The Sir John Colfox Academy Mathematics Mastery Scheme of Work Overview Year 8

Autumn 1

Number

Autumn 2

Algebraic expressions

Spring 1

2D Geometry

Spring 2

Proportional reasoning

Summer 1

3D Geometry

Summer 2

Statistics

Supporting content from year 7/ KS2…

Factors, multiples,

primes

Multiplication and

division

Fraction equivalence and

calculations

Problem solving with

fractions

Order of operations

(BIDMAS)

Form algebraic

expressions

Substitution

Angle types

Angle facts

(triangle,

quadrilateral,

straight line, full

turn)

Rectangle & triangle

areas & perimeters

x/÷ by powers of 10

Negative numbers

Rounding

Bar modelling with

fractions

FDP equivalence

Rectilinear areas

Fractions

Percentage

increase/ decrease

Substitution with

negatives

Statistical diagrams

Ratio and rate

Mean average

Calculator skills and

rounding

Core content for Year 8 (all students should have access to)…

a) Primes, squares and

cubes, indices, prime

factorisation and Venn

diagrams to find LCM,

HCF. 7.1, 8.1

b) Venn diagrams and

enumerating sets

c) All 4 operations with

fractions (+, -, x, ÷)

a) Substitute into and

evaluate

expressions 7.6, 8.5

b) Linear equations

7.13, 8.11

c) Expressions and

equations from real-

world situations

d) Linear sequences,

nth term rule 8.8

e) Negative numbers

and inequality

statements

a) Construct/ draw

accurate triangles

quadrilaterals 7.4

b) Unknown angles (incl

parallel lines) 8.9

c) Areas and

perimeters of

parallelograms,

trapezia and

composite figures

plus length and area

conversions 7.10

a) Convert between

percentages, vulgar

fractions and

decimals 8.6

b) Percentage

increase/decrease,

find the whole given

part and percentage

7.12, 8.10

c) Ratio (equivalent, of

quantity), rate 8.7

d) Speed distance time

a) Rounding, significant

figures and

estimation 7.15

b) Circumference and

area of a circle 8.13

c) Visualise and

identify 3-D shapes

and their nets 7.5

d) Volume of a cuboid,

prism, cylinder,

composite solids

7.10, 8.12

a) Collect and organise

data

b) Interpret and

compare statistical

representations

7.17, 8.15

c) Mean, median and

mode averages 7.18,

8.16

d) The range and

outliers 7.18, 8.16

Extension content (highest attaining students may be stretched through depth by consideration of the following)…

Egyptian fractions

HCF and LCM

generalisation

Explore non-linear

sequences

T-totals

Similarity and ratio

Complex

constructions

Simple angle proofs

Density

Area scale factors

Loan repayment

Platonic solids

Percentage errors

Plans and elevations

Misleading graphs

Equal width

histograms

Sampling methods

Autumn 1a: Primes, squares and cubes, indices, prime factorisation and Venn diagrams to find LCM, HCF. 7.1, 8.1

Key concepts

use the concepts and vocabulary of prime numbers, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem

use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5

Support material Core material

?? Extension material

Identify factors and multiples of numbers

Define the concept ‘prime’ number and identify prime numbers up to 100

Secure written strategies for multiplication and division of integers

Recall and use tests for divisibility

Understand the use of notation for powers

Know the meaning of the square root symbol (√)

Use a calculator to calculate powers and roots

Recall the first 15 square numbers

Recall the first 5 cube numbers

Recognise when a problem involves using the highest common factor of two numbers

Recognise when a problem involves using the lowest common multiple of two numbers

Understand the meaning of prime factor

Write a number as a product of its prime factors, simplified to index form

Use a Venn diagram to sort information

Use prime factorisations to find the highest common factor of two numbers

Use prime factorisations to find the lowest common multiple of two numbers

HCF and LCM generalisation

Suggested activities Pedagogical notes

KM: Exploring primes activities: Factors of square numbers; Mersenne primes; LCM sequence; n² and (n + 1)²; n² and n² + n; n² + 1; n! + 1; n! – 1; x2 + x +41 KM: Use the method of Eratosthenes' sieve to identify prime numbers, but on a grid 6 across by 17 down instead. What do you notice? KM: Square number puzzle KM: History and Culture: Goldbach’s Conjectures NRICH: Factors and multiples NRICH: Powers and roots NRICH Perfect numbers investigation Learning review KM: 7M1 BAM Task

Pupils need to know how to use a scientific calculator to work out powers and roots. Note that while the square root symbol (√) refers to the positive square root of a number, every positive number has a negative square root too. NCETM: Departmental workshop: Index Numbers NRICH: Divisibility testing NCETM: Glossary Common approaches The following definition of a prime number should be used in order to minimise confusion about 1: A prime number is a number with exactly two factors. Every classroom has a set of number classification posters on the wall

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

When using Eratosthenes sieve to identify prime numbers, why is there no need to go further than the multiples of 7? If this method was extended to test prime numbers up to 200, how far would you need to go? Convince me.

Kenny says ’20 is a square number because 102 = 20’. Explain why Kenny is wrong. How could he change his statement so that it is fully correct?

Always / Sometimes / Never: The lowest common multiple of two numbers is found by multiplying the two numbers together.

Show me two (three-digit) numbers with a highest common factor of 18. And another. And another…

Show me two numbers with a lowest common multiple of 240. And another. And another…

Power (Square and cube) root Prime Prime factor Prime factorisation Product Venn diagram Highest common factor Notation Index notation: e.g. 53 is read as ‘5 to the power of 3’ and means ‘3 lots of 5 multiplied together’

Many pupils believe that 1 is a prime number – a misconception which can arise if the definition is taken as ‘a number which is divisible by itself and 1’

A common misconception is to believe that 53 = 5 × 3 = 15

Autumn 1b: Venn diagrams and enumerating sets

Key concepts

To organise information into a Venn diagram and be familiar with the notation used with Venn diagrams

Support material Core material

?? Extension material

Organise numbers and words into Venn diagrams

Use curly brackets to list sets of objects numbers

Organise numbers and words into Venn diagrams

Use set notation including union and intersection

Use set notation including union, intersection, difference, complement, empty set and universal

Suggested activities Pedagogical notes

Folder of resource ideas from TES http://www.mathsisfun.com/sets/venn-diagrams.html https://app.mymaths.co.uk/1731-resource/venn-diagrams-1 https://app.mymaths.co.uk/1730-resource/venn-diagrams-2

This is a new addition to the GCSE and leads into probability. For year 8 the main focus should be on understanding the relationship between Venn diagrams and the intersection and union notation, without extending to probability. To help students remember the difference between ∪ and ∩ encourage them to think of them as cups, which way up would hold the most? The ∪ and so this refers to the most things.

