Objectives By the end of the module students should be able to …

Grade

Whole numbers

1 – 3

Understand and order integers

multiply and divide positive integers

multiply and divide negative integers

Round whole numbers to the nearest, 10, 100, 1000, …

Multiply and divide whole numbers by a given multiple of 10

Check their calculations by rounding, e.g. 29 ´ 31 » 30 ´ 30

G F E G D

Decimals

3 – 5

Put digits in the correct place in a decimal number

Write decimals in ascending order of size

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 dp), e.g. 266.22 ¸ 0.34

Know that e.g. 13.5 ¸ 0.5 = 135 ¸ 5

Check their answer by rounding, know that e.g. 2.9 ´ 3.1 » 3.0 ´ 3.0

F F F E D D D

Fractions: Addition and subtraction

1 – 3

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 number, and visa versa

Add and subtract mixed numbers

F D D C

Fractions: Multiplication and division

1 – 3

Convert a fraction to a decimal, or a decimal to a fraction

Find the reciprocal of whole numbers, fractions, and decimals

Multiply and divide a fraction by an integer, by a unit fraction and by a general fraction (expressing the answer in its simplest form)

Convert a fraction to a recurring decimal (and visa versa)

Use fractions in contextualised problems

D C C B

Coordinates

1 – 3

Plot and reading coordinates on a coordinate grid (in all four quadrants)

Understand that one coordinate identifies a point on a line, two coordinates identify a point in a plane and three coordinates identify a point in space, and use the terms ‘1-D’, ‘2-D’ and ‘3-D’

Find the coordinates of the fourth vertex of a parallelogram

Identify the coordinates of the vertex of a cuboid on a 3-D grid

Writing down the coordinates of the midpoint of the line connecting two points

Calculate the length of the line segment joining to point in the plane (all four quadrants)

F A D A C A

Introduction to algebra

1 – 3

Simplify algebraic expressions in one or more like terms by addition and subtraction

Multiply and divide with letters and numbers

Multiply and divide powers of the same letter

Understand and use the index rules to simplify algebraic expressions

Use brackets to expand and simplify simple algebraic expressions

E D C C/B C

Angles

5 – 7

Distinguish between acute, obtuse, reflex and right angles

Use angle properties on a line and at a point to calculate unknown angles

Use angle properties of triangles and quadrilaterals to calculate unknown angles

Use parallel lines to identify alternate and corresponding angles

Calculate interior and exterior angles in a polygon

Understand and use bearings

F F E D C D

Collecting data

3 – 5

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

Choose and justify an appropriate sampling scheme, including random and systematic sampling

Deal with practical problems in data collection, such as non-response, missing and anomalous data

C B D A D

Charts and graphs

1 – 3

Represent data as:

Pie charts (for categorical data)

Bar charts and histograms (equal class intervals)

Frequency polygons

Choose an appropriate way to display discrete, continuous and categorical data

Understand the difference between a bar chart and a histogram

Compare distributions shown in charts and graphs

E B D/C B A-E

2-D shapes

1 – 3

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 regular hexagon inside a circle, etc

A region bounded by a circle and an intersecting line

A path equidistant from 2 points or 2 line segments, etc

E C C C D C C

Properties of triangles and quadrilaterals

3 – 5

Mark parallel lines in a diagram

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 missing

Identify and list the properties of quadrilaterals (including kites)

Name all quadrilaterals that have a pair of opposite sides that are equal

G D D

Factors and multiples

1 – 3

Find: squares; cubes; square roots; cube roots of numbers, with and without a calculator (including the use of trial and improvement)

Understand odd and even numbers, and prime numbers

Find the HCF and the LCM of numbers

Write a number as a product of its prime factors, e.g. 108 = 22 ´ 33

E G C C

Percentages

5 – 7

Understand that a percentage is a fraction in hundredths

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

G F D E C B D C

Perimeter and area

1 – 3

Use Pythagoras’ theorem to find unknown lengths, e.g. the height of an isosceles triangle given the lengths of all three sides

