Tutor’s Toolkit: Application of Number

Interpreting Information

I3	Read and understand numbers used in different ways, 
including negative numbers

Useful Web Links:

BBC Bitesize Revision: Maths, Fractions
http://www.bbc.co.uk/schools/gcsebitesize/maths/numberih/fractionsrev1.shtml

BBC Bitesize Revision: Maths, Negative Numbers
http://www.bbc.co.uk/schools/gcsebitesize/maths/numberf/negativenumbersrev1.shtml

BBC Skillswise: Numbers, Comparing fractions and percentages
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/comparing/fractionspercentages/

BBC Skillswise: Numbers, Negative numbers
http://www.bbc.co.uk/skillswise/numbers/wholenumbers/whatarenumbers/negativenumbers/index.shtml

I3.1	Fractions and percentages

Scope:

Students need to be able to:

•	read a fraction as a number of chosen items from a total number of items – and simplify,
•	work out an amount as a fraction of another amount (e.g. work out 1.5 as a fraction of 12),
•	approximate an amount as a fraction of another amount (e.g. 148 out of 151 as an approximate fraction),
•	work out and approximate amounts as percentages of given amounts. (e.g. 300 out of 800 as a percentage).

What students need to know:

We can read  in many different ways:
“Three eighths”
“Three parts out of eight”
“Three over eight”
“Three divided by eight”

They all mean the same thing.

If a factory finds that 12 toys are faulty in a batch of 32, then the fraction that is faulty is . 
We could read this as twelve thirty-twoths!

Here is the twelve out of 32. 
It is exactly the same as 3 out of 8.
We can write the same fractions in many different ways, either with larger numbers or smaller numbers.
The trick is to increase or decrease both parts by the same factor.

We normally write fractions in their simplest form. This means using the smallest possible numbers. This happens when we cannot divide top and bottom by the same amount any more.

One special larger number we sometimes aim to get on the bottom of a fraction is 100. We give this fraction a special name – it’s called a percentage. (Percentage is French for “out of 100”.)
For example: . 

Example Questions: 
Write these amounts as fractions :

1.	14 out of 30.
	14 out of 30 is .
	We can simplify: . 
	Answer: 

2.	0.4 out of 1.2.
	0.4 out of 1.2 is . (It is not good style to write fractions with decimal parts, so we aim to get rid of the decimal first!)
	We can simplify: . 
	Answer: 

3.	 out of 18.
	 out of 18 is . (Get rid of the fraction on top quickly!)
	We can simplify: 
	Answer: 

4.	596 out of 1487.
	These numbers suggest an approximate fraction will be appropriate!
	596 is roughly 600.
	1487 is roughly 1500.
	We can simplify: 

	Answer: 596 out of 1487 is roughly 	

Example Questions:
Write these amounts as percentages:

1.	500 students pass a test and 300 fail. What percentage passed?

	Notice that 500 + 300 = 800 students took the test, so the fraction that passed is .
	As a percentage make the fraction out of 100:

	Answer: 62.5%

2.	36 out of 60.
	This will have to be done in two stages. We can divide by 6 to make ten and then multiply by ten to make 100.

	 Answer: 60%

For information on converting between fractions and percentages, have a look at Section C3

I3.2	Negative Numbers

Scope: 

Students need to be able to:
•	recognise a negative quantity and read it as less than zero,
•	find the difference between two amounts in words (commonly in the context of range).

What students need to know:

We can count down:
Five, four, three, two one, (Not “blast off”, but:) zero, then negative one, negative two, negative three, ….
Numbers less than zero are negative.
(Don’t use the word “minus” because that means to subtract).
The negative symbol should be smaller than a ‘minus’ symbol and raised a little:
Negative 7 is 7

It is helpful to imagine these numbers as rungs on a ladder.

For Example:

•	5 is smaller than 2
•	0.05 is bigger than 231
•	The difference between 3 and 5 is 8 (Count 8 rungs up the ladder from 3 to 5).

Example Question:
Find the range of the numbers 12, 7, 7, 10, 5, 14.

Solution:

The range of a set of numbers is the difference between the largest and the smallest.
We need the difference between 10 (the smallest number in the list) and 14 (the largest number in the list).

Start 10 rungs below zero then go up to 14 rungs above zero a total of 24 rungs.
The range is 24.

The requirements of this section are covered in Carrying Out Calculations, Section C9.