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Set Union Intersection Difference Complement Empty Universal Venn diagram

Students will confuse when to use ∪ and ∩

Autumn 1c: All four operations with fractions

Key concepts

Add, subtract, multiply and divide with fractions

Interleave: Area/ perimeter where side lengths are fractions

Support material Core material

?? Extension material

Understand equivalent fractions

Simplify fractions to lowest terms

Convert between mixed numbers and improper fractions

Appreciate fractions as an operator and calculate divisions where the answer is not whole by writing as a top heavy fraction and converting to a mixed number

Add and subtract fractions with a common denominator

Calculate fractions of quantities

Add and subtract fractions with different denominators

Appreciate the sense of converting a mixed number to a top heavy fraction prior to a subtraction, in case the fraction part becomes negative, or be aware of how to deal with this in writing the final answer.

Multiply integers by fractions

Divide integers by fractions

Multiply fractions with other fractions

Convert improper fractions to mixed numbers prior to multiplication.

Divide fractions by fractions

Egyptian fractions investigation

Challenge those that master concepts early with algebraic

fractions

Suggested activities Pedagogical notes

NRICH: Fraction Match NRICH: Matching Fractions NCETM: Activity F - Comparing Fractions KM: Crazy cancelling, silly simplifying NRICH: Rod fractions KM: Mixed numbers: mixed approaches NRICH: Would you rather? NRICH: Keep it simple NRICH: Egyptian fractions NRICH: The greedy algorithm NRICH: Fractions jigsaw NRICH: Countdpwn fractions

It is important that pupils are clear that the methods for addition and subtraction of fractions are different to the methods for multiplication and subtraction. A fraction wall is useful to help visualise and re-present the calculations. Use of a fraction wall to visualise multiplying fractions and dividing fractions by a whole number. For example, pupils need to

read calculations such as 1

4×

1

2 as

1

4 multiplied by

1

2 and therefore,

1

2 of

1

4=

1

8;

4

10÷ 2 as

4

10 divided by 2 and therefore

2

10.

NRICH: Teaching fractions with understanding NCETM: Departmental workshop: Fractions NCETM: The Bar Model, Teaching fractions, Fractions videos NCETM: Glossary Common approaches When multiplying a decimal by a whole number pupils are taught to use the corresponding whole number calculation as a general strategy When adding and subtracting mixed numbers pupils are taught to convert to improper fractions as a general strategy Teachers use the horizontal fraction bar notation at all times

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me a proper (improper) fraction. And another. And another.

Show me a mixed number fraction. And another. And another.

Jenny thinks that you can only multiply fractions if they have the same common denominator. Do you agree with Jenny? Explain your answer.

Benny thinks that you can only divide fractions if they have the same common denominator. Do you agree with Jenny? Explain.

Kenny thinks that 6

10÷

3

2=

2

5 .Do you agree with Kenny?

Mixed number Equivalent fraction Simplify, cancel, lowest terms Proper fraction, improper fraction, vulgar fraction Notation Mixed number notation Horizontal / diagonal bar for fractions

Some pupils may think that you simply can simply add/subtract the whole number part of mixed numbers and

add/subtract the fractional art of mixed numbers when adding/subtracting mixed numbers, e.g. 31

3 - 2

1

2= 1

−1

6

Some pupils may make multiplying fractions over complicated by applying the same process for adding and subtracting of finding common denominators

Autumn 2a: Substitute into and evaluate expressions 7.6, 8.5

Key concepts

use and interpret algebraic notation, including: ab in place of a × b, 3y in place of y + y + y and 3 × y, a² in place of a × a, a³ in place of a × a × a, a/b in place of a ÷ b, brackets, a²b in place of a × a × b, coefficients written as fractions rather than as decimals

substitute numerical values into expressions and scientific formulae

use conventional notation for priority of operations, including brackets

Support material Core material

?? Extension material

Know basic algebraic notation (the rules of algebra)

Use letters to represent variables

Identify like terms in an expression

Simplify an expression by collecting like terms

Substitute positive numbers into expressions and formulae

Use the order of operations correctly in algebraic situations

Know how to write products algebraically

Use fractions when working in algebraic situations

Simplify an expression involving terms with combinations of variables (e.g. 3a²b + 4ab2 + 2a2 – a2b)

Substitute positive and negative numbers into formulae

Be aware of common scientific formulae

Know the multiplication (division, power, zero) law of indices

Understand that negative powers can arise

Suggested activities Pedagogical notes

KM: Pairs in squares. Prove the results algebraically. KM: Algebra rules KM: Use number patterns to develop the multiplying out of brackets KM: Algebra ordering cards KM: Spiders and snakes. See the ‘clouding the picture’ approach NRICH: Your number is … NRICH: Crossed ends NRICH: Number pyramids and More number pyramids

Pupils will have experienced some algebraic ideas previously. Ensure that there is clarity about the distinction between representing a variable and representing an unknown. Note that each of the statements 4x, 42 and 4½ involves a different operation after the 4, and this can cause problems for some pupils when working with algebra. NCETM: Algebra NCETM: Glossary Common approaches

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me an example of an expression / formula / equation

Always / Sometimes / Never: 4(g+2) = 4g+8, 3(d+1) = 3d+1, a2 = 2a, ab = ba

What is wrong?

Jenny writes 2a + 3b + 5a – b = 7a + 3. Kenny writes 2a + 3b + 5a – b = 9ab. What would you write? Why?

Algebra Expression, Term, Formula (formulae), Equation, Function, Variable Mapping diagram, Input, Output Represent Substitute Evaluate Like terms Simplify / Collect Notation

Some pupils may think that it is always true that a=1, b=2, c=3, etc.

A common misconception is to believe that a2 = a × 2 = a2 or 2a (which it can do on rare occasions but is not the case in general)

When working with an expression such as 5a, some pupils may think that if a=2, then 5a = 52.

