Nodehill Maths

Multiplying and Dividing by 10, 100 and 1000

This is an important skill which children need to master, especially now that we live in a metric, rather than an Imperial measuring world. You will see how important when you look at the section on converting metric units.

Nevertheless I find that many children are not taught how to do this properly at an early age and as such learn bad habits. It is amazing how many children tell me that they have been taught to multiply by 10 by adding a zero onto the end of the number, and two zeros when multiplying by 100 etc. This is fine when dealing with whole numbers, but when we start multiplying decimals we get responses such as this:-  2.7 x 10 = 2.70  or  0.56 x 100 = 0.5600

And you can't blame the children, as they are only doing what they have been taught to do! This is the danger of children learning rules without understanding the mathematical processes behind them, and of teachers not knowing the correct rules themselves.

The easiest way to demonstrate the rules is to get children to write out the standard mathematical column headings starting with the thousands column, then the hundreds, tens, units, tenths, hundredths and thousandths. Into these columns they write the number 0.018. Next get a calculator and multiply 0.018 by 10. They will get the answer 0.18, which they also write in the correct columns. Now multiply this answer by 10 again to get 1.8, again to get 18, again to get 180 etc. The children should thus notice that when numbers are multiplied by 10 the digits simply stay together, but all move 1 column to the left.

You can start again, but multiply by 100 or 1000 instead. Can they say what the rules are this time? How about starting with 4500 and dividing by 10 repeatedly. They should see that this time the digits stay together but move one column to the right. The rules are therefore:

To multiply by 10 the digits stay together and move 1 column to the left.

To multiply by 100 the digits stay together and move 2 columns to the left.

To multiply by 1000 the digits stay together and move 3 columns to the left.

To divide by 10 the digits stay together and move 1 column to the right.

To divide by 100 the digits stay together and move 2 columns to the right.

To divide by 1000 the digits stay together and move 3 columns to the right.

To help remember the rules there is a poster below, which can be downloaded and printed off. It was designed and created for me by J Foreman.

Download 10_100_1000_poster.pdf

Rules of Divisibility

The rules of divisibilty are rules which allow us to look at a number and straight away see if you can divide the number equally by others, leaving no remainder. The rules are as follows:

• If a number divides by 10 without a remainder then there will be a zero in the units column. So 340 will divide by 10 with no remainder, but 578 won't.

• If a number divides by 5 with no remainder then there will be either a 5 or a 0 in the units column. So 365 and 670 can be divided exactly by 5, but 408 can't.

• If a number divides exactly by 2 then it will have a 0, 2, 4, 6 or 8 in the units column (it will be even). So 364, 256, 48 can be divided exactly by 2, but 345, 567 and 649 can't.

• If a number divides exactly by 4 then its last 2 digits will be in the 4 times table. So 324 and 572 can divide exactly by 4 because 24 and 72 are in the 4 times table, whereas 234 and 458 can't, as 34 and 58 are not in the 4 times table.

• If a number divides by 8 then you take its last 3 digits, halve them and then see if the last 2 digits are in the 4 times table  e.g. 3456 is in the 8 times table because if you take the last 3 digis (456) and halve them you get 228, and 28 is in the 4 times table, so 3456 is in the 8 times table.

• If a number divides by 3, then the total of its digits will be a multiple of 3. So 456 can be divided by 3 because 4 + 5 + 6 = 15, and 15 is in the 3 times table i.e. a multiple of 3. 1003 cannot be divided by 3 because 1 + 0 + 0 + 3 = 4, and 4 is not a multiple of 3.

• If a number divides by 6 then firstly see if it is an even number, then see if it can be divided by 3. e.g. 276 can be divided by 6 because it is an even number and its digits add up to 15, so it can be divided by 3.

• If a number divides equally by 9 then the total of all its digits will be a multiple of 9. e.g. 8757 can be divided by 9 because 8 + 7 + 5 + 7 = 27, which is a multiple of 9.

Solving Missing Number Problems.

Often children are asked in tests to solve problems where numbers are missing e.g. ? + 234 = 782, 7 x ? = 371 etc. It is therefore important that they remember the following rules which will enable them to solve this type of question.

1. Where the missing number is at the start of a calculation.  Here in order to solve this type of problem you start with the answer and perform the opposite operation to it to find the missing number.

e.g.  ? + 78 = 235  To solve it start with the answer 235 and do the opposite of adding 78, which is subtracting 78. The missing number is therefore 235 - 78 = 157, so 157 + 78 = 235.

? x 9 = 477  To solve take the answer 477 and do the opposite of multiplying by 9, which is dividing by 9. 477 divided by 9 is 53, so 53 x 9 = 477

2. Where the missing number is in the middle of an add or subtract sum. Here in order to solve the problem you need to find the difference between the 2 numbers given i.e. subtract the smaller number from the larger one.

e.g. 234 + ? = 578   To solve do 578 - 234 which is 344, so 234 + 344 = 578

793 - ? = 562   To solve do 793 - 562 which is 231, so 793 - 231 = 562

3. Where the missing number is in the middle of a multiply or divide calculation. To solve this type of problem you need to divide the larger number given by the smaller one.

e.g.  513 divided by ? = 9   to solve do 513 divided by 9 which is 57, so 513 divided by 57 = 9

7 x ? = 476  to solve do 476 divided by 7 which is 68, so 7 x 68 = 476.

 

In order to practice using these rules I have a worksheet which may be downloaded below. If further questions are required for consolidation then just make up you own questions in a similar way and get your child to show what working they have done to arrive at the answer.

Download Missing Number Problems