Dalziel High School 2014-2015

Numeracy across the Curriculum

Numeracy booklet

S1 – S3

Name: Class:

1

Introduction

This booklet has been developed to help pupils and parents gain a better understanding

of the numeracy concepts pupils will be expected to use across the curriculum in S1 - S3.

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 departments 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.

For numeracy websites go to the maths page on the school website

or try www.mathsrevision.com (see QR code).

Contents:

1. Estimation and Rounding page 2

2. Subtraction page 3

3. Rules of Operators – BODMAS page 3

4. Fractions page 4

5. Percentages page 5-6

6. Time page 7

7. Scientific Notation (Standard Form) page 7

8. Money page 8

9. Proportion page 9

10. Ratio page 9-10

11. Measurement page 11

12. Information Handling

Pie Charts page 12,13,15

Bar Graphs /Histograms page 14,15

Line Graphs page 16-18

Averages and Range page 19

Front cover designed by Scott Rankin (S1)

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 30

36 becomes 40

To the nearest 100 327 becomes 300

352 becomes 400

To the nearest whole number: 86.2 becomes 86

86.5 becomes 87

To 1 decimal place: 7.52 becomes 7.5

7.96 becomes 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).

2 6 1 3 0 0

- 2 8 - 6 3

2 3 3 2 3 7

We encourage pupils 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. Rules of Operators – BODMAS

Pupils 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 pupils to use the correct

sequence of carrying out number operations. Pupils are taught to recognise that basic

(four function) calculators will work differently from scientific calculators.

B Brackets

O Of

D Division

M Multiplication

A Addition

S Subtraction

Here are a few examples to illustrate this:

(a) 2 + 3 x 4 (b) 6 x 2 + 3 x 5 (c) 6 + 5(4 – 1) (d) 4

1 of 12 - 1

= 2 + 12 = 12 + 15 = 6 + 5 x 3 = 3 - 1

=14 = 27 = 6 + 15 = 2

= 21

5 1 2 1 1 9

4

4. Fractions

Fractions of a quantity

3

1 of 12 = 4

5

1 of 40 = 8

4

3 of 120 = 90

Addition and Subtraction Multiplication Division

we make the we multiply top and bottom we invert the second

denominators equal and then simplify fraction and multiply

e.g. 2

1 +

3

1 e.g.

3

2 x

4

3 e.g.

4

3 ÷

5

2

= 6

3 +

6

2 =

12

6 =

4

3 x

2

5

= 6

5 =

2

1 =

8

15

= 18

7

In Music, pupils 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

we do 12 ÷ 3

we do 40 ÷ 5

we do 120 ÷ 4 and then multiply by 3

5

5. Percentages

Pupils are expected to have a sense of common percentages and their equivalent

fractions and decimals.

All pupils should learn the following table:

Percentage 100% 50% 333

1% 66

3

2% 25% 75% 20% 40% 60% 80% 10% 30% 70% 90%

Fraction 1 2

1

3

1

3

2

4

1

4

3

5

1

5

2

5

3

5

4

10

1

10

3

10

7

10

9

Decimal 1.0 0.5 0.33.. 0.67 0.25 0.75 0.2 0.4 0.6 0.8 0.1 0.3 0.7 0.9

Pupils are expected to find more complex percentages with the use of a calculator.

Pupils should recognise the word ‘of’ as meaning multiply and

% as meaning “divide by 100”.

For example 24% of 400 means calculate 100

24x400 = 96

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

so 65% of 40 = 20 + 4 + 2 = 26

Express 5

2 as a percentage.

5

2 =

10

4 =

100

40 = 40%

We separate the 65% into a

combination of simple percentages

that are much easier to calculate

6

Percentages in context:

An electrical shop has a 25% off sale. How much would a kettle cost if its

original price was £24?

Solution: 4

1 of £24 = £6.

Then sale price = £24 - £6

= £18

A £75 vacuum cleaner has been reduced by £15. Calculate the discount as a

percentage.

Solution: discount = 75

15 x 100

= 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 = 5000

1500 x 100

= 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 = 5

2.11 X 100

= 224%

Here we are being asked

to consider £24 less 25%

(or a quarter)

Percentage = difference x 100

increase/decrease original

Notice the answer is greater

than 100% because it has

increased by more than twice the

original quantity of fertiliser

7

6. 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 0942

5 mins + 2 hrs + 42 mins = 2hrs 47mins

Total time is 2hrs 47mins

7. 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 = 5.7 x 10 6

23,400,000 = 2.34 x 10 7

1,425,000,000 = 1.425 x 10 9

0.000025 = 2.5 x 10 5

0.000766 = 7.66 x 104

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

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

8

8. 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

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 pupils 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:

£ £

Opening Balance 36.00

CASH IN

Pocket Money 10.00

Present 6.00

Car Washing 5.00

Cash Available to Spend 57.00

CASH OUT

Game 42.00

Cinema 5.45

Drinks & Snacks 5.50

Closing Balance £ 4.05

Taylor has enough money and has £4.05 to spare.