The convention of not writing a coefficient of 1 (i.e. ‘1x’ is written as ‘x’ may cause some confusion. In particular some pupils may think that 5h – h = 5

Autumn 2b: Linear equations 7.13, 8.11

Key concepts

Solve linear equations, including those with brackets and unknown values on both sides

Support material Core material

?? Extension material

Expand single brackets

Identify the correct order of undoing the operations in an equation

Solve linear equations with an unknown on one side

Check the solution to an equation by substitution

Solve linear equations with the unknown on both sides when the solution is an integer, fraction, or negative number

Solve linear equations with the unknown on both sides when the equation involves brackets

Recognise that the point of intersection of two graphs corresponds to the solution of a connected equation

Suggested activities Pedagogical notes

KM: Solving equations KM: Stick on the Maths: Constructing and solving equations NRICH: Think of Two Numbers Learning review KM: 8M10 BAM Task

This unit builds on the wok solving linear equations with unknowns on one side. It is essential that pupils are secure with solving these equations before moving onto unknowns on both sides. Encourage pupils to ‘re-present’ the problem using the Bar Model. NCETM: The Bar Model NCETM: Algebra NCETM: Glossary Common approaches All pupils should solve equations by balancing:

4x + 8 = 14 + x - x - x

3x + 8 = 14 - 8 - 8 3x = 6

÷ 3 ÷ 3 x = 2

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me an (one-step, two-step) equation with a solution of -8 (negative, fractional solution). And another. And another …

Show me a two-step equation that is ‘easy’ to solve. And another. And another …

What’s the same, what’s different: 2x + 7 = 25, 3x + 7 = x + 25, x + 7 = 7 – x, 4x + 14 = 50 ?

Convince me how you could use graphs to find solutions, or estimates, for equations.

Algebra, algebraic, algebraically Unknown Equation Operation Solve Solution Brackets Symbol Notation The lower case and upper case of a letter should not be used interchangeably when worked with algebra Juxtaposition is used in place of ‘×’. 2a is used rather than a2. Division is written as a fraction

Some pupils may think that you always have to manipulate the equation to have the unknowns on the LHS of the equal sign, for example 2x – 3 = 6x + 6

Some pupils think if 4x = 2 then x = 2.

When solving equations of the form 2x – 8 = 4 – x, some pupils may subtract ‘x’ from both sides.

x x x x 8

x 14

x x x 8

14

x x x

6

x

2

Autumn 2c: Expressions and equations from contextual and real-world situations (Not in Kangaroo)

Key concepts

Form and solve expressions and equations to help solve problems

Interleaving: Opposite angles in triangles, straight lines, quadrilaterals and full turns

Interleaving: Perimeter and area

Support material Core material

?? Extension material

Recall angle facts

Select the appropriate angle fact for a problem and use it to write an equation to represent the problem

Write simple expressions and equations to represent real life problems

Construct more complex expressions and equations

Recognise that the point of intersection of two graphs corresponds to the solution of a connected equation

Suggested activities Pedagogical notes

NRICH: Create expressions to always get same answer NRICH: Perimeter expressions NRICH: Lots of short expressions problems KM: Stick on the Maths: Constructing and solving equations Forming equations - angles Forming and solving eq ppt Forming and solving eq ws Forming and solving eq perimeter EASY Forming and solving eq perimeter MED Forming and solving eq perimeter HARD Learning review KM: 8M10 BAM Task

This unit builds on the wok solving linear equations with unknowns on one side. It is essential that pupils are secure with solving these equations before moving onto unknowns on both sides. Encourage pupils to ‘re-present’ the problem using the Bar Model. NCETM: The Bar Model NCETM: Algebra NCETM: Glossary Common approaches All pupils should solve equations by balancing:

4x + 8 = 14 + x - x - x

3x + 8 = 14 - 8 - 8 3x = 6

÷ 3 ÷ 3 x = 2

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me an expression to represent this problem

What information would we need to be able to turn this into an equation?

Why do we write equations to help us solve problems?

Algebra, algebraic, algebraically Unknown Equation Operation Solve Solution Brackets Symbol Notation The lower case and upper case of a letter should not be used interchangeably when worked with algebra Juxtaposition is used in place of ‘×’. 2a is used rather than a2. Division is written as a fraction

Some pupils may think that you always have to manipulate the equation to have the unknowns on the LHS of the equal sign, for example 2x – 3 = 6x + 6

Some pupils think if 4x = 2 then x = 2.

When solving equations of the form 2x – 8 = 4 – x, some pupils may subtract ‘x’ from both sides.

x x x x 8

x 14

x x x 8

14

x x x

6

x

2

Autumn 2d: Linear sequences, nth term rule 8.8

Key concepts

generate terms of a sequence from either a term-to-term or a position-to-term rule

deduce expressions to calculate the nth term of linear sequences

Support material Core material

?? Extension material

Use a term-to-term rule to generate a sequence

Find the term-to-term rule for a sequence

Describe a sequence using the term-to-term rule

Understand the meaning of a position-to-term rule

Use a position-to-term rule to generate a sequence

Find the position-to-term rule for a given sequence

Use algebra to describe the position-to-term rule of a linear sequence (the nth term)

Use the nth term of a sequence to deduce if a given number is in a sequence

Recognise different types of sequence (eg arithmetic and

geometric)

Investigate quadratic sequences

Investigate real life sequences such as Fibonacci

Suggested activities Pedagogical notes

NRICH: Frogs Frogs intro ppt KM: Spreadsheet sequences KM: Generating sequences KM: Maths to Infinity: Sequences KM: Stick on the Maths: Linear sequences NRICH: Charlie’s delightful machine NRICH: A little light thinking NRICH: Go forth and generalise Learning review KM: 8M9 BAM Task

Using the nth term for times tables is a powerful way of finding the nth term for any linear sequence. For example, if the

pupils understand the 3 times table can be described as ‘3n’ then the linear sequence 4, 7, 10, 13, … can be described as the 3

times table ‘shifted up’ one place, hence 3n + 1.

Exploring statements such as ‘is 171 is in the sequence 3, 9, 15, 21, 27, ..?’ is a very powerful way for pupils to realise that ‘term-to-term’ rules can be inefficient and therefore ‘position-to-term’ rules (nth term) are needed. NCETM: Algebra NCETM: Glossary Common approaches Teachers refer to a sequence such as 2, 5, 8, 11, … as ‘the three times table minus one’, to help pupils construct their understanding of the nth term of a sequence. All students have the opportunity to use spreadsheets to generate sequences

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me a sequence that could be generated using the nth term 4n ± c. And another. And another …

What’s the same, what’s different: 4, 7, 10, 13, 16, …. , 2, 5, 8, 11, 14, … , 4, 9, 14, 19, 24, …. and 4, 10, 16, 22, 28, …?

