PILOT OF A LINKED PAIR OF GCSEs in MATHEMATICS
GCSE Applications of Mathematics
Further notes on the Specification
NOVEMBER 2013
Unit 1 & 2 - Further notes and examples
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PILOT OF A LINKED PAIR OF GCSEs in MATHEMATICS
GCSE Applications of Mathematics – Unit 1
Unit 1 - Further notes and examples
Page 3
Number – Unit 1
FOUNDATION
HIGHER
Understand and use number operations and the
relationships between them, including inverse
operations and hierarchy of operation.
Understand and use number operations and the
relationships between them, including inverse
operations and hierarchy of operations.
Numbers and their representations including
powers, roots, indices (integers).
8 = 2 3 , 32 = 2 5, 25 , 3 64 .
Numbers and their representations including
powers, roots, indices (integers, fractional and
negative), and standard index form.
2
Simplify 81 , 8 3 .
Use the concepts and vocabulary of factor (divisor),
multiple, common factor, common multiple and
prime number.
Approximate to specified or appropriate degrees of
accuracy including a given power of ten, number of
decimal places and significant figures.
Divide a quantity in a given ratio.
Divide £1520 in the ratio 5 : 3 : 2.
Use the concepts and vocabulary of factor (divisor),
multiple, common factor, common multiple and
prime number.
Approximate to specified or appropriate degrees of
accuracy including a given power of ten, number of
decimal places and significant figures.
Divide a quantity in a given ratio.
Divide £1520 in the ratio 5 : 3 : 2.
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FURTHER NOTES / CLARIFICATION
Including BIDMAS.
Including addition, subtraction, multiplication and
division of whole numbers, decimals, fractions and
negative numbers and place value.
Foundation – Multiplying fractions with a whole
number, simple multiplying of fractions.
Including reciprocals.
Including the terms product and sum.
Including estimation.
Including questions such as:
White and black paint are mixed in the ratio 5:2 to
make grey paint. How much black paint is mixed
with 800ml of white paint?
Questions regarding recipes could be asked.
Measures – Unit 1
FOUNDATION
HIGHER
FURTHER NOTES / CLARIFICATION
Standard metric units of length, mass and capacity.
The standard units of time; the 12- and 24- hour
clock.
(The notation for the 12- and 24- hour clock will be
1:30 p.m. and 13:30.)
Convert measurements from one unit to another.
Convert measurements from one unit to another.
The use of common measures of time, length,
mass, capacity and temperature in the solution of
practical problems.
Knowledge and use of the relationship between
metric units.
Conversion between the following metric and
Imperial units:
km - miles; cm, m - inches, feet; kg - lb; litres - pints,
gallons.
Candidates will be expected to know and use the
following approximate equivalences.
8km ≈ 5 miles; 1kg ≈ 2.2 lb; 1 litre ≈ 175 pints
Interpret scales on a range of measuring
instruments and recognise the inaccuracy of
measurements.
Understand and use bearings.
Only three figures bearings will be used e.g. 009°,
065°, 237°.
Interpret scales on a range of measuring
instruments and recognise the inaccuracy of
measurements.
Understand and use bearings.
Only three figures bearings will be used e.g. 009˚,
065˚, 237˚.
Measure and draw lines and angles.
Lengths are accurate to 2mm and angles accurate
to 2°.
Measure and draw lines and angles.
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Compass points must not be used e.g. N 330 W
Algebra – Unit 1
Manipulate algebraic expressions by collecting like
terms, by multiplying a single term over a bracket,
and by taking out common factors.
Simplify 3a – 4b + 4a + 5b.
Expand 7(x – 3).
Simplify 2(3x – 1) – (x – 4).
Simplify x(x – 1) + 2(x2 – 3).
Factorise 6x + 4.
HIGHER
Manipulate algebraic expressions by collecting like
terms, by multiplying a single term over a bracket,
and by taking out common factors.
Simplify 3a – 4b + 4a + 5b.
Expand 7(x – 3).
Simplify 2(3x – 1) – (x – 4).
