**Detailed unit content**

Content that is Higher Tier only is indicated in **bold type**.

The content of Higher Tier subsumes the content of Foundation Tier.

**1 Number**

**What students need to learn:**

**Number**

Add, subtract, multiply and divide any number

Add, subtract, multiply and divide whole numbers, integers, fractions, decimals and

numbers in index form

Add, subtract, multiply and divide negative numbers

Multiply or divide by any number between 0 and 1

Solve a problem involving division by a decimal (up to 2 decimal places)

** **Order rational numbers

Order integers, decimals and fractions

Understand and use positive numbers and negative integers, both as positions and

translations on a number line

** **

Use the concepts and vocabulary of factor (divisor), multiple, common factor, Highest Common Factor, Least Common Multiple, prime number and prime factor decomposition

Identify factors, multiples and prime numbers

Find the prime factor decomposition of positive integers

Find the common factors and common multiples of two numbers

Find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two

numbers

** **

Use the terms square, positive and negative square root, cube and cube root

Recall integer squares from 2 2 to 15 15 and the corresponding square roots

Recall the cubes of 2, 3, 4, 5 and 10

** **

Use index notation for squares, cubes and powers of 10

Use index notation for squares and cubes

Use index notation for integer powers of 10

Find the value of calculations using indices

** **

Use index laws for multiplication and division of integer, **fractional and negative **powers

Use index laws to simplify and calculate the value of numerical expressions involving

multiplication and division of integer, **fractional and negative **powers, and powers of a power

**Recall that ***n***0 ****= 1 and ***n *1 = 1 *n*

— **for positive**

**integers ***n ***as well as **** n **2

**1**

*n***and**3 31

****

*n*

*n***for any positive number**

*n***Higher**

**Interpret, order and calculate with numbers written in standard index form**

**Use standard form, expressed in conventional notation**

**Be able to write very large and very small numbers presented in a context in standard form**

**Convert between ordinary and standard form representations**

**Interpret a calculator display using standard form**

**Calculate with standard form**

** **

Understand equivalent fractions, simplifying a fraction by cancelling all common factors

Find equivalent fractions

Write a fraction in its simplest form

Convert between mixed numbers and

improper fractions

** **

Add and subtract fractions Add and subtract fractions

** **

Use decimal notation and recognise that each terminating decimal is a fraction

Recall the fraction-to-decimal conversion of

familiar simple fractions

Convert between fractions and decimals

** **

Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals

Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals

Convert between recurring decimals and fractions

**Understand a recurring decimal to fraction proof**

**Higher**

Understand that ‘percentage’ means ‘number of parts per 100’ and use this to compare proportions

Convert between fractions, decimals and percentages

** **

Use percentage, **repeated proportional change**

Use percentages to solve problems

Use percentages in real-life situations

– VAT

– Simple Interest

– Income tax calculations

– **Compound interest**

– **Depreciation**

– Find prices after a percentage increase or decrease

– **Percentage profit and loss**

**Calculate an original amount when given the transformed amount after a percentage change**

**Calculate repeated proportional change**

** **

**Understand and use direct and indirect proportion**

**Calculate an unknown quantity from quantities that vary in direct or inverse proportion**

** **

** **Interpret fractions, decimals and percentages as operators

Find a fraction of a quantity

Express a given number as a fraction of another number

Find a percentage of a quantity

Use decimals to find quantities

Express a given number as a percentage of another number

Understand the multiplicative nature of percentages as operators

**Represent repeated proportional change using a multiplier raised to a power**

**Use compound interest**

Use a multiplier to increase or decrease by a

percentage in any scenario where percentages are used

** **

** **Use ratio notation, including reduction to its simplest form and its various links to fraction notation

Use ratios

Write ratios in their simplest form

Understand and use number operations and the relationships between them, including inverse operations and hierarchy of operations

Multiply and divide numbers, using the commutative, associative, and distributive

laws and factorisation where possible, or place value adjustments

Use brackets and the hierarchy of operations

Use one calculation to find the answer to another

Understand ‘reciprocal’ as multiplicative inverse, knowing that any non-zero number

multiplied by its reciprocal is 1 (and that zero has no reciprocal, because division by zero

is not defined)

