Download KS4 Learning Plan - GCSE Maths Objectives Higher Tier

Higher Tier – Order of Topics (2013-15)
The order is an approximate guide to the order of topics taught in Years 10
and 11 by term and will depend on teaching group. This order and timings
can change over the course of the year.
Time will also be left over for revision of topics over the course of the two
years and for completion of past exam papers.
Written homework will be set along with some online homework.
Assessments take place throughout the year and dates will be placed on the
GCSE Connect pages during the term prior to these.
Year 10: Autumn Term
Topic
Objectives
N1. Integers
(Whole Numbers)
N2. Rounding
Grade
Add, subtract, multiply and divide negative integers
Understand simple instances of BIDMAS, e.g. work out 12 × 5 – 24
÷8
Write a number as a product of primes in index form
Find the HCF and LCM
Approximations, e.g. 29 × 31 ≈ 30 × 30
Check answers by reverse calculation, e.g. if 9 × 23 = 207 then 207
÷ 9 = 23
Estimate an answer using approximations e.g. 99.12× 2.03 ≈ 400
F
E
D
C
C
D
D
C
0.501
Round to a given number of decimal places, significant figures
Find the upper and lower bounds of a number (decimal
places/significant figures) and apply in context.
Simplify algebraic expressions by collecting like terms e.g. 7p + 2p =
9p
Simplify algebraic expressions by collecting like terms (two terms
2
2
2
and with powers) e.g. x + 4x = 5x
Multiply and divide with letters and numbers
Expand single brackets e.g. 3(x+5) = 3x + 15
Expand two single brackets and collect like terms
e.g. 5(x+1)+2(x+3) = 7x + 11
2
Expand double brackets and simplify e.g. (x+5)(x+3)=x +8x+15
2
Factorise an expression (put in to brackets) e.g. x – 3x = x (x – 3)
Factorise quadratic expressions (including the difference of two
2
squares) e.g. x +10x – 24 = (x+12)(x-2)
Simplify algebraic fractions e.g. x 2 + 4 x + 3 ( x + 3)( x +1) ( x +1)
A1. Use of Symbols
x2 + 5x + 6
=
=
( x + 3)( x + 2)
C
C – A*
F
E
D
E
D/C
C
C
B/A
A*
( x + 2)
A
Add/subtract fractions in algebraic form e.g.
SSM1. Coordinates
Plot/Identify coordinates in one and four quadrants
Identify 3-dimensional coordinates
Find the midpoint of a line segment/Work backwards from a
midpoint.
F
C
D-C
A2. Graphs of
Linear Functions
Sketch simple straight line graphs e.g. y = 2x + 5 with and without a
table.
Find gradients and y-intercepts using y = mx + c
Know and use the properties of parallel lines. (i.e. same gradient)
Know that the line perpendicular to y = mx + c has gradient −1/m
(negative reciprocal)
Find the equations of perpendicular lines.
D/C
1
2
3x + 7
+
=
x + 5 x − 3 ( x + 5)( x − 3)
C
C
B
A
Autumn Assessment: Non-Calculator test (1 lesson) testing the topics covered so far. Date will be
announced during the term and placed on the GCSE Higher tier Connect page.
N3. Fractions
Write a fraction in its simplest form and recognise equivalent
fractions
Compare the sizes of fractions using a common denominator
Add and subtract fractions by using a common denominator
Write an improper fraction as a mixed fraction
Add and subtract fractions in mixed form e.g. 1 1 + 2 1
E
D
D
C
2
4
Multiply and divide a number with a fraction, e.g. 3× 1 = 3
4 4
C
Multiply/divide a fraction with a fraction (expressing the answer in its
C
7 1 35
3
or ÷ =
=4
8 5
8
8
N4. Decimals
simplest form) e.g. 1 × 2 = 2
3 5 15
Simplify multiplication of fractions by first cancelling common factors
Convert a fraction to a decimal, or a decimal to a fraction
Find the reciprocal of whole numbers, fractions, and decimals, e.g.