The 4th term of a linear sequence is 15. Show me the nth term of a sequence with this property. And another. And another …

Convince me that the nth term of the sequence 2, 5, 8, 11, … is 3n -1 .

Kenny says the 171 is in the sequence 3, 9, 15, 21, 27, … Do you agree with Kenny? Explain your reasoning.

Sequence

Linear Term Difference Term-to-term rule Position-to-term rule Ascending Descending Notation T(n) is often used when finding the nth term of sequence

Some pupils will think that the nth term of the sequence 2, 5, 8, 11, … is n + 3.

Some pupils may think that the (2n)th term is double the nth term of a linear sequence.

Some pupils may think that sequences with nth term of the form ‘ax ± b’ must start with ‘a’.

Spring 1a: Construct/ draw accurate triangles and quadrilaterals 7.4

Key concepts

use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries

use the standard conventions for labelling and referring to the sides and angles of triangles

draw diagrams from written description

Support material Core material

?? Extension material

Use a ruler to measure and draw lengths to the nearest millimetre

Use a protractor to measure and draw angles to the nearest degree

Know the meaning of faces, edges and vertices

Use notation for parallel lines

Know the meaning of ‘perpendicular’ and identify perpendicular lines

Know the meaning of ‘regular’ polygons

Identify line and rotational symmetry in polygons

Use AB notation for describing lengths

Use ∠ABC notation for describing angles

Use ruler and protractor to construct triangles from written descriptions

Use ruler and compasses to construct triangles when all three sides known

Suggested activities Pedagogical notes

KM: Shape work (selected activities) NRICH: Notes on a triangle Learning review KM: 7M13 BAM Task

NCETM: Departmental workshop: Constructions The equals sign was designed (by Robert Recorde in 1557) based on two equal length lines that are equidistant NCETM: Glossary Common approaches Dynamic geometry software to be used by all students to construct and explore dynamic diagrams of perpendicular and parallel lines.

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Given SSS, how many different triangles can be constructed? Why? Repeat for ASA, SAS, SSA, AAS, AAA.

Always / Sometimes / Never: to draw a triangle you need to know the size of three angles; to draw a triangle you need to know the size of three sides.

Convince me that a hexagon can have rotational symmetry with order 2.

Edge, Face, Vertex (Vertices)

Plane Parallel Perpendicular Regular polygon Rotational symmetry Notation The line between two points A and B is AB

The angle made by points A, B and C is ∠ABC

The angle at the point A is  Arrow notation for sets of parallel lines Dash notation for sides of equal length

Two line segments that do not touch are perpendicular if they would meet at right angles when extended

Pupils may believe, incorrectly, that: - perpendicular lines have to be horizontal / vertical - only straight lines can be parallel - all triangles have rotational symmetry of order 3 - all polygons are regular

Spring 1b: Find unknown angles (including parallel lines)

Key concepts

understand and use alternate and corresponding angles on parallel lines

Support material Core material

?? Extension material

Use angles at a point, angles at a point on a line and vertically opposite angles to calculate missing angles in geometrical diagrams

Know that the angles in a triangle total 180°

Know that the angles in a quadrilateral total 360°

Find missing angles in triangles and quadrilaterals

Identify fluently angles at a point, angles at a point on a line and vertically opposite

Find missing angles in isosceles triangles

Explain reasoning using vocabulary of angles

Identify alternate angles and know that they are equal

Identify corresponding angles and know that they are equal

Use knowledge of alternate and corresponding angles to calculate missing angles in geometrical diagrams

Identify known angle facts in more complex geometrical diagrams

Use angles in a straight line/ triangle fact to derive proof for

alternate angles being equal

Suggested activities Pedagogical notes

KM: Maths to Infinity: Lines and angles KM: Stick on the Maths: Angles NRICH: Triangle problem NRICH: Square problem NRICH: Two triangle problem KM: Alternate and corresponding angles KM: Perplexing parallels KM: Maths to Infinity: Lines and angles KM: Stick on the Maths: Alternate and corresponding angles

It is important to make the connection between the total of the angles in a triangle and the sum of angles on a straight line by encouraging pupils to draw any triangle, rip off the corners of triangles and fitting them together on a straight line. However, this is not a proof and this needs to be revisited using alternate angles to prove the sum is always 180°. The word ‘isosceles’ means ‘equal legs’. What do you have at the bottom of equal legs? Equal ankles! The KM: Perplexing parallels resource is a great way for pupils to discover practically the facts for alternate and corresponding angles. Common approaches Teachers insist on correct mathematical language (and not F-angles or Z-angles for example)

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me possible values for a and b. And another. And another.

Convince me that the angles in a triangle total 180°

Convince me that the angles in a quadrilateral must total 360°

What’s the same, what’s different: Vertically opposite angles, angles at a point, angles on a straight line and angles in a triangle?

Kenny thinks that a triangle cannot have two obtuse angles. Do you agree? Explain your answer.

Jenny thinks that the largest angle in a triangle is a right angle? Do you agree? Explain your thinking.

Show me a pair of alternate (corresponding) angles. And another. And another …

Degrees Right, Acute, Obtuse, Reflex angle Protractor Vertically opposite Geometry, geometrical Parallel Alternate angles, corresponding angles Notation Right angle notation Arc notation for all other angles The degree symbol (°)

Some pupils may think it’s the ‘base’ angles of an isosceles that are always equal. For example, they may think that a = b rather than a = c.