Find reciprocals

Use inverse operations

**Understand that the inverse operation of raising a positive number to a power ***n ***is raising the result of this operation to the power ***n *1

Understand and use unit fractions as multiplicative inverses

Solve word problems

**Use reverse percentage calculations N r Use surds and ***π ***in exact calculations**

**Use surds and ** **in exact calculations, without a calculator**

**Give an answer to a question involving the area of a circle as ****25***π*

**Give an answer to use of Pythagoras’ theorem as ****√13**

**Write ****(3 – √3)****2 ****in the form ***a ***+ ***b***√3**

**Rationalise a denominator**

**N s Calculate upper and lower bounds**

**Calculate the upper and lower bounds of calculations, particularly when working with measurements**

**Find the upper and lower bounds of calculations involving perimeter, areas and volumes of 2-D and 3-D shapes**

**Find the upper and lower bounds in real life situations using measurements given to appropriate degrees of accuracy**

**Give the final answer to an appropriate degree of accuracy following an analysis of the upper and lower bounds of a calculation **

** **

Divide a quantity in a given ratio Divide a quantity in a given ratio

Solve a ratio problem in a context

** **

Approximate to specified or appropriate degrees of accuracy including a given power of ten, number of decimal places and significant figures

Round numbers to a given power of 10

Round to the nearest integer and to any number of significant figures

Round to a given number of decimal places

Estimate answers to calculations, including use of rounding

Use calculators effectively and efficiently, including **trigonometrical **functions

Enter a range of calculations, including those involving time and money

Know how to enter complex calculations

Use an extended range of calculator

functions, including +, –, ×, ÷, *x*², √*x*, memory, *x**y*, *x*1/*y*, brackets and **trigonometric functions**

Understand, and interpret, the calculator display

Understand that premature rounding can cause problems when undertaking calculations with more than one step

**Calculate the upper and lower bounds of calculations, particularly when working with measurements**

**Use standard form display and know how to enter numbers in standard form**

**Calculate using standard form**

**Use calculators for reverse percentage calculations by doing an appropriate division**

**Use calculators to explore exponential growth and decay**

** **

**2 Algebra**

**What students need to learn:**

Distinguish the different roles played by letter symbols in algebra, using the correct notation

Use notation and symbols correctly

** **

Distinguish in meaning between the words ‘equation’, ‘formula’, **‘identity’ **and ‘expression’

Write an expression

Select an expression/**identity**/equation/formulae from a list

** **

Manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors, **multiplying two linear expressions, factorise quadratic expressions including the difference of two squares and simplify rational expressions**

Manipulate algebraic expressions by collecting like terms

Multiply a single term over a bracket

Use instances of index laws, including use of **fractional, zero and negative **powers, and

powers raised to a power

Factorise algebraic expressions by taking out common factors

Write expressions to solve problems

Use algebraic manipulation to solve problems

**Expand the product of two linear expressions**

**Factorise quadratic expressions**

**Factorise quadratic expressions using the difference of two squares**

**Simplify rational expressions by cancelling, adding, subtracting, and multiplying**

Set up and solve simple equations **including simultaneous equations in two unknowns**

Set up simple equations

Rearrange simple equations

Solve simple equations

Solve linear equations, with integer coefficients, in which the unknown appears

on either side or on both sides of the equation

Solve linear equations that include brackets,

those that have negative signs occurring anywhere in the equation, and those with a

negative solution

Solve linear equations in one unknown, with integer or fractional coefficients

**Find the exact solutions of two simultaneous equations in two unknowns**

**Use elimination or substitution to solve simultaneous equations**

**Interpret a pair of simultaneous equations as a pair of straight lines and their solution as the point of intersection**

**Set up and solve a pair of simultaneous equations in two variables**

** **

**Solve quadratic equations **

**Solve simple quadratic equations by using the quadratic formula**

**Solve simple quadratic equations by factorisation and completing the square**

** **

** **Derive a formula, substitute numbers into a formula and change the subject of a formula

Derive a formula

Use formulae from mathematics and other subjects

Substitute numbers into a formula

Substitute positive and negative numbers into expressions such as 3*x*2 + 4 and 2*x*3