find the reciprocal of 0.4
Know that 0 does not have reciprocal, and that a number multiplied
by its reciprocal is 1
Use fractions in real-life problems (functional skills)
Convert a fraction to a recurring decimal (and vice versa)
Approximate decimals to a given number of decimal places or
significant figures
Multiply and divide decimal numbers by whole numbers and decimal
numbers (up to 2 decimal places), e.g. 266.22 ÷ 0.34
Know that e.g. 13.5 ÷ 0.5 = 135 ÷ 5
Convert a recurring decimal to a fraction. e.g. 0.43
ɺ ɺ = 43
99
e.g. 0.063
ɺ ɺ = 63
990
N5. Percentages
Write a percentage as a decimal; or as a fraction in its simplest
terms
Write one number as a percentage of another number
Calculate the percentage of a given amount
Find a percentage increase/decrease, of an amount
Find a reverse percentage, e.g. find the original cost of an item
given the cost after a 10% deduction
Use a multiplier to increase by a given percent, e.g. 1.1 × 64
increases 64 by 10%
Calculate simple and compound interest for two, or more, periods of
time
C
E
D
D
C
C
B
F/E
D
D
B
A
F
D/C
E
D
C
B
C/B
B
SSM2. Properties
of Polygons
Use angle properties on a line and at a point to calculate unknown
angles.
Measure a bearing
Calculate bearings
Mark parallel lines in a diagram
Use angle properties of triangles and quadrilaterals to find missing
angles (including in algebraic form)
Find missing angles using properties of corresponding angles and
alternate angles, giving reasons
Find the three missing angles in a parallelogram when one of them
is given
Identify and list the properties of quadrilaterals (including kites)
Name all quadrilaterals that have a pair of opposite sides that are
equal
Calculate and use the sums of the interior angles of polygons of
sides 3, 4, 5, 6, 8, 10
Know, or work out, the relationship between the number of sides of
a polygon and the sum of its interior angles
Know that the sum of the exterior angles of any polygon is 360
degrees
Find the size of each exterior/interior angle of a regular polygon
Solve linear equations with one, or more, operations
e.g. Solve x – 5 = 9 or 7x + 5 = 10
Solve linear equations involving a single pair of brackets
e.g. Solve 2(x+5) = 10
Solve an equation with unknowns on both sides
Solve an equation in fractional form e.g. x − 5 = 10
Solve equations in fractional form e.g.
A3. Equations
2
x x
+ =5
3 2
F
D/C
C
F
D-C
D
D
C
C
C
C
C
C
F–D
D
C
C
B
Year 10: Spring Term
Topic
A4. Inequalities
Objectives
SSM3. Area and
Volume
Grade
Solve linear inequalities in one variable and present the solution set
on a number line. e.g. Solve 2x + 1 > 5 (Ans: x > 2)
Find the integer solutions e.g. -4 < x ≤ 5
Draw the graphs of linear inequalities in two variables and interpret
the solution sets given by regions in the coordinate plane, or to
identify all the integer coordinates with crosses. (shading regions)
Find inequalities that represent the shaded area of a region.
C
Find the perimeters and areas of shapes made up from triangles
and rectangles
Use formulae to find the area of shapes made up of rectangles and
triangles
Find the surface area of cuboids and prisms
Solve a range of problems involving areas (parallelograms,
trapeziums, squares and rectangles)
Find when numbers are given to a specific degree of accuracy, the
upper and lower bounds of perimeters and areas
Solve more complex problems, e.g. given the surface area of a
sphere find the volume
Find the volume of a frustum
Work backwards from volume of spheres/cones
Solve problems involving the circumference and area of a circle (and
simple fractional parts of a circle)
Solve problems involving the volume of a cylinder
Find exact answers by leaving answers in terms of π
Calculate the length of an arc
Calculate the area of a segment/sector
Work backwards from the area of a sector/arc length to find the
radius/angle
F-D
C
B
B
G
D
C/B
E
C
B
A/A*
A*
A*
C
C
C
B
A
A
SSM4. Reflections
and Rotations
Rotate a shape anywhere and from a point
E-D
Understand translation as a combination of a horizontal and vertical
shift (including vector notation)
Reflect shapes in a given mirror line. Initially line parallel to the
D
coordinate axes and then y = x or y = –x
E
o
Describe a single transformation fully e.g. rotation, 90 clockwise,
C
centre (0, 1)
Spring Assessment: Calculator test (1 lesson) testing the topics covered so far. Date will be announced
during the term and placed on the GCSE Higher tier Connect page.
SSM5.