Some pupils may think that alternate and/or corresponding angles have a total of 180° rather than being equal.

a b 40°

a b c

Spring 1c: Areas and perimeters of parallelograms, trapezia and composite figures plus length and area conversions 7.10

Key concepts

use standard units of measure and related concepts (length, area, volume/capacity)

calculate perimeters of 2D shapes

know and apply formulae to calculate area of triangles, parallelograms, trapezia

Support material Core material

?? Extension material

Understand the meaning of area and perimeter

Know how to calculate areas of rectangles and triangles using the standard formulae

Know that the area of a triangle is given by the formula area = ½ × base × height = base × height ÷ 2 = 𝑏ℎ

2

Know appropriate metric units for measuring perimeter and area

Recognise that the value of the perimeter can equal the value of area

Find missing lengths in 2D shapes when the area is known

Know that the area of a trapezium is given by the

formula area = ½ × (a + b) × h = (𝑎+𝑏

2) ℎ =

(𝑎+𝑏)ℎ

2

Calculate the area of a parallelogram, trapezium and composite shapes

Know conversions for metric units of lengths

Know conversions for metric units of area

Calculate missing lengths in similar shapes

Calculate missing areas in similar shapes

Suggested activities Pedagogical notes

KM: Equable shapes (for both 2D and 3D shapes) KM: Triangle takeaway

Ensure that pupils make connections with the area of rectangles and area of parallelograms, in particular the importance of the perpendicular height. NCETM: Glossary Common approaches Pupils have already derived the formula for the area of a parallelogram. They use this to derive the

formula for the area of a trapezium as (𝑎+𝑏)ℎ

2 by copying and rotating a trapezium as shown above.

Pupils use the area of a triangle as given by the formula area = 𝑏ℎ

2.

Every classroom has a set of area posters on the wall.

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Convince me that the area of a triangle = ½ × base ×

height = base × height ÷ 2 = 𝑏ℎ

2

(Given a right-angled trapezium with base labelled 8 cm, height 5 cm, top 6 cm) Kenny uses the formula for the area of a trapezium and Benny splits the shape into a rectangle and a triangle. What would you do? Why?

Perimeter, area, Square, rectangle, parallelogram, triangle, trapezium (trapezia) Polygon Square millimetre, square centimetre, square metre, square kilometre Formula, formulae Length, breadth, depth, height, width Notation Abbreviations of units in the metric system:

km, m, cm, mm, mm2, cm2, m2, km2

Some pupils may use the sloping height when finding the areas of parallelograms, triangles and trapezia

Some pupils may think that the area of a triangle is found using area = base × height

Some pupils may think that you multiply all the numbers to find the area of a shape

Spring 2a: Convert between percentages, vulgar fractions and decimals 8.6

Key concepts

work interchangeably with percentages, terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 or 3/8)

Support material Core material

?? Extension material

Understand that fractions, decimals and percentages are different ways of representing the same proportion

Convert between mixed numbers and top-heavy fractions

Write one quantity as a fraction of another

Identify if a fraction is terminating or recurring

Recall some decimal and fraction equivalents (e.g. tenths, fifths, eighths)

Write a decimal as a fraction

Write a fraction in its lowest terms by cancelling common factors

Identify when a fraction can be scaled to tenths or hundredths

Convert a fraction to a decimal by scaling (when possible)

Use a calculator to change any fraction to a decimal

Write a decimal as a percentage

Write a fraction as a percentage

Suggested activities Pedagogical notes

KM: FDP conversion. Templates for taking notes. KM: Fraction sort. Tasks one and two only. KM: Maths to Infinity: Fractions, decimals, percentages, ratio, proportion NRICH: Matching fractions, decimals and percentages Learning review KM: 8M4 BAM Task

The diagonal fraction bar (solidus) was first used by Thomas Twining (1718) when recorded quantities of tea. The division symbol (÷) is called an obelus, but there is no name for a horizontal fraction bar. NRICH: History of fractions NRICH: Teaching fractions with understanding NCETM: Glossary Common approaches All pupils should use the horizontal fraction bar to avoid confusion when fractions are coefficients in algebraic situations

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Without using a calculator, convince me that 3/8 = 0.375

Show me a fraction / decimal / percentage equivalent. And another. And another …

What is the same and what is different: 2.5, 25%, 0.025, ¼ ?

Fraction Mixed number Top-heavy fraction Percentage Decimal Proportion Terminating Recurring Simplify, Cancel Notation Diagonal and horizontal fraction bar

Some pupils may make incorrect links between fractions and decimals such as thinking that 1/5 = 0.15

Some pupils may think that 5% = 0.5, 4% = 0.4, etc.

Some pupils may think it is not possible to have a percentage greater than 100%.

Spring 2b: Percentage increase and decrease, finding the whole given the part and the percentage 7.12, 8.10

Key concepts

interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively

compare two quantities using percentages

solve problems involving percentage change, including original value problems, and simple interest including in financial mathematics

Support material Core material

?? Extension material

Convert between fractions, decimals and percentages

Use calculators to find a percentage of an amount using multiplicative methods

Calculate percentages of amounts without a calculator, using 1% and 10% as a base

Identify the multiplier for a percentage increase or decrease

Use calculators to increase (decrease) an amount by a percentage using multiplicative methods

Compare two quantities using percentages

Know that percentage change = actual change ÷ original amount

Calculate the percentage change in a given situation, including percentage increase / decrease

Recognise when a fraction (percentage) should be interpreted as a number

Recognise when a fraction (percentage) should be interpreted as a operator

Identify the multiplier for a percentage increase or decrease when the percentage is greater than 100%

Use calculators to increase an amount by a percentage greater than 100%

Solve problems involving percentage change

Solve original value problems when working with percentages

Solve financial problems including simple interest

Suggested activities Pedagogical notes

KM: Stick on the Maths: Percentage increases and decreases KM: Maths to Infinity: FDPRP KM: Percentage methods KM: Stick on the Maths: Proportional reasoning KM: Stick on the Maths: Multiplicative methods KM: Percentage identifying NRICH: One or both NRICH: Antiques roadshow Learning review KM: 8M6 BAM Task

The bar model is a powerful strategy for pupils to ‘re-present’ a problem involving percentage change. Only simple interest should be explored in this unit. Compound interest will be developed later. NCETM: The Bar Model NCETM: Glossary

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Always/Sometimes/Never: To reverse an increase of x%, you decrease by x%

Lenny calculates the % increase of £6 to £8 as 25%. Do you agree with Lenny? Explain your answer.

Convince me that the multiplier for a 150% increase is 2.5

Kenny buys a poncho in a 25% sale. The sale price is £40. Kenny thinks that the original is £50. Do you agree with Kenny? Explain your answer.

Jenny thinks that increasing an amount by 200% is the same as multiplying by 3. Do you agree with Jenny? Explain your answer.

Percent, percentage Multiplier Increase, decrease Original amount Multiplier (Simple) interest

Some pupils may think that the multiplier for a 150% increase is 1.5

Some pupils may think that increasing an amount by 200% is the same as doubling.