Change the subject of a formula **including cases where the subject is on both sides of the original formula, or where a power of the subject appears**

** **

Solve linear inequalities in one or **two **variables, and represent the solution set on a number line **or coordinate grid**

Solve simple linear inequalities in one variable, and represent the solution set on

a number line

Use the correct notation to show inclusive and exclusive inequalities

**Show the solution set of several inequalities in two variables on a graph**

** **Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them

Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them

Understand the connections between changes of sign and location of roots

** **

Generate terms of a sequence using term-to-term and positionto-term definitions of the sequence

Recognise sequences of odd and even numbers

Generate simple sequences of numbers, squared integers and sequences derived from diagrams

Describe the term-to-term definition of a sequence in words

Find a specific term in a sequence using the position-to-term and term-to-term rules

Identify which terms cannot be in a sequence

** **

Use linear expressions to describe the *n*th term of an arithmetic sequence

Find the *n*th term of an arithmetic sequence

Use the *n*th term of an arithmetic sequence

** **

Use the conventions for coordinates in the plane and plot points in all four quadrants,

including using geometric information

Use axes and coordinates to specify points in all four quadrants in 2-D **and 3-D**

Identify points with given coordinates

Identify coordinates of given points (NB: Points may be in the first quadrant or all four quadrants)

Find the coordinates of points identified by geometrical information in 2-D **and 3-D**

Find the coordinates of the midpoint of a line segment

Calculate the length of a line segment

** **Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradients

Draw, label and scale axes

Recognise that equations of the form

*y *= *mx *+ *c *correspond to straight-line graphs in the coordinate plane

Plot and draw graphs of functions

Plot and draw graphs of straight lines with equations of the form *y *= *mx *+ *c*

Find the gradient of a straight line from a graph

**Find the gradient of lines given by equations of the form ***y ***= ***mx ***+ ***c*

**Analyse problems and use gradients to interpret how one variable changes in relation to another**

** **

**Understand that the form ***y = mx + c ***represents a straight line and that ***m ***is the gradient of the line and ***c ***is the value of the ***y**– ***intercept**

**Interpret and analyse a straight line graph**

**Understand that the form ***y = mx + c ***represents a straight line**

**Find the gradient of a straight line from its equation**

** **

**Understand the gradients of parallel lines**

**Explore the gradients of parallel lines and lines perpendicular to each other**

**Write down the equation of a line parallel or perpendicular to a given line**

**Select and use the fact that when ***y = mx + c ***is the equation of a straight line then the gradient of a line parallel to it will have a gradient of ***m ***and a line perpendicular to this line will have a gradient of **1*m*

**Interpret and analyse a straight line**

**graph and generate equations of lines**

**parallel and perpendicular to the given**

**line**

**Find the intersection points ofthe graphs of a linear and quadratic function, knowing that these are the approximate solutions of the corresponding simultaneous equations representing the linear and quadratic functions**

**Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other, or where the second equation is of the form ***x***2 ****+ ***y***2 ****= ***r***2**

**Find approximate solutions to simultaneous equations formed from one linear function and one quadratic function using a graphical approach**

**Select and apply algebraic and graphical techniques to solve simultaneous equations where one is linear and one quadratic**

** **

** Draw, sketch, recognise graphs of simple cubic functions, the reciprocal function ***y ***=***x*1 **with **** x **

**0**

**, the function**

*y***=**

*k*

*x***for integer values of**

*x***and simple positive values of**

*k***,**

**the trigonometric functions**

*y***= sin**

*x***and**

*y***= cos**

*x* **Plot graphs of simple cubic functions, the reciprocal function ***y = x ***1 ****with **** x **

