RWM101: Foundations of Real World Math

1a. Apply the commutative law of addition and multiplication

• For which operations does the commutative property apply?
• For which operations does the commutative property NOT apply?

The commutative property states that we can reverse the order of addition or multiplication, without changing the outcome. For example, 2 + 3 = 3 + 2, therefore the commutative property applies to addition. Since 3 × 5 = 5 × 3, the commutative property also applies to multiplication.

The commutative property does NOT apply to subtraction and division, since 3 - 5 ≠ 5 - 3 and 4 ÷ 2 ≠ 2 ÷ 4.

To review, see Commutative Law of Addition and Commutative Law of Multiplication.

1b. Apply the associative law of addition and multiplication

• For which operations does the associative property apply?
• For which operations does the associative property NOT apply?

The associative property states that we can do addition and multiplication in any order or with any grouping of numbers. For example, when adding 4 + 5+ 6, you can add 4 and 6 first, or 5 and 6, or 4 and 5, and you will get the same answer.

This is helpful when we are adding several numbers together, as grouping makes finding the answer easier. For example, when we add 4 + 7 + 16 + 3, it is much easier to add 4 + 16 = 20, then 7 + 3 = 10, and finally 20 + 10 = 30.

Similarly, we can regroup multiplication problems to find the answer more easily. For example, with 5 × 3 × 20 × 6, it is much easier to calculate 5 × 20 = 100, and 3 × 6 = 18, then finish with 100 × 18 = 1800.

The associative property does NOT apply to subtraction or division. For example, 5 - 3 - 2 ≠ 2 - 3 - 5 and 5 ÷ 2 ÷ 3 ≠ 2 ÷ 3 ÷ 5.

To review, see Commutative and Associative Properties.

1c. Apply the identity property of addition and multiplication

• What number is the identity for addition?
• What number is the identity for multiplication?

The identity property is based on the mathematical concept of an identity, which is a specific number that can be added to or multiplied by another number and not change its value. The identity for addition is 0, since 0 added to any number does not change its value. For example, 5 + 0 = 5.

In the same way, there is an identity for multiplication, but it is 1, not 0. The idea is the same, 1 multiplied by any number does not change the value of that number. For example, 5 × 1 = 5.

To review, see Identity Property of Zero and Identity Property of 1.

1d. Apply the inverse property of addition and multiplication

• What does a number added to its inverse always equal?
• What does a number multiplied by its inverse always equal?

The inverse property of addition simply states that a number added to its "opposite" always equals 0. For example, 5 + (-5) = 0. In this case, (-5) would be the additive inverse of 5. The additive inverse is often called the opposite.

The inverse property of multiplication similarly states that a number multiplied by its reciprocal is always 1. If we have a number, a, then its multiplicative inverse is 1/a, also known as the reciprocal. For example, the multiplicative inverse of 2 is ½.

To review, see Inverse Property of Addition and Inverse Property of Multiplication.

1e. Apply the zero property of multiplication and division

• What happens when we multiply a number by 0?
• What happens when we divide a number by 0?

The zero property defines what happens when we multiply or divide by zero. In multiplication, any number multiplied by 0 is always 0. For example, 5 × 0 = 0.

When dividing with 0, we have to consider both dividing 0 by another number, and dividing a number by 0. When dividing 0 by another number, such as 0/5, the answer is always 0, since you can break 0 things into as many groups as you want, and have 0 in each group.

The issue comes when dividing by 0, such as 5/0. In this case, you cannot break 5 things into 0 groups, therefore the answer is undefined. Undefined means it cannot be computed, or has no value we can determine. Note that 0/0 is also undefined.

To review, see Why Dividing by Zero is Undefined and Multiplication by Zero.

1f. Apply the distributive property

• How do you apply the distributive property?

The distributive property shows us that we can multiply a number outside parentheses to every element inside the parentheses, provided that the elements inside the parentheses are connected with addition or subtraction.

For example:

An example involving subtraction:

To review, see Using the Distributive Property.

Unit 1 Vocabulary

This vocabulary list includes terms listed above that students need to know to successfully complete the final exam for the course.

• additive inverse
• associative property
• commutative property
• distributive property
• identity
• identity property
• inverse property
• multiplicative inverse
• reciprocal
• undefined
• zero property