In isolation, pupils may be able to solve original value problems confidently. However, when it is necessary to identify the type of percentage problem, many pupils will apply a method for a more simple percentage increase / decrease problem. If pupils use models (e.g. the bar model, or proportion tables) to represent all problems then they are less likely to make errors in identifying the type of problem.

Spring 2c: Ratio (equivalent, of quantity) and rate 8.7

Key concepts

express the division of a quantity into two parts as a ratio; apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)

express a multiplicative relationship between two quantities as a ratio or a fraction

use compound units (such as speed, rates of pay, unit pricing)

change freely between compound units (e.g. speed, rates of pay, prices) in numerical contexts

Support material Core material

?? Extension material

Understand and use ratio notation

Divide an amount in a given ratio

Identify ratio in a real-life context

Write a ratio to describe a situation

Identify proportion in a situation

Find a relevant multiplier in a situation involving proportion

Use fractions fluently in situations involving ratio or proportion

Understand the connections between ratios and fractions

Understand the meaning of a compound unit

Identify when it is necessary to convert quantities in order to use a sensible unit of measure

Suggested activities Pedagogical notes

KM: Proportion for real KM: Investigating proportionality KM: Maths to Infinity: Fractions, decimals, percentages, ratio, proportion NRICH: In proportion NRICH: Ratio or proportion? NRICH: Roasting old chestnuts 3 Standards Unit: N6 Developing proportional reasoning Learning review KM: 8M5 BAM Task

The Bar Model is a powerful strategy for pupils to ‘re-present’ a problem involving ratio. NCETM: The Bar Model NCETM: Multiplicative reasoning NCETM: Departmental workshops: Proportional Reasoning NCETM: Glossary Common approaches All pupils are taught to set up a ‘proportion table’ and use it to find the multiplier in situations involving proportion

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me an example of two quantities that will be in proportion. And another. And another …

(Showing a table of values such as the one below) convince me that this information shows a proportional relationship

6 9

10 15

14 21

Which is the faster speed: 60 km/h or 10 m/s? Explain why.

Ratio

Proportion Proportional Multiplier Speed Unitary method Units Compound unit

Many pupils will want to identify an additive relationship between two quantities that are in proportion and apply this to other quantities in order to find missing amounts

Some pupils may think that a multiplier always has to be greater than 1

When converting between times and units, some pupils may base their working on 100 minutes = 1 hour

Spring 2d: Speed, distance, time

Key concepts

To use understanding of proportion and compound units to calculate speed, distance and time

Support material Core material

?? Extension material

Understand the meaning of a compound unit

Know the common units for speed

Be able to convert between units of distance and time

Know that 5 miles = 8km and 1 miles = 1.6km and use proportional reasoning to convert miles to km and vice versa

Use proportional reasoning to calculate speeds, distances and times, taking speed to be the distance covered in one unit of time

Solve problems involving speed

Identify when it is necessary to convert quantities in order to use a sensible unit of measure

Suggested activities Pedagogical notes

NRICH: Walk and ride NRICH: Overtake NRICH: Tummy ache NRICH: Unhappy end

Teach speed as ‘distance covered per one unit of time’ and encourage students to write speed as a unitary Distance:Time ratio, which can then be used to find related distances and times.

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Speed Units Compound unit Notation Kilometres per hour is written as km/h or kmh-1 Metres per second is written as m/s or ms-1

When a time answer is given as hours in decimal form, students may struggle to convert to minutes. E.g. 1.25 hours students may interpret as 1 hour and 25 minutes

Summer 1a: Rounding, significant figures and estimation 7.15

Key concepts

round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures)

estimate answers; check calculations using approximation and estimation, including answers obtained using technology

Support material Core material

?? Extension material

Approximate any number by rounding to the nearest 10, 100 or 1000, 10 000, 100 000 or 1 000 000

Approximate any number with one or two decimal places by rounding to the nearest whole number

Approximate any number with two decimal places by rounding to the one decimal place

Approximate by rounding to any number of decimal places

Know how to identify the first significant figure in any number

Approximate by rounding to the first significant figure in any number

Understand estimating as the process of finding a rough value of an answer or calculation

Use estimation to predict the order of magnitude of the solution to a (decimal) calculation

Estimate calculations by rounding numbers to one significant figure

Use inverse operations to check solutions to calculations

Suggested activities Pedagogical notes

KM: Approximating calculations KM: Stick on the Maths: CALC6: Checking solutions Learning review KM: 7M6 BAM Task

This unit is an opportunity to develop and practice calculation skills with a particular emphasis on checking, approximating or estimating the answer. Pupils should be able to estimate calculations involving integers and decimals. Also see big pictures: Calculation progression map and Fractions, decimals and percentages progression map NCETM: Glossary Common approaches All pupils are taught to visualise rounding through the use a number line

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Convince me that 39 652 rounds to 40 000 to one significant figure

Convince me that 0.6427 does not round to 1 to one significant figure

What is wrong: 11 × 28.2

0.54≈

10 × 30

0.5= 150. How can you

correct it?

Approximate (noun and verb) Round Decimal place Check Solution Answer Estimate (noun and verb) Order of magnitude Accurate, Accuracy Significant figure Inverse operation Notation

The approximately equal symbol () Significant figure is abbreviated to ‘s.f.’ or ‘sig fig’

Some pupils may truncate instead of round

Some pupils may round down at the half way point, rather than round up.

Some pupils may think that a number between 0 and 1 rounds to 0 or 1 to one significant figure

Some pupils may divide by 2 when the denominator of an estimated calculation is 0.5

Summer 1b: Circumference and area of a circle

Key concepts

identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference

know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr²

calculate circumference and areas of circles

Support material Core material

?? Extension material

Know how to use formulae to find the area of rectangles, parallelograms, triangles and trapezia

Know how to find the area and perimeter of compound shapes

Calculate the circumference of a circle when radius (diameter) is given

Calculate the area of a circle when radius (diameter) is given

Know the formulas circumference = πd and area of a circle = πr²

Calculate the radius (diameter) of a circle when the circumference is known

Calculate the perimeter of composite shapes that include sections of a circle

Know the formula area of a circle = πr²

Calculate the radius (diameter) of a circle when the area is known

Calculate the area of composite shapes that include sections of a circle

Suggested activities Pedagogical notes

KM: Circle connections, Circle connections v2 KM: Circle circumferences, Circle problems KM: Stick on the Maths: Circumference and area of a circle Circumference and area of circle ppt Circumference perimeter treasure hunt Circles revision ws Learning review KM: 8M12 BAM Task

C = πd can be established by investigating the ratio of the circumference to the diameter of circular objects (wheel, clock, tins, glue sticks, etc.) Pupils need to understand this formula in order to derive A = πr². NCETM: Glossary Common approaches The area of a circle is derived by cutting a circle into many identical sectors and approximating a parallelogram Every classroom has a set of area posters on the wall

Pupils use area of a trapezium = (𝑎+𝑏)ℎ

2 and area of a triangle = area =

𝑏ℎ

2

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Convince me C = 2πr = πd.