**0**

**, the exponential function**

*y = k*

*x***for integer values of**

*x***and simple positive values of**

*k***,**

**the circular functions**

*y***= sin**

*x***and**

*y***= cos**

*x***, within the range -360º to +360º**

**Recognise the characteristic shapes of all these functions**

**Draw and plot a range of mathematical functions**

**Interpret and analyse a range of mathematical functions and be able to draw them, recognising that they were of the correct shape**

** **

**Construct the graphs of simple loci**

**Construct the graphs of simple loci including the circle ***x***2 ****+ ***y***2 ****= ***r***2 ****for a circle of radius ***r ***centred at the origin of the coordinate plane**

**Find graphically the intersection points of a given straight line with this circle**

**Select and apply construction techniques and understanding of loci to draw graphs**

**based on circles and perpendiculars of lines**

** **Construct linear, **quadratic and other **functions from real-life problems and plot their

corresponding graphs

Draw straight line graphs for real-life situations

– ready reckoner graphs

– conversion graphs

– fuel bills

– fixed charge (standing charge) and cost per unit

– distance time graphs

Draw distance-line graphs

**Generate points and plot graphs of simple quadratic functions, then more general quadratic functions**

**Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function**

**Find the intersection points of the graphs of a linear and quadratic function, knowing that these are the approximate solutions of the corresponding simultaneous equations representing the linear and quadratic functions**

** **

Discuss, plot and interpret graphs (which may be non-linear) modelling real situations

Plot a linear graph

Interpret straight line graphs for real-life situations

– ready reckoner graphs

– conversion graphs

– fuel bills

– fixed charge (standing charge) and cost per unit

Interpret distance-time graphs

Interpret information presented in a range of

linear and non-linear graphs

** **

Generate points and plot graphs of simple quadratic functions, and use these to find approximate solutions

Generate points and plot graphs of simple quadratic functions, then more general

quadratic functions

Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function

**Select and use the correct mathematical techniques to draw quadratic graphs**

**Direct and indirect proportion**

**Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relate algebraic solutions to graphical representation of the equations**

** **

**Transformation of functions ** **Apply to the graph of ***y ***= f(***x***) ****the transformations ***y ***= f(***x***) + ***a***, ***y ***= f(***ax***)****, ***y ***= f(***x ***+ ***a***)****, ***y ***= ***a***f(***x***) ****for linear, quadratic, sine and cosine functions ****f(***x***)**

**Select and apply the transformations of reflection, rotation, enlargement and translation of functions expressed algebraically**

**Interpret and analyse transformations of functions and write the functions algebraically**

**3 Geometry**

**What students need to learn:**

Recall and use properties of angles at a point, angles on a straight line (including right

angles), perpendicular lines, and opposite angles at a vertex

Recall and use properties of angles

– angles at a point

– angles at a point on a straight line, including right angles

– perpendicular lines

– vertically opposite angles

** **

Understand and use the angle properties of parallel lines, triangles and quadrilaterals

Distinguish between scalene, isosceles, equilateral, and right-angled triangles

Understand and use the angle properties of triangles

Use the angle sum of a triangle is 180°

Understand and use the angle properties of intersecting lines

Understand and use the angle properties of parallel lines

Mark parallel lines on a diagram

Use the properties of corresponding and alternate angles

Understand and use the angle properties of quadrilaterals

Give reasons for angle calculations

Explain why the angle sum of a quadrilateral is 360°

Understand the proof that the angle sum of a triangle is 180°

Understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices

Use the size/angle properties of isosceles and equilateral triangles

**Recall and use these properties of angles in more complex problems**

Calculate and use the sums of the interior and exterior angles of polygons

Calculate and use the sums of the interior angles of polygons

Use geometric language appropriately and recognise and name pentagons, hexagons,

heptagons, octagons and decagons

Use the angle sums of irregular polygons

Calculate and use the angles of regular polygons

Use the sum of the interior angles of an *n*-sided polygon

Use the sum of the exterior angles of any polygon is 360o

Use the sum of the interior angle and the exterior angle is 180o

Find the size of each interior angle or the size of each exterior angle or the number of sides of a regular polygon

