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Numeracy in Sciencea common approach
A resource to support the development of numeracy skills required for KS3 & 4 Sciences
Using the resource• The areas of mathematics that arise naturally from
the science content in science GCSEs are listed in slide 3.
• This is not a checklist for each question paper or Controlled Assessment, but assessments reflect these mathematical requirements, covering the full range of mathematical skills over a reasonable period of time.
• Each area is hyperlinked to a series of slides that models a strategy
Numerical skills required for GCSE Sciences1. Understand number size and scale and the quantitative relationship between units.2. Understand when and how to use estimation.3. Carry out calculations involving +, – , x, ÷, either singly or in combination, decimals
, fractions, percentages and positive whole number powers.4. Provide answers to calculations to an appropriate number of significant figures.5. Understand and use the symbols =, <, >, ~.6. Understand and use direct proportion and simple ratios.7. Calculate arithmetic means.8. Understand and use common measures and simple compound measures such as speed.9. Plot and draw graphs (line graphs, bar charts, pie charts, scatter graphs, histograms
) selecting appropriate scales for the axes.10. Substitute numerical values into simple formulae and equations using appropriate units.11. Translate information between graphical and numeric form.12. Extract and interpret information from charts, graphs and tables.13. Understand the idea of probability.14. Calculate area, perimeters and volumes of simple shapes. In addition, higher level candidates must also
15. Interpret, order and calculate with numbers written in standard form.16. Carry out calculations involving negative powers (only -1 for rate).17. Change the subject of an equation.18. Understand and use inverse proportion.19. Understand and use percentiles and deciles.
Maths grades of the numerical skills required1. Understand number size and scale and the quantitative relationship between units./grade G+2. Understand when and how to use estimation./grade F-E3. Carry out calculations involving +, – , x, ÷, either singly or in combination, decimals, fractions, percentages
and positive whole number powers./grade G-C4. Provide answers to calculations to an appropriate number of significant figures./ grade D-C5. Understand and use the symbols =, <, >, ~./grade G+6. Understand and use direct proportion and simple ratios./grade D+7. Calculate arithmetic means./grade E+8. Understand and use common measures/ grade G-F and simple compound measures/grade D+ such as
speed.9. Plot and draw graphs (line graphs/grade E+, bar charts/ grade G+, pie charts/ grade E+, scatter
graphs/grade D+, histograms/ grade A) selecting appropriate scales for the axes.10. Substitute numerical values into simple formulae and equations using appropriate units./grade F+11. Translate information between graphical and numeric form./grade D+12. Extract and interpret information from charts, graphs and tables./ dependant on diagram13. Understand the idea of probability./ grade F+14. Calculate area, perimeters and volumes of simple shapes. / grade dependant on shape calculated In addition, higher level candidates must also
15. Interpret, order and calculate with numbers written in standard form./ grade B+16. Carry out calculations involving negative powers (only -1 for rate)./grade B+?17. Change the subject of an equation. /grade C+18. Understand and use inverse proportion. /grade B+19. Understand and use percentiles and deciles./ not formally covered at KS4 maths
Writing one number as a percentage of another
E.g. What percentage is 48 boys out of a year group of 180?
48 out of 180 = 48 ÷180 = 0·27
= 27%
To calculate the percentage of an amount
E.g. 35% of 72
= 0·35 × 72 = 25·2
Convert to decimal
of means times
48
180
Calculating percentages3
When the answer to your calculation has many decimal places it might be appropriate to round
the number
To 1 d.p. = 6·24823567
So we write 6·2
The number after the line is less than 5 so the number at
1 decimal place does not need changing
E.g. If your answer was 6·24823567…
To 2 d.p. = 6·24823567
So we write 6·25
The number after the line is more than 5 so the number at 2 decimal place needs to
be rounded up
Rounding4
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Significant figures
Digits of a number are kept in place by zeros where necessary.
Zeros at the beginning or end don’t usually count, but zeros ‘inside’ the number do.
The rounded answer should be a suitable reflection of the original number
E.g. 24,579 to 1 s.f could not possibly be 224,579 to 1 s.f is 20,000
Further examples on next page…
4
Deciding on Significant figures4
Number 1 sf 2sf3456 3000 3500345·5 300 35034·56 30 353·456 3 3·50·3456 0·3 0·350·03456 0·03 0·0350·003456 0·003 0·0035
Example 2A steel alloy rod is made up of 60% iron and 40% copper.
Ratio: what does this mean?
So the ratio of iron to copper is 60:40.
We could simplify this to give 3:2
Ratio is used to compare the sizes of two (or more) quantities
6
Example 1:A drink is made by mixing two parts orange juice with five parts water.
This relationship of 2 to 5 can be written as the ratio 2:5.
next
Simplifying a ratio6
Oil : Petrol
4 : 100
2 : 50
1 : 25
Use a similar approach to simplifying fractions
What would divide into both sides?
