Download Algebra – Unit 1

PILOT OF A LINKED PAIR OF GCSEs in MATHEMATICS
GCSE Methods in Mathematics
NOVEMBER 2013
Unit 1 & 2 - Further notes and examples
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PILOT OF A LINKED PAIR OF GCSEs in MATHEMATICS
GCSE Methods in 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 operations.
Arithmetic of real numbers.
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).
Numbers and their representations including
powers, roots, indices (integers, fractional and
negative), and standard index form.
Arithmetic of real numbers, including exact
calculation with surds and pi.
Simplify 81 , 8 3 .
Approximate to specified or appropriate degrees of
accuracy including a given power of ten, number of
decimal places and significant figures.
Use the concepts and vocabulary of factor (divisor),
multiple, common factor, common multiple, highest
common factor, least common multiple, prime
number and prime factor decomposition.
Write 360 as the product of its prime factors in index
form.
Candidates may be required to find the LCM and
HCF of numbers written as the product of their
prime factors.
Understand that factors of a number can be derived
from its prime factorisation.
FURTHER NOTES / CLARIFICATION
Including whole numbers, decimals, fractions and
negative numbers and understanding place value.
Including BIDMAS
Including addition, subtraction, multiplication and
division of whole numbers, decimals, fractions and
negative numbers and estimation.
Foundation – Multiplying fractions with a whole
number, simple multiplying of fractions.
Higher – Distinguish between rational and irrational
numbers.
Simplify numerical expressions involving surds and
π.
Including reciprocals.
Use of the rules of indices
Foundation – positive indices only.
2
Approximate to specified or appropriate degrees of
accuracy including a given power of ten, number of
decimal places and significant figures.
Use the concepts and vocabulary of factor (divisor),
multiple, common factor, common multiple, highest
common factor, least common multiple, prime
number and prime factor decomposition
Write 360 as the product of its prime factors in index
form.
Candidates may be required to find the LCM and
HCF of numbers written as the product of their
prime factors.
Understand that factors of a number can be derived
from its prime factorisation.
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Including the terms product and sum.
Number – Unit 1
FOUNDATION
Understand and use the relationship between
fractions and decimal representations, including
recurring and terminating decimals.
HIGHER
Understand and use the relationship between
fractions and decimal representations including
recurring and terminating decimals.
0.2  0.2222.....
 2  0.12121212.....
0.1
1
 23  0.123123......
0.1
 0.3333333...........
3
0.142857142857 
12
99
0.1010010001....
FURTHER NOTES / CLARIFICATION
1
7
0.121212 
cannot be expressed as a fraction
Divide a quantity in a given ratio.
Divide £1520 in the ratio 5 : 3 : 2.
Divide a quantity in a given ratio.
Divide £1520 in the ratio 5 : 3 : 2.
Understand and use Venn diagrams to solve
problems.
Understand and use Venn diagrams to solve
problems.
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Including questions such as:
A number is divided in the ratio 3:4. If the smaller
number is 15, what is sum of the two numbers?
Candidates may need to draw their own Venn
diagrams.
To include three events.
Algebra – Unit 1
FOUNDATION
Distinguish between the different roles played by
letter symbols in algebra, using the correct notation.
Distinguish in meaning between the words equation,
inequality, formula, and expression
HIGHER
Distinguish the different roles played by letter
symbols in algebra, using the correct notation.
Distinguish in meaning between the words equation,
inequality, formula, identity and expression.
FURTHER NOTES / CLARIFICATION
Higher - Use of the symbol ≡
Show that
(3x – 1)(3x + 1) – (1 – x)(1 + x) + 3(1 – 2x)(1 + 2x) ≡ 1– 2x2
Show that
(x – 1)(x2 + 2x + 3) – x(x + 1) ≡ x3 – 3
Manipulate algebraic expressions by collecting like
terms, by multiplying a single term over a bracket,
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.
Manipulate algebraic expressions by collecting like
terms, by multiplying a single term over a bracket,
taking out common factors, multiplying two linear
expressions, factorising quadratic expressions
including the difference of two squares, and
simplifying rational expressions.
Simplify 3a – 4b + 4a + 5b.
Expand 7(x – 3).
Simplify 2(3x – 1) – (x – 4).
Simplify x(x – 1) + 2(x2 – 3).
Expand and simplify (2x – y)(3x + 4y).
Expand and simplify (3x – 2y)2.
Factorise 6x + 4.
Factorise
i) 3x2 – 6x,
ii) x2 + 3x + 2,
iii) x2 – 5x – 6,
iv) x2 – 9,
v) 3m2 – 48,
vi) 3m2 – 10m + 3,
vii) 12d 2 + 5d – 2,
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Including
Higher Simplification of expressions involving surds
(√3 + √2)2 – (√3 – √2)2 = 4√6
Excluding the rationalisation of the denominator of a
1
fraction such as
.
