Numeracy Skills
1
Introduction
This booklet has been developed to give a better understanding of the numeracy
concepts students will be expected to use across the curriculum.
Numeracy is the ability to reason using numbers and other
mathematical concepts. We are numerate if we can use numbers
to solve problems, analyse information and make informed
decisions based on calculations.
Numeracy is a skill for life, learning and work. Having well-developed
numeracy skills allows young people to be more confident in social
settings and enhances their enjoyment in a large number of leisure
activities.
Numeracy is developed in Maths but is reinforced in other subject areas
across the school. It is more than an ability to do basic arithmetic and requires
understanding of a range of techniques. The concepts of numbers and measures,
number systems and problem solving can be approached in a range of different
contexts, such as calculations in Science, map scales in Geography or
representing musical notes as fractions. Numeracy also requires understanding
of the ways in which data can be collected by counting and measuring and can be
presented in graphs, charts and tables. These skills are taught across the
school in different settings and contexts and, as such, it is important that
there is a consistent approach by all teachers to avoid confusion for our young
people.
The new National Curriculum for mathematics aims to ensure that all students :
 become fluent in the fundamentals of Mathematics, including through
varied and frequent practice, so that students develop conceptual
understanding and recall and apply knowledge
 reason mathematically by following a line of enquiry, conjecturing
relationships and generalisations as well as using mathematical language
 can solve problems by applying their mathematics and numeracy skills to
a variety routine and non-routine problems
2
1. Estimation and Rounding
An estimate is an approximation of a quantity that has been based on judgement
rather than guessing. Rounding is used to obtain this approximation.
Rounding to the nearest ten, hundred or thousand:
Remember the rule, ‘five or more’. Look at the next digit after the one to which you
are adjusting. If this is five or more, the digit you are adjusting goes up.

To the nearest 10
32 becomes
36 becomes
30
40

To the nearest 100
327 becomes
352 becomes
300
400

To the nearest whole number:
86.2 becomes
86.5 becomes
86
87

To 1 decimal place:
7.52 becomes
7.96 becomes
7.5
8.0

More decimal place values:
3.141592
= 3.14 (2 dec places)
= 3.142 (3 dec places)
= 3.1416 (4 dec places)
17.45695
= 17.46 (2 dec places)
= 17.457 (3 dec places)
= 17.4570 (4 dec places)
Using rounding to estimate:
At a concert in Wembley stadium, there were 64,880 fans. Here we would say
there were approximately 65,000 fans.
The number of passengers on board 197 flights from Glasgow Airport was
48,976. Approximately how many were on each plane?
48,976 ÷ 197 ≈ 50,000 ÷ 200
= 250 passengers
We round both numbers to
“1 figure” accuracy first
3
2. Subtraction
We use the standard decomposition method (illustrated below).
5 1
261
- 2 8
233
2
1
9
1
3 0 0
- 6 3
2 3 7
We encourage students to check answers by addition.
We actively promote varied mental strategies as appropriate, for example:
counting on
e.g. to solve 51 – 24, count on from 24 until you reach 51
breaking up the number being subtracted
e.g. to solve 51 – 24, subtract 20 then subtract 4
3. Long multiplication
4
4. Division
h t u
h t u
0
4)248
062
4)248
4 does not go into 2. You can put zero in the quotient in the hundreds place or omit it. But 4 does go
into 24, six times. Put 6 in the quotient.
Explanation:
The 2 of 248 is of course 200 in reality. If you divided 200 by 4, the result would be less than 100,
so that is why the quotient won't have any whole hundreds.
But then you combine the 2 hundreds with the 4 tens. That makes 24 tens, and you CAN divide 24
tens by 4. The result 6 tens goes as part of the quotient.
Check the final answer: 4 × 62 = 248.
h t u
th h t u
0 4 1 r1
4) 165
0 4 0 0 r7
8) 3207
4 does not go into 1 (hundred). So combine the
1 hundred with the 6 tens (160).
8 does not go into 3 of the thousands. So
combine the 3 thousands with the 2 hundreds
(3,200).
4 goes into 16 four times.
