Download AQA GCSE Mathematics Linked Pair Topics to be assessed in the

AQA
GCSE Mathematics
Linked Pair
Topics to be assessed in the M1 (Methods 1)
examination
Page 1
Foundation Tier Content
M1.N1
Understand and use
number operations and
the relationship
between them,
including inverse
operations and
hierarchy of
operations.
Candidates’ ability to
perform number
operations and show
knowledge of
BODMAS will be
assessed in Section
B (non-calculator). In
Section A the need to
perform number
operations will occur
in the context of
problems in number,
algebra or probability
and candidates will
be expected to use a
calculator.
Section A:
1. Complete the shopping bill
0.8 kg apples @ 85 p per kg .….
….. kg pears @ 95 p per kg .….
Total
£1.82
2. Complete the magic square
24
5
7
15
6
Section B:
1. Put brackets in the following calculation to
make it true
2 + 3  4 = 20
2. Work out
(a) 173 + 45
(b) 602 – 278
(c) 452 ÷ 4
(d) 23  17
M1.N2
Page 2
Arithmetic of real
numbers: add,
subtract, multiply and
divide any number.
Non calculator
arithmetic
competency will be
assessed in this unit.
In Section B (noncalculator):
Calculations will be
restricted to 3-digit
integers and
decimals up to two
decimal places.
Multiplication of
integers will be
limited to 3-digit
integers by
2-digit integers.
Multiplication and
division involving
decimals will be
limited to multiplying
or dividing by a single
digit integer or
decimal number to 1
significant figure.
Addition and
Section A:
A four-sided home-made spinner
is tested.
The table shows the results
Number
Frequency
1
56
2
72
3
49
4
23
(a) Work out the relative
frequency of a 2
(b) Is the coin fair?
Give a reason for your
answer
Section B:
Work out
(a) 42 ÷ 0.3
(b)
6.8  0.7
subtraction of
fractions will be
assessed.
In Section A
(calculator allowed):
The four rules will be
assessed in the
context of problems
in number, algebra
and probability.
M1.N3
Numbers and their
representation
including powers,
roots, indices
(integers).
M1.N5
Use the concepts and
vocabulary of factor
(divisor) multiple and
prime numbers.
M1.N6
Understand that
‘percentage’ means
‘number of parts per
100’ and use this to
compare proportions.
Page 3
Knowledge of
powers, roots and
indices will be
assessed in Section
B (non calculator)
although they can
occur in number,
algebra and
probability questions
in Section A.
Candidates should
also understand that
an integer is a
positive or negative
whole number and
would, for example, if
describing the integer
members of two
overlapping
inequalities or sets,
include zero.
Candidates should
know the squares
and corresponding
roots up to 15  15
and the cubes and
corresponding roots
of the cubes of 1, 2,
3, 4, 5 and 10.
These terms will not
be explicitly assessed
in this unit but their
meaning should be
known as they could
be used in a question
on number, algebra
or probability. These
terms could appear in
any section.
Section A:
Questions explicitly
assessing knowledge
of percentage will be
assessed in this unit
and in M2.
In Section B (non
calculator) questions
will be based on a
starting point of
calculating 50%,
25%, 10% or 1%.
Section A:
In bag A the probability of selecting a blue ball
at random is 45%.
Bag B contains 50 balls of which 23 are blue.
Which bag has the higher probability of
selecting a blue ball at random?
Use your calculator to work out 4.5 4
Section B:
(a) Write down √196
(b) Work out 17 2
Two dice are thrown and their combined scores
added together.
What is the probability that the score is
(a) a multiple of 5
(b) a prime number
Section B:
An examination for cricket umpires is marked
out of 60.
Candidates need to score 80% on the test to
pass.
What is the pass mark?
M1.N7
M1.N8
M1.N9
Use multipliers for
percentage change.
Interpret fractions,
decimals and
percentages as
operators.
Understand and use
the relationship
between ratio, fractions
and decimal
representations.
This reference will be
explicitly assessed in
this unit.
In Section B (non
calculator)
percentage change
will be based on a
starting point of 50%,
25%, 10% or 1%.
Use of a multiplier will
give a decimal
product. Calculating
the percentage
change and adding or
subtracting may be a
more efficient
method.
Section A:
Increase £ 80 by 62%
Candidates should be
able to: use
percentages to
interpret or compare
relative frequencies;
interpret a
percentage as a
multiplier when
solving problems.
Candidates should be
able to convert
between fractions,
decimals and
percentages to find
the most appropriate
method of
calculation.
Section A:
A jacket costs £ 56.
The price is reduced by 12%.
What is the new price of the jacket?
(Calculate 0.88  56)
Candidates should
know that the ratio 3 :
7 means division into
1. Bag A contains only red and
white marbles.
The ratio of red to white
marbles in bag
A is 2 : 3
In Bag B there are also
coloured
marbles.
The probability of taking a red marble
3
10
and
7
10
for
example
