Table Of ContentMaths Is Not A Mystery
The Maths Is Not A Mystery programmes of tutorial material for KS2 and for
GCSE are written by Peter Jeffcock, who is a successful and experienced
maths teacher, former head teacher, education leadership consultant and
author.
The books are available in single topic PracticeBooks, Multi-topic TopicBooks
and BigBooks, covering a wider spread of the mathematics curriculum.
Maths Is Not A Mystery Primary Tutor Book PD2
© 2021 Panda Publications
Maths Is Not A Mystery
Welcome!
The purpose of the Maths Is Not A Mystery series is to demystify maths. We
don’t make any assumptions about what you already know and we don’t
make huge jumps between difficulty levels as we go along.
We start with the basics and give you full, common sense explanations. We
move forward a small step at a time and we give you plenty of practice so
that you can develop your skills and become genuinely confident.
PARENT INVOLVEMENT
A child of primary school age should not be asked to work through a tutor
book alone – they need to talk about things with adults to make sure they
are getting the right end of the stick! By using this book with your child you
can eliminate the need for an expensive tutor because the book explains to
you, as the adult, how the concepts work, what your child is expected to
demonstrate knowledge of and how to approach the questions.
There has been a lot of confusion about how maths is taught in modern
schools and this has deterred parents from wanting to help. No parent wants
to add to any insecurities their child has in their school work by teaching
them things ‘the wrong way’. Our programme helps you to explain things the
correct way to your child so that the work you do using this book will
complement the work your child is doing in school – no confusion, just
confidence and consolidation.
HOW TO USE THIS BOOK
Read through the explanations with your child (have a quick read beforehand
if you want, so that you already know what’s coming!). Everything that you
ought to be reading and discussing with your child is written in BLACK
Do the example questions together and check your answers with the worked
solution – solutions are given to absolutely every question or example.
Ask your child to attempt the practice questions on their own (there are
practice questions within the book and also at the end). The work your child
should aim to do alone is written in BLUE
Go through the answers together, using the worked solutions.
If your child has made a mistake, don’t simply show them the right answer –
get them to discover where they went wrong, by referring to the worked
solution. The solutions, which are for you to talk about with your child, are
written in GREEN
Come back to the book at a later date, after covering different topics. It’s
important to keep making sure that the skills haven’t been lost.
In this book we are learning about averages. Most of the book is based on
looking at the ‘mean’ average, because that is the thing you must know
about at KS2. But we will also look at the other types of averages as well –
just briefly – because they are important, and you will need to study them at
secondary school, so it won’t hurt to be familiar with them.
Averages
Averages can be very useful to us if they are used in appropriate situations.
They can also be very misleading if they are used in the wrong situations. We
will spend most of the time looking at useful averages, but at the end of the
section we will look at how averages can be unhelpful.
The idea of an average is to ‘level out’ a series of numbers. We will start with
small groups of numbers in order to make the idea of an average make sense.
Later, we will look at much larger groups of numbers.
The Average of 2 Numbers
Q: Jill and Bill each have some money. Jill has £80 and Bill has £50.
What is the average of the two amounts of money?
Basically, what the question is asking is ‘if they both had the same amount of
money, how much would they each have?’
Between them they have £130 (80 + 50)
If they shared that £130 equally, then 130 ÷ 2 = 65
They would each have £65 if they shared the money equally.
So the average of £80 and £50 is £65.
And that’s what an average is. It levels out the differences between all the
amounts and makes it so that every amount is the same.
The total amount of money is the same (£130) but the average tells us what
each person would have if their shares were levelled out.
Let’s look at another example.
Q: Sue and Lou have measured their heights.
Sue is 145cm tall and Lou is 153cm tall.
What is their average height?
The two people are different sizes. If we add up their total height (145
+ 153) and then share it between both of them, that will tell us the
average.
145 + 153 = 298cm
If they stood on top of each other, they would reach a height of 298cm!
If we now split the 198cm between the two people (298 ÷ 2 = 149) their
average height is 149cm.
149cm is a bit taller than Sue and a bit shorter than Lou – somewhere
in-between.
The average will always be somewhere in between. It can never be smaller
than the smallest number in the list, and it can never be bigger than the
biggest number in the list. But it isn’t always exactly in the middle in big lists
of numbers, as we will see later.
So let’s recap. When we find the average of 2 numbers, we add them up and
divide the answer by 2 to share the total equally.
What happens if we need to find the average of 3 numbers?
Q: Wayne, Shane and Jane have some money.
Wayne has £20, Shane has £80 and Jane has £110
What is the average amount of money?
As we did before, we start by adding up all the money to see how much
there is altogether.
20 + 80 + 110 = £210
They have a total of £210 between them. The average is the amount each
person would have if the £210 was shared equally.
This time we have to share it by 3, because there are 3 people.
210 ÷ 3 = £70
If the money was shared equally between the three people, they would
each have £70 (and it would still add up to £210)
So the average of £20, £80 and £110 is £70.
Have you noticed that £70 is not midway between the smallest and the
biggest number? 70 is more than half way between 20 and 110 but it is the
average of the three numbers.
Why is that?
In the list of numbers, 80 and 110 are both at the high end of the scale, and
20 is on its own as a much lower number. So the average is balanced towards
where most of the numbers are – and that is at the higher end.
Let’s demonstrate by doing another average.
This time, Shane has spent some of his money and he now only has £50.
Wayne and Jane still have their original amounts of money. So let’s find the
new average.
Add up all the numbers: 20 + 50 + 110 = 180
Between them, they now have £180.
Share £180 by 3: 180 ÷ 3 = 60
The average is now £60.
The smallest and biggest numbers haven’t changed, but the middle number is
no longer at the higher end of the scale, so the average has dropped. The
average has dropped because the total amount of money has dropped but
we are still dividing it by 3.
Let’s now find the average of a group of 4 numbers.
Q: The cost of an ice cream at four seaside kiosks is:
£1.50, £2.80, £1.20, £2.50
What is the average price of an ice cream in the seaside town?
Begin by adding up all the prices, to see what it would cost if you bought
an ice cream from each kiosk.
£1.50 + £2.80 + £1.20 + £2.50 = £8.00
The total cost would be £8.00
Now share that amount between all four kiosks.
£8.00 ÷ 4 = £2.00
The average cost of an ice cream is £2.00
By now, hopefully you have seen that the way to find an average is to:
Add up all the numbers in the list
Divide your total by however many numbers you added up
So to find the average of 2 numbers, add them up and divide by 2.
To find the average of 3 numbers, add them up and divide by 3.
To find the average of 6 numbers, add them up and divide by 6.
And so on.
Have a go at finding some averages on your own.
Q: Bob and Rob have some sweets.
Bob has 70 sweets and Rob has 44 sweets.
What is the average number of sweets?
Q: Bill, Jill and Phil have been on long bike rides.
Bill rode 55 miles, Jill rode 46 miles and Phil rode 19 miles.
What was the average length of their bike rides?
Q: Five pupils did an exam. The times they each took to complete it were:
12mins 15mins 8mins 16mins 14mins
What was the average time taken to complete the exam?