Enlarge a shape from anywhere
D
Enlargements and
Enlarge shapes by a given scale factor from a given point; using
C
Translations
positive whole number scale factors, then positive fractional scale
factors
Enlarge shapes by a negative scale factor either integer or fractional
A
1
e.g. -2, - /2
Describe a single transformation fully e.g. enlargement, centre (2,8),
C
scale factor 2
N6. Indices and
Understand and use the index laws for multiplication and division
B
3
3
6
15
4
11
Standard Form
e.g. g x g = g and t ÷ t = t
4 2
4x2
8
Understand and use the index laws for brackets e.g. (3 ) = 3 = 3
B
0
Know that a = 1 (zero index) for any non-zero value of a
B
1/3
5/2
Work with fractional powers e.g. 8 = 2 or 4 = 32
A/A*
Understand the standard form convention
B
Convert numbers to, and from, standard form
B
B
Calculate with numbers given in standard form with, and without, a
calculator
B
Round numbers given in standard form to a given number of
significant figures
2
N7. Surds
A*
Simplify surds, such as (3 – 5 ) in the form a + b 5
A5. Quadratics
2
5
Rationalise the denominator of a surd such as
Solve quadratic equations by factorising (including values of a not
equal to 1)
Use the quadratic formula to solve quadratic equations giving the
answers to 1 d.p.
Use the quadratic formula to solve quadratic equations leaving the
answer in surd form
Complete the square of a quadratic function (using this to write
down the max/min of the function)
Find graphically the solutions of quadratic equations by considering
the intercept on the x-axis
A
B/A
A
A
A*
A
Year 10: Summer Term
Topic
A6. Quadratic
Graphs
HD1. Collecting
Data
Objectives
HD2. Statistical
Measures
Substitute values of x into a quadratic function to find the
corresponding values of y
2
2
Draw graphs of quadratic functions such as y=x +2 or y = x - 5x + 3
Solve simple quadratic equations using a quadratic graph.
Design a suitable question for a questionnaire
Understand the difference between: primary and secondary data;
discrete and continuous data
Design suitable data capture sheets for surveys and experiments
Understand about bias in sampling
Use sampling methods including random and stratified sampling.
Understand that increasing sample size generally leads to better
estimates
Find the mode or the median for (small) sets of data
Find the mean and the range for (small) sets of data
Use a stem and leaf diagram to sort data
Grade
C
C/B
B/A
D
D
G–D
C
A
C
G
F
D
Know the advantages/disadvantages of using the different measure
D
of average (e.g. mean is affected by extreme values)
Identify the modal class interval (group) in grouped and ungrouped
F
frequency distributions
D
Find the mean of an ungrouped frequency distribution
Find an estimate for the mean of a grouped frequency distribution by
C
using the mid-interval value
End of year Assessment: Non-Calculator and Calculator tests (2 lessons in total) testing the topics
covered so far. Date will be announced during the term and placed on the GCSE Higher tier Connect page.
HD3. Representing
Represent data as:
Data
Frequency polygons
C
Choose an appropriate way to display discrete, continuous and
D
categorical data
Represent categorical data in a pie chart
C
Interpret categorical data in a pie chart
C
Cumulative Frequency Diagrams
B
Box Plot
B
A histogram
A
Histograms:
Complete a histogram from a frequency table
A
Complete a frequency table from a histogram
A*
Use a histogram to work out the frequency in part of a class interval
A*
Cumulative Frequency:
Find the median and quartiles for large sets of ungrouped data
B
Draw a cumulative frequency table for grouped data (using the
B
upper class boundary)
Draw a cumulative frequency curve for grouped data
B
Use a cumulative frequency diagram to find estimates for the
B
median and quartiles of a distribution (IQR – Interquartile range)
Use a cumulative frequency diagram to solve problems, e.g. how
B
many greater than a particular value
Draw a box plot to summarise information given in cumulative
B
frequency diagrams
Compare cumulative frequency diagrams and box lots to make
B
inferences about distributions
HD4. Scatter
Graphs and
Correlation
HD5. Probability
A7. Simultaneous
Equations
Plot points to produce a scatter graph
Appreciate that correlation is a measure of the strength of
association between two variables
Distinguish between positive, negative and zero correlation using a
line of best fit
Appreciate that zero correlation does not necessarily imply ‘no
correlation’ but merely ‘no linear relationship’
Draw lines of best fit by eye and understand what it represents
Write down the theoretical probability for an equally likely event
Estimate a probability by relative frequency
Know that a better estimate for a probability is achieved by
increasing the number of trials
Complete two-way tables and find probabilities
Draw/complete tree diagrams
Understand independent and non-independent events
Find probabilities of successive independent events.