Candidates should
know that the fraction
3
5
M1.N10
Page 4
Understand and use
direct proportion.
represents one
Section B (non calculator)
Decrease £ 80 by 15%.
Section B:
A savings bond pays 5% annual interest.
Karen invests £ 400.
How much will it be worth after a year?
from bag B is
9
20
.
part of the ratio of 3 :
2 for example.
Candidates should
know how to convert
a fraction to a
decimal.
Which bag has the higher
taking a red
marble?
probability of
2. Put these probabilities in
with the
smallest.
order starting
Candidates should
know that if two
quantities are in
direct proportion,
then as one quantity
increases so does
the other.
Candidates could be
asked to find the
scaling factor for two
quantities in direct
Section A:
65%, 0.7,
2
3
Two quantities x and y are in direct proportion.
When x = 28, y = 35
What is the value of y when
x = 21?
Section B:
Two quantities x and y are in direct proportion.
When x = 30, y = 63
proportion.
What is the value of y when
x = 20?
M1.N11
Divide a quantity in a
given ratio.
Candidates should be
able to interpret a
ratio that enables the
correct proportion of
an amount to be
calculated.
The ratio of blue balls to red balls in a bag is 5 :
7.
Some more blue balls are added so that the
ratio of blue balls to red balls is now 6 : 8.
What is the least number of blue balls that
could have been added?
M1.N12
Use calculators
effectively and
efficiently.
Candidates should be
able to use a
calculator for:
calculations involving
the four rules; to
check answers; enter
complex calculations;
calculations using the
four rules with
fractions; calculations
using the functions x
2
, x 3, x n, √x, 3√x, 1/x.
The term reciprocal
need not be known at
Foundation tier.
Candidates should be
able to interpret the
calculator display (for
example, values that
have been rounded)
and understand that
3.6 as a money
answer should be
written as £ 3.60
Calculators are only
allowed in Section A
120 people take their driving test in one week.
71 of these people pass.
What percentage passed the test?
M1.A1
Distinguish the
different roles played
by letter symbols in
Algebra, using the
correct notation.
Candidates should be
able to: use notations
and symbols
correctly; understand
that letters represent
variables in
equations, functions
and formulae.
Section A or B:
Write an expression for the total cost of 6
apples at a pence each and 9 pears at p pence
each.
M1.A3
Manipulate algebraic
expressions by
collecting like terms,
by multiplying a single
term over a bracket,
taking out common
factors.
Candidates should;
understand that the
manipulation of
algebraic expressions
obeys the general
rules of arithmetic;
manipulate an
expression by
collecting like terms;
multiply a single term
over a bracket; write
expressions using
squares and cubes
and the
corresponding roots;
factorise algebraic
expressions by taking
out common factors.
Section A or B:
Page 5
1. Simplify 3a – 2b + 5a + 9b
2. Expand and simplify
3(a – 4) + 2(2a + 5)
3. Factorise 6w – 8y
M1.A4
M1.A8
Set up, and solve
simple equations and
inequalities.
Derive a formula,
substitute numbers
into a formula and
change the subject of a
formula.
Candidates should be
able to: solve simple
linear equations;
solve simple linear
equations where the
unknown appears on
both sides of the
equals sign; set up
linear equations to
solve problems in
number or probability.
Candidates should
know the difference
between <, >, ≤ and
≥; solve simple
inequalities and
represent the solution
on a number line,
knowing the
convention of an
open circle for a strict
inequality and a
closed circle for an
inclusive inequality.
Section A or B:
1. Solve 4x – 11 = 3
Candidates should be
able to: use formulae
from mathematics;
use formulae
expressed in words
and symbols;
substitute numbers
into a formula;
change the subject of
a formula which will
involve at most two
letters and two
inverse operations to
rearrange and will not
include any terms
containing a power.
1. When a = 5, b = –7 and
c = 8, work out the value of
a(b 3)
c
2. Bill is twice as old as Will and
Will is three years older than
Phil. The sum of their ages is
29. If Will is x years old, form
an equation and use it to work out the ages
of Will, Bill and Phil.
3. A 4-sided spinner has the
numbers 1, 2, 3, 4 on its
faces.
The table shows the
probability of each number
1
2
3
4
x
2x
3x
4x
Work out the value of x.
2. Rearrange y = 2x + 3 to
subject.
make x the
M1.A10
Use the conventions
for coordinates in the
plane and plot points in
all four quadrants.
In this unit plotting
points will be used
when drawing
straight line graphs
Draw the line y = 2x + 3 using values of x from
–3 to 3
M1.A11
Recognise and plot
equations that
correspond to straightline graphs in the
coordinate plane.
Candidates should be
able to: recognise
that equations of the
form y = mx + c
correspond to straight
line graphs; plot
graphs in which y is
given explicitly in