(2  3 )
2x  4
Simplify 2
.
x  9x  14
x 2  25
Simplify 2
.
x  3x  10
2x 2  4x  16
Simplify 2
.
x  7x  10
Including writing rational expressions as a single
fraction in its simplest form.
e.g.
8
5

f 4 3f 2
Algebra – Unit 1
FOUNDATION
Solve quadratic equations approximately using a
graph.
Derive a formula, substitute numbers into a formula
and change the subject of 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.
HIGHER
Solve quadratic equations approximately using a
graph, exactly by factorising, completing the
square.
Solve x2 + 7x + 12 = 0.
Derive a formula, substitute numbers into a formula
and change the subject of 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.
FURTHER NOTES / CLARIFICATION
Foundation – further example –
Complete the table of values and plot the graph of
y = x2 – 2x. Use your graph to find the value of y
when x = 2.6. Use your graph to solve x2 – 2x =5.
Higher – factorising and solving quadratics where
the coefficient x2  1.
Completing the square - Candidates may be asked
to write a quadratic expression in the form
(x + m)2 + n (m and could be positive or negative).
Alternatively candidates could be asked to
‘complete the square’. The coefficient of x2 = 1.
Higher – some further examples ,
Make x the subject of the equation 3x + 5 = y – ax.
Make x the subject of the equation: 10 
ax  12
bx  1
Find the value of 6f + 7g when f = – 3 and g = 2.
Given that m = 7n – 3, find n in terms of m.
Given that
1 1 1
 
a b c
find b in terms of a and c.
Generate terms of a sequence using term-to-term
and position-to-term definitions.
Generate terms of a sequence using term-to-term
and position-to-term definitions.
Form linear expressions to describe the nth term of a
sequence.
Form linear and quadratic expressions to
describe the nth term of a sequence.
Use the conventions for coordinates in the plane
and plot points in all four quadrants.
Use the conventions for coordinates in the plane
and plot points in all four quadrants.
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If questions ask for answers ‘in words’ then ‘+2’
must be expressed as ‘add two each time’.
Quadratic sequence will be in from an2 + b or
an2 + bn (a ≥ 1)
Algebra – Unit 1
FOUNDATION
Recognise and plot equations that correspond to
straight-line graphs in the co-ordinate plane.
Use geometric information to complete diagrams on
a co-ordinate grid.
Find the coordinates of the fourth vertex of a
parallelogram with vertices at (2, 1), (– 7, 3) and (5,
6).
HIGHER
Recognise and plot equations that correspond to
straight-line graphs in the co-ordinate plane.
Use geometric information to complete diagrams on
a co-ordinate grid.
Find the coordinates of the fourth vertex of a
parallelogram with vertices at (2,1), (–7, 3) and (5,
6).
Use y = mx + c and understand the relationship
between gradients of parallel and perpendicular
lines.
Draw, sketch, recognise graphs of linear,
quadratic, simple cubic functions, the reciprocal
function
y 
1
x
with x  0, the function y = k x for
integer values of x and simple positive values of
k.
Sketch simple transformations of a given
function.
Recognise and use equivalence in numerical,
algebraic and graphical representations.
Recognise and use equivalence in numerical,
algebraic and graphical representations.
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FURTHER NOTES / CLARIFICATION
Including finding the equation of a line given the
gradient and one point. Finding the equation of a
line given two points on the line and giving
examples of lines that are parallel or perpendicular
to a given line.
Knowledge and use of m1  m2 = -1
Equations may be in the form ax + by + c = 0.
Including graphs in the form
a
y = mx + c, y 
x
c
y = ax2 + b, y = ax2 + bx + c, y  ax 2  bx 
with
x
x 0
y = ax3, y = ax3 + b, and y = ax3 + bx2 + cx + d
Including y = f(x+a), y = f(kx) and y =f(x)+a, applied
to y=f(x), y=kf(x) where a and k could be positive or
negative.
Including matching graphs and equations and
equivalent forms e.g. a  a = a2.
Fractional and decimal equivalences.
Geometry – Unit 1
FOUNDATION
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.
Recall the properties and definitions of special
types of quadrilateral, including square, rectangle,
parallelogram, trapezium, kite and rhombus.
Calculate and use the sums of the interior and
exterior angles of polygons.
Solve problems in the context of tiling patterns and
tessellation.
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.
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.
Calculate and use the sums of the interior and
exterior angles of polygons.
Solve problems in the context of tiling patterns and
tessellation.
Understand, prove and use circle theorems,
intersecting chords.
FURTHER NOTES / CLARIFICATION
Including problems involving algebra.
