1.1 Identifying Properties

Learning Objectives:

You will learn to identify the commutative, associative, inverse, and identity properties of addition and multiplication for rational numbers.

Introduction

The Shark Dive

Lemon Sharks and Divers by Willy Volk / CC-BY-NC-SA

The next day, Hector decided to take the day off from diving and play volleyball on the beach. His Dad decided to do a deep dive to hopefully see some sharks. Hector would love to do a deep dive, but he isn’t old enough yet, so this gave his Dad a chance to dive on his own.

When Hector’s Dad returned, he told Hector the story of how he went down to a depth of 80 feet with hopes of seeing a shark. After ten minutes or so, he spotted a beautiful shark swimming above him. Hector’s Dad went up about 20 feet to try to take a picture of the shark.

He did get a few good shots before the shark swam away.

“What depth did you see the shark at?” Hector asked.

Do you know? To figure this out, you will need to subtract integers. Subtracting integers is the focus of this lesson. By the end of it, you will know how to figure out at what depth Hector's dad saw the shark.

Guided Learning

In this lesson we will learn how to identify properties with rational numbers.

Next, let's review some properties of numbers. You may recall these properties from the work you have done with whole numbers. In this section, we will see how these properties can help us compute with rational numbers, too.

Here are the properties that we will be using in this section.

The Commutative Property of Addition states that numbers being added can be added in any order. The Commutative Property of Multiplication states that numbers being multiplied can be multiplied in any order.

Example A

$0.3+ 7.5 & = 7.5+0.3\\\frac{1}{2} \times (-3) & = -3 \times \frac{1}{2}$

The Associative Property of Addition states that the grouping of numbers that are being added does not matter. The Associative Property of Multiplication states that the grouping of numbers being multiplied does not matter.

Example B

$\left ( \frac{3}{10} + \frac{11}{5} \right ) + \frac{1}{5} & = \frac{3}{10} + \left ( \frac{11}{5} + \frac{1}{5} \right )\\(-3 \times 4) \times 10 & = -3 \times (4 \times 10)$

The Inverse Property of Addition states that when a number is added to its opposite (or additive inverse), the sum is zero.

Example C

4+(-4)=0

The Inverse Property of Multiplication states that when a number is multiplied by its reciprocal (or multiplicative inverse), the product is 1. You can find the reciprocal of a fraction by flipping it. For example, the reciprocal of $\frac{7}{5}$ can be found by flipping the fraction to get its reciprocal, $\frac{5}{7}$ .

Example D

$\frac{7}{5} \cdot \frac{5}{7} = 1$

The Identity Property of Addition states that when zero is added to any number, the sum is that number.

Example E

$3 \frac{1}{25} + 0 = 3 \frac{1}{25}$

The Identity Property of Multiplication states that when a number is multiplied by 1, the product is that number.

Example F

$0.16 \times 1 = 0.16$

Try a few on your own:

Practice the problems below and then check them with a partner:

Identify the number property that each equation illustrates.

a. -159 + 0 = -159

b. (0.3+1.2)+0.8=0.3+(1.2+0.8)

c. $8 \times \frac{3}{4} = \frac{3}{4} \times 8$

d. $6 \cdot \frac{1}{6}=1$

Check your answers with a peer.

Consider the equation in a.

In -159+0=-159 , a negative integer is being added to zero and the sum is equal to the negative integer.

This is an example of the Identity Property of Addition.

Consider the equation in b.

In (0.3+1.2)+0.8=0.3+(1.2+0.8) , the parentheses show that the sums remain equal even when the numbers are grouped in different ways.

This is an example of the Associative Property of Addition.

Consider the equation in c.

In $8 \times \frac{3}{4} = \frac{3}{4} \times 8$ , the order of the two factors being multiplied is reversed. Both sides of the equation result in the same value.

This is an example of the Commutative Property of Multiplication.

Consider the equation in d.

In $6 \cdot \frac{1}{6}=1$ , two numbers that are reciprocals are being multiplied together, resulting in a product of one.

This is an example of the Inverse Property of Multiplication.

Real Life Example Completed

The Shark Dive

Here is the original problem once again. Reread it and underline any important information.

He did get a few good shots before the shark swam away.

“What depth did you see the shark at?” Hector asked.

To find the depth that Hector’s Dad saw the shark, we need to write a subtraction problem and solve it. Remember that depth has to do with below the surface, so we use negative integers to represent different depths.

-80 was his starting depth, then he went up -20 so we take away 20 feet.

$-80 - (-20) = -60 \ feet$