## 2.1: A Division Expression

Here is an expression: $20\div 4$.

What are some ways to think about this expression? Describe at least two meanings you think it could have.

Let’s explore ways to think about division.

Here is an expression: $20\div 4$.

What are some ways to think about this expression? Describe at least two meanings you think it could have.

A baker has 12 pounds of almonds. She puts them in bags, so that each bag has the same weight.

- Clare and Tyler drew diagrams and wrote equations to show how they were thinking about $12 \div 6$.
How do you think Clare and Tyler thought about $12 \div 6$? Explain what each diagram and each part of each equation (especially the missing number) might mean in the context of the bags of almonds.

Pause here for a class discussion. -
Explain what each division expression could mean in the context of the bags of almonds. Then draw a diagram and write a multiplication equation to show how you are thinking about the expression.

- $12 \div 4$
- $12 \div 2$
- $12 \div \frac12$

A loaf of bread is cut into slices.

- If each slice is $\frac12$ of a loaf, how many slices are there?
- If each slice is $\frac15$ of a loaf, how many slices are there?
- What happens to the number of slices as each slice gets smaller?
- Interpret the meaning of dividing by 0 in the context of slicing bread.

Suppose 24 bagels are being distributed into boxes. The expression $24 \div 3$ could be understood in two ways:

- 24 bagels are distributed equally into 3 boxes, as represented by this diagram:
- 24 bagels are distributed into boxes, 3 bagels in each box, as represented by this diagram:

In both interpretations, the quotient is the same ($24 \div 3 = 8$), but it has different meanings in each case. In the first case, the 8 represents the number of bagels in each of the 3 boxes. In the second, it represents the number of boxes that were formed with 3 bagels in each box.

These two ways of seeing division are related to how 3, 8, and 24 are related in a multiplication. Both $3 \boldcdot 8$ and $8 \boldcdot 3$ equal 24.

- $3 \boldcdot 8 =24$ can be read as “3 groups of 8 make 24.”
- $8 \boldcdot 3 = 24$ can be read as “8 groups of 3 make 24.”

If 3 and 24 are the only numbers given, the multiplication equations would be: $$3 \boldcdot {?} =24$$ $${?} \boldcdot 3 =24$$

In both cases, the division $24 \div 3$ can be used to find the value of the “?” But now we see that it can be interpreted in more than one way, because the “?” can refer to *the size of a group* (as in “3 groups of what number make 24?”), or to *the number of groups* (as in “How many groups of 3 make 24?”).