Find the perimeter and area of shapes made up from triangles and rectangles

Find when numbers are given to a specific degree of accuracy, the upper and lower bounds of perimeters and areas

Convert between units of area

C D B D

3-D shapes

1 – 3

Count the vertices, faces and edges of 3-D shapes

Draw nets of solids and recognise solids from their nets

Draw and interpret plans and elevations

Draw planes of symmetry in 3-D shapes

Recognise and name examples of solids, including prisms, in the real world

D/G D D E

Solving linear equations

5 – 7

Solve linear equations with one, or more, operations (including fractional coefficients)

Solve linear equations involving a single pair of brackets

C D

Patterns and sequences

3 – 5

Find the missing numbers in a number pattern or sequence

Find the nth term of a number sequence as an algebraic expression

Explain why a number is, or is not, a member of a given sequence

Use a calculator to produce a sequence of numbers

E C

Brackets

1 – 3

Expand or factorise algebraic expressions involving one pair of brackets

Expand and simplify expressions involving two pairs of brackets

Factorise quadratic expressions (including the difference of two squares)

D B B

Formulae

5 – 7

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. convert the formula for converting Centigrade into Fahrenheit into a formula that converts Fahrenheit into Centigrade

Generate a formula from given information, e.g. find the formula for the perimeter of a rectangle given its area A and the length of one side

D C D – A* B D – A*

Circle theorems

5 – 7

Understand, prove and use circle theorems (see below)

Use circle theorems to find unknown angles and explain their method- quoting the appropriate theorem(s)

Understanding that the tangent at any point on a circle is perpendicular to the radius at that point

Understanding and using the fact that tangents from an external point are equal in length

Explaining why the perpendicular from the centre to a chord bisect the chord

Proving and using the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference

Proving and using the fact that the angle subtended at the circumference by a semicircle is a right angle

Proving and using the fact that angles in the same segments are equal

Proving and using the fact that opposite angles of a cyclic quadrilateral sum to 180 degrees

Proving and using the alternate segment theorem

B B B B B A A A A

Linear functions y = mx + c

7 – 9

Substitute values of x into linear functions to find corresponding values of y

Plot points for linear functions on a coordinate grid and draw the corresponding straight lines

Interpret m and c as gradient and y-intercept in linear functions

Understand that the graphs of linear functions are parallel if they have the same value of m

Know that the line perpendicular to y = mx + c has gradient -1/m

Understand linear functions in practical problems, e.g. distance-time graphs

C C B A

Similar shapes

5 – 7

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

B/C E A

Perimeter and area of circles

3 – 5

Find the perimeter and area of shapes made up from triangles, rectangles and parts of circles

Use and recall formulae to calculate perimeters and areas of circles, and parts of circles

C/D D

Scatter graphs and correlation

3 – 5

Draw and 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 a line of best fit by eye and understand what it represents

Use a line of best fit to interpolate/ extrapolate

D C C D

Ratio and scale

1 – 3

Appreciate that e.g. the ratio 1:2 represents 1/3 and 2/3 of a quantity

Divide quantities in a given ratio, e.g. divide £20 in the ratio 2:3

Solve word problems involving ratios, e.g. Find the cost of 8 pencils given that 6 cost 78p

Work out the real distance from a map, e.g. Find the real distance represented by 4 cm on a map with scale 1:25 000

Work out the distance on a map for a given real distance and scale

D/C C E

Direct and inverse proportion

5 – 7

Interpret direct and inverse proportions as algebraic functions, e.g. yµx2 as y = kx2

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 (yµx, yµx2, yµx3, yµ 1/x, yµ 1/x2)

A A A A

The mean (large data sets)

1 – 3

Find the mean of data given in an ungrouped frequency distribution

Use the mid interval value to find an estimate for the mean of data given in a grouped frequency distribution