REMEMBER

£ to foreign MULTIPLY

foreign to £ DIVIDE

Answer has 2 decimal places since money!

9

9. 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 cost 80p

1 costs 80p ÷ 5 = 16p

3 cost 16p x 3 = 48p

10. 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

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:

grey: white

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

For every 5 red marbles, there are 4 blue marbles

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.

Both of these numbers can

be divided by 5!

Both of these numbers

can be divided by 2!

10

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 blue

2 3

6 9

Quantity of green paint = 6 + 9

= 15 litres

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

x 3 x 3

Always answer the question!

There are 100cm in a metre.

There are 1000m in a kilometre.

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.

11

11. Measurement

We always use the metric system in maths but pupils should be made

aware of imperial units. Some useful information is shown below:

Metric Units Equivalence

Length Volume Mass 10mm = 1 cm 1000ml = 1 litre 1000mg = 1 g

100cm = 1 m 100cl = 1 litre 1000g = 1 kg

1000m = 1 km 1cm 3 = 1 ml 1000kg = 1 tonne

Imperial Units Equivalence

Length Volume Mass 1 inch = 2.5 cm 8 pints = 1 gallon 16 ounces = 1 pound

1 mile = 1.6 km 14 pounds = 1 stone

Approximations

Length Volume Mass

12 inches = 1 foot 1 litre = 14

3 pints 1kg = 2.2 pounds

Pupils can use the following diagram to help them with unit conversions within the

metric system.

Within technical, pupils will always measure in

millimetres. In graphic communications, pupils

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).

In some subjects

the term mass and

weight are used to

mean the same

thing but in science

you would be

expected to know

the difference

between the two.

Mass

All the matter objects

are made up of.

Weight

A force measured in

Newtons.

kilometres

(km)

metres

(m)

centimetres

(cm)

millimetres

(mm)

x 1000

x 100

x 10 ÷ 1000

÷ 100

÷ 10

12

12. Information handling

Pupils 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.

The pie is divided into

20 equal sections, so

each section is worth

5% of the pie

Unknown

Mammals

Frost

Insects & fungi

13

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

pencil and a RULER to

draw neat lines for

each section :0) Chicken wings

Chicken curry

Fish & chips

salad

If the slice is too thin

you can put the label

outside the pie

14

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

Displaying this information as a bar graph:

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).

Eye colour Frequency

Highest number to plot is 16.

Numbers must go up evenly

from zero on your axis.

Remember to use

pencil and a RULER to

draw neat bars :0)

Remember to label each axis

using the titles from your

table.

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.

15

This data could also be displayed as a pie chart. In maths pupils 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

Blue 16

Green 12

Brown 8

TOTAL 36

Convert the data into degrees:

Blue 36

16 x 360 = 160o

Brown 36

12 x 360 = 120o

Green 36

8 x 360 = 80o

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

but a histogram groups numbers into

ranges, which you decide on.

Notice: The bars of a histogram are

right next to each other and do not

have gaps between them.

Top tip: make sure you

leave gaps between the bars

of a Bar Graph, so it doesn't

look like a Histogram.

Age range (years)

Num

ber

of

chil

dre

n

Eye colour Frequency

Numbers are grouped

into an age range

No gaps are

left between

the bars

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)

16

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) 0 1 2 3 4 5

Volume of gas released (cm3)

0 18 38 62 80 80

0 1 2 3 4 5

Time (minutes)

80

70

60

50

40

30

20

10

0

Notice there are 5

small boxes

between 0 and 10

so each small box

is worth 2

We can see from the results

that as the time increases the

volume of gas released also

increases until 4 minutes.

After 4 minutes the volume of

gas produced remains constant.

Volu

me

of

gas

rel

ease

d (

cm3)

Units must be

included beside the

label on each axis

17

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 50

20 62

30 68

40 70

50 75

60 85

The results are used to draw a line graph:

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

measured by the pupil (i.e. the

results of the experiment) goes on

the Y axis

The thing (or ‘variable’) that

was changed by the pupil

goes on the X axis

18

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:

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:

Months

Rise in Sales Rise in Sales

A zig zag

should be

shown at

the bottom

of this axis

The rise in

sales should be

illustrated like

this

misleading

data

19

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.

Example:

Find the mean, median and mode for these numbers.

1 1 1 1 2 3 26

mean = 1 + 1+ 1 + 1 + 2 + 3 + 26 median = 4th number mode = 1

7

= 35 = 1

7

= 5

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: 4, 5, 7, 7, 9, 12.

range = highest – lowest

= 12 – 4

= 8

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).

Note in Science you would be expected

to state the range as 4 to 12 (lowest

to highest number).