*How would you clarify the mathematical expression 12 ÷ 3 = 4 to your youngster?*

The idea of division normally brings to thoughts pictures of sharing sweets amongst a bunch of buddies. This can be a easy sufficient approach of giving context to expressions like *12 ÷ 3 = 4* (and let’s be sincere, any excuse to get the sweets out is to be welcomed). The extra hands-on youngsters can get with mathematical objects, the higher outfitted they’ll be to make sense of all these symbols.

However sharing is only one approach to consider division. When educating division to your youngster, it’s additionally useful to produce other representations at hand.

### Find out how to educate division as sharing and grouping

In class, youngsters are normally taught division by way of sharing and grouping. They’ll be requested to ‘share an quantity equally between’ or to ‘group an quantity into equal units’. There’s a delicate distinction.

Let’s return to *12 ÷ 3 = 4*.

**Sharing**

Pete has 12 grapes.

He shares them between 3 of his buddies. What number of grapes does every buddy get? After sharing out the grapes one by one into three piles, Pete will find yourself with this:

We are saying 12 grapes shared between 3 individuals to present 4 grapes every. The reply, on this case, is the worth of every equal share (portion dimension). Let’s spell this out a bit extra by way of the terminology round division:

The dividend is the factor to be divided (12 grapes), the divisor is the variety of teams (3 buddies). The quotient is the variety of gadgets in every group (portion dimension), which is 4.

Now let’s take a look at the identical calculation by way of grouping.

**Grouping**

Pete has 12 grapes.

He desires to place them into luggage of three. What number of luggage will he have? Pete fills up the baggage one by one, ending up with:

This time, we are saying that when 12 grapes are put into luggage (teams), we find yourself with 4 luggage, every containing 3 grapes. Notice that the reply right here shouldn’t be the variety of grapes in a bunch, however the variety of luggage.

The dividend is identical as earlier than (12 grapes) however now the divisor is the variety of gadgets in every group, 3, and the quotient is the variety of teams/luggage, 4.

The grouping technique of division exhibits clearly that division is the inverse (reverse) of multiplication. The expression *12 ÷ 3 = 4* holds true as a result of *4 x 3 = 12* (and vice versa). We are able to actually see 4 luggage every containing 3 grapes when grouping. Put one other approach, after we clear up the baggage of grapes drawback, we’re asking *what number of 3s are there in 12?* This is identical as asking *what do I multiply 3 by to get 12?*

### Find out how to educate division (and multiplication) with arrays

Arrays supply a strong approach of visualising the hyperlink between multiplication and division. They make no distinction between grouping and sharing. An array merely arranges objects in columns and rows and by doing so, they provide not less than 6 interpretations of multiplication and division:

Saying **‘3 teams of 4’** is represented by the calculation **4 x 3 = 12**

Saying **‘4 teams of three’** is represented by the calculation **3 x 4 = 12**

Saying **’12 splits into 3 equal shares of 4′** is represented by the calculation **12 ÷ 3 = 4**

Saying **’12 splits into 4 equal shares of three′** is represented by the calculation **12 ÷ 4 = 3**

Saying **’12 splits into 4 teams of three′** is represented by the calculation **12 ÷ 3 = 4**

Saying **’12 splits into 3 teams of 4′** is represented by the calculation** 12 ÷ 4 = 3**

### Why does all this matter?

A cynic may marvel why we hassle with all these representations, when absolutely the one factor that issues is getting the reply.

To assist youngsters develop as mathematical thinkers we have to dig into the buildings that underlie all of the calculations they’re anticipated to do. Fairly than simply rattling off *12 ÷ 3 = 4* from reminiscence, it helps to connect such calculations to concrete representations as a result of that’s how these calculations might be encountered in the true world. Grouping and sharing are each legitimate methods of eager about the identical calculation. Arrays prolong our pondering even additional. As we’ve seen, by finding out these representations we carry out necessary connections – just like the hyperlink between division and multiplication. These symbols make much more sense when studied in context.

Language additionally issues: it’s simple to combine up sharing with grouping, however we’ve seen that the reply (or quotient) represents a special amount in every case. Mathematical calculations are very exact, so our alternative of phrases have to be too.

With a wealthy repository of representations and exact language at hand, your youngster can have the foundations to interact in mathematical dialogue and discover patterns and connections amongst all these calculations.