Understand tessellations of regular and irregular polygons

Tessellate combinations of polygons

Explain why some shapes tessellate when other shapes do not

** **

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 symmetry properties

List the properties of each, or identify (name) a given shape

Classify quadrilaterals by their geometric properties

** **

** **Recognise reflection and rotation symmetry of 2-D shapes

Recognise reflection symmetry of 2-D shapes

Identify and draw lines of symmetry on a shape

Recognise rotation symmetry of 2-D shapes

Identify the order of rotational symmetry of a 2-D shape

Draw or complete diagrams with a given number of lines of symmetry

State the line symmetry as a simple algebraic equation

Draw or complete diagrams with a given order of rotational symmetry

** **Understand congruence and similarity

Recognise that all corresponding angles in similar figures are equal in size when the

lengths of sides are not

**Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and a pair of compasses constructions**

**Understand similarity of triangles and of other plane figures, and use this to make**

**geometric inferences**

**Complete a formal geometric proof of similarity of two given triangles**

** **

Use Pythagoras’ theorem in 2-D **and 3-D**

Understand, recall and use Pythagoras’ theorem in 2-D, **then in 3-D problems**

**Understand the language of planes, and recognise the diagonals of a cuboid**

**Calculate the length of a diagonal of a cuboid**

** **

**Use the trigonometric ratios and the sine and cosine rules to solve 2-D and 3-D**

**problems**

**Use the trigonometric ratios to solve 2-D and 3-D problems**

**Understand, recall and use trigonometric relationships in right-angled triangles, and use these to solve problems in 2-D and in 3-D configurations**

**Find the angle between a line and a plane (but not the angle between two planes or between two skew lines)**

**Find angles of elevation and angles of depression**

**Use the sine and cosine rules to solve 2-D and 3-D problems**

** **

Distinguish between centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

Recall the definition of a circle and identify (name) and draw the parts of a circle

Understand related terms of a circle

Draw a circle given the radius or diameter

** Understand and construct geometrical proofs using circle theorems**

**Understand and use the fact that the tangent at any point on a circle is perpendicular to the radius at that point**

**Understand and use the fact that tangents from an external point are equal in length**

**Find missing angles on diagrams**

**Give reasons for angle calculations involving the use of tangent theorems**

**Prove and 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 in a semicircle is a right angle**

– **angles in the same segment are equal**

– **opposite angles of a cyclic quadrilateral sum to 180****o**

– **alternate segment theorem**

– **the perpendicular from the centre of a circle to a chord bisect the chord**

** **

Use 2-D representations of 3-D shapes

Use 2-D representations of 3-D shapes

Use isometric grids

Draw nets and show how they fold to make a 3-D solid

Understand and draw front and side elevations and plans of shapes made from simple solids

Given the front and side elevations and the plan of a solid, draw a sketch of the 3-D solid

** **Describe and transform 2-D shapes using single or combined rotations, reflections,

translations, or enlargements by a positive, **fractional or negative **scale factor and

distinguish properties that are preserved under particular transformations

Describe and transform 2-D shapes using single rotations

Understand that rotations are specified by a centre and an (anticlockwise) angle

Find the centre of rotation

Rotate a shape about the origin, or any other point

Describe and transform 2-D shapes using single reflections

Understand that reflections are specified by a mirror line

Identify the equation of a line of symmetry

Describe and transform 2-D shapes using single translations

Understand that translations are specified by a distance and direction (using a vector)

Translate a given shape by the vector

Describe and transform 2-D shapes using enlargements by a positive and **a negative**

**or fractional **scale factor

Understand that an enlargement is specified by a centre and a scale factor

Enlarge shapes using (0, 0) as the centre of enlargement

Enlarge shapes using centre other than (0, 0)

Find the centre of enlargement

Describe and transform 2-D shapes using combined rotations, reflections, translations, or enlargements

Distinguish properties that are preserved under particular transformations

Recognise that enlargements preserve angle but not length

Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations

Understand that distances and angles are preserved under rotations, reflections and translations so that any shape is congruent to its image

Describe a transformation

** **

Use straight edge and a pair of compasses to carry out constructions

Use straight edge and a pair of compasses to do standard constructions

Construct a triangle

Construct an equilateral triangle

Understand, from the experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not