Simplify x(x – 1) + 2(x2 – 3).
Factorise 6x + 4.
Derive a formula, substitute numbers into a formula.
Wage earned = hours worked  rate her hour.
Find the wage earned if a man worked for 30 hours
and was paid at the rate of £4.50 per hour.
Derive a formula, substitute numbers into a formula.
Wage earned = hours worked  rate her hour.
Find the wage earned if a man worked for 30 hours
and was paid at the rate of £4.50 per hour.
Find the value of 6f + 7g when f = – 3 and g = 2.
Find the value of 6f + 7g when f = – 3 and g = 2.
Use the conventions for coordinates in the plane
and plot points in all four quadrants.
Recognise and plot equations that correspond to
straight-line graphs in the coordinate plane.
Use the conventions for coordinates in the plane
and plot points in all four quadrants.
Recognise and plot equations that correspond to
straight-line graphs in the coordinate plane.
Find approximate solutions of equations using
graphical methods and systematic trial and
improvement.
Find, by trial and improvement, the solution of the
equation x3 – 5x = 80 which lies between 4 and 5.
Give your answer correct to 1 decimal place.
Find approximate solutions of equations using
graphical methods and systematic trial and
improvement.
Find, by trial and improvement, the solution of the
equation x3 – 5x = 80 which lies between 4 and 5.
Give your answer correct to 1 decimal place.
FOUNDATION
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FURTHER NOTES / CLARIFICATION
The examples listed in the previous column are
skills needed by candidates. The questions
assessing these skills will be set in real-life
contexts.
e.g.
Owen has a pile of 5 pence and 10 pence coins. He
counts the number of 10 pence coins and finds that
he has x of them.
(a) Write down, in terms of x, the total value of 10
pence coins.
(b) Owen has four more 5 pence coins than 10
pence coins. Write down, in terms of x, how many 5
pence coins he has.
(c) Write down, in terms of x, the total value of 5
pence coins.
(d) Write down, in terms of x, the total value of all
the coins that Owen has. You must simplify your
answer as far as possible.
Including deriving expressions e.g.
Boxes of sweets each contain 2x – 3 sweets. Aled
buys 6 boxes and Yasmin buys 9 boxes of sweets.
Write down an expression, in terms of x, for the
total number of sweets bought by Aled and
Yasmin.
Including function machines.
Sometimes the interval in which the solution lies will
not be given.
Further examples – When the stone has been in the
air for t seconds, its height above sea level, h
metres, is given by h = 24t – 5t2 + 50. Complete the
table of values, draw the graph and write down the
times when the stone is 10 metres above the cliff.
Algebra – Unit 1
FOUNDATION
Find and interpret gradients and intercepts of
straight line graphs in practical contexts.
Construct linear functions from real-life problems
and plot their corresponding graphs.
e.g. conversion graphs.
HIGHER
Find and interpret gradients and intercepts of
straight line graphs in practical contexts.
Construct linear, quadratic and other functions
from real-life problems and plot their corresponding
graphs. e.g. conversion graphs.
Interpret the gradient at a point on a curve as the
rate of change.
Recognise and use graphs that illustrate direct
proportion.e.g. conversion graphs.
Discuss, plot and interpret graphs (which may be
non-linear) modelling real situations, including
journeys / travel graphs.
Recognise and use graphs that illustrate direct and
inverse proportion. e.g. conversion graphs.
Discuss, plot and interpret graphs (which may be
non-linear and/or periodic) modelling real
situations, including journeys / travel graphs.
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FURTHER NOTES / CLARIFICATION
Including travel graphs.
Some useful resources http://www.cimt.plymouth.ac.uk/projects/mepres/ale
vel/pure_ch8.pdf
Geometry – Unit 1
FOUNDATION
HIGHER
Recall and use properties of angles at a point,
angles at a point on a straight line (including right
angles), perpendicular lines, and vertically opposite
angles.
Recall and use properties of angles at a point,
angles at a point on a straight line (including right
angles), perpendicular lines, and vertically opposite
angles.
Understand and use the angle properties of parallel
and intersecting lines, triangles and quadrilaterals.