4 goes into 5 once, leaving a remainder of 1.
1. Divide.
t u
8 goes into 32 four times (3,200 ÷ 8 = 400)
8 goes into 0 zero times (tens).
8 goes into 7 zero times, and leaves a
remainder of 7
2. Multiply & subtract.
t u
3. Drop down the next digit.
t u
2
2 )5 8
2
2)58
-4
1
29
2)58
-4↓
18
Two goes into 5 two times, or 5
tens ÷ 2 = 2 whole tens -- but
there is a remainder!
To find it, multiply 2 × 2 = 4,
write that 4 under the five, and
subtract to find the remainder
of 1 ten.
Next, drop down the 8 of the
ones next to the leftover 1 ten.
You combine the remainder ten
with 8 ones, and get 18.
5
1. Divide.
2. Multiply & subtract.
3. Drop down the next digit.
t u
t u
29
2)58
-4
18
29
2)58
-4
18
-18
0
29
2)58
-4
18
-18
0
Multiply 9 × 2 = 18, write that
18 under the 18, and subtract.
The division is over since there
are no more digits in the
dividend. The quotient is 29.
Divide 2 into 18. Place 9 into
the quotient.
t u
5. Rules of Operators – BIDMAS
Students are taught to know that multiplication and division have priority over addition
and subtraction and that brackets have an even higher precedence.
BODMAS is the memory aid we teach in maths to enable students to use the correct
sequence of carrying out number operations. Students are taught to recognise that
basic (four function) calculators will work differently from scientific calculators.
B
I
D
M
A
S
Brackets
Indices
Division
Multiplication
Addition
Subtraction
Here are a few examples to illustrate this:
(a) 2 + 3 x 4
= 2 + 12
=14
(b) 6 x 2 + 3 x 5
= 12 + 15
= 27
(c) 6 + 5(4 – 1)
=6+5x3
= 6 + 15
= 21
(d)
1
of 12 - 1
4
=3-1
=2
(e) 4 x 23
4 x 8= 32
6
6. Fractions
Fractions of a quantity
1
of 12 = 4
3
1
of 40 = 8
5
3
of 120 = 90
4
we do 12 ÷ 3
we do 40 ÷ 5
Addition and Subtraction
we make the
denominators equal
e.g.
1
1
+
2
3
3
2
=
+
6
6
5
=
6
we do 120 ÷ 4 and then multiply by 3
Multiplication
we multiply top and bottom
and then simplify
e.g.
2
3
x
3
4
6
=
12
1
=
2
Division
we invert the second
fraction and multiply
e.g.
3
2
÷
4
5
3
5
=
x
4
2
15
=
8
7
= 1
8
In Music, students are asked to compile 2, 3 or 4 beats in the bar. To do this they may
use a variety of different notes, all carrying different fractional amounts, but must
ensure that their fractions add up to the amount of beats they have been given.
For example:
Crotchet
1 beat
Quaver
½ beat
Semi-quaver
¼ beat
Demi-semi-quaver
⅛ beat
7
7. Percentages
Students are expected to have a sense of common percentages and their equivalent
fractions and decimals.
All students should learn the following table:
Percentage
100%
50%
Fraction
1
1
2
Decimal
1.0
0.5
33
1
2
% 66 % 25%
3
3
1
2
1
3
3
4
0.33..
0.67
0.25
75%
20%
40%
60%
80%
10%
30%
70%
90%
3
4
1
5
2
5
3
5
4
5
1
10
3
10
7
10
9
10
0.75
0.2
0.4
0.6
0.8
0.1
0.3
0.7
0.9
Students are expected to find more complex percentages with the use of a calculator.
Students should recognise the word ‘of’ as meaning multiply and
% as meaning “divide by 100”.
For example 24% of 400 means calculate
24
x400 = 96
100
We tend not to use the % button on calculators because of inconsistencies and
increased error risk. To calculate 24% of 100 we type:
24
÷
100
x
400
=
into the calculator.
Some mental strategies:
Calculate 65% of 40
50% = 20
10% = 4
5% = 2
We separate the 65% into a
combination of simple percentages
that are much easier to calculate
so 65% of 40 = 20 + 4 + 2 = 26
Express
2
as a percentage.