What is wrong with this statement? How can you correct it? The area of a circle with radius 7 cm is approximately 441 cm2 because (3 × 7)2 = 441.

Convince me the area of a semi-circle = 𝜋𝑑2

4

Circle Centre Radius, diameter, chord, circumference Pi Polygon, polygonal Notation π Abbreviations of units in the metric system: km, m, cm, mm, mm2, cm2, m2, km2

Some pupils will work out (π × radius)2 when finding the area of a circle

Some pupils may use the sloping height when finding areas that are parallelograms, triangles or trapezia

Some pupils may think that the area of a triangle = base × height

e) Summer 1c: Visualise and identify 3-D shapes and their nets 7.5

Key concepts

identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

Support material Core material

?? Extension material

Know the names of common 3D shapes

Know the meaning of face, edge, vertex

Know the connection between faces, edges and vertices in 3D shapes

Visualise a 3D shape from its net

Draw a net for a cube and cuboid on squared paper

Make 3D drawings of cubes and cuboids using isometric paper

Accurately draw the net for a variety of prisms on squared paper

Draw plans and elevations of prisms and composite prisms

Accurately construct the nets for 3D shapes with and

without squared paper using construction techniques

Suggested activities Pedagogical notes

KM: Euler’s formula KM: Visualising 3D shapes KM: Dotty activities: Shapes on dotty paper KM: What's special about quadrilaterals? Constructing quadrilaterals from diagonals and summarising results. KM: Investigating polygons. Tasks one and two should be carried out with irregular polygons. NRICH: Property chart NRICH: Quadrilaterals game Learning review www.diagnosticquestions.com

Ensure that pupils do not use the word ‘diamond’ to describe a kite, or a square that is 45° to the horizontal. ‘Diamond’ is not the mathematical name of any shape. A cube is a special case of a cuboid and a rhombus is a special case of a parallelogram A prism must have a polygonal cross-section, and therefore a cylinder is not a prism. Similarly, a cone is not a pyramid. NCETM: Departmental workshop: 2D shapes NCETM: Glossary Common approaches Every classroom has a set of triangle posters and quadrilateral posters on the wall Models of 3D shapes to be used by all students during this unit of work

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me an example of a trapezium. And another. And another …

Always / Sometimes / Never: The number of vertices in a 3D shape is greater than the number of edges

Which quadrilaterals are special examples of other quadrilaterals? Why? Can you create a ‘quadrilateral family tree’?

What is the same and what is different: Rhombus / Parallelogram?

Face, Edge, Vertex (Vertices) Cube, Cuboid, Prism, Cylinder, Pyramid, Cone, Square, Rectangle, Parallelogram, (Isosceles) Trapezium, Kite, Rhombus Delta, Arrowhead Diagonal Notation Dash notation to represent equal lengths in shapes and geometric diagrams Right angle notation

Some students may struggle to visualise which faces must be joined together to make a net that ‘works’

Summer 1d: Volume of a cuboid, prism, cylinder, composite solids 7.10, 8.12,

Key concepts

know and apply formulae to calculate volume of prisms, cylinders and composite solids

Support material Core material

?? Extension material

Recognise and use units of measure for length, area and volume

Understand that volume is a measure of the space within a 3D shape, measured in cubes.

Calculate the area of rectangles and triangles

Calculate the volume of a cuboid made from cubes by counting cubes

Be able to distinguish prisms and right prisms from other 3D solids

Be able to identify the cross-sectional face of a prism

Know that the formula for volume of a prism is cross-sectional area multiplied by the depth of the prism

Calculate volume of a cuboid

Calculate volume of a right triangular prism

Calculate the volume of composite right prisms

Calculate percentage errors in calculations by considering bounds

of calculations

Suggested activities Pedagogical notes

KM: Maths to Infinity: Area and Volume KM: Stick on the Maths: Right prisms NRICH: Blue and White NRICH: Efficient Cutting NRICH: Cola Can Sustainable design project ppt Sustainable design project ws Learning review KM: 8M12 BAM Task

A prism is a solid with constant polygonal cross-section. A right prism is a prism with a cross-section that is perpendicular to the ‘length’. NCETM: Glossary Common approaches Every classroom has a set of area posters on the wall The formula for the volume of a prism is ‘area of cross-section × length’ even if the orientation of the solid suggests that height is required

Pupils use area of a trapezium = (𝑎+𝑏)ℎ

2 and area of a triangle = area =

𝑏ℎ

2

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Name a right prism. And another. And another …

Convince me that a cylinder is not a prism

(Right) prism Cross-section Cylinder Polygon, polygonal Solid Notation π Abbreviations of units in the metric system: km, m, cm, mm, mm2, cm2, m2, km2, mm3, cm3, km3

Some pupils may think that you multiply all the numbers to find the volume of a prism

Some pupils may confuse the concepts of surface area and volume

Summer 2a: Collect and organize data

Key concepts

Support material Core material

?? Extension material

Suggested activities Pedagogical notes

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

(

Summer 2b: Interpret and compare statistical representations 7.17, 8.15

Key concepts

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate graphical representation involving discrete, continuous and grouped data

use and interpret scatter graphs of bivariate data

Support material Core material

?? Extension material

Construct and interpret a pictogram

Construct and interpret a bar chart

Construct and interpret a line graph

Understand that pie charts are used to show proportions

Use a template to construct a pie chart by scaling frequencies

Know the meaning of categorical data

Know the meaning of discrete data

Interpret and construct frequency tables

Construct and interpret pictograms (bar charts, tables) and know their appropriate use