Understand and use the angle properties of parallel
and intersecting lines, triangles and quadrilaterals.
Recall the properties and definitions of special types
of quadrilateral, including square, rectangle,
parallelogram, trapezium, kite and rhombus.
Recall the properties and definitions of special types
of quadrilateral, including square, rectangle,
parallelogram, trapezium, kite and rhombus.
Use 2D representations of 3D shapes.
Use 2D representations of 3D shapes.
Use and interpret maps and scale drawings
Use and interpret maps and scale drawings.
Draw triangles and other 2D shapes using a ruler,
pair of compasses and protractor.
Draw triangles and other 2D shapes using a ruler,
pair of compasses and protractor.
Use straight edge and a pair of compasses to do
constructions.
Use straight edge and a pair of compasses to do
constructions.
Construct loci.
Estimate areas of irregular shapes.
Calculate perimeters and areas of shapes made
from triangles and rectangles.
Find circumferences of circles and areas enclosed
by circles.
FURTHER NOTES / CLARIFICATION
Including reflection and rotational symmetry
Including knowledge of 3D shapes, nets, plans and
elevations.
Scales may be written in the form 1cm represents
5m or 1:500
Including quadrilaterals and circles.
Including bisecting lines and angles.
Constructing angles of 60, 30, 90 and 45.
Candidates will be asked to show their construction
lines.
Constructing the locus of a point which moves such
that it is
(i) a given distance from a fixed point or line,
Construct loci.
(ii) equidistant from two fixed points or lines.
Solving problems involving intersecting loci in two
dimensions.
Estimate areas of irregular shapes and areas under Higher - including use of the Trapezium rule or
curves.
methods of splitting the area from first principles.
Calculate perimeters and areas of shapes made
Including perimeters and areas of squares,
from triangles and rectangles and other shapes.
rectangles, triangles, parallelograms, trapezium and
composite shapes.
Including surface area
Find circumferences of circles and areas enclosed
To include shapes made from semicircles and
by circles.
quarter circles and reverse problems.
Higher - to include area of sectors and
segments.
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Statistics and Probability – Unit 1
FOUNDATION
Understand and use the vocabulary of probability
and the probability scale.
Understand and use theoretical models for
probabilities including the model of equally likely
outcomes.
Understand and use estimates of probability from
relative frequency.
HIGHER
Understand and use the vocabulary of probability
and the probability scale.
Understand and use theoretical models for
probabilities including the model of equally likely
outcomes.
Understand and use estimates of probability from
relative frequency.
FURTHER NOTES / CLARIFICATION
Including expected values
Including knowledge of the cycle:
Understand and use the statistical problem solving
process/handling data cycle.
Understand and use the statistical problem solving
process/handling data cycle.
Useful resource http://www.censusatschool.org.uk/resources/relevant-aengaging-stats/257-chap1
The term ‘hypothesis’
Higher - discuss sampling methods (candidates’ will
not be expected to generate samples by using
various methods e.g. stratified sampling, however,
they should be aware of the need for sampling
Awareness of bias.
Design an experiment or survey, identifying possible
sources of bias.
Design an experiment or survey, identifying possible
sources of bias.
Produce and interpret diagrams for grouped
discrete data and continuous data, including
histograms with unequal class intervals.
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Statistics and Probability – Unit 1
FOUNDATION
HIGHER
FURTHER NOTES / CLARIFICATION
Produce and use cumulative frequency graphs
and box-and-whisker plots.
Work with time series, including their graphical
representation.
Calculate the median, mean, range, mode and
modal class.
Understand that when a statistical experiment or
survey is repeated there will usually be different
outcomes, and that increasing sample size
generally leads to better estimates of probability
and population characteristics.
Discuss and start to estimate risk.
Work with time series and moving averages,
including their graphical representation.
Scale will not always be included.
Understand and use the term ‘skew’.
Including comparing distributions using box-andwhisker plots
Plot, interpret time series graphs.