Find probabilities of successive dependent events.
Solve algebraically two simultaneous equations
Interpret the solution of two simultaneous equations as the point of
intersection the corresponding lines
Solve simultaneous equations graphically
Find graphically the approximate solutions of linear and quadratic
simultaneous equations
Find the exact solutions of linear and quadratic simultaneous
equations
2
2
Draw a circle of radius r centred at the origin e.g. x + y = 25
Find graphically the approximate solutions of linear and circular
F
D
D
D
C
E
C
D
D
B
A
A
A*
B
B
A
A
A
A*
A*
simultaneous equations
Find the exact solutions of linear and circular simultaneous
equations
A*
Year 11: Autumn Term
Topic
Objectives
SSM6. Similarity
and Congruence
Use integer and non-integer scale factors to find the length of a
missing side in each of two similar shapes, given the lengths of a
pair of corresponding sides
Know the relationship between linear, area and volume scale factors
of similar shapes
Prove formally geometric properties of triangles, e.g. that the base
angles of an isosceles triangle are equal
Prove formally that two triangles are congruent
Use letters or words to state the relationship between different
quantities
Substitute positive and negative numbers into simple algebraic
formulae
Substitute positive and negative numbers into algebraic formulae
involving powers
Find the solution to a problem by writing an equation and solving it
Change the subject of a formula, e.g. change the formula for
converting Centigrade into Fahrenheit into a formula that converts
Fahrenheit into Centigrade
Change the subject of a formula with the unknown variable on both
sides.
Solve cubic functions by successive substitution of values of x
3
e.g. Solve x + 3x = 20 using trial and improvement
A8. Formulae
A9. Trial and
Improvement
Value of x
2
3
N8. Ratio and
Proportion
B
A
A*
A*
E
D
C
D/C
D/C
B/A
C
Comment
Too small
Too big
14
36
Apply trial and improvement to volume problems
B
Understand what is meant by ratio
Write a ratio in its simplest form; and find an equivalent ratio
Share a quantity in a given ratio (e.g. Divide £10 in to the ratio 2: 3)
Understand and use examples in direct proportion
Interpret map/model scales as a ratio
Interpret direct and inverse proportions as algebraic functions, e.g. y
2
2
∝ x as y = kx
Use given information to find the value of the constant of
proportionality
Use algebraic functions for direct and inverse proportionality, with
their value of k, to find unknown values
Recognise and sketch the graphs for direct and inverse proportions
2
3
2
(y ∝ x, y ∝ x , y ∝ x , y ∝ 1/x, y ∝ 1/x )
F
F
C
D
E
A
Find the missing numbers in a number pattern or sequence
Find the nth term of a number/picture sequence
Find whether a number is part of a given sequence
Use a calculator to produce a sequence of numbers
Find the nth term of a fractional sequence e.g. 1 , 3 , 5 , 7
G
C
E
E
B
Find the nth term of a simple non-linear sequence (quadratic/cubic)
B
Use a ruler and compass to draw accurate triangles, and other 2-D
shapes, given information about their side lengths and angles.
Construct triangles using a compass given all three sides (SSS), two
C
A10. Sequences
Grade
A
A
A
3 5 7 9
SSM7.
Constructions
C
SSM8. Loci
SSM9. Measures
SSM10. Real-Life
Graphs
SSM11.
Pythagoras’
Theorem
SSM12.
Trigonometry
SSM13. Properties
of Circles (Circle
Theorems)
sides and an angle (SAS) , two angles and a side (ASA).
Recall and use angle properties of equilateral, isosceles and rightangled triangles
Appreciate why some shapes tessellate and why some shapes do
not tessellate.
Construct and recognise the nets of 3-D solids such as pyramids
and triangular prisms
Draw plans and elevations of 3-D solids
Given the front and side elevations and the plan of a solid, draw a
sketch of the 3-D solid.
