site map

sums will set you free

how to teach your child numbers arithmetic mathematics

division

Just as multiplication is constant addition, so division is constant subtraction, or if you prefer division is the reverse of multiplication.

For example, let’s do the division sum of dividing fourteen [ten plus four] into collections of two.

Now start taking away collections of two:

And another two:

And continue...

... until all fourteen blocks are divided into seven groups of two.

So fourteen can have two constantly subtracted seven times, or the shorthand description is ‘fourteen divided by seven equals two’, or 14÷2=7 (or 14/2=7).

Now look at the division sum of dividing fourteen [ten plus four] into collections of seven. A collection of seven is taken away, leaving....

... a collection of seven.

So fourteen can have seven constantly subtracted two times (twice), or the shorthand description is ‘fourteen divided by two equals seven’, or 14÷7=2 (or 14/7=2).

Dividing a collection of objects does not always give such a neat result as in the sum above. What to do when the sum is not so neat? Now look at the division sum of seven divided by two, 7÷2, or subtracting collections of two from seven.

Now start taking away collections of two:

After taking away three collections of two, there is one block left. That ‘surplus’ block is called the remainder, so two goes into seven three times remainder one, or 7÷2=3 r.1.

a very special case - dividing by 0 (nothing)

There come times in maths when we just don’t know what to do. People are often reluctant to say, “I don’t know”. In maths, you will hear such situations being described as ‘undefined’ or ‘meaningless’. Dividing by zero is one of those situations.

Myths sometimes tend to spread, such as dividing by zero results in ‘infinity’, or perhaps that zero divided by zero equals one. But this is not how mathematicians treat this situation. They could have done, but they don’t; a big reason being that such decisions would result in more problems down the road.

When mathematicians hit this sort of road block, they make up a definition, or decision, as to what is to be done if you meet it. In this case, the rule is that no result can be found. Thus, the sum becomes illegitimate Any answer takes the form of, “Can’t be done, sir” or “This sum is an outlaw”. You don’t try to stumble on, you’ve hit a brick wall.

In computing, a marker will be set and a message printed such as, “Computation error”. If you actually find such a sum in your homework or text book, either someone has made a mistake ( it may be you!) or someone trying to trick you. Stay alert, your country needs lerts.

and now for something completely different - a piece of cake

A cake (or, in this case, a French gateau) can be divided evenly into four pieces by cutting it many different ways. Here are three ways to divide it into four.

advertising disclaimer

packing in sets - convenient numbers

Multiple packs of food and drinks are pre-divided into smaller numbers of objects for convenience, carryability and to encourage sales. A supermarket is a good place for the young to gain experience of counting.

Water is often divided into a pack of six bottles.

The pack of water bottles can be seen as divided into threes, or into twos.
Two lots of three are six,
or three lots of two are six.
2x3=6, 3x2=6; 6÷3=2, 6÷2=3.

Help on writing division sums.

a story of twelve cans

A row of twelve tins
Two rows of six tins		Four rows of three tins

Twelve is a very useful number, because it divides several ways.

• Twelve lots of one,
• one lot of twelve,
• six lots of two,
• two lots of six,
• three lots of four,
• four lots of three.

Do you have one pack, or do you have twelve cans?

Further up the page, do you have one cake or, if you cut it, four bits of cake, or is each piece now a cake?

abelard.org maths educational counter

[This counter functions with javascript, you need to ensure that javascript is enabled for the counter to work.]

On this page is a more concise version of the Brilliant educational maths counter. For the full version with more detailed instructions, go to the introduction page.

So, to practise doing divisions, for example 12 ÷ 3,

• Reset Counter Value to 12;
• Change Step to 3;
• Switch Direction to Decreasing;
• Now click on the Manual Step button four times. The red number counts to 4.

The counter counts down (decreasing): 12, 9, 6, 3, 0. So 12 ÷ 3 = 4.

Now help the learner to try other division sums. Each time, click the red Reset button to return Manual Steps (the red number) to zero.

remainder division sums and the abelard.org counter

Now, to do division sums with a remainder, for example 73 ÷ 9.

• Click the red Reset button to return Manual Steps (the red number) to zero.
• Reset Counter Value to 73;
• Change Step to 9;
• Switch Direction (if necessary) to Decreasing;
• Now click on the Manual Step button.

The counter will count down (decreasing): 73, 64, 55, 46, 37, 28, 19, 10, 1.
The red number counts to 8. Now the big counter number shows 1.
If you click again, the big number becomes negative (-8), indicating that more has taken away than was there in the first place.
The red 8 shows how many times 9 goes into or can be taken away from 73.
The big 1 is the remainder - what is left after 9 has been taken away from 73 as many times as possible.
Thus 73 ÷ 9 = 8 r.1.

With this last example, you are coming close to understanding how a computer can do sums.

Remember when trying much bigger sums, such as 873 ÷ 74, that this would mean wearing your finger out trying to clicking the Manual Step button 74 times. So when you reach sums and numbers beyond clickability, now is the time to start introducing a calculator. At first, work using the calculator with small number division sums, and match the results on the abelard.org counter with those on the calculator.

some notes

Help on writing down multiplication sums.

Another set can be seen on a clock, a set of twelve, or in some cases a set of twenty-four.

Again, a calculator, or similar device, is useful for the learner to become familiar with the click-over into higher units, which will be tens [enter 1, then +, then keep hitting = ]; or by winding a clock round twelve times.

A second hand, or a minute hand, can also be thought as clicking over every sixty seconds or minutes.

Computers do their collections in twos (binary). [Note: these collections of two, tens, twelves or sixties are referred to in the jargon as bases. For instance, the normal/usual/common system collections, in the jargon are called sets.]

The size of a convenient collection varies according to purpose. Packing boxes in tens would leave you with long thin boxes (1x1x10), or at best with boxes that are five times as long as they are wide (5x2x1 - five rows of 2, or two rows of 5). Thus packing in dozens (12) improves handling because as well as 1 x12 or 2 x 6, items can be packed as 3 x 4 or 4 x 3.

Several of these notes are to give an idea of where we are going. The learner is not expected to grasp all this at once, but to gradually build up familiarity.

Similar remarks apply to some of the yellow links from these pages, where I become even more technical/detailed/careful. Your job is to assist the learner in such a way that you do not introduce confusions which have to be undone later. The house must be built on realistic foundations, not on jelly or sand or false assumptions.

Our sets/collections are driven by convenience - carrying home a pound or a kilogram of apples or sugar is easier than carrying a hundredweight (112lb, approximately 50kg) - and by conformity - it helps if everyone is using the same measure, including for communication.

end notes

1. Reversing an operation is often loosely referred to as ‘doing the opposite’. For example, the opposite of riding your bicycle from Oxford to London, in clement weather, when you’re fit and fresh, is riding back at night, in pouring rain, after a hard day’s partying.
   
   

email email_abelard [at] abelard.org

© abelard, 2009, 24 July

all rights reserved

the web address for this page is https://www.abelard.org/sums/teaching_number_arithmetic_mathematics_division.php

585 words
prints as 4 A4 pages (on my printer and set-up)