Tutor’s Toolkit: Application of Number

Carrying Out Calculations

C1	Show clearly your methods of carrying out calculations
and give the level of accuracy of your results

Useful Web Links:

BBC Bitesize Revision: Approximating and estimating
http://www.bbc.co.uk/schools/gcsebitesize/maths/numberf/approximatingandestimatingrev1.shtml

BBC Skillswise: Numbers: Rounding and estimating
http://www.bbc.co.uk/skillswise/numbers/wholenumbers/whatarenumbers/rounding/

C1.1	Presenting calculations

Scope: 

Students need to be able to:
•	Show and present methods of calculation. [NOE]

What students need to know:

It is important that calculations can be read easily by someone reading them. There are various things that will help:

•	Annotate the method 
Add some notes to explain how the calculation is done and most especially why that particular calculation has been chosen.

•	Underline the answer 
It is important to be able to see easily what the answer actually is. It should be written separately, in the terms of the question. It is helpful to underline the answer to make it stand out.

•	State the chosen accuracy 
Always say what accuracy an answer has been given to, if you have done any rounding. For example, to the nearest whole penny or to 2 decimal places would be appropriate. (Refer to Carrying Out Calculations, Section C1.2 and Interpreting Information, Section I4.1). 

•	State the units 
Always make sure that units are used if they are in the question. Be especially careful with areas and volumes to include (for example) the cm² or cm³.

C1.2	Levels of accuracy

Scope: 

Students need to be able to:
•	Choose a suitable level of accuracy. [NOE]
•	Round a quantity to a given number of decimal places. [NOE]
•	Recognise the amount that quantities have been rounded to (e.g. recognise that 3.8 given as £ is to the nearest 10p).
•	Round a quantity to a given number of significant figures. [NOE]

What students need to know:

Choose a suitable level of accuracy

In an exam, it is common for a question to tell you what level of accuracy is required. For example, it may say, “Write your answers to 2 decimal places”. In this case, it is important to do as the question says. In practical work, a sensible level of accuracy should be chosen. (Refer to Interpreting Information, Section I4.1).

When you are working with units that round naturally, make sure they are used appropriately. Money is the best example. When working in pounds, always use 2 decimal places. For example, £3.45, that is 3 pounds and 45p. Writing £3.5 is never good style! When a rough answer is fine, write it to the nearest pound i.e. £3.

For practical experiments, rounding to one decimal place is most common. For especially accurate work, 2 decimal places can be used. It would be rare to need a higher level of accuracy than this.

Where there is no decimal involved, work to 2 significant figures or, for rough approximation or checking work, to 1 significant figure (Refer to Interpreting Information, Section I4.2).

Round a quantity to a given number of decimal places

The trick with decimal places is to look at the next digit. If we want to round to 2 decimal places, look at the third digit. If it is 5 or above, then we round the second digit up one. 
(Notice that we can abbreviate decimal places to d.p.)

Example Questions:

Round 3.478 to 2 decimal places. 
In 3.478, the 3rd digit is 8 so we round up. 
The 7 rounds up to 8.
3.478 = 3.48 (to 2 d.p.)

Round 12.79621 to 2 decimal places.
In 12.79621, the 3rd digit is 6 so we round up.
The 9 rounds up to 10. 
That means the 79 becomes 80.
12.79621 = 12.80 (to 2 d.p.)

Recognise the amount that quantities have been rounded to

Pounds and pence are a very common special case of using decimals. It is important to be able to read decimal pounds as pounds and pence.

•	£3.20 is 3 pounds and 20p
•	£3.02 is 3 pounds and 2p
•	3.2 pounds is 3 pounds and 20p. It is written to the nearest 10p.

Round a quantity to a given number of significant figures

Significant figures are needed to choose the accuracy of larger numbers.

For example:

A newspaper headline reports the amount of money that was stolen in a major robbery. Saying “£3,456,092 stolen” isn’t very punchy! Instead we would say “nearly £3,500,000”. This is the amount written to 2 significant figures. It works the same as decimal places. We look at the third number to decide whether the second number should be rounded up or not.

To arrive at this, we follow the following steps:

•	3,456,092 – the third digit is 5 so we round the 4 up to a 5.

•	We have to put in zeroes to make the number the same order of magnitude, i.e. roughly the same size.

•	So we write 3,500,000.

We can also do this for very small numbers. The only difference is that we start counting the significant figures at the first non-zero digit.

•	0.000437819 to 2 significant figures is 0.00044.

•	The third digit is 7 so we round the 3 up to a 4.