5
2
4
40
= 40%
=
=
5
10
100
8
Percentages in context:
An electrical shop has a 25% off sale. How much would a kettle cost if its
original price was £24?
Here we are being asked
1
of £24 = £6.
Solution:
to consider £24 less 25%
4
(or a quarter)
Then sale price = £24 - £6
= £18
A £75 vacuum cleaner has been reduced by £15. Calculate the discount as a
percentage.
Solution: discount =
15
x 100
75
= 20%
Percentage increase/decrease (or profit/loss):
This is when we express an increase or decrease as a percentage of the original
quantity. First we must calculate the difference between the original and final
values.
A car is purchased for £5000. It is sold a year later for £3500. Calculate the
percentage loss (decrease).
Loss (difference) = 5000 – 3500
= 1500
Percentage Loss =
Percentage
=
increase/decrease
difference
original
x 100
1500
x 100
5000
= 30%
Farmer Jones added 5 tonnes of fertiliser to his field. The next year this
increased to 16.2 tonnes of fertiliser. Calculate the percentage increase in
fertiliser over the period.
Increase = 16.2 – 5
= 11.2 tonnes
It follows then % increase =
11.2
X 100
5
= 224%
Notice the answer is greater
than 100% because it has
increased by more than twice the
original quantity of fertiliser
9
8. Time
Conversion of time between 12 and 24 hour clock is reinforced in S1 maths.
Calculation of duration in hours and minutes is taught by counting on to the next hour
and then on to the required time.
We do not teach time as a subtraction.
How long is it from 0655 to 0942?
0655
0700
0900
5 mins + 2 hrs +
0942
42 mins = 2hrs 47mins
Total time is 2hrs 47mins
9. Scientific Notation (A.K.A. Standard Form)
Scientific notation is a method for writing very large or very small numbers in a
manageable way. They are rewritten as a number between one and ten and multiplied
by 10 to a power of a value. The power is how many places we have to move the
largest place value to get this number into the units column.
Examples:
5,700,000
23,400,000
1,425,000,000
0.000025
=
=
=
=
5.7 x 10 6
2.34 x 10 7
1.425 x 10 9
2.5 x 10 5
0.000766 = 7.66 x 10
4
The 5 has been moved 6 places to the
right. Note large numbers (greater than
10) have a positive power
The 7 has been moved 4 places to the
left. Note small numbers (less than 1)
have
a
negative
power
Dinosaurs roamed the Earth 228 million years ago. Write this figure in
scientific notation.
228 million = 228 000 000
= 2.28 x 108
The wavelength of red light is 6.65 x 10-7 metres. Write this number out in full.
6.65 x 10-7 metres = 0.000000665 metres
10
10. Money
Foreign Exchange
Katie is going on holiday to Spain and has
managed to save £650. How many Euros
will she receive if the exchange rate is
1 Pound = 1.17 Euros?
Solution: £650 = 650 x 1.23
= £799.50
REMEMBER
£ to foreign
MULTIPLY
foreign to £
DIVIDE
Answer has 2 decimal places since money!
Tommy returns from Florida with $1200. The Post office exchange rate is
1 Pound = 1.68 Dollars. How much will he receive in pounds?
Solution: $1200 = 1200 ÷ 1.68
= £714.29 (to 2 decimal places)
Budgeting
We encourage students to plan ahead when working out their finances. This allows
them to manage their money efficiently and effectively.
Taylor has £36 in his piggy bank. He received his weekly pocket money of £10
and a present of £6 from his Gran. He got £5 for washing the cars.
He plans to buy a new game costing £42. He also wants to spend £5.45 going to
the cinema and £5.50 on drinks & snacks. He needs to make sure he has enough
money before he goes out.
In business, we represent this information in a table:
£
£
36.00
Opening Balance
CASH IN
Pocket Money
Present
Car Washing
Cash Available to Spend
CASH OUT
Game
Cinema
Drinks & Snacks
10.00
6.00
5.00
57.00
42.00
5.45
5.50
Closing Balance
£
4.05
Taylor has enough money and has £4.05 to spare.
