The Sir John Colfox Academy Mathematics Mastery Scheme of Work Overview Year 7

Autumn 1

Solve word problems

(add and subtract)

Autumn 2

Explain and investigate

(multiply and divide)

Spring 1

Geometry

Spring 2

Fractions

Summer 1

Applications of algebra

Summer 2

Percentages and

statistics

Supporting content from KS2 (or previous year 7 content)…

Number bonds

Convert units

Money +/-

Measurement

Mental strategies

Multiplication facts

Written

multiplication

strategies

Lengths and units

Parallel and

perpendicular

Work with angles

Equal parts

Factors, multiples

Tenths, hundredths

Fractional areas

Area or rectangles

and triangles

Number patterns

Algebraic notation

Triangles and

quadrilaterals

Decimals and

problem solving

Fractions of shapes

Equivalence

BIDMAS

Core content for Year 7 (all students should have access to)…

a) Place value (integers and

decimals) 6.1, 7.2

b) Add and subtract

(integers, directed and

decimals) 6.2, 7.3

c) Rounding and estimation

7.15

d) Perimeter 7.14

a) Factors, HCF,

multiples, LCM 7.1

b) Multiply and divide

(including directed

and decimals) 6.2,

7.3

c) Area of rectangle

and triangle 6.14,

7.14

d) Calculate the mean

7.18

a) Draw, measure and

name acute, obtuse

and reflex angles

6.11

b) Find unknown angles

(straight lines, at a

point, vertically

opposite) 7.11

c) Properties of

triangles and

quadrilaterals 7.4,

7.5

a) Equivalent

fractions, mixed

numbers and

improper 7.7

b) Compare and order

fractions &

decimals 7.2

c) Fraction of a

quantity 7.12

d) Add, subtract,

multiply and divide

fractions 7.12

a) BIDMAS,

substitution and

expressions 7.6

b) Sequences (term-

to-term, not nth

term) 7.9

a) Construct and

interpret statistical

diagrams including

pie charts 7.17

b) Convert between

FDP 7.7, 8.6

c) Percentage of a

quantity, plus find

the whole, given the

part and the

percentage 7.12

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

Different counting

systems or bases

Generalisation

Upper and lower bounds

Shikaku puzzles

Different counting

systems or bases

Alternative

methods of

multiplication

Generalisation

Tessellating

triangles and

quadrilaterals

Tangram

investigations

Rigid shapes

Terminating and

recurring decimals

Fractions of

tangrams

Shape block

challenges

Four fours

Patterns and

generalising

Algebraic mean

Comparing and

converting between

different

representations

Applications of

percentages

Autumn 1a: Place value (integers and decimals) 6.1, 7.2

Key concepts

identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places

read, write, order and compare numbers up to 10 000 000 and determine the value of each digit

order positive and negative integers and decimals use the symbols =, ≠, <, >, ≤, ≥

Support material Core material Extension material

Understand and use place value in numbers with up to seven digits

Understand that negative numbers are numbers less than zero

Order a set of decimals with a mixed number of decimal places (up to a maximum of three)

Understand place value in numbers with up to three decimal places

Understand (order, write, read) place value in numbers with up to eight digits

Understand and use negative numbers when working with temperature

Understand and use negative numbers when working in other contexts

order positive and negative integers and decimals

use the symbols =, ≠, <, >, ≤, ≥

Place a set of negative numbers in order

Place a set of mixed positive and negative numbers in order

Use inequality symbols to compare numbers

Different counting systems or bases

Suggested activities Pedagogical notes

KM: Use Powers of ten to demonstrate connections. NRICH: The Moons of Vuvv NRICH: Round and round the circle NRICH: Counting cogs KM: Farey Sequences KM: Decimal ordering cards 2 KM: Maths to Infinity: Directed numbers NRICH: Greater than or less than? Learning review www.diagnosticquestions.com

Zero is neither positive nor negative. When multiplying and dividing by powers of ten, the decimal point is fixed and it is the digits that move. Ensure that pupils can deal with large numbers that include zeros in the HTh and/or H column (e.g. 43 006 619) .The set of integers includes the natural numbers {1, 2, 3, …}, zero (0) and the ‘opposite’ of the natural numbers {-1, -2, -3, …}. Pupil must use language correctly to avoid reinforcing misconceptions: for example, 0.45 should never be read as ‘zero point forty-five’; 5 > 3 should be read as ‘five is greater than 3’, not ‘5 is bigger than 3’. NCETM: Glossary Common approaches Teachers use the language ‘negative number’ to avoid future confusion with calculation that can result by using ‘minus number’ Every classroom has a vertical negative number washing line on the wall

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Jenny writes 2.54 × 10 = 25.4. Kenny writes 2.54 × 10 = 25.40. who do you agree with? Explain why.