Higher – Calculate, plot and interpret moving
averages. Plotting and interpreting trend lines
Calculate, and for grouped data estimate, median,
mean, range, quartiles and inter-quartile range,
mode and modal class.
Understand that when a statistical experiment or
survey is repeated there will usually be different
outcomes, and that increasing sample size generally
leads to better estimates of probability and
population characteristics.
Discuss and start to estimate risk.
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Understanding risk as a measure of the uncertainty
of an event with its utility or consequences.
Useful resources –
http://understandinguncertainty.org/
http://understandinguncertainty.org/node/604
http://www.bowland.org.uk/case_studies/how_risky
_is_life/start.htm
PILOT OF A LINKED PAIR OF GCSEs in MATHEMATICS
GCSE Applications of Mathematics Unit 2
Further notes on the Specification
Unit 2 - Further notes and examples
Page 11
Number – Unit 2
FOUNDATION
Understand and use number operations and the
relationships between them, including inverse
operations and hierarchy of operations.
Approximate to specified or appropriate degrees of
accuracy including a given power of ten, number of
decimal places and significant figures.
Understand that 'percentage' means 'number of
parts per 100' and use this to compare proportions.
Use multipliers for percentage change.
HIGHER
Understand and use number operations and the
relationships between them, including inverse
operations and hierarchy of operations.
Standard index form.
Approximate to specified or appropriate degrees of
accuracy including a given power of ten, number of
decimal places and significant figures.
Understand and use upper and lower bounds.
The lower and upper bounds of 140 (to the nearest
10) are 135 and 145 respectively.
FURTHER NOTES / CLARIFICATION
Including whole numbers, decimals, fractions and
negative numbers and understanding place value.
Including BIDMAS.
Foundation – Multiplying fractions with a whole number,
simple multiplying of fractions.
With positive and negative powers of 10.
Including estimation.
Including calculating the upper and lower bounds in
calculations involving addition, subtraction, multiplication
and division of numbers expressed to given degrees of
accuracy.
Understand that 'percentage' means 'number of
parts per 100' and use this to compare proportions.
Use multipliers for percentage change; work with
repeated percentage change; solve reverse
percentage problems.
Given that a meal in a restaurant costs £36 with
VAT at 17·5%, its price before the VAT is calculated
as £ 36 .
1  175
Interpret fractions, decimals and percentages as
operators.
Find proportional change.
Understand and use direct and proportion.
Use calculators effectively and efficiently, including
statistical functions.
Interpret fractions, decimals and percentages as
operators.
Find proportional change and repeated
proportional change.
Exponential growth/decay, its relationship with
repeated proportional change including
financial and scientific applications.
Understand and use direct and inverse proportion.
Use calculators effectively and efficiently, including
trigonometrical and statistical functions.
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Including equivalences between decimals, fractions,
ratios and percentages.
Including compound interest and depreciation.
Financial and Business Applications – Unit 2
FOUNDATION
Carry out calculations relating to enterprise, saving
and borrowing, appreciation and depreciation.
The value of a car is £12,000. Each year its value
decreases by 10%. Find the value of the car at the
end of three years.
Use mathematics in the context of personal and
domestic finance including loan repayments,
budgeting, RPI and CPI, exchange rates and
commissions.
e.g. fuel and other fuel bills, hire purchase, VAT,
taxation, discount, best buys, wages and salaries.
HIGHER
Carry out calculations relating to enterprise, saving
and borrowing, appreciation and depreciation and
understand AER.
The value of a car is £12,000. Each year its value
decreases by 10%. Find the value of the car at the
end of three years.
Use mathematics in the context of personal and
domestic finance including loan repayments,
budgeting, RPI and CPI, exchange rates and
commissions.
FURTHER NOTES / CLARIFICATION
Language and concepts at Foundation & Higher Tier
Basic ideas of banking – savings, including maintaining a
simple 3 column bank account sheet
Bank account, credit, debit, balance, withdrawal, deposit,
brought forward, carried forward
Investment – savings and investments – how they grow.
Investment, interest rate, compound interest, simple interest,
per annum, loan, gross rate, net rate
Personal Finance – basic money management, buying using
a credit plan, wages and tax.
