property shown in the examples below.

1. 4 + 5 = 5 + 4 __________________________

2. 4 + (2 + 8) = (4 + 2) + 8 __________________________

3. 10 (4 + 7) = 10 (4) + 10 (7) __________________________

4. (2 x 3) x 4 = 2 x (3 x 4) __________________________

5. p • q = q • p __________________________

6. x (y + z) = xy + xz __________________________

Practice Level 2: Tell whether each statement is true or false. For each true statement, list the property being represented.

1. 23 – 12 = 12 – 23

2. 4 x 10 = 10 x 4

3. 1 ÷ 2 = 2 ÷ 1

4. 13 + 27 + 8 = 13 + 8 + 27

5. 3 + 5 = 5 + 3

6. (7 + 4) + 3 = 7 + (4 + 3)

7. 4(w – 5) = 4w – 5

8. 9 ÷ 3 = 3 ÷ 9

9. (8 – 3) – 1 = 8 – (3 – 1)

10. 1 x 16 x 2 = 1 x 2 x 16

11. 8 + (3 + 1) = (8 + 3) + 1

12. xy = yx

13. 3(x + 2) = 3x + 6

14. 3 – 5 = 5 – 3

Practice Level 3: Match each example with the property it is illustrating. 1. Which property of real numbers is illustrated by the equation 52+(27 + 36) = (52 + 27) + 36? a) commutative property c) distributive property b) associative property d) identity property of addition 2. Which property is illustrated by the equation 4(2 – 1) = 8 – 4 a) commutative property c) distributive property b) associative property d) identity property of addition 3. Which property is illustrated by the equation 6 + (4 + x) = 6 + (x + 4)? a) associative property of addition c) associative property of multiplication b) distributive property d) commutative property of addition

4. Which property is represented by the statement

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12(6a + 4b) = 3a + 2b?

a) commutative property c) associative property b) distributive property d) identity property 5. If M and A represent integers, M + A = A + M is an example of which property? a) commutative property c) distributive property b) associative property d) closure 6. While solving the equation 4(x + 2) = 28, Becca wrote 4x + 8 = 28. Which property did she use? a) distributive property c) commutative property b) associative property d) identity property 7. Tori computes the value of 8 x 95 in her head by thinking 8 x 90 + 8 x 5. Which property is she using? a) commutative property c) associative property b) distributive property d) closure

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“Like terms” are terms that have the same variable and the same exponent. Like terms can be added or subtracted, unlike terms cannot.

• Identify the like terms for the expression below: 9a + 4b + 4 + 4a + 5b + 7

When you are being asked to simplify an expression, you are combining like terms. • Simplify the expression above.

Let’s try some: Level 1: Simplify each expression.

1) 5x + 6x

2) 2x + 7 + 3x

3) 4x + x + 8

4) 2x + 6x + 9

5) 9y + 6y – 2

6) 25x – x + 2y

7) -2m + 16 – 13m

8) -6k + 7k

9) n – 10 + 9n – 3

10) –r – 12r

11) -8x – 11x

12) 5n + 11n Level 2: Simplify each expression. 1) 4a + 8b + 11a – 10b

2) 13xy + 18 xy – 20 xy

3) 16x + -18y + 10x – 7y

4) 6c – 8ab + 9c – 10

5) 18ab + -6a + -7b +26ab + -7b

6) 5x – 3x + 2xy + 31x – 18xy

7) 8x + y + 4x + 2y

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Level 3: Simplify each expression. 1) 11a + 4b2 + 14b2

2) c2 + 4d2 – d2

3) 3a2 + 6a + 2b2

4) 4x + 3x2 – 3x

5) 7x2 – 7x + 7x2 – 1

6) 5x2 + 7x + 2 – 2x2 + 7 + x2

7) 3d5 + 9d + 15d – 9d5 + 3

8) 3c + 15c + 2d4 – 15d – 7d – 9d – 7d4

9) 17m2 + 4n2 + -4n2 – 17m2 + 9

10) 9 + 2x – 5x + 7x2 + x

Geometry Applications for Combining Like Terms. For #1 – 4, write an expression that represents the sum of the angles of the triangle below in

simplest form. 1) 2)

3) 4)

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A square has side lengths of x centimeters. One dimension of the square is increased by 3 centimeters to create a new rectangle.