Look at this number (24 054 028). Show me another number (with 4, 5, 6, 7 digits) that includes a 5 with the same value. And another. And another …

NCETM: Place Value Reasoning

Place value Digit Positive number Negative number Integer Decimal point Notation The ‘equals’ sign: = The ‘not equal’ sign: ≠ The inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (more than or equal to)

Some pupils can confuse the language of large (and small) numbers since the prefix ‘milli- means ‘one thousandth’ (meaning that there are 1000 millimetres in a metre for example) while one million is actually a thousand thousand.

Some pupils may believe that 0.400 is greater than 0.58

Some pupils may believe that -6 is greater than -3. For this reason ensure pupils avoid saying ‘bigger than’

Autumn 1b: Add and subtract (integers, directed and decimals) 6.2, 7.3

Key concepts

solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

solve worded problems involving addition and subtraction apply addition and subtraction, including formal written methods, to integers, decimals and directed numbers

Support material Core material Extension material

Use column addition and subtraction for numbers with more than four digits

Add a three-digit number to a two-digit number mentally (when bridging of hundreds is required)

Identify when addition or subtraction is needed as part of solving multi-step problems

Explain why addition or subtraction is needed at any point when solving multi-step problems

Solve multi-step problems involving addition and subtraction

Know that addition and subtraction have equal priority

Know how to add and subtract with directed numbers

Selecting the most appropriate method for the question and

justifying method

Problem 1

Problem 6

Generalisation

Addition equation sudoku

Suggested activities Pedagogical notes

NRICH: Eleven NRICH: Read this page NRICH: Addition equation sudoku NRICH: Counting cogs KM: Decimal ordering cards 2 KM: Maths to Infinity: Directed numbers NRICH: Greater than or less than? Learning review www.diagnosticquestions.com

Do NOT teach ‘tricks’ such as “2 negatives make a plus” for subtractions such as 3 - - 2¸as students confuse this and try to apply it to -3 – 2. Instead spend a long time looking at addition as counting up a number line, and subtracting counting down a number line, before moving on to subtracting a minus number and developing the concept of reversing the subtraction. KM: Progression: Addition and Subtraction NCETM: The Bar Model, Subtraction, Multiplication, Multiplicative reasoning, Glossary Common approaches To avoid confusion with language, all teachers use ‘sum’ to refer only to the result of an addition. Teachers say ‘complete these calculations’ instead of ‘complete these sums’

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Why have you chosen to add / subtract ? NCETM: Addition and Subtraction Reasoning

Addition Subtraction Sum, Total Difference, Minus, Less Column addition Column subtraction Operation

Some pupils may write statements such as 140 - 190 = 50

When subtracting mentally some pupils may deal with columns separately and not combine correctly; e.g. 180 – 24: 180 – 20 = 160. Taking away 4 will leave 6. So the answer is 166.

Autumn 1c: Rounding 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

Upper and lower bounds

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 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

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

Autumn 1d: Perimeter 7.14

Key concepts

use standard units of measure for length and have an appreciation of the size of each

calculate perimeters of 2D shapes

Support material Core material Extension material

Addition and subtraction methods with integers and decimals

Names for 2D shapes

Use a ruler to accurately measure line segments to the nearest millimetre

Calculate the perimeter of rectangles

Calculate the perimeter of triangles

Calculate the perimeter of compound shapes

Calculate the perimeter of compound shapes where the side lengths are decimal number

Upper and lower bounds

Suggested activities Pedagogical notes

Outside measuring! NCETM: Glossary Common approaches Get students outside measuring real perimeters! Get out the meter rulers, trundle wheels, etc. Ask students to estimate perimeters before measuring – how accurate were they?

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Which types of shapes have the biggest perimeters? Why? Where might you want to maximise or minimise a perimeter in real life?