Including the vocabulary of triangles - isosceles,
equilateral and scalene.
Including reflection and rotational symmetry
Use angle and tangent properties of circles.
Understand that the tangent at any point on a circle is
perpendicular to the radius at that point.
Use the facts that the angle subtended by an arc at the
centre of a circle is twice the angle subtended at any
point on the circumference, the angle subtended at the
circumference by a semicircle is a right angle, that
angles in the same segment are equal, and that opposite
angles of a cyclic quadrilateral sum to 180°.
Use the alternate segment theorem.
Understand and use the fact that tangents from an
external point are equal in length.
Candidates will not be expected to prove the above circle
theorems directly, but the applications of these proofs will
be needed to understand and construct geometrical
proofs.
Intersecting chords - Chords can intersect inside or
outside the circle. Candidates will not be asked to prove
the intersecting chord theorems directly but the
applications of the proofs will be assessed.
http://www.mathopenref.com/chordsintersecting.html
http://www.cimt.plymouth.ac.uk/projects/mepres/allgcse/pr3es.pdf (page 48)
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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.
Use of sample spaces for situations where
outcomes are single events and for situations where
outcomes are two successive events.
Identify different mutually exclusive and exhaustive
outcomes and know that the sum of the probabilities
of all these outcomes is 1.
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.
Use of sample spaces for situations where
outcomes are single events and for situations where
outcomes are two successive events.
Identify different mutually exclusive and exhaustive
outcomes and know that the sum of the probabilities
of all these outcomes is 1.
Understand and use set notation to describe events
and compound events.
Understand and use set notation to describe events
and compound events.
Use Venn diagrams to represent the number of
possibilities and hence find probabilities.
Use Venn diagrams to represent the number of
possibilities and hence find probabilities.
Use tree diagrams to represent outcomes of
compound events, recognising when events are
independent or dependent.
Know when to add or multiply probabilities: if A
and B are mutually exclusive, then the
probability of A or B occurring is P(A) + P(B); if
A and B are independent events, the probability
of A and B occurring is P(A) × P(B).
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FURTHER NOTES / CLARIFICATION
Including expected values
Foundation - Knowing and using set notation S, A, B A’,
B’, AB, AB, A’B, AB’. Identify the above on Venn
diagrams.
Higher - Knowing and using set notation S, A, B, A’, B’
AB, AB, A’B, AB’ ABC etc. Identify the above
on Venn diagrams.
Use of the notation P(A), P(A’) etc
Three events may be used.
http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i1/b
k7_1i3.htm
Candidates may be required to complete branches and a
third event may be introduced.
Compare experimental data and theoretical
probabilities, and make informal inferences about
the validity of the model giving rise to the theoretical
probabilities.
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.
Compare experimental data and theoretical
probabilities, and make informal inferences about
the validity of the model giving rise to the theoretical
probabilities.
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.
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PILOT OF A LINKED PAIR OF GCSEs in MATHEMATICS
GCSE Methods in Mathematics Unit 2
Further notes on the Specification
Unit 2 - Further notes and examples
Page 13
Number – Unit 2
FOUNDATION
Understand and use number operations and the
relationships between them, including inverse
operations and hierarchy of operations.
Arithmetic of real numbers.
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.
Arithmetic of real numbers, including exact
calculation with pi.
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 that 'percentage' means 'number of
parts per 100' and use this to compare proportions.
Understand and use direct proportion.
Use calculators effectively and efficiently.
Including addition, subtraction, multiplication and
division of whole numbers, decimals, fractions and
negative numbers and estimation.
Foundation – Multiplying fractions with a whole
number, simple multiplying of fractions.
Higher – Distinguish between rational and irrational
numbers.
Simplify numerical expressions involving surds and .
With positive and negative powers of 10.
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
£
Interpret fractions, decimals and percentages as
operators.
Understand and use the relationship between ratio
and fractions.
Find proportional change, using fractions, decimals
and percentages.
FURTHER NOTES / CLARIFICATION
Including whole numbers, decimals, fractions and
negative numbers and understanding place value.
Including BIDMAS
36
1  175
.
Interpret fractions, decimals and percentages as
operators.
Understand and use the relationship between ratio
and fractions.
Find proportional change and repeated
proportional change, using fractions, decimals
and percentages.
Understand and use direct and inverse proportion.
Use calculators effectively and efficiently, including
trigonometric functions.
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Including equivalences between decimals, fractions,
ratios and percentages.
Algebra – Unit 2
HIGHER
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.
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.
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,
Solve
3
x
12
,
x
12
,
3
5x + 2 = 17,
10x + 9 = 6x + 11,
3(1 – x) = 5(2 + x),
(x – 1) = 3x + 1.