terms of x.
Plot the graph of y = 3x – 1 for
–4 ≤ x ≤ 4
M1.P1
Understand and use
the vocabulary of
probability and the
probability scale.
This reference will be
explicitly assessed in
this unit.
Candidates should be
able to: use words to
indicate the chances
of an outcome for an
event; use fractions,
Here is a list of probability words
Impossible unlikely
evens
likely
certain
Write one of the words by the following events
(a) The sun will rise tomorrow.
(b) Getting a head when a coin is tossed.
Page 6
M1.P2
Understand and use
theoretical models for
probabilities including
the model of equally
likely outcomes.
decimals or
percentages to put
values to
probabilities; place
events with equally
likely outcomes on a
probability scale from
0 to 1.
(c) Throwing a 6 on an ordinary fair dice.
In this unit candidates
should be able to
work out probabilities
by counting or listing
equally likely
outcomes.
A bag contains blue, red and green counters.
The probability of a blue counter is equal to the
probability of a red counter.
The probability of taking a green counter is 0.3
Complete this table to show the number of
each counter in the bag.
Colour
Number of
counters
Blue
14
Red
Green
M1.P3
Understand and use
estimates of probability
from relative
frequency.
Candidates should be
able to estimate
probabilities by
considering relative
frequency.
A home-made four sided spinner numbered 1
to 4 is spun twice.
This is repeated 160 times.
This table shows the number of times each
combination occurred.
1
2
3
4
1
6
8
7
12
2
8
7
8
13
3
7
8
6
14
4
12
14
11
19
(a) The spinner biased. Explain how you can
tell this from the table
(b) Which number on the spinner do you think
it will land on most.
Give a reason for your answer.
M1.P11
Page 7
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.
Candidates should
understand that
random processes
rarely give the same
outcomes and
appreciate the lack of
‘memory’ in a random
situation. This
reference will be
assessed in this unit
in the context of a
situation that has
equally likely
outcomes.
A fair dice is rolled 10 times.
These are the results
4 6 2 4 3 1 1 1 1 1
What is the probability of a 1 on the next
throw?
Higher Tier Content
M1.N2
M1.N3
Exact calculation with
surds and .
Simplification of surds
including rationalising
a denominator.
Fractional and negative
indices, and use of
standard index form.
Questions that assess knowledge
of surds will be in section B.
Questions will not ask for
answers in terms of π in this unit.
Section B:
In this unit these properties will
be assessed explicitly.
Questions on standard form could
appear in either section but
questions that assess knowledge
of fractional and negative indices
will be assessed in section B.
1. Work out the value of
6 -2 × 144 0.5.
Give your answer in its
simplest form.
Expand (3 – √5)(2 + √5).
Give your answer in the form
a + b√5
2..Work out the value of
(a) 2 -4
(b) 16 ½
(c) 81 ¾
M1.N4
Approximate to
appropriate degrees of
accuracy
Questions in Section A may ask
for an answer to be given to ‘an
appropriate/suitable/sensible
degree of accuracy.
Candidates should give their
answer to the same degree of
accuracy as the values in the
question or to 3 sf.
A car cost £ 13 399 when
new.
In the first year the value
decreased by 18%.
In the second year the value
decreased by 12%.
How much was the car worth
after 2 years?
Give your answer to an
appropriate degree of
accuracy.
[Full answer is 9668.7184
which should be given as
9670]
M1.N7
Work with repeated
percentage change;
solve reverse
percentage problems.
Candidates should be able to
calculate compound interest and
reverse percentage problems.
Section A:
1. How many years will it
take
to double an investment if
the
compound interest rate is
5%?
2. A quantity increases by
15%.
The new value is 322.
What is the original
quantity?
Section B:
1. A quantity X is decreased
by
12% to give a value Y.
Which of the following
calculations will give the
value of the original
quantity?
Y
0.88
0.88Y
100Y
88
Y
1 .12
2. What single percentage
Page 8
increase is equal to
a
10% increase
followed by a
10% increase?
M1.N10
Understand and use
inverse proportion.
M1.A6
Set up and solve
simultaneous
equations in two
unknowns where on of
the equations might
include squared terms
in one or both
unknowns.
This reference will be assessed
explicitly in this unit.
y is inversely proportional to
the square root of x.
When y = 4, x = 16.
Find the value of y when x =
32
Page 9
In this unit questions asking for
the solution of two linear
simultaneous equations or one
linear equation and one of the
form y = ax2 + bx + c or
x2 + y2 = r
Solve the simultaneous
equations
y = 2x + 3
x2 + y2 = 4
Content specific to the linked pair.
M1.N.9 and 9h
Understand and use the relationship between ratio, fractions
and decimal representations, including recurring and
terminating decimals.
Assessment Guidance
Candidates should be able to:


Convert from a rational number to a terminating or recurring decimal.
Convert from a terminating or recurring decimal to a rational number.
Notes
This will be assessed in section B (non-calculator)
Examples
Foundation Tier
1
Write 0.3 and 0.6 as fractions
2
Write the recurring decimal 0.429 429 429 using recurring decimal notation.
3
Write as recurring decimals
4
(a)
1
6
(b)
4
11
Which one of
5 7
9
, , and
is a recurring decimal?
6 8
10
Show clearly how you made your decision.
Foundation and Higher
5
6
5
is equivalent to 0.555
9
(a)
Show that
(b)
Use the answer to part a. to write the decimal 0.4555 as a fraction.
(a)
The nth term of a sequence is given by
2n  1
n 1
The first two terms are
3
1
= 0.5 and
=1
3
2
Show that the 6th term is the first term of the sequence that is not a terminating decimal.
Higher Tier

7
Express the recurring decimal 0.0 7 2 as a fraction.
Page 10
Give your answer in its simplest form.
You must show your working.

8
Page 11
Show that the recurring decimal 0.6 7 =
61
90
M1.A.6h
Set up, and solve simultaneous equations in two unknowns where one
of the equations might include squared terms in one or both unknowns.
Assessment Guidance
Candidates should be able to:


Solve quadratic equations by factorisation
Solve simultaneous equations where one of them is linear and one is non-linear.
Notes
The linear equation will be of the form y = ax + b or cx + dy = e where a, b, c, d and e are positive or
negative integers and at least one of c or d will be 1.
The non-linear equation will be of the form y = ax 2 + bx + c or x 2 + y 2 = r, where a and r are positive
integers and b and c are positive or negative integers.
Example
1
Solve the simultaneous equations
2x + y = 5 and x 2 + y 2 = 10
2
Solve the simultaneous equations
y = 3x – 1 and y = 2x 2 – 2x + 1
Page 12
M1.A.14h
Understand and use the Cartesian equation of a circle centred at the
origin and link to the trigonometric functions.
Assessment Guidance
Candidates should be able to:

Know that x 2 + y 2 = r 2 is the equation of a circle centred at the origin with a radius of r

Know that the coordinate of any point on this circle will be of the form
(± r cos , ± r sin ), where  is the angle between the line joining the point to the
origin and the x axis.
y
(–r cos , r sin )
r

O
x
Notes
Candidates could use the convention of measuring the angle anti-clockwise from the positive x axis.
Hence if this angle is called , the coordinates will be (r cos , r sin ).
Example
1
The circle x 2 + y 2 = 16 is shown.
The point X is such that the angle between OX and the x-axis is 60°.
y
Not drawn
accurately
X
60
O
Work out the coordinates of the point X.
Page 13
x
M1.P.6
Understand and use set notation to describe events and
compound events.
M1.P.7
Use Venn diagrams to represent the number of possibilities and hence
find probabilities.
Assessment Guidance
Candidates should be able to:


Understand that P(A) means the probability of event A
Understand that P(A’) means the probability of not event A



Understand that P(A  B) means the probability of event A or B
Understand that P(A  B) means the probability of event A and B.
Understand a Venn diagram consisting of a universal set and at most two sets, which may or
may not intersect.
Shade areas on a Venn diagram involving at most two sets which may or may not intersect.

Notes
The diagrams will be restricted to the universal set  and two sets.
The symbol  should be known
Questions involving P(A  B) and P(A  B) will always be linked with a Venn diagram.
The addition law for probability will not be specifically required, but students should be able to understand
and use probabilities, such as P(A) and P(A'), from the Venn diagram.
The two sets may be referred to as two capital letters, for example A and B.
Candidates should know the following notations and the subsequent shaded area on the Venn diagram.

A
B
A  B to mean the intersection of A and B.

A
B
A’ to mean everything not in A
Page 14

A
B
A  B to mean the union of A and B.

A
B
A’  B to mean everything not in A that is in B.

A

B
A’  B to mean the union of A’ and B.

A
A
A
B
B
(A  B)’ = A’  B’ to mean everything
not in the intersection.
B
(A  B)’ = A’  B’ to mean everything
not in the union.
Examples
1
You are given that P(A) = 0.7
Work out P(A’)
2
The Venn diagram shows the number of left handed students in a year group (set A) and the
number of vegetarians in the same year group (set B)

B
A
25
15
45
115
Page 15
(a)
How many students are in the year group altogether?
(b)
Work out P(A  B)
(c)
A student from the year group is chosen at random.
What is the probability that the student is a right-handed vegetarian?
3
On the Venn diagram shaded the area that represents P’  Q
P
Page 16
Q