Perimeter Square, rectangle, parallelogram, triangle, trapezium (trapezia) Polygon millimetre, centimetre, metre, kilometer inch, foot, yard, mile Formula, formulae Length, breadth, depth, height, width Notation Abbreviations of units in the metric system: km, m, cm, mm,

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

Autumn 2a: Factors, HCF, multiples, LCM 7.1

Key concepts

use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor and lowest common multiple

Support material Core material Extension material

Know how to find common multiples of two given numbers

Know how to find common factors of two given numbers

Recall multiplication facts to 12 × 12 and associated division facts

Recall prime numbers up to 50

Know how to test if a number up to 150 is prime

Know the meaning of ‘highest common factor’ and ‘lowest common multiple’

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

Different counting systems or bases

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: History and Culture: Goldbach’s Conjectures NRICH: Factors and multiples Number bonds website Learning review KM: 7M1 BAM TaskFactor

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.

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

Multiple Prime ((Lowest) common) multiple and LCM ((Highest) common) factor and HCF

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’

Students will confuse factor and multiple

Autumn 2b: Multiply and divide, including directed and decimals 6.2, 7.3

Key concepts

multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

solve problems involving multiplication

understand and use place value (e.g. when working with very large or very small numbers, and when calculating with decimals)

apply multiplication, including formal written methods, to directed integers and decimals

Support material Core material Extension material

Recall multiplication facts for multiplication tables up to 12 × 12

Recall division facts for multiplication tables up to 12 × 12

Understand the commutativity of multiplication and addition

Multiply a two-digit number by a single-digit number mentally

Multiply a four-digit number by a two-digit number using long multiplication

Identify multiplication is needed as part of solving multi-step problems

Solve multi-step problems involving multiplication

Use knowledge of place value to multiply with decimals

Use knowledge of place value to divide a decimal

Use knowledge of place value to divide by a decimal

Use knowledge of inverse operations when dividing with decimals

Be fluent at multiplying a three-digit or a two-digit number by a two-digit number

Be fluent when using the method of short division

Different counting systems or bases

Problem 2

Problem 7

Suggested activities Pedagogical notes

KM: Long multiplication template KM: Maximise, minimise. Adapt ideas to fit learning intentions. KM: Maths to Infinity: Complements KM: Maths to Infinity: Multiplying and dividing NRICH: Become Maths detectives NRICH: Exploring number patterns you make NRICH: Reach 100 KM: Dividing (lots) KM: Interactive long division KM: Misplaced points NRICH: Skeleton NRICH: Long multiplication Learning review KM: 7M2 BAM Task www.diagnosticquestions.com

The grid method is promoted as a method that aids numerical understanding and later progresses to multiplying algebraic statements Progression: Multiplication and Division NCETM: The Bar Model, Multiplication, Multiplicative reasoning, Glossary Common approaches All classrooms display a times table poster with a twist To avoid confusion with language, all teachers use ‘sum’ to refer only to the result of an addition. Teachers say ‘complete these calculations’ instead of ‘complete these sums’ Long multiplication is promoted as the ‘most efficient method’. Short division is promoted as the ‘most efficient method’.

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Find missing digits in otherwise completed long multiplication / short division calculations

Show me a calculation that is connected to 14 × 26 =

364. And another. And another …Convince me that 2472 × 12 = 29664

Why have you chosen to multiply/divide? NCETM: Multiplication and Division Reasoning

Operation Multiply, Multiplication, Times, Product Commutative Factor Short multiplication Long multiplication Short division Long division Remainder Estimate

Many pupils will want to multiply the two component part of a decimal and forget they need to multiply by the other parts too.e.g. 1.4 x 3.6 they will want to do 1 x 3 plus 0.4 x 0.6. Spend time exploring why this is an incomplete answer.

Pupils may incorrectly apply place value when dividing by a decimal for example by making the answer 10 times bigger when it should be 10 times smaller.

Some pupils may have inefficient methods for multiplying and dividing numbers

Autumn 2c: Area of rectangle and triangle 6.14, 7.14

Key concepts

recognise that shapes with the same areas can have different perimeters and vice versa

calculate the area of rectangles, parallelograms and triangles

recognise when it is possible to use formulae for area of shapes

solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places where appropriate

Support material Core material Extension material

Be able to multiple and divide integers and decimals

Understand the concept of perimeter

Understand the concept of area as space inside a 2D shape

Know that to calculate area you can count the number of squares inside the shape

Use standard formulae for area

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

Understand the meaning of areaKnow how to calculate areas of rectangles, parallelograms 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 area

Recognise that shapes with different areas can have the same perimeters and vice versa

Understand the meaning of surface area

Find the surface area of cuboids (including cubes) when lengths

are known

Suggested activities Pedagogical notes

KM: Another length KM: Fibonacci’s disappearing squares KM: Dissections deductions NRICH: Disappearing square NRICH: Triangle in