Rent, (electricity) bill, credit plan, timesheet, basic rate,
overtime rate, tax free, tax rate, income tax, taxable pay,
household budget, repayments
Enterprise - basic money management, depreciation of
equipment, as personal finance
Depreciation, VAT, expenses, commission, profit
Currency transactions - changing sterling into a foreign
currency and vice versa.
Exchange, price comparisons, commission
Comparison of ‘true’ price by converting to a common
currency.
Inflation calculating the effect of inflation on prices and
wages
Inflation, index number, cost of living (index), base year.
Language and concepts at Higher Tier
Issues are those given at Foundation tier with the following
additions
Investment
Principal, AER, Mortgage
Compound interest
The AER formulae will be given.
Inflation - calculating the effect of inflation on prices and
wages
Retail Price Index (RPI), Consumer Price Index (CPI)
Questions will not be set which require and knowledge of
the difference between the two indices.
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Use spreadsheets to model financial, statistical and
other numerical situations.
Use spreadsheets to model financial, statistical and
other numerical situations.
Construct and use flow charts.
Construct and use flow charts.
Basic spreadsheet formula notation. Use BIDMAS
e.g. =A2+B2
=B2-A2
=A2*B2
=A2*5 + B3*8
=sum(A2:..)
=Average(A2:..)
The following flowchart notation should be used.

to indicate the flow

process

input/ output


a rounded rectangle for
start/stop
decision box
Candidates should construct the most efficient
flowchart.
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Measures – Unit 2
FOUNDATION
Convert measurements from one unit to another.
HIGHER
Convert measurements from one unit to another.
Make sensible estimates of a range of measures.
Make sensible estimates of a range of measures.
Understand and use compound measures in
familiar contexts.
Understand and use compound measures in
familiar and unfamiliar contexts.
Page 15
FURTHER NOTES / CLARIFICATION
Standard metric units of length, mass and capacity.
The standard units of time; the 12- and 24- hour clock.
(The notation for the 12- and 24- hour clock will be 1:30
p.m. and 13:30.)
The use of common measures of time, length, mass,
capacity and temperature in the solution of practical
problems.
Knowledge and use of the relationship between metric
units.
Conversion between the following metric and Imperial
units:
km - miles; cm, m - inches, feet; kg - lb; litres - pints,
gallons.
Candidates will be expected to know and use the
following approximate equivalences.
8km ≈ 5 miles; 1kg ≈ 2.2 lb; 1 litre ≈ 175 pints
Use compound measures including speed and density.
Use of compound measures such as m/s, km/h, mph,
mpg, kg/m3, g/cm3
Algebra – Unit 2
FOUNDATION
Set up, and solve simple equations and inequalities
The angles of a quadrilateral are x˚, 49˚, 3x˚ and
111˚.Form an equation in x, and use your equation
to find the value of x.
Three times a number n plus 6 is less than 27.
Write down an inequality which is satisfied by n
and rearrange it in the form n < a where a is a
rational number.
Solve x + 6 = 15,
12
,
x
12
x
,
3
3
5x + 2 = 17,
10x + 9 = 6x + 11,
3(1 – x) = 5(2 + x),
(x – 1) = 3x + 1.
Solve linear inequalities in one variable, and
represent the solution set on a number line.
Solve 3x + 1  7.
Solve 4 – x  5.
HIGHER
Set up, and solve simple equations and inequalities.
Three times a number n plus 6 is less than 27. Write
down an inequality which is satisfied by n and
rearrange it in the form n < a where a is a rational
number.
Solve
FURTHER NOTES / CLARIFICATION
At Foundation candidates will be required to solve simple
linear inequalities with whole numbers and fractional
coefficients.
x + 6 = 15,
12
,
x
12
x
,
3
3
5x + 2 = 17,
10x + 9 = 6x + 11,
3(1 – x) = 5(2 + x),
(x – 1) = 3x + 1.
Solve 3x + 1  7.
Solve 4 – x  5.