Understand and use the sigma notation for the mean of ungrouped, and grouped, data

D C

Simultaneous equations

5 – 7

Solve algebraically two simultaneous equations

Interpret the solution of two simultaneous equations as the point of intersection the corresponding lines

B

Probability

7 – 9

List all the outcomes from mutually exclusive events, e.g. from two coins, and sample space diagrams

Write down the probability associated with equally likely events, e.g. the probability of drawing an ace from a pack of cards

Know that if the probability of an event occurring is p than the probability of it not occurring is 1 – p

Find the missing probability from a list or table

Know that the probability of A or B is P(A) + P(B)

Know that the probability of A and B is P(A) ´ P(B)

Draw and use tree diagrams to solve probability problems (including examples of non-replacement)

Find estimates of probabilities by considering relative frequency in experimental results (including two-way tables)

Know that the more an experiment is repeated the better the estimate of probability

E E B C C

Standard form

1 – 3

Understand the standard form convention

Convert numbers to, and from, standard form

Calculate with numbers given in standard form with, and without, a calculator

Round numbers given in standard form to a given number of significant figures

B B B B

Inequalities

3 – 5

Rearrange and solve linear inequalities in one variable and show the solution set on a number line, or to write down all the integer solutions.

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

C B

Quadratic functions

3 – 5

Plot the graphs of quadratic functions for positive and negative values of x

Find graphically the solutions of quadratic equations by considering the intercept on the x-axis

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 dp

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)

F C B A A A*

Speed and density

3 – 5

Use the relationship between distance, speed and time to solve problems

Convert between metric units of speed e.g. km/h to m/s

Know that density is found by mass ÷ volume

Use the relationship between density, mass and volume to solve problems, e.g. find the mass of an object with a given volume and density

Convert between metric units of density e.g. kg/m to g/cm

G

Trial and improvement

1 – 3

Solve cubic functions by successive substitution of values of x

C

Angle properties of polygons

1 – 3

Calculate and use the sums of the interior angles of convex 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

D/C C

Surface area and volume

1 – 3

Find volumes of shapes by counting cubes

Use formulae to calculate the surface areas and volumes of cuboids, right-prisms and cylinders

Solve a range of problems involving surface area and volume, e.g. given the volume and length of a cylinder find the radius

Convert between units of volume

G C A

Transformations

5 – 7

Understand translation as a combination of a horizontal and vertical shift including signs for directions

Understand rotation as a (clockwise) turn about a given origin

Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = –x

Enlarge shapes by a given scale factor from a given point; using positive and negative scale factors greater than one

Enlarge shapes by a given fractional scale factor, e.g. 2/3

Understand that shapes produced by translation, rotation and reflection are congruent to its image

D D/C C D C C

Further simultaneous equations

5 – 7

Find graphically the approximate solutions of linear and quadratic simultaneous equations

Find the exact solutions of linear and quadratic simultaneous equations

Draw a circle of radius r centred at the origin

Find graphically the approximate solutions of linear and circular simultaneous equations

Find the exact solutions of linear and circular simultaneous equations

B/A A A A A*

Surface area and volume

5 – 7

Find the surface area and the volume of more complex shapes, e.g. find the volume of an equilateral triangular prism

Solve more complex problems, e.g. given the surface area of a sphere find the volume

C – A* A/A*

Median and interquartile range (large data sets)

5 – 7

Find the median and quartiles for large sets of ungrouped data

Draw a cumulative frequency table for grouped data (using the upper class boundary)

Draw a cumulative frequency curve for grouped data

Use a cumulative frequency diagram to find estimates for the median and quartiles of a distribution

Use a cumulative frequency diagram to solve problems, e.g. how many greater than a particular value

Draw a box plot to summarise information given in cumulative frequency diagrams

Compare cumulative frequency diagrams and box lots to make inferences about distributions