Keep going until no further simplification
is possiblenext
Dividing a quantity in a ratio
Divide 560kg in the ratio 3:4
To find 1 part, divide the amount by the total number of parts: 560 ÷ 7 = 80kg
Multiply to calculate each share:3 × 80 = 240kg4 × 80 = 320kg
So dividing 560kg in the ratio 3:4 is 240kg and 320kg
6
Total number of parts = 7
(3 + 4)
There are 3 different averages commonly used.MEAN, MODE and MEDIAN
You also have RANGEThe range of a set of data is the difference between
the highest and the lowest data values.
In an examination the highest mark is 80% and the lowest mark is 45%
Range is 35% because 80% - 45% = 35% It is not acceptable to leave the answers as an interval.
eg 45% → 80%
Calculating averages7
next
Mean = adding up all the values and dividing by the number of values.
For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Mean = 3 + 2 + 5 + 8 + 4 + 3 +6 + 3 + 2 + 8 = 44 = 4·4 10 10
Calculating mean7
next
Mode is the most common value. It is sometimes called the modal group.
For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Mode is 3because 3 is the value which occurs most often
Calculating mode7
next
Median is the middle value when a set of values has been arranged in order.
For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Median – is 3·5because 3·5 is in between 3 and 4 when the values are put
in order (2, 2, 3, 3, 3, 4, 5, 6, 8, 8)
If there are an even number of values it is the mean of the centre two).
Calculating median7
Rearranging formula
The ‘triangle trick’ is used to help rearrange formula for speed/distance/time and density/mass/volume
d
s t
Speed = distance ÷ time
Time = distance ÷ speed
Distance = speed × time
8
next
Gradient =
Draw a triangle on your graph
Y
X
Calculating gradient of a line8
6 squares ÷ 3 units = 2
0
2
4
6
8
10
12
Brow n Blue Green
If your data is discrete (in categories) the bars must
not touch
Eye Colour Frequency
Brown 10
Blue 5
Green 2
0
2
4
6
8
10
12
0 5 10 15 20 25
Height Frequency
0 ≤ h < 5 12
5 ≤ h < 10 4
10 ≤ h < 15 3
15 ≤ h < 20 2
If your data is continuous the bars should be next to each
other so that all possible values are included
9 Bar Charts
next
Pie charts9
next
Mode of Transport Frequency Angle
calculation Angle
Walk 10 10 × 12 120˚Train 3 3 × 12 36˚Car 5 5 × 12 60˚Bus 6 6 × 12 72˚
Other 6 6 × 12 72˚Total 30 360˚
As a pie chart is based on a circle if the numbers involved are simple it will be possible to calculate simple fractions of 360˚.
1 person’s share of the pie chart 360 ÷ total frequency
360 ÷ 30 = 12°
walk
traincar
bus
other
Pie Chart showing pupil modes of transport
Pie charts9
next
With more difficult numbers we would expect to use a calculator. However, the approach remains the same.
Any calculations of angles should be rounded to the nearest degree only at the final stage of the calculation
Subject Number of pupils Pie Chart AngleEnglish 53 53 × 2·4 = 127·2˚
Mathematics 65 65 × 2·4 = 156˚Science 32 32 × 2·4 = 76·8˚
Total 150 360˚
1 person’s share of the pie chart 360 ÷ total frequency
360 ÷ 150 = 2·4°
Scatter graphsThese are used to compare two sets of numerical data. The two
values are plotted on two axes labelled as for continuous data.
9
next
8 10 12 14 16 18 20 22 24 260
10
20
30
40
50
60
70
A simple Scatter Graph
Temperature (degrees C)
Nu
mb
er o
f ic
e cr
ems
sold
The graph shows a positive correlation between the two
variables.However you need to ensure
that there is a reasonable connection between the two,
e.g. ice cream sales and temperature. Plotting use of
mobile phones against cost of houses will give two increasing
sets of data but are they connected?
Scatter graphs9
0 2 4 6 8 10 12 140
2,000
4,000
6,000
8,000
10,000
12,000
Negative Correlation
Number of pets
0
2
4
6
8
10
0 20 40 60 80 100
Height (cm)
Nu
mb
er o
f P
ets
Negative correlation depicts one variable increasing as the other
decreases.
No correlation comes from a random distribution of points
If possible a ‘line of best fit’ should be drawn. If students are expected to draw the line of best fit, this is done 'by eye'
next
Histograms9
Frequency density = Frequency of class interval Width of class interval
The area of each bar represents the class or group frequency of the bar. This means each bar can have a different width.
Price (P) in pounds, £ Frequency Frequency density
0 < P ≤ 5 40 40 ÷ 5 = 8
5 < P ≤ 10 60 60 ÷ 5 = 12
10 < P ≤ 20 60 60 ÷ 10 = 6
20 < P ≤ 40 40 40 ÷ 20 = 2
next
Histograms9
The horizontal axis has a continuous scale since it represents continuous data, such as time, weight or length.
