Tutor’s Toolkit: Application of Number

Carrying Out Calculations

C11	Check your methods in ways that pick up faults 
and make sure your answers make sense

C11.1	Checking methods

Scope: 

Students need to be able to:

•	use an approximate calculation to comment on the reasonableness of an accurate calculation. [NOE],
•	check that calculations are the right order of magnitude [NOE],
•	make reverse calculations.

What students need to know:

Use an approximate calculation to comment on the reasonableness of an accurate calculation

If you do a calculation using a calculator, it is a good idea to have a rough idea what the answer should be. One way of doing this is to round all of the numbers in the calculation and get an approximate answer. Use the methods shown in Interpreting Information, Section I4.2

Check that calculations are the right order of magnitude.

The other way is to check that your answer is the correct order of magnitude. This means that it starts in the correct ‘tens’ column. 

For example, 32 is a different order of magnitude to 426. The 32 starts with a digit in the 10’s and the 426 starts with a digit in the 100’s. So, if we expected an answer starting in the 10’s and the calculator gave us an answer in the 100’s we would guess it was wrong and repeat the calculation.

We refer to the units column as order one, the 10’s column as order 2, the 100’s column as order 3, and so on.

Example Question:
If we want to work out 45612  7362 what order of magnitude should the answer be?

Solution:

45612 has order 5 and 7362 has order 4.
We would expect an answer with an order of magnitude of 4 + 5 = 9.

On a calculator you get 45612  7362 = 335795544.
Counting up the digits, you can see it does in fact have an order of magnitude of 9.

Note: 
If the first digits multiply to less than 10, then we will get an order of magnitude one less than expected.
For example: 256  35 = 8960.
This has order of magnitude 4 not the 2 +  3 = 5 expected.

Make reverse calculations

If you worked out 183 – 57 and got the answer 134, you could quickly check it by reversing the calculation.

The reverse of  –  is + (and vice-versa)
The reverse of    is  (and vice-versa).

Let’s check: 
We worked out 183 – 57 = 134.
So, if we calculate 134 + 57, we should get 183.
134 + 50 = 184
184 + 7 = 191. Oops – we were wrong!

Try again:
183 – 57 = 183 – 60 + 3 = 123 + 3 = 126. 
Check again:
126 + 57 = 126 + 50 + 7 = 176 + 7 =183. That’s better!