Division

Have you ever shared something fairly with friends? Or have you ever put your toys away into equal groups? When you do this, you are using division! Division is a powerful tool for sharing and grouping equally.

What is Division?

Discover

Hugo, Aisha, and Louis have 6 apples. They want to share them equally. How can they do it so that it's fair?

Answer

We can give one apple to each friend until all the apples are gone. Each friend gets $\textcolor{colorprop}{2}$ apples. This process of fair sharing is what we call division.

Definition Division

Division is the process of splitting a total amount into equal groups. We use the division sign ($\div$) to write a division sentence.
We can show "six divided by three equals two" in many ways:

With Numbers: $$\textcolor{olive}{6}\div \textcolor{colordef}{3}=\textcolor{colorprop}{2}$$
In Groups$\textcolor{olive}{6}$ shared into $\textcolor{colordef}{3}$ equal groups is $\textcolor{colorprop}{2}$ in each group
With Cubes:
With a Part-Whole Model:

Two Kinds of Division Questions

Method Question 1: How many in each group?

Sometimes you know the $\textcolor{olive}{\text{total}}$ and the $\textcolor{colordef}{\text{number of groups}}$. Division helps you find the $\textcolor{colorprop}{\text{number of items in each group}}$.$$\begin{aligned}[t]\textcolor{olive}{\text{Total}} \div \textcolor{colordef}{\text{Number of groups}} &= \textcolor{colorprop}{\text{Number in each group}}\end{aligned}$$For example, we have $\textcolor{olive}{12}$ apples to share equally among $\textcolor{colordef}{4}$ friends.

Each friend gets $\textcolor{olive}{12} \div \textcolor{colordef}{4} = \textcolor{colorprop}{3}$ apples.

Method Question 2: How many groups can be made?

Other times, you know the $\textcolor{olive}{\text{total}}$ and the $\textcolor{colorprop}{\text{number of items for each group}}$. Division helps you find the $\textcolor{colordef}{\text{number of groups}}$ you can make.$$\begin{aligned}[t]\textcolor{olive}{\text{Total}} \div \textcolor{colorprop}{\text{Number in each group}} &= \textcolor{colordef}{\text{Number of groups}}\end{aligned}$$For example, we have $\textcolor{olive}{15}$ apples and we put them into boxes of $\textcolor{colorprop}{5}$.

We can make $\textcolor{olive}{15} \div \textcolor{colorprop}{5} = \textcolor{colordef}{3}$ boxes.

The Multiplication and Division Connection

Discover

This array of cubes shows how multiplication and division are connected.

Multiplication builds the total:
3 columns of 2 cubes each make 6 cubes in total. $$\textcolor{colordef}{3} \times \textcolor{colorprop}{2} = \textcolor{olive}{6}$$
Division finds the group size:
6 cubes shared among 3 columns give 2 cubes per column. $$\textcolor{olive}{6} \div \textcolor{colordef}{3} = \textcolor{colorprop}{2}$$

Proposition The Multiplication and Division Connection

Multiplication and division are inverse operations. This means they are partners that "undo" each other.

The fact family for $\textcolor{colordef}{3}$, $\textcolor{colorprop}{2}$, and $\textcolor{olive}{6}$ is:$$\begin{aligned}[t]\textcolor{colordef}{3} \times \textcolor{colorprop}{2} &= \textcolor{olive}{6} \\ \textcolor{colorprop}{2} \times \textcolor{colordef}{3} &= \textcolor{olive}{6}\end{aligned}\qquad \qquad\begin{aligned}[t]\textcolor{olive}{6} \div \textcolor{colordef}{3} &= \textcolor{colorprop}{2} \\ \textcolor{olive}{6} \div \textcolor{colorprop}{2} &= \textcolor{colordef}{3}\end{aligned}$$