# 3.8: Properties of Operations (Part 1)

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So far, you have seen a couple of different *models* for the operations: addition, subtraction, multiplication, and division. But we haven’t talked much about the operations themselves — how they relate to each other, what properties they have that make computing easier, and how some special numbers behave. There’s lots to think about!

The goal in this section is to use the models to understand why the operations behave according to the rules you learned back in elementary school. We’re going to keep asking ourselves “Why does it work this way?”

Think / Pair / Share

Each of these models lends itself to thinking about the operation in a slightly different way. Before we really dig in to thinking about the operations, discuss with a partner:

- Of the models we discussed so far, do you prefer one of them?
- How well do the models we discussed match up with how you usually think about whole numbers and their operations?
- Which models are useful for computing? Why?
- Which models do you think will be useful for explaining how the operations work? Why?

## Connections Between the Operations

We defined addition as combining two quantities and subtraction as “taking away.” But in fact, these two operations are intimately tied together. These two questions are exactly the same:

\[27 - 13 = \_\_\_\_ \qquad \qquad 27 = 13 + \_\_\_\_\_\_ \ldotp\]

More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you choose for a, b, and c!)

\[c - b = a \qquad \qquad c = a + b \ldotp\]

In other words, we can think of every subtraction problem as a “missing addend” addition problem. Try it out!

Problem 11

Here is a strange addition table. Use it to solve the following problems. Justify your answers. Important: Don’t try to assign numbers to A, B, and C. Solve the problems just using what you know about the operations!

+ | A | B | C |
---|---|---|---|

A | C | A | B |

B | A | B | C |

C | B | C | A |

\[A + C \qquad B + C \qquad A - C \qquad C - A \qquad A - A \qquad B - C\]

Think / Pair / Share

How does an addition table help you solve subtraction problems?

We defined multiplication as repeated addition and division as forming groups of equal size. But in fact, these two operations are also tied together. These two questions are exactly the same:

\[27 : 3 = \_\_\_\_\_ \qquad \qquad 27 = \_\_\_\_\_ \times 3 \ldotp\]

More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you choose for a, b, and c!)

\[c : b = a \qquad \qquad c = a \cdot b \ldotp\]

In other words, we can think of every division problem as a “missing factor” multiplication problem. Try it out!

Problem 12

Rewrite each of these division questions as a “missing factor” multiplication question. Which ones can you solve and which can you not solve? Explain your answers.

\[9 : 3 \qquad 100 : 25 \qquad 0 : 3 \qquad 9 : 0 \qquad 0 : 0\]

Problem 13

Here’s a multiplication table.

\(\times\) | A | B | C | D | E |
---|---|---|---|---|---|

A | A | A | A | A | A |

B | A | B | C | D | E |

C | A | C | E | B | D |

D | A | D | B | E | C |

E | A | E | D | C | B |

- Use the table to solve the problems below. Justify your answers. Important: Don’t try to assign numbers to the letters. Solve the problems just using what you know about the operations! $$C \times D \qquad C \times A \qquad A \times A \qquad C : D \qquad D : C \qquad D : E$$
- Can you use the table to solve these problems? Explain your answers. Recall that \(x^{n}\) means copies of multiplied together, \(x \cdot x \cdot x \cdot \ldots \cdot x\) $$D^{2} \qquad C^{3} \qquad A : C \qquad A : D \qquad D : A \qquad A : A$$

Think / Pair / Share

How does a multiplication table help you solve division (and exponentiation) problems?

Throughout this course, our focus is on explanation and justification. As teachers, you need to know what is true in mathematics, but you also need to know *why* it is true. And you will need lots of ways to explain *why*, since different explanations will make sense to different students.

Think / Pair / Share

**Arithmetic Fact:** *a + b = c* and *c – b = a* are the same mathematical fact.

Why is this *not* a good explanation?

- “I can check that this is true! For example, 2+3 = 5 and 5 – 3 = 2. And 3 + 7 = 10 and 10 – 7 = 3. It works for whatever numbers you try.”

*Addition and Subtraction: Explanation 1*

###### Arithmetic Fact:

*a + b = c* and *c – b = a* are the same mathematical fact.