Construct the perpendicular bisector of a given line

Construct the perpendicular from a point to a line

Construct the perpendicular from a point on a line

Construct the bisector of a given angle

Construct angles of 60o, 90o, 30o, 45o

Draw parallel lines

Draw circles and arcs to a given radius

Construct a regular hexagon inside a circle

Construct diagrams of everyday 2-D situations involving rectangles, triangles, perpendicular and parallel lines

Draw and construct diagrams from given information

Construct loci

Construct:

– a region bounded by a circle and an intersecting line

– given distance from a point and a given distance from a line

– equal distances from two points or two line segments

– regions which may be defined by ‘nearer to’ or ‘greater than’

Find and describe regions satisfying a combination of loci (NB: All loci restricted to two dimensions only)

** **

Calculate perimeters and areas of shapes made from triangles, rectangles **and other shapes**

Measure shapes to find perimeter or area

Find the perimeter of rectangles and triangles

Calculate perimeter and area of compound shapes made from triangles, rectangles **and other shapes**

Recall and use the formulae for the area of a triangle, rectangle and a parallelogram

Calculate areas of shapes made from triangles and rectangles

Calculate perimeters of compound shapes made from triangles and rectangles

Find the area of a trapezium

Find the area of a parallelogram

Find the surface area of simple shapes (prisms) using the formulae for triangles and

rectangles, **and other shapes**

** **

** Calculate the area of a triangle using **2 1 *ab ***sin ***C*

**Calculate the area of a triangle given the**

**length of two sides and the included angle**

**Higher**

Find circumferences and areas of circles

Find circumferences of circles and areas enclosed by circles

Recall and use the formulae for the circumference of a circle and the area enclosed by a circle

Use 3.142 or use the button on a calculator

Find the perimeters and areas of semicircles and quarter circles

**Calculate the lengths of arcs and the areas of sectors of circles**

**Answers in terms of ** **may be required**

Find the surface area of a cylinder

** **

Calculate volumes of right prisms and shapes made from cubes and cuboids

Calculate volumes of right prisms, including the triangular prism, and shapes made from cubes and cuboids

Recall and use the formula for the volume of a cuboid

Find the volume of a cylinder

Use volume to solve problems

** **

**Solve mensuration problems involving more complex shapes and solids**

**Solve problems involving more complex shapes and solids, including segments of circles and frustums of cones**

**Find the surface area and volumes of compound solids constructed from cubes, cuboids, cones, pyramids, spheres, hemispheres, cylinders ***Examples:*

**Solve problems including examples of solids in everyday use**

**Find the area of a segment of a circle given the radius and length of the chord**

** Use vectors to solve problems**

**Understand and use vector notation**

**Calculate, and represent graphically, the sum of two vectors, the difference of two vectors and a scalar multiple of a vector**

**Calculate the resultant of two vectors**

**Solve geometrical problems in 2-D using vector methods**

**Apply vector methods for simple geometrical proofs**

**4 Measures**

**What students need to learn:**

Use and interpret maps andscale drawings

Use and interpret maps and scale drawings

Read and construct scale drawings

Draw lines and shapes to scale

Estimate lengths using a scale diagram

** **

Understand **and use **the effect of enlargement for perimeter, area and volume of shapes and solids

Understand the effect of enlargement for perimeter, area and volume of shapes and

solids

Understand that enlargement does not have the same effect on area and volume

Use simple examples of the relationship between enlargement and areas and volumes

of simple shapes and solids

**Use the effect of enlargement on areas and volumes of shapes and solids**

**Know the relationships between linear,**

**area and volume scale factors of mathematically similar shapes and solids**

** **

** **Interpret scales on a range of measuring instruments and recognise the inaccuracy of

measurements

Know that measurements using real numbers depend upon the choice of unit

Recognise that measurements given to the nearest whole unit may be inaccurate by

up to one half in either direction

** **Convert measurements from one unit to another

Convert between units of measure in the same system (NB: Conversion between imperial units will be given. Metric equivalents should be known)