11
11. Proportion
We use the unitary method of proportion, which means that we find the value of one
item and then multiply by the required number.
E.g. If 5 apples cost 80p, what do 3 apples cost?
5
1
3
cost
costs
cost
80p
80p ÷ 5 = 16p
16p x 3 = 48p
12. Ratio
A ratio shows how much of one thing there is compared to another thing.
In the diagram below there are 3 grey squares and 1 white square.
3:1
We would say the ratio of grey to white squares is “3 to 1” or 3 : 1.
In other words for every 3 grey squares there is 1 white square.
For a ratio we need give the simplest WHOLE NUMBER of grey squares compared to
white. To do this you have to find the largest number which both sides can be
divided by.
What is the ratio of 6 grey squares and 2 white
We would write the ratio like this:
Both of these numbers
grey: white
can be divided by 2!
6:2
3:1, so our ratio is 3 : 1
For every 3 grey squares, there is 1 white square 
Work out the ratio of red marbles (25) to blue (20).
red marbles : blue marbles
25 : 20
5 : 4 , so our ratio is 5 :
4
Both of these numbers can
be divided by 5!
For every 5 red marbles, there are 4 blue marbles
12
The art department need 15 litres of green paint for the school show set.
To make green, the ratio of yellow to blue is 2:3. They only have 6 litres
of yellow paint but plenty of blue paint. Do they have enough to make green
paint for the set?
We set the sum out like this:
yellow
x3
2
6
Quantity of green paint = 6 + 9
= 15 litres
blue
3
9
x3
We multiply by 3 to get from
2 to 6 for the yellow part, so
we must multiply by 3 for
the blue part too.
Always answer the question!
Yes the art department have exactly enough green paint for the school show set.
Divide £1000 in the ratio of 7:3.
Solution: number or parts = 7 + 3
= 10
Divide 1000 by 10 to get 100 which means £100 per part
For 7 parts 7 x 100 = £700
For 3 parts 3 x 100 = £300
In Home Economics, ratio can be used in recipes.
For example, in making a sponge cake, scaling up can be used as follows:
1 egg to 50g of flour, 50g of sugar, 50g of margarine
2eggs to 100g of flour, 100g of sugar, 100g of margarine
If on a map the scale is 1:50 000. What distance is 10cm on the map in real life?
1cm (map) =
=
=
50 000 cm (real)
50 000 ÷ 100
500m
10cm (map) =
=
=
=
10x 500m
5000m
5000 ÷ 1000
5km
There are 100cm in a metre.
There are 1000m in a kilometre.
13
13. Measurement
We always use the metric system in maths but students should be made
aware of imperial units. Some useful information is shown below:
Metric Units Equivalence
Length
Volume
Mass
10mm = 1 cm
100cm = 1 m
1000m = 1 km
1000mg = 1 g
1000g = 1 kg
1000kg = 1 tonne
1000ml = 1 litre
100cl = 1 litre
1cm 3 = 1 ml
Imperial Units Equivalence
Length
Volume
1 inch = 2.5 cm
1 mile = 1.6 km
Approximations
Length
12 inches = 1 foot
Mass
8 pints = 1 gallon
16 ounces = 1 pound
14 pounds = 1 stone
Volume
expected to know
the difference
between the two.
Mass
All the matter objects
are made up of.
Mass
3
1 litre = 1 pints
4
In some subjects
the term mass and
weight are used to
mean the same
thing but in science
you would be
1kg = 2.2 pounds
Weight
A force measured in
Newtons.
Students can use the following diagram to help them with unit conversions within the
metric system.
x 1000
kilometres
(km)
x 100
metres
(m)
÷ 1000
x 10
centimetres
(cm)
÷ 100
millimetres
(mm)
÷ 10
Within technical, students will always measure in
millimetres. In graphic communications,
students will be expected to produce both 2D
and 3D drawings using a ruler with a millimetre
scale, for example this isometric view of a
sports podium (shown to the right).