Solve linear inequalities in one or two variables,
and represent the solution set on a number line or
suitable diagram.
Set up and solve problems in linear
programming, finding optimal solutions.
Set up and solve linear simultaneous equations
in two unknowns.
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Higher - including the use of straight line graphs to locate
regions given by linear inequalities.
Candidates may be required to set up the inequalities
required from the question, locate regions and find
optimal solutions.
Candidates may be required to set up the equations
using given information.
Both equations may be of the form ax + by = c
Geometry – Unit 2
FOUNDATION
HIGHER
Recognise reflection and rotation symmetry of 2D
shapes.
Recognise reflection and rotation symmetry of 2D
shapes.
Understand congruence and similarity, including the
relationship between lengths, in similar figures.
Understand congruence and similarity, including the
relationship between lengths, areas and volumes
in similar figures.
Use Pythagoras’ theorem in 2D.
Use Pythagoras’ theorem in 2D and 3D.
Distinguish between centre, radius, chord, diameter,
circumference, tangent, arc, sector and segment.
Calculate perimeters and areas of shapes made
from triangles and rectangles.
Use the trigonometric ratios to solve 2D and 3D
problems.
Distinguish between centre, radius, chord, diameter,
circumference, tangent, arc, sector and segment.
Calculate perimeters and areas of shapes made
from triangles and rectangles and other shapes.
Calculate volumes of right prisms and of shapes
made from cubes and cuboids.
Calculate volumes of right prisms and of shapes
made from cubes and cuboids.
Solve mensuration problems involving more
complex shapes and solids.
Page 17
FURTHER NOTES / CLARIFICATION
Simple description of symmetry in terms of reflection in a
line/plane or rotation about a point.
Order of rotational symmetry.
Including reverse problems
e.g. proving that a triangle is right angled.
Calculating a side or an angle of a right-angled triangle.
Including angles of elevation and depression.
Including perimeters and areas of squares, rectangles,
triangles, parallelograms, trapezium, and composite
shapes.
Including surface area
Including the area of circles, semicircles quarter circles
Higher – to include the area of sectors and segments
Including volumes of spheres, hemispheres, cylinder,
cone, truncated cone (frustum) and pyramids.
Statistics and Probability – Unit 2
FOUNDATION
HIGHER
Design data-collection sheets distinguishing
between different types of data.
Design data-collection sheets distinguishing between
different types of data.
Extract data from publications, charts, tables
and lists.
Extract data from publications, charts, tables and lists.
Design, use and interpret two-way tables for
discrete and grouped data.
Look at data to find patterns and exceptions.
Compare distributions and make inferences.
Design, use and interpret two-way tables for discrete and
grouped data.
Look at data to find patterns and exceptions.
Compare distributions and make inferences
Produce and interpret charts and diagrams
for categorical data including bar charts, pie
charts and pictograms.
Produce and interpret diagrams for
ungrouped discrete numerical data, including
vertical line charts and stem-and-leaf
diagrams.
Recognise correlation and draw and/or use
lines of best fit by eye, understanding and
interpreting what these represent, and
appreciating that correlation does not imply
causality.
Discuss and start to estimate risk.
FURTHER NOTES / CLARIFICATION
Including use of the range, mode and median.
Including using/calculating of the mean, range,
mode and median.
Produce and interpret charts and diagrams for categorical
data including bar charts, pie charts and pictograms.
Produce and interpret diagrams for ungrouped discrete
numerical data, including vertical line charts and stem-andleaf diagrams.
Including use of the range, mode, median and
modal group when interpreting stem-and-leaf
diagrams.
Recognise correlation and draw and/or use lines of best fit
by eye, understanding and interpreting what these
represent, and appreciating that correlation does not imply
causality.
Including plotting and interpreting scatter graphs
Discuss and start to estimate risk.
Understanding risk as a measure of the uncertainty of an
event with its utility or consequences.
Useful resources –
http://understandinguncertainty.org/
http://understandinguncertainty.org/node/604
http://www.bowland.org.uk/case_studies/how_risky_is_lif
e/start.htm
Page 18