Using Models to Divide with Bar Models and Arrays

The equal groups model is usually the first one students meet. A total number of objects is drawn, then circled into equal-sized groups until nothing is left over. For \(12 \div 3\), 12 counters are separated into 3 circles, and counting the items in one circle shows the answer.

Each of the 3 dashed groups holds exactly 4 dots, so \(12 \div 3 = 4\). This same picture also shows the related multiplication fact, \(3 \times 4 = 12\).

An array arranges the total into equal rows and equal columns, which makes the link between division and multiplication very visible: the number of rows times the number of columns equals the total. Arrays are especially useful once numbers get bigger, because counting rows and columns is faster than counting scattered dots. For a full walkthrough of building and reading arrays, see Dividing with Arrays.

A bar model represents the whole amount as a single rectangle, then splits that rectangle into equal sections. The length of the whole bar stands for the total, and each equal section stands for one group. Bar models are especially helpful for word problems, since the picture makes it clear what the total is and how many equal parts it should be split into.

Since the bar of 12 splits evenly into 3 sections of 4, the bar shows \(12 \div 3 = 4\) just as clearly as the equal groups picture, but it is often faster to draw for bigger numbers because there is no need to draw every single object.

A number line model shows division as repeated subtraction. Starting at the total, equal-sized jumps are taken backward toward zero, and the number of jumps needed is the answer. For \(12 \div 3\), jumping back by 3 from 12 lands on 9, then 6, then 3, then 0, which is exactly 4 jumps, so \(12 \div 3 = 4\). This model connects nicely to skip counting and to the idea of division as "how many groups of 3 fit inside 12".

Every model above can be checked the same way: multiply the group size by the number of groups, and the result should be the original total, \(b \times c = a\). Division is also closely tied to repeated addition, since adding a group size the same number of times as there are groups builds the total right back up. For more on how addition connects to division, see Relating Addition with Division.

Models can represent two slightly different situations. Sharing means the number of groups is already known, and the model finds how many go in each group, like sharing 12 stickers equally among 3 friends. Grouping means the size of each group is already known, and the model finds how many groups can be made, like putting 12 stickers into bags of 4. Both situations use the same division fact, but the models help students see the difference. For more practice telling these apart, check out Sharing & Grouping Up to 999.

A baker has 15 muffins and wants to pack them into boxes that each hold 5 muffins. How many boxes are needed?

Using a bar model, draw one bar for 15 muffins split into equal sections of 5. Counting the sections shows 3 equal parts, so \(15 \div 5 = 3\) boxes are needed. The same answer appears with an equal groups picture (3 circles of 5 muffins each) or a number line (three jumps of 5 back from 15 to 0).

No single model works best for every problem. Equal groups pictures are great for small totals, arrays make the multiplication connection obvious, bar models are quick for word problems with larger numbers, and number lines highlight the repeated subtraction idea. Being comfortable switching between models makes it much easier to move on to writing full division statements and working with remainders in later lessons.

Related lessons

• Math 3