Know rough metric equivalents of pounds, feet, miles, pints and gallons:

**Metric Imperial**

1 kg 2.2 pounds

1 *l *14 3 pints

4.5 *l *1 gallon

8 km 5 miles

30 cm 1 foot

Convert between imperial and metric measures

Convert between metric area measures

Convert between metric volume measures

Convert between metric speed measures

Convert between metric units of volume and units of capacity measures, eg 1 m*l *= 1cm³

** **

Make sensible estimates of a range of measures

Make sensible estimates of a range of measures in everyday settings

Choose appropriate units for estimating or carrying out measurements

** **

Understand and use bearings Use three-figure bearings to specify direction

Mark on a diagram the position of the point *B *given its bearing from point *A*

Measure or draw a bearing between the points on a map or scaled plan

Given the bearing of a point *A *from point *B*, work out the bearing of *B *from *A*

** **

Understand and use compound measures

Understand and use compound measures, including speed and **density**

** **

Measure and draw lines and angles

Measure and draw lines, to the nearest mm

Measure and draw angles, to the nearest degree

Draw triangles and other 2-D shapes using ruler and protractor

Make accurate drawing of triangles and other 2-D shapes using a ruler and a protractor

Make an accurate scale drawing from a diagram

Use accurate drawing to solve a bearings problem

**5 Statistics**

**What students need to learn:**

** **Understand and use statistical problem solving process/handling data cycle

Specify the problem and plan

Decide what data to and what statistical analysis is needed

Collect data from a variety of suitable primary and secondary sources

Use suitable data collection techniques

Process and represent the data

Interpret and discuss the data

** **

Identify possible sources of bias

**Discuss how data relates to a problem, identify possible sources of bias and plan to minimise it**

**Understand how different sample sizes may affect the reliability of conclusions drawn**

** **

Design an experiment or survey Identify which primary data they need to collect and in what format, including grouped data

Consider fairness

Understand sample and population

Design a question for a questionnaire

Criticise questions for a questionnaire

**Design an experiment or survey**

**Select and justify a sampling scheme and a method to investigate a population, including random and stratified sampling**

**Use stratified sampling**

** **

Design data-collection sheets distinguishing between different types of data

Design and use data-collection sheets for grouped, discrete and continuous data

Collect data using various methods

Sort, classify and tabulate data and discrete or continuous quantitative data

Group discrete and continuous data into class intervals of equal width

** **Extract data from printed tables and lists

Extract data from lists and tables

** **

** **Design and use two-way tables for discrete and grouped data

Design and use two-way tables for discrete and grouped data

Use information provided to complete a twoway table

** **

Produce charts and diagrams for various data types

Produce:

– Composite bar charts

– Comparative and dual bar charts

– Pie charts

– Histograms with equal class intervals

– Frequency diagrams for grouped discrete data

– Scatter graphs

– Line graphs

– Frequency polygons for grouped data

– **Grouped frequency tables for continuous data**

– Ordered stem and leaf diagrams

– **Cumulative frequency tables**

– **Cumulative frequency graphs**

– **Box plots from raw data and when given quartiles, median**

– **Histograms from class intervals with unequal width**

**Use and understand frequency density**

Calculate median, mean, range, **quartiles and interquartile range, **mode and modal class

Calculate:

– mean,

– mode,

– median,

– range,

– modal class,

– the interval which contains the median

Estimate the mean of grouped data using the mid-interval value

Find the median, **quartiles and interquartile range **for large data sets with grouped data

Estimate the mean for large data sets with grouped data

Understand that the expression ‘estimate’ will

be used where appropriate, when finding the mean of grouped data using mid-interval

values

**Use cumulative frequency graphs to find median, quartiles and interquartile range**

**Interpret box plots to find median, quartiles, range and interquartile range**

Interpret a wide range of graphs and diagrams and draw conclusions

Interpret:

– composite bar charts

– comparative and dual bar charts

– pie charts

– stem and leaf diagrams

– scatter graphs

– frequency polygons

– **box plots**

– **cumulative frequency diagrams**

– **histograms**

Recognise simple patterns, characteristics and relationships in line graphs and frequency polygons

