22 Properties of Operations
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 = ____ 27 = 13 + _____.
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 c = a + b.
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 + C B + C A – C C – A A – A 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 = _____ 27 = _____ × 3.
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 c = a · b.
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 ÷ 3 100 ÷ 25 0 ÷ 3 9 ÷ 0 0 ÷ 0
Problem 13
Here’s a multiplication table.
- Use the table to solve the problems below. Justify your answers. Important: Don’t try to assign numbers to the letters.