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sums will set you freehow to teach your child numbers arithmetic mathematicsdivisionbeta release |
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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÷2=7 (or 14/2=7).
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 reminder, so two goes into seven three times reminder one, or 7÷2=3 r.1.
Some examples of division in the everyday world. and now for something completely different - a piece of cakeA 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.
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packing in sets - convenient numbersMultiple 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.
Help on writing division sums. a story of twelve cans
Twelve is a very useful number, because it divides several ways.
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. The full version with more detailed instructions, go to the introduction page. So, to practise doing divisions, for example 12 ÷ 3,
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.
The counter will count down (decreasing): 73,
64, 55, 46,
37, 28, 19, 10, 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 notesHelp 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.
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| moneybookers information | e-gold information | fiat money and inflation | |
| calculating moving averages | the arithmetic of fractional banking | ||
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