This handy **Spectrum Math Grade 7 Answer Key**** Chapter 3 Lesson 3.1 Mathematical Properties & Equivalent Expressions** provides detailed answers for the workbook questions.

## Spectrum Math Grade 7 Chapter 3 Lesson 3.1 Mathematical Properties & Equivalent Expressions Answers Key

**Commutative Property:** The order in which numbers are added does not change the sum. The order in which numbers are multiplied does not change the product.

a + b = b + a

a × b = b × a

**Associative Property:** The grouping of addends does not change the sum. The grouping of factors does not change the product.

a + (b + c) = (a + b) + c

a × (b × c) = (a × b) × c

**Identity Property:** The sum of an addend and 0 is the addend. The product of a factor and 1 is the factor.

a + 0 = a

a × 1 = a

**Properties of Zero:** The product of a factor and 0 is 0. The quotient of the dividend 0 and any divisor is 0.

a × 0 = 0

0 ÷ a = 0

**Distributive Property:** If two addends or the minuend and subtrahend in an equation are being multiplied by the same factor, the equation can be rewritten by factoring out the common factor.

a × (b + c) = (a × b) + (a × c)

a × (b – c) = (a × b) – (a × c)

**Rewrite each expression using the property indicated.**

Question 1.

a.

associative: (7 + 6) + y =

___________

Answer: 7 + (6 + y)

associative: (7 + 6) + y = 7 + (6 + y)

According to the associative principle of addition, when adding three integers, the outcome will always be the same regardless of how the numbers are grouped. If there are three numbers, x, y and z, the associative property of addition implies that x + (y + z) = (x + y) + z. The grouping of addends does not change the sum.

b. identity: 724 + 0 =

___________

Answer: 724

identity: 724 + 0 = 724

An identity in mathematics is a number, n, that results in the same number, n, when other numbers are added to it. The identity of the additive is always zero. This brings us to the identity property of addition, which simply states that when you add zero to any number, it equals the number itself.

Question 2.

a. commutative: z × 8 =

___________

Answer: 8 × z

commutative: z × 8 = 8 × z

According to the commutative property of multiplication, changing the order of the numbers we are multiplying does not change the product. If there are two numbers, x and y, the commutative property of multiplication implies that x × y = y × x.

b. zero: 61 × 0 =

___________

Answer: 0

zero: 61 × 0 = 0

According to the zero property of multiplication, if a number is multiplied by 0 (zero), the result will be zero. This property is applied when numbers are multiplied by 0. If there is a number, x then the identity property implies that x × 0 = 0.

Question 3.

a. distributive: 6 × (a + b) =

___________

Answer: (6 × a) + (6 × b)

distributive: 6 × (a + b) = (6 × a) + (6 × b)

The distributive property states that multiplying the sum of two or more addends by a number yields the same outcome as multiplying each addend separately by the number and combining the resulting products.

b. zero: 0 ÷ 5 =

___________

Answer: 0

zero: 0 ÷ 5 = 0

According to the zero property of division, if 0(zero) is divided by any other number, the result will be zero. If there is a number, x then the zero property of division implies that 0 ÷ x = 0.

Question 4.

a. commutative: 7 + y =

___________

Answer: y + 7

commutative: 7 + y = y + 7

According to the commutative property of addition, the sum is unaffected by changes in the order of the numbers being added. The commutative property of addition can be defined as the fact that adding two integers in any sequence results in the same result. Therefore, the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

b. associative: 5 × (6 × 3) =

___________

Answer: ( 5 × 6) × 3

associative: 5 × (6 × 3) = ( 5 × 6) × 3

According to the associative principle of multiplication, when multiplying three integers, the outcome will always be the same regardless of how the numbers are grouped. If there are three numbers, x, y and z, the associative property of multiplication implies that x × (y × z) = (x × y) × z. The grouping of factors does not change the product.

Question 5.

a. identity: 45 × 1 =

___________

Answer: 45

identity: 45 × 1 = 45

According to the identity property of multiplication, if a number is multiplied by 1 (one), the result will be the original number. This property is applied when numbers are multiplied by 1. If there is a number, x then the identity property implies that x × 1 = x.

b. distributive: (7 × 3) + (7 × 7) =

___________

Answer: 7 × (3 + 7)

distributive: (7 × 3) + (7 × 7) = 7 × (3 + 7)

The distributive property states that multiplying the sum of two or more addends by a number yields the same outcome as multiplying each addend separately by the number and combining the resulting products.

Use phrases to help you understand which operations to use in word problems.

Addition Phrases

more than

the sum of

Subtraction Phrases

less than

decreased by

Multiplication Phrases

the product of

times

Division Phrases

the quotient of

divided by

**Write each phrase as an expression or equation.**

Question 1.

a. three increased by d

___________________

Answer: 3 + d

The expression for ‘three increased by d’ can be given as 3 + d

b. the product of eight and w

___________________

Answer: 8 × w

The expression for ‘the product of eight and w’ can be given as 8 × w

Question 2.

a. seven less than 12

___________________

Answer: 7 < 12

The expression for ‘seven less than 12’ can be given as 7 < 12

b. two more than a number is nine

___________________

Answer: 2 + x = 9

The equation for ‘two more than a number is nine’ can be given as 2 + x = 9

Question 3.

a number divided by 6 is 8

___________________

Answer: x ÷ 6 = 8

The equation for ‘a number divided by 6 is 8’ can be given as x ÷ 6 = 8

b. nine more than 15

___________________

Answer: 9 + 15

The expression for ‘nine more than 15’ can be given as 9 + 15

Question 4.

a. the sum of five and six is eleven

___________________

Answer: 5 + 6 =11

The equation for ‘the sum of five and six is eleven’ can be given as 5 + 6 =11

b. the quotient of twelve and s is 4

___________________

Answer: 12 ÷ s = 4

The equation for ‘the quotient of twelve and s is 4’ can be given as 12 ÷ s = 4

Question 5.

a. three less than t is five

___________________

Answer: t – 3 = 5

The equation for ‘three less than t is five’ can be given as t – 3 = 5

b. the product of two and b is 4

___________________

Answer: 2 x b = 4

The equation for ‘the product of two and b is 4’ can be given as 2 x b = 4

Question 6.

a. the product of five and three is y

___________________

Answer: 5 x 3 = y

The equation for ‘the product of five and three is y’ can be given as 5 x 3 = y

b. twenty divided by a number is five

___________________

Answer: 20 ÷ x = 5

The equation for ‘twenty divided by a number is five’ can be given as 20 ÷ x = 5

Question 7.

a. 12 more than 20

___________________

Answer: 12 + 20

The expression for ’12 more than 20′ can be given as 12 + 20

b. the sum of 4 and 11 is 15

___________________

Answer: 4 + 11 = 15

The equation for ‘ the sum of 4 and 11 is 15’ can be given as 4 + 11 = 15

Question 8.

a. the quotient of 30 and f is 3

___________________

Answer: 30 ÷ f = 3

The equation for ‘the quotient of 30 and f is 3’ can be given as 30 ÷ f = 3

b. 7 times b is 63

___________________

Answer: 7 x b = 63

The equation for ‘7 times b is 63’ can be given as 7 x b = 63