Construct and interpret comparative bar charts

Interpret pie charts

Construct a pie chart where the total frequency is a factor of 360

Plot a scatter diagram of bivariate data

Understand the meaning of ‘correlation’

Interpret a scatter diagram using understanding of correlation

Construct pie charts when the total frequency is not a factor of 360

Know the meaning of continuous data

Interpret a grouped frequency table for continuous data

Construct a grouped frequency table for continuous data

Construct histograms for grouped data with equal class intervals

Interpret histograms for grouped data with equal class intervals

Construct and use the horizontal axis of a histogram correctly

Suggested activities Pedagogical notes

NRICH: Picturing the World NRICH: Charting Success KM: Make a ‘human’ scatter graph by asking pupils to stand at different points on a giant set of axes. KM: Gathering data KM: Spreadsheet statistics KM: Stick on the Maths HD2: Selecting and constructing graphs and charts KM: Stick on the Maths HD3: Working with grouped data Learning review www.diagnosticquestions.com

The word histogram is often misused and an internet search of the word will usually reveal a majority of non-histograms. The correct definition is ‘a diagram made of rectangles whose areas are proportional to the frequency of the group’. If the class widths are equal, as they are in this unit of work, then the vertical axis shows the frequency. It is only later that pupils need to be introduced to unequal class widths and frequency density. Lines of best fit on scatter diagrams are not introduced until Stage 9, although pupils may well have encountered both lines and curves of best fit in science by this time. NCETM: Glossary Common approaches All students collect data about their class’s height and armspan when first constructing a scatter diagram

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me a pie chart representing the following information: Blue (30%), Red (50%), Yellow (the rest). And another. And another.

Kenny says ‘If two pie charts have the same section then the amount of data the section represents is the same in each pie chart.’ Do you agree with Kenny?

Show me a scatter graph with positive (negative, no) correlation. And another. And another.

Show me a histogram. And another. And another.

Kenny thinks that histogram is just a ‘fancy’ name for a bar chart. Do you agree with Kenny? Explain your answer.

What’s the same and what’s different: histogram, scatter diagram, bar chart, pie chart?

Categorical data, Discrete data Continuous data, Grouped data Table, Frequency table, Frequency Histogram Scale, Graph, Axis, axes Scatter graph (scatter diagram, scattergram, scatter plot) Bivariate data (Linear) Correlation Positive correlation, Negative correlation Notation Correct use of inequality symbols when labeling groups in a frequency table

Some pupils may think that a line graph is appropriate for discrete data

Some pupils may think that each square on the grid used represents one unit

Some pupils may confuse the fact that the sections of the pie chart total 100% and 360°

Some pupils may not leave gaps between the bars of a bar chart

Some pupils may label the bar of a histogram rather than the boundaries of the bars

Some pupils may leave gaps between the bars in a histogram

Some pupils may misuse the inequality symbols when working with a grouped frequency table

Summer 2c: Mean, median and mode averages 7.18, 8.16

Key concepts

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean and mode)

apply statistics to describe a population

Support material Core material

?? Extension material

Understand the mode and median as measures of typicality (or location)

Find the mode of set of data

Find the median of a set of data

Calculate the mean of a set of data

Find the median of a set of data when there are an even number of numbers in the data set

Use the mean to find a missing number in a set of data

Find the mode from a frequency table

Calculate the mean from a frequency table

Find the median from a frequency table

Analyse and compare sets of data

Appreciate the limitations of different statistics

Find the modal class of set of grouped data

Find the class containing the median of a set of data

Find the midpoint of a class

Calculate an estimate of the mean from a grouped frequency table

Choose appropriate statistics to describe a set of data

Justify choice of statistics to describe a set of data

Suggested activities Pedagogical notes

KM: Maths to Infinity: Averages KM: Maths to Infinity: Averages, Charts and Tables KM: Stick on the Maths HD4: Averages NRICH: M, M and M NRICH: The Wisdom of the Crowd KM: Swillions KM: Lottery project NRICH: Half a Minute Learning review www.diagnosticquestions.com

The word ‘average’ is often used synonymously with the mean, but it is only one type of average. In fact, there are several different types of mean (the one in this unit properly being named as the ‘arithmetic mean’). NCETM: Glossary Common approaches Every classroom has a set of statistics posters on the wall Always use brackets when writing out the calculation for a mean, e.g. (2 + 3 + 4 + 5) ÷ 4 = 14 ÷ 4 = 3.5

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me a set of data with a mean (mode, median, range) of 5.

Always / Sometimes / Never: The mean is greater than the mode for a set of data

Always / Sometimes / Never: The mean is greater than the median for a set of data

Convince me that a set of data could have more than one mode.

What’s the same and what’s different: mean, mode, median?

Average Spread Consistency Mean Median Mode Measure Calculate an estimate Grouped frequency

If using a calculator some pupils may not use the ‘=’ symbol (or brackets) correctly; e.g. working out the mean of 2, 3, 4 and 5 as 2 + 3 + 4 + 5 ÷ 4 = 10.25.

Some pupils may think that the range is a type of average

Some pupils may think that a set of data with an even number of items has two values for the median, e.g. 2, 4, 5, 6, 7, 8 has a median of 5 and 6 rather than 5.5

Some pupils may not write the data in order before finding the median.

Some pupils may incorrectly estimate the mean by dividing the total by the numbers of groups rather than the total frequency.

Some pupils may incorrectly think that there can only be one model class.

Summer 2d: The range and outliers 7.18, 8.16

Key concepts

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of spread (range)

apply statistics to describe a population

Support material Core material

?? Extension material

Understand the range as a measure of spread (or consistency)

Calculate the range of a set of data presented as a list of numbers

Understand the range as a measure of spread (or consistency)

Calculate the range of data presented in a frequency table

Understand the effects of an outlier on the range

Appreciate the limitations of the range

Estimate the range from a grouped frequency table

Calculate the interquartile range for a set of data

Know when to consider the range, IQR, and the strengths/

weaknesses of each

Suggested activities Pedagogical notes

Range or IQR ppt

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Convince me how to estimate the range for grouped data. Spread Consistency Range Statistic Statistics Approximate, Round Calculate an estimate Grouped frequency Midpoint

Some pupils may refer to the range as an ‘average’

Some pupils may incorrectly estimate the range of grouped data by subtracting the upper bound of the first group from the lower bound of the last group.