###### Why It’s True, Explanation 1:

First we’ll use the definition of the operations.

Suppose we know *c – b = a* is true. Subtraction means “take away.” So

\[c - b = a\]

means we start with quantity *c* and take away quantity *b*, and we end up with quantity *a*. Start with this equation, and imagine adding quantity *b* to both sides.

On the left, that mans we started with quantity *c*, took away *b*things, and then put those *b* things right back! Since we took away some quantity and then added back the exact same quantity, there’s no overall change. We’re left with quantity *c*.

On the right, we would be combining (adding) quantity *a* with quantity *b*. So we end up with: c = a + b.

On the other hand, suppose we know the equation *a + b = c* is true. Imagine taking away (subtracting) quantity *b* from both sides of this equation: a + b = c.

On the left, we started with *a* things and combined that with *b *things, but then we immediately take away those *b* things. So we’re left with just our original quantity of *a*.

On the right, we start with quantity *c* and take away *b* things. That’s the very definition of *c – b*. So we have the equation:

\[a = c - b \ldotp\]

###### Why It’s True, Explanation 2:

Let’s use the measurement model to come up with another explanation.

The equation *a + b = c* means Zed starts at 0, walks forward *a* steps, and then walks forward *b* steps, and he ends at *c*.

If Zed wants to compute *c – b*, he starts at 0, walks forward *c* steps, and then walks backwards *b* steps. But we know that to walk forward *c* steps, he can first walk forward *a* steps and then walk forward *b* steps. So Zed can compute *c – b* this way:

- Start at 0.
- Walk forward
*a*steps. - Walk forward
*b*steps. (Now at*c*, since*a + b = c*.) - Walk backwards
*b*steps.

The last two sets of steps cancel each other out, so Zed lands back at *a*. That means *c – b = a*.

On the other hand, the equation *c – b = a* means that Zed starts at 0, walks forward *c* steps, then walks backwards *b* steps, and he ends up at *a*.

If Zed wants to compute *a + b*, he starts at 0, walks forward *a* steps, and then walks forwards *b* additional steps. But we know that to walk forward *a* steps, he can first walk forward *c *steps and then walk backwards *b* steps. So Zed can compute *a + b* this way:

- Start at 0.
- Walk forward
*c*steps. - Walk backwards
*b*steps. (Now at*a*, since*c – b = a*.) - Walk forward
*b*steps.

The last two sets of steps cancel each other out, so Zed lands back at *c*. That means *a + b = c*.

Think / Pair / Share

- Read over the two explanations in the example above. Do you think either one is more clear than the other?
- Come up with your own
**explanation**(not examples!) to explain: $$c : b = a \quad \text{is the same fact as} \quad c = a \times b \ldotp$$

## Properties of Addition and Subtraction

You probably know several properties of addition, but you may never have stopped to wonder: *Why is that true?! *Now’s your chance! In this section, you’ll use the definition of the operations of addition and subtraction and the models you’ve learned to explain why these properties are always true.

Here are the three properties you’ll think about:

- Addition of whole numbers is
*commutative*. - Addition of whole numbers is
*associative*. - The number 0 is an
*identity*for addition of whole numbers.

For each of the properties, we don’t want to confuse these three ideas:

- what the property is called and what it means (the definition),
- some examples that
*demonstrate*the property, and - an explanation for
*why*the property holds.

Notice that *examples* and *explanations* are not the same! It’s also very important not to confuse the *definition* of a property with the *reason* it is true!

These properties are all universal statements — statements of the form “for all,” “every time,” “always,” etc. That means that to show they are true, you either have to check every case or find a *reason why* it must be so.

Since there are infinitely many whole numbers, it’s impossible to check every case. You’d never finish! Our only hope is to look for *general **explanations*. We’ll work out the explanation for the first of these facts, and you will work on the others.

### ADDITION IS COMMUTATIVE

Example: Commutative Law

###### Property:

Addition of whole numbers is commutative.

###### What it Means (words):

When I add two whole numbers, the order I add them doesn’t affect the sum.

