## Tuesday, October 16, 2018

### Bar Modelling For Two Types of Division

Many people aren’t aware that there are two different types of division. Even if you search the web, only one type mostly comes up. It’s the type we learn first: that when we share one quantity, we split it into equal parts. This is ​partitive​ division, or dividing a quantity into parts. When we divide a whole amount, partitive division tells us how many items there are in each group. Let’s look at a couple of examples.

Jeanine bought 24 buns for a party. She wanted to put an equal amount on each of 6 tables.
How many buns should she put on each table?

Jeanine should put 4 buns on each table.

Armando earned \$56 for a day’s work.
If he worked 7 hours, how much was his hourly wage?

Armando’s hourly wage was \$8.

The other type is ​quotative​ division, and it’s less familiar to most people. In quotative division, we find the number of groups, not how many in each group. It is also known as measurement division, or finding how many of a certain unit it will take to measure something. Let’s look at a couple of examples.

When the party planner was setting up, he found he had 32 place cards. He would like to put four on each table. How many tables would he need to set up?

The party planner would need to set up 8 tables.

For a pizza party, a parent has a maximum budget of 6 pizzas. He plans on each child being allowed three slices. This pizza parlour slices each pizza into 8 slices. How many children can be invited to the party?

16 children can be invited to the party.

The above example is as interesting way to model dividing a whole number by a fraction (an upper primary standard in most countries). Although bar modelling may not be as useful as a tool to calculate quotative division problems, it can be very helpful to help students visualize the problem.

Often when teaching such problems, many teachers resort to multiplying by the reciprocal, or 6 × 8/3 (We need to avoid teaching Keep-Change-Flip though, as this 3 mnemonic can lead to lots to misconceptions.).

Did you know that you can also divide using common denominators? You can
change 6 to 48/8  and solve 48/8 ÷ 3/8 . See if you can figure out why this works.