I can give the area of the original square in two ways; factored form or length x width, and expanded form, the sum of the individual areas.

Area of Square: Area of Rectangle: Factored form A = x • x Factored Form: Expanded Form A = x2 Expanded Form:

Let’s try it out: A) Each diagram shows a large rectangle divided into two smaller rectangles. Write

two expressions, one in factored form and one in expanded form, for the area of the shaded part. 1) 2) 3) A = A = A = A = A = A =

Practice: Level 1: Use the distributive property to write an equivalent expression.

1. 4(x + 7) = 2. 7(x – 3) =

3. -2(x + 4) = 4. x(x + 9) =

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5. a(a – 1) = 6. 3m(m – 2) =

7. -4(a – 4) = 8. a(a – 12) =

9. -6(a + 8) = 10. 4(1 + 9x) =

11. 6(-5n + 7) = 12. (9m + 10) • -2 =

Level 2: Use the distributive property to write an equivalent expression.

1. x(y – 4)

2. (3y + -6)2x

3. -5y(6z – 10)

4. y(-16 + 2x)

5. -3x(-7 + 8y)

6. 3b(b – 5)

7. 2x2(x + 9)

The Distributive Property, Part 2 A new rectangle is created by increasing one side of a square with sides of length x by 2 centimeters and increasing the other sides by 3 centimeters.

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• Write two expressions for the new rectangle, one in factored form and one in expanded

form, that represents the area of the rectangle. Lets try some more: A) Each rectangle below has been divided into four smaller rectangles. Write two expressions for the area of the shaded rectangle, one in factored form and one in expanded form

Distributive property practice: Level 1: Use the distributive property to write an equivalent expression.

1. 2(x + 6)

2. -5(8 – b)

3. 4(-x + 7)

4. -4(12 – 6b)

5. 3.5(3x – 8)

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Level 2: Use the distributive property to write an equivalent expression.

1. x(y – 4)

2. 2x(3y + -6)

3. -5y(6z – 10)

4. y(-16 + 2x)

Level 3: Use the distributive property to write an equivalent expression.

1. (x + 2) (x + 3)

2. (x + 5) (x + 1)

Putting it all together: Use order of operations, the distributive property, and combining like terms to simplify each expression.

1. 3(a + b) + 2c

2. 6m + 7(7m + 9)

3. 5c + 2c(5 – 6)

4. 8 – 3(6 – 6x)

5. 4y + 6(y + 8)

6. 3(d + 2e) – 11d

7. -3a – 2b(a + 8) – 6ab + 31b

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3 7x −4(x −3)[ ] +1

MCA Practice: 1. Is the equation 3(2x – 4) = -18 equivalent to 6x – 12 = -18?

a. Yes, the equations are equivalent by the Associative Property of Multiplication. b. Yes, the equations are equivalent by the Commutative Property of Multiplication. c. Yes, the equations are equivalent by the Distributive Property of Multiplication. d. No, the equations are not equivalent.

2. Which expression is equivalent to 3x – 3y? a. 3xy b. 3 (x – y) c. 3x – y d. x – 3y 3. Which equation is equivalent to = 16? a. 9x – 2 = 16 b. 9x + 37 = 16 c. 17x – 2 = 16 d. 17 + 13 = 16 4. Which equation is equivalent to 5x – 2(7x + 1) = 14x a. -9x – 2 = 14x b. -9x + 1 = 14x c. -9x + 2 = 14x d. 12x – 1 = 14x 5. What is the perimeter of the figure shown below, which is NOT drawn to scale?