14
14. Handling data - Statistics
Students should be able to interpret and construct various types of
statistical information such as graphs and charts. Let’s look at some
examples…
Pie charts
Pie charts use different-sized sectors of a circle to represent data.
A pie chart represents 100%.
½ a pie = 50%
¾ of a pie = 75%
¼ of a pie = 25 %
Example
The following table shows the frequency of plants with different types of damage:
Cause of damage
Frequency of damage (%)
Mammals
30
Insects & fungi
10
Weather
5
Frost
15
Unknown
40
A pie chart can be constructed to display this information.
Mammals
Insects &
fungi
Frost
The pie is divided into
20 equal sections, so
each section is worth
5% of the pie
Unknow
15
Sometimes the data you are asked to present as a pie chart is not given as a percentage.
In this case you must convert the figures into fractions first.
Example
The results from a survey on popular lunchtime meals was carried out. Out of the 50 people
surveyed 25 people preferred chicken wings. 10 preferred chicken curry.
10 preferred fish and chips and 5 preferred salad.
Steps…
1. Take each type of meal in turn and work out the size of its ‘slice’ of pie by
converting into a percentage.
25 people prefer chicken wings out of 50 = 25/50 x 100
= 50% -> 1/2 of the pie
10 people prefer chicken curry out of 50 = 10/50 x 100
= 20% -> 1/5 of the pie
10 people prefer fish and chips out of 50 = 10/50 x 100
= 20% -> 1/5 of the pie
5 people prefer salad out of 50 = 5/50 x 100
= 10% -> 1/10 of the pie
2. Now label each slice to show what it represents.
Remember to use
Chicken
curry
Fish &
chips
pencil and a RULER to
draw neat lines for
Chicken
wings
each section :0)
salad
If the slice is too thin
you can put the label
outside the pie
16
Drawing Bar Graphs
A bar graph (AKA a bar chart) is a graph that uses rectangular bars to represent different
values. This shows comparisons among categories e.g. pocket money received by different
year groups, or frequency of blood groups in Scotland. Bar graphs are most commonly drawn
vertically (although sometimes they can be drawn horizontally).
Example
36 students compared the colours of their eyes and recorded the results on the following
table:
Eye colour
Frequency
Blue
16
Green
12
Brown
8
TOTAL
36
Why type of graph should we draw
for this data? A bar graph would
be appropriate since the data is
given in the form of numbers
(frequency) and words (eye colour).
Displaying this information as a bar graph:
Highest number to plot is 16.
Numbers must go up evenly
from zero on your axis.
Eye colour Frequency
The order the bars
are put in does not
matter but each
individual bar must
have a label and
they must all be the
same width of bar.
Remember to label each axis
using the titles from your
table.
Remember to use
pencil and a RULER to
draw neat bars :0)
17
This data could also be displayed as a pie chart. In maths students will learn to construct
pie charts and they will use the 360o rotation within a circle to make their sections
accurate.
Eye colour data:
Eye colour Frequency
Eye colour
Frequency
Blue
16
Green
12
Brown
8
TOTAL
36
Convert the data into degrees:
Blue
16
x 360 = 160o
36
12
x 360 = 120o
Brown
36
8
x 360 = 80o
Green
36
A key is used in this example to show
what each section of the pie represents
(instead of labelling each section of the
pie as in the previous examples)
Histograms
Bar graphs are ideal when your data is in categories (such as "Brown", "Blue", etc).
But when you have continuous data (such as a person's height or weight) you should draw a
histogram.
Histograms are similar to bar graphs
Notice: The bars of a histogram are
right next to each other and do not
have gaps between them.
Top tip: make sure you
No gaps are
left between
Number of children
but a histogram groups numbers into
ranges, which you decide on.
the bars
leave gaps between the bars
of a Bar Graph, so it doesn't
look like a Histogram.
Age range (years)
Numbers are grouped
into an age
e
range
e
18
Drawing Line Graphs
When data is given as two sets of numbers, a line graph is usually used to display the
information. A line graph uses points and lines on a grid to show change over a period of time.
Key points to remember when drawing a line graph:
The horizontal axis is called the X axis and the vertical axis is
called the Y axis.