FURTHER NOTES / CLARIFICATION
At Foundation candidates will be required to solve
simple linear inequalities with whole numbers and
fractional coefficients.
x + 6 = 15,
3
x
12
,
x
12
,
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.
Use algebra to support and construct arguments.
Set up and use equations that describe direct
and inverse proportion.
Set up, and solve simultaneous equations in two To include finding the coordinates of the points of
unknowns where one of the equations might
intersection of a straight line ax + by + c = 0 with a
include squared terms in one or both unknowns. circle x2 + y2 = a2 , or a parabola of the form
y = ax2 + by + c.
Solve quadratic equations using the formula.
Find, correct to 2 decimal places, the roots of the
equation 3x2 - 7x - 2 = 0.
Use algebra to support and construct arguments
For example using algebraic and geometric proofs
and proofs.
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Algebra – Unit 2
FOUNDATION
HIGHER
Draw, sketch, recognise graphs of the
trigonometric functions y = sin x, y = cos x and
y = tan x.
FURTHER NOTES / CLARIFICATION
Understand and use the Cartesian equation of a
circle centred at the origin and link to the
trigonometric functions.
To include:
 Writing the equation of a circle, centred at the
origin
 Finding the co-ordinates of the points of
intersection of a circle, centred at the origin,
and a straight line including the case where the
line is a tangent to the circle
 Writing expressions for the trigonometric
functions using a circle centred at the origin
 Deriving the equation of the circle centred at
origin
Construct the graphs of simple loci.
Expressing simple loci as equations of straight lines
and circles and constructing the corresponding
graphs.
e.g.
 Write an expression for the locus of a point
that moves such that its distance from the
origin is 5 units.
 Write expressions for the locus of a point
that moves such that its distance from the xaxis is 3 units.
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Geometry – Unit 2
FOUNDATION
Recognise reflection and rotation symmetry of 2D
shapes.
HIGHER
Recognise reflection and rotation symmetry of 2D
shapes.
Understand and use the midpoint and the
intercept theorems.
Understand and construct geometrical proofs
using formal arguments, including proving the
congruence, or non congruence of two triangles
in all possible cases.
Describe and transform 2D shapes using single or
combined rotations, reflections, translations, or
enlargements by a positive scale factor and
distinguish properties that are preserved under
particular transformations.
Use 2D vectors to describe translations.
Use Pythagoras’ theorem in 2D.
Candidates will not be expected to prove the above
theorems directly, but the applications of these
proofs will be needed to understand and construct
other geometrical proofs.
Understand and use SSS, SAS, ASA and RHS
conditions to prove the congruence of triangles
using formal arguments. Reasons may be required
in the solution of problems involving congruent
triangles.
Describe and transform 2D shapes using single or
combined rotations, reflections, translations, or
enlargements by a positive scale factor then use
positive fractional and negative scale factors
and distinguish properties that are preserved under
particular transformations.
Use 2D vectors to describe translations.
Use vectors to solve simple geometric problems
and construct geometric arguments.
Understand congruence and similarity, including the
relationship between lengths, in similar figures.
FURTHER NOTES / CLARIFICATION
As in the legacy specifications
Including
 Magnitude of a vector
 Sum and difference of vectors
 Commutative and associative properties of
vector addition
 Multiplication of vector by a scalar
 Simple additions to geometry in twodimensions
Understand congruence and similarity, including the
relationship between lengths, areas and volumes
in similar figures.
Use the trigonometric ratios to solve 2D and 3D
problems.
Use the sine and cosine rules to solve problems
in 2D and 3D.
Including angles of elevation and depression.
Use Pythagoras’ theorem in 2D and 3D.
Including reverse problems e.g. proving a triangle
isn’t a right-angled triangle.
Page 17
Distinguish between centre, radius, chord, diameter,
circumference, tangent, arc, sector and segment.
Distinguish between centre, radius, chord, diameter,
circumference, tangent, arc, sector and segment.
Find circumferences of circles and areas enclosed
by circles.
Find circumferences of circles and areas enclosed
by circles.
Calculate perimeters and areas of shapes made
from triangles and rectangles.
Calculate perimeters and areas of shapes made
from triangles and rectangles and other shapes.
Calculate the area of a triangle using
Calculate volumes of right prisms and of shapes
made from cubes and cuboids
1
ab
2
Including semicircles and quarter circles.
Higher – lengths of circular arcs and area of sectors
and segments.
Including reverse problems
Including perimeters and areas of squares,
rectangles, triangles, parallelograms, trapezium and
composite shapes.
Including surface area
sin C.
Calculate volumes of right prisms and of shapes
made from cubes and cuboids.
Solve mensuration problems involving more
complex shapes and solids.
Page 18
Including volumes of spheres, hemispheres,
cylinder, cone, truncated cone (frustum) and
pyramids.