The vertical axis is always labelled
‘Frequency density’
A histogram never has gaps between the bars
P rice ( ) in p o u n d s (£ )P0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
F req u en cyd en sity(n u m b ero f b o o k sp e r £ )
2 0
1 6
1 2
8
4
0
Substitution
Need to know the order of operations:BracketsIndicesDivisionMultiplicationAdditionSubtraction
This equally applies to all number as well as algebra
10
next
These have the same precedence
These have the same precedence
Substitution – replacing the letter representing a variable by a number
SubstitutionThis formula allows you to substitute any °F temperature to find its equivalent temperature in °C
c = 5(f – 32) where f represents the temperature in °F. c represents the temperature in °C
ExampleWhen f = 68
5(f - 32) ÷ 9 = 5(68 - 32) ÷ 9
5(36) ÷ 9 = (5 × 36) ÷ 9 = 180 ÷ 9 = 20 So 68°F = 20°C
A number next to anything in brackets means the contents of
the brackets should be multiplied, so 5(36) means
5 × 36:
Work out the brackets first: (68 – 32) = 36
9
10
Line graphs
Line graphs are used in Science and maths to show how data changes over a period of time.
In maths, these are also called ‘trend graphs’
12
next
The horizontal scale across the bottom
and the vertical scale along the side tell us how much or
how many. These must go up in equal
jumps
The points or dots on the graph show
us the facts
The title of the line graph tells us what the graph is about
The horizontal label across the bottom and the
vertical label along the side tells us
what kinds of facts are listed
The lines connecting the points give estimates of the values between the
points
12
Area of a rectangle = length × width
Area of a triangle = (base × height) ÷ 2
Separate the area under the graph into rectangles and
triangles
Calculating area under a graph12
Total area = A + B + C
ProbabilityBased on the concept that the probability scale runs from 0
(impossible) to 1 (certain)
13
next
Can be expressed as a fraction, a decimal or a percentage
Theoretical probability
P(event) = Number of success Total number of outcomes
E.g.: when tossing a coin P(head) = ½
Relative frequency
Relative Freq = Number of successful trials Total number of trials
E.g.: an experiment found that a tossed coin landed on heads 12 out of 20 throws. P(head) = 12⁄20
Simple Shapes – area and perimeter14
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Simple Shapes – area and perimeter14
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Simple Shapes – area and perimeter14
next
Simple Shapes – surface area and volume14
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Simple Shapes – surface area and volume14
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Simple Shapes – surface area and volume14
Negative Powers16
Example
52 ÷ 54 = 5(2-4) = 5-2 = 1 = 1 52 125
A negative power is best shown by:
a2 ÷ a5 = a × a = 1 a × a × a × a × a a3
We also know that:
a2 ÷ a5 = a 2-5 = a-3. So 1 = a-3
a3
Standard Index Form15
next
The ‘Standard’ refers to the fact that all numbers have a similar appearance when written in ‘standard form’. The ‘Index’ refers to the index or power. All numbers in Standard Form take the form: a × 10x
Where 1 ≤ a < 10 and x is an integer (positive or negative whole number)
Standard (Index) Form is a simple way to write very large and very small numbers.
Standard (Index) Form15
next
Adding and subtracting numbersConvert them into ordinary numbers, do the calculation, then change them back if you want the answer in standard form.
Example 14·5 × 104 + 6·45 × 105
= 45,000 + 645,000= 690,000= 6·9 × 105
Standard (Index) Form15
Multiplying and dividing numbersHere you can use the rules for multiplying and dividing powers.
Remember these rules:To multiply powers you add,E.g. 105 × 103 = 108 To divide powers you subtractE.g. 105 ÷ 103 = 102
Example 2Simplify (2 × 103) × (3 × 106)Solution(2 × 103) × (3 × 106) = 6 × 109
Multiply together to give 6
Use rule to multiply powers
Changing the subject of the formula
a + b = c
To rearrange the formula to make b the subject:
The method
is the same as solving
equations
-a
b = c - a
17
c is the subject
a + b = c -a -a from both sides
next
Changing the subject of the formula
To rearrange so that b is on the right
bx + c = a-c-c
bx = a - c÷x ÷x
b = a – c x
17
bx + c = a
-c from both sides
÷x from both sides
next
Changing the subject of the formula
n = m - 3s
n + 3s = m 3s = m - n
s = m - n 3
17
To rearrange to make s the subject
n = m - 3s +3s to both sides +3s +3s
-n from both sides
Divide both sides by 3
Inverse proportionInverse proportion is when one value increases as the
other value decreases
18
Exampley is inversely proportional to x. When y = 3, x = 12 .Find the constant of proportionality, and the value of x when y = 8.If y is inversely proportional to x we can write it as y ∝ 1/x Or, y = k/x where k is a constant
So xy = k Substituting the values of x and y gives:3 × 12 = 36 So k = 36
To find the value of x when y = 8, substitute k = 36 and y = 8 into xy = k 8x = 36 So x = 4·5