**Find the median from a histogram or any other information from a histogram, such as the number of people in a given interval**

From line graphs, frequency polygons and frequency diagrams

– read off frequency values

– calculate total population

– find greatest and least values

From pie charts:

– find the total frequency

– find the size of each category

Find the mode, median, range and

**interquartile range**, **as well as the greatest and least values from stem and leaf diagrams**

**From cumulative frequency graphs:**

– **estimate frequency greater/less than a given value**

– **find the median and quartile values and interquartile range**

**From histograms:**

– **complete a grouped frequency table **– **understand and define frequency density **(NB: No pictograms or bar charts at higher)

Look at data to find patterns and exceptions

Present findings from databases, tables and charts

Look at data to find patterns and exceptions

**Explain an isolated point on a scatter graph**

Recognise correlation and draw and/or use lines of best fit by eye, understanding what these represent

Draw lines of best fit by eye, understanding what these represent

Distinguish between positive, negative and zero correlation using lines of best fit

Understand that correlation does not imply causality

Use a line of best fit, **or otherwise, **to predict values of one variable given values of

the other variable

**Appreciate that correlation is a measure of the strength of the association between two variables and that zero correlation does not necessarily imply ‘no relationship’ but merely ‘no linear relationship’**

** **

Compare distributions and make inferences

**Compare distributions and make inferences, using the shapes of distributions and measures of average and spread, including median and quartiles**

Compare the mean and range of two distributions, **or median and interquartile range**, as appropriate

Understand that the frequency represented by corresponding sectors in two pie charts is dependent upon the total populations represented by each of the pie charts

Use dual or comparative bar charts to compare distributions

Recognise the advantages and disadvantages between measures of average

**Compare the measures of spread between a pair of box plots/cumulative**

**frequency graphs**

** **Use calculators efficiently and effectively, including statistical functions

Calculate the mean of a small data set, using the appropriate key or a scientific calculator

**Use ****Σ***x ***and ****Σ***fx ***or the calculation of the line of best fit**

**6 Probability**

**What students need to learn:**

Understand and use the vocabulary of probability and probability scale

Distinguish between events which are; impossible, unlikely, even chance, likely, and

certain to occur

Mark events and/or probabilities on a probability scale of 0 to 1

Write probabilities in words or fractions, percentages or decimals

** **

** **Understand and use estimates or measures of probability from theoretical models (including equally likely outcomes), or from relative frequency

Understand and use estimates or measures of probability, including relative frequency

Use theoretical models to include outcomes using dice, spinners, coins

Find the probability of successive events, such as several throws of a single dice

Estimate the number of times an event will occur, given the probability and the number of trials

** **

** **List all outcomes for single events, and for two successive events, in a systematic way and derive relative probabilities

List all outcomes for single events, and for two successive events, systematically

Use and draw sample space diagrams

** **

Identify different mutually exclusive outcomes and know that the sum of the probabilities

of all these outcomes is 1

Add simple probabilities

Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1

Use 1 − *p *as the probability of an event not occurring where *p *is the probability of

the event occurring

Find a missing probability from a list or table

** **

**Know when to add or multiply two probabilities: when ***A ***and ***B ***are mutually exclusive, then the**

**probability of ***A ***or ***B ***occurring is ****P(***A***) ****+ ****P(***B***)****, whereas when ***A ***and ***B ***are independent vents, the probability of ***A ***and ***B ***occurring is ****P(***A***) ****× ****P(***B***)**

**Understand conditional probabilities**

**Understand selection with or without replacement**

** Use tree diagrams to represent outcomes of compound events, recognising when events are**

**independent**

**Draw a probability tree diagram based on given information (no more than 3 branches per event)**

**Use a tree diagram to calculate conditional probability**

** **

Compare experimental data and theoretical probabilities

Compare experimental data and theoretical probabilities

Understand that if they repeat an experiment, they may – and usually will − get different outcomes, and that increasing sample size generally leads to better estimates of probability and population characteristics