When data is given in the form of a table use the headings in
the table to label each axis of your graph.
Remember to include appropriate units in brackets beside each label e.g. Length (mm);
Temperature (oC); Mass (g); Time (s) etc.
A small cross or dot should be used for each point plotted.
The scale on each axis should be even e.g. 0, 2, 4, 6, 8, 10
0, 5, 10, 15, 20, 25
0, 250, 500, 750, 1000
0, 0.2, 0.4, 0.6, 0.8, 1.0
To decide on the scale look at the highest number which needs to be plotted then make
sure your numbers go up evenly using as much graph paper as possible.
A single line should go through the centre of each point to join them together
(an exception to this is when a line of best fit is drawn).
Note that the ends of the line do not need to join the axes.
Example 1 - Science
During a chemistry experiment chalk was added to acid to find out the volume of carbon
dioxide gas released over a period of time. The results are given in the table:
Time (minutes)
Volume of gas
released (cm3)
0
1
2
3
4
5
0
18
38
62
80
80
80
Units must be
included beside the
label on each axis
Notice there are 5
small boxes
between 0 and 10
so each small box
Volume of gas released (cm3)
70
60
50
40
We can see from the results
30
that as the time increases the
20
volume of gas released also
10
increases until 4 minutes.
After 4 minutes the volume of
0
0
1
2
3
4
Time (minutes)
5
gas produced remains constant.
is worth 2
16
Example 2 – Science
An experiment was set up by a pupil to investigate the response of maggots to different
intensities of light. A maggot was placed in the dish with a lamp positioned above it. The
brightness of the lamp was altered using a dimmer switch.
Here are the results:
Light intensity (units)
Rate of movement (mm/minute)
10
20
30
40
50
60
50
62
68
70
75
85
The results are used to draw a line graph:
The thing (or ‘variable’) that was
measured by the pupil (i.e. the
results of the experiment) goes on
the Y axis
Notice there are
5 small boxes
between 0 and
20 so each small
box is worth 4
A point is not plotted at
a light intensity of zero
since the data for this
result is not given
The thing (or ‘variable’) that
was changed by the pupil
goes on the X axis
17
Example 3 – Geography: Climate Graphs
Climate graphs show two types of information on the same graph, so it has two Y axes!
The Y axis of a climate graph shows temperature and rainfall. Rainfall is shown in a bar
graph and temperature is shown in a line graph.
The X axis shows the months of the year.
We measure temperature in degrees celsius (oC) and rainfall is measured in millimetres (mm).
Make sure these units are included on the Y axis labels of the graph.
An example of a climate graph:
Months
In some data sets it might be appropriate to start your axis at a number other than zero.
This is called a break in the data and we use a zig zag symbol to illustrate it. Sometimes in
the media data can be misleading. An example is shown below:
Rise in Sales
A zig zag
should be
shown at
the bottom
of this axis
Rise in Sales
The rise in
sales should be
illustrated like
this
misleading
data
18
Averages
There are 3 different ways to calculate the average number within a set of data.
Mean – We add up all the numbers and divide by how many numbers there are.
Median - The value which appears in the middle of an ORDERED list. When there are
two middle values the median is half way between them.
Mode - The value which appears most often.
It is possible to have two modes but no more
than that. If more than two values have the
highest frequency, we say there is no mode.
Numbers have to be
in ascending order
(lowest to highest).
Example:
Find the mean, median and mode for these numbers.
1
1
1
mean = 1 + 1+ 1 + 1 + 2 + 3 + 26
7
= 35
7
= 5
1
2
3
26
median = 4th number
mode = 1
=1
This data set illustrates that the mean is not always the best average. This
is due to one number being much larger than the rest. The average of this data
set is best represented by either the median or the mode.
Range
The RANGE is a measure of the SPREAD of a data set.
In maths it is the difference between the highest and lowest numbers in the list.
Range = highest – lowest
Calculate the range for the data:
range = highest – lowest
= 12 – 4
=8
4, 5, 7, 7, 9, 12.
Note in Science you would be expected
to state the range as 4 to 12 (lowest
to highest number).
19