C B B B B B B

Histograms

3 – 5

Complete a histogram from a frequency table

Complete a frequency table from a histogram

Use a histogram to work out the frequency in part of a class interval

A A A

Sine, cosine and tangent

5 – 7

Use trigonometric ratios (sin, cos and tan) to calculate angles in right-angled triangles

Use the trigonometric ratios to calculate unknown lengths in right-angled triangles

B B

Trigonometry for non right-angled triangles

3 – 5

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

A

Applications of trigonometry in 3-D

3 – 5

Calculate the length of a diagonal of a rectangle given the lengths of the sides of the rectangle

Calculate the diagonal through a cuboid, or across the face of a cuboid

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

C B A* A* A

Further functions

5 – 7

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)

Find the values of p and q in the function y = pqx given the graph of y = pqx

Match equations with there graphs

Sketch graphs of given functions

Vectors

5 – 7

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 mid-points of the sides of any quadrilateral forms a parallelogram

A A A A A A*

Transformations of graphs

5 – 7

§Represent translations in the x and y direction, reflections in the x-axis and the y‑axis, and stretches parallel to the x-axis and the y-axis

§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 the coordinates of the minimum of y=x2 – 2x

Scheme of work

General TopicNo. of Hours*ObjectivesBy the end of the module students should be able to …GradeWhole numbersF

E

G

D

DecimalsF

F

E

D

D

D

Fractions: Addition and subtractionD

D

C

Fractions: Multiplication and divisionC

C

B

CoordinatesA

D

A

C

A

Introduction to algebraD

C

C/B

C

AnglesF

E

D

C

D

Collecting dataB

D

A

D

Charts and graphsB

D/C

B

A-E

2-D shapesC

C

C

D

C

C

Properties of triangles and quadrilateralsD

D

Factors and multiplesG

C

C

PercentagesF

D

E

C

B

D

C

Perimeter and areaD

B

D

3-D shapesD

D

E

Solving linear equationsD

Patterns and sequencesnth term of a number sequence as an algebraic expressionC

BracketsB

B

FormulaeC

D – A*

B

D – A*

Circle theoremsB

B

B

B

A

A

A

A

Linear functionsy=mx+cxinto linear functions to find corresponding values of ymandcas gradient andy-intercept in linear functionsmy=mx+chas gradient -1/mC

B

A

Similar shapesE

A

Perimeter and area of circlesD

Scatter graphs and correlationC

C

D

Ratio and scaleC

E

Direct and inverse proportionyµx2 asy=kx2k, to find unknown valuesyµx,yµx2,yµx3,yµ 1/x,yµ 1/x2)A

A

A

The mean (large data sets)C

Simultaneous equationsProbabilitypthan the probability of it not occurring is 1 –pE

B

C

C

Standard formB

B

B

InequalitiesB

Quadratic functionsxx-axisanot equal to 1)C

B

A

A

A*

Speed and densityTrial and improvementxAngle properties of polygonsC

Surface area and volumeC

A

Transformationsy=xory= –xD/C

C

D

C

C

Further simultaneous equationsA

A

A

A*

Surface area and volumeA/A*

Median and interquartile range (large data sets)B

B

B

B

B

B

HistogramsA

A

Sine, cosine and tangentB

Trigonometry for non right-angled trianglesApplications of trigonometry in 3-DB

A*

A*

A

Further functionsxfindy(and visa versa)y=pqxgiven the graph ofy=pqxVectorsais parallel toaand twice its lengthais parallel to -aand in the opposite directionAB+BCACanda+bcA

A

A

A

A*

Transformations of graphsxandydirection, reflections in thex-axis and they‑axis, and stretches parallel to thex-axis and they-axisy= 3 sin 2x, given the graph ofy=sinxy= f(x+ 2),y= f(x) + 2,y=2f(x),y= f(2x) given the shape of the graphy= f(x)y= f(x+ 3),y= f(x) + 3 given the coordinates of the minimum ofy=x2 – 2xA*

A*