Construct:
An equilateral triangle with a given side
The mid-point and perpendicular bisector of a line segment
The perpendicular from a point on a line
The bisector of an angle
The angles 60, 30 and 45 degrees
A region bounded by a circle and an intersecting line
A path equidistant from 2 points or 2 line segments, etc
Find the locus of points e.g. the locus of points equidistant to two
given points
Construct:
The mid-point and perpendicular bisector of a line segment
The perpendicular from a point on a line
The bisector of an angle
A region bounded by a circle and an intersecting line
A path equidistant from 2 points or 2 line segments, etc
Use the relationship between distance, speed and time to solve
problems
Convert between metric units of speed e.g. km/h to m/s
Solve problems involving mass, density and volume
Understand linear functions in practical problems, e.g. distance-time
graphs
Discuss and interpret graphs modelling real situations (including fuel
bills, fixed charge/standing charge and cost per unit)
Use and draw simple conversion graphs.
Find missing sides (hypotenuse or shorter side) using Pythagoras’
Theorem in right-angled triangles.
Apply Pythagoras’ Theorem to find the length of line segments.
Find the length of a diagonal in a rectangle.
Apply Pythagoras’ Theorem to Isosceles Triangles and Trapezium
problems.
Use Pythagoras’ Theorem in 3-dimensional problems including
those where trigonometry is required.
Use trigonometric ratios (sin, cos and tan) to calculate angles in
right-angled triangles.
Use the trigonometric ratios to calculate unknown lengths in rightangled triangles.
Find the unknown lengths, or angles, in non right-angle triangles
using the sine and cosine rules.
Find the area of triangles given two lengths and an included angle
Find the angle between the diagonal through a cuboid and the base
of the cuboid.
Find the angle between a sloping edge of a pyramid and the base of
the pyramid.
Identify when to use the sine or cosine rule and adapt the relevant
formula to the given triangle.
Sketch the trigonometric graphs for sin and cos (and solve simple
trigonometric equations e.g. sinx = 0.5)
Know that:
o
Angle at the circumference of a semi-circle is 90 .
Angle at centre is double the angle at the circumference.
E
C
F
D
D
D
C
C
C
C
C
C
C
C
C
C
C
C
C
D-C
C
C
C
B
D
C
B
B
B
A/A*
B
B
A
A
B
A*
A*
A/A*
B
B
Use:
Angles in the same segment are equal.
Opposite angles in a cyclic quadrilateral are equal.
o
The angle between the tangent and radius is 90
Tangents from an external point are equal in length
Angles in the alternate segment are equal (Alternate segment
theorem)
Circle theorems to find unknown angles and explain their methodquoting the appropriate theorem(s).
B
B
A
A
A
B/A
Mock GCSE Exam:
Full GCSE mock exam to be taken which consists of two papers (Non-Calculator
and Calculator). Each paper will last 1 hour 45 minutes. Date will be announced during the term and placed on
the GCSE Higher Connect page.
Year 11: Spring Term
A11. Other
Functions
A12. Transforming
Functions
SSM14. Vectors
Plot and recognise cubic, reciprocal, exponential and circular
functions
Use the graphs of these functions to find approximate solutions to
equations, e.g. given x find y (and visa versa)
Represent translations in the x and y direction, reflections in the xaxis and the y-axis, and stretches parallel to the x-axis and the yaxis
Sketch the graph of y = 3 sin 2x, given the graph of y=sin x
Sketch the graph of y = f(x + 2), y = f(x) + 2, y=2f(x), y = f(2x) given
the shape of the graph y = f(x)
Find the coordinates of the minimum of y = f(x + 3), y = f(x) + 3 given
2
the coordinates of the minimum of y=x – 2x
Understand that 2a is parallel to a and twice its length
Understand that a is parallel to −a and in the opposite direction
Use and interpret vectors as displacements in the plane (with an
associated direction)
Use standard vector notation to combine vectors by addition, e.g.
AB + BC = AC and a + b = c
Represent vectors, and combinations of vectors, in the plane
Solve geometrical problems in 2-D, e.g. show that joining the midpoints of the sides of any quadrilateral forms a parallelogram.
A13. Algebraic
Decide with a reason whether a harder statement is true or false
Proof
Identify a counter example
Understand the difference between a demonstration and a proof
Show step-by-step deductions in providing a basic algebraic
explanation Show step-by-step deductions in providing a full
mathematical explanation Derive simple algebraic proofs using
reasoning
Derive harder algebraic proofs using reasoning and logic
Time left over for revision of topics and completion of past exam papers.
GCSE Exams: To be completed in the summer of Year 11
B/A
A
A*
A*
A*
A*
A
A
A
A
A
A*
D
C
B
A
A*