###### What it Means (symbols):

For any two whole numbers *a* and *b*,

\[a + b = b + a \ldotp\]

Now we need a *justification*. *Why* is addition of whole numbers commutative?

###### Why It’s True, Explanation 1:

Let’s think about addition as combining two quantities of dots.

- To add
*a + b*, we take*a*dots and*b*dots, and we combine them in a box. To keep things straight, lets imagine the*a*dots are colored red and the*b*dots are colored blue. So in the box we have*a*red dots,*b*blue dots and*a + b*total dots. - To add
*b + a*, let’s take*b*blue dots and*a*red dots, and put them all together in a box. We have*b*blue dots,*a*red dots and*b +**a*total dots. - But the total number of dots are the same in the two boxes! How do we know that? Well, there are
*a*red dots in each box, so we can match them up. There are*b*blue dots in each box, so we can match them up. That’s it! If we can match up the dots one-for-one, there must be the same number of them!

• That means *a + b = b + a*.

###### Why It’s True, Explanation 2:

We can also use the measurement model to explain why *a + b = b + a* no matter what numbers we choose for *a* and *b*. Imagine taking a segment of length *a* and combining it linearly with a segment of length *b*. That’s how we get a length of *a + b*.

But if we just rotate that segment so it’s upside down, we see that we have a segment of length *b* combined with a segment of length *a*, which makes a length of *b + a*.

But of course it’s the same segment! We just turned it upside down! So the lengths must be the same. That is, *a + b = b + a*.

### ADDITION IS ASSOCIATIVE

Your turn! You’ll answer the question, “Why is addition of whole numbers associative?”

**Property:** Addition of whole numbers is associative.

**What it Means (words):** When I add three whole numbers in a given order, the way I group them (to add two at a time) doesn’t affect the sum.

**What it Means (symbols):** For any three whole numbers *a*, *b*, and *c*, $$(a + b) + c = a + (b + c) \ldotp\]

Problem 14

- Come up with at least three
*examples*to demonstrate associativity of addition. - Use our models of addition to come up with an
*explanation*. Why does associativity hold in*every case*?**Note:**your explanation should not use specific numbers. It is not an example!

### 0 IS AN IDENTITY FOR ADDITION

**Property: **The number 0 is an *identity* for addition of whole numbers.

**What it Means (words):** When I add any whole number to 0 (in either order), the sum is the very same whole number I added to 0.

**What it Means (symbols):** For any whole numbers *n*, $$n + 0 = n \quad \text{and} \quad 0 + n = n \ldotp\]

Problem 15

- Come up with at least three
*examples*to demonstrate that 0 is an identity for addition. - Use our models of addition to come up with an
*explanation*. Why does this property of 0 hold in*every possible case*?

### PROPERTIES OF SUBTRACTION

Since addition and subtraction are so closely linked, it’s natural to wonder if subtraction has some of the same properties as addition, like commutativity and associativity.

Example: Is subtraction commutative?

Justin asked if the operation of subtraction is commutative. That would mean that the difference of two whole numbers doesn’t depend on the order in which you subtract them.

In symbols: *for every choice* of whole numbers *a* and *b* we would have *a – b = b – a*.

Jared says that subtraction is *not* commutative since 4 – 3 = 1, but 3 – 4 ≠ 1. (In fact, 3 – 4 = -1.)

Since the statement “subtraction is commutative” is a *universal statement*, one counterexample is enough to show it’s not true. So Jared’s counterexample lets us say with confidence:

Subtraction is **not** commutative.

Think / Pair / Share

Can you find any examples of whole numbers *a* and *b* where *a – b = b – a* is true? Explain your answer.

Problem 16

Lyle asked if the operation of subtraction is associative.

- State what it would mean for subtraction to be associative. You should use words and symbols.
- What would you say to Lyle? Decide if subtraction is associative or not. Carefully explain how you made your decision and
*how you know you’re right*.

Problem 17

Jess asked if the number 0 is an identity for subtraction.

- State what it would mean for 0 to be an identity for subtraction. You should use words and symbols.
- What would you say to Jess? Decide if 0 is an identity for subtraction or not. Carefully explain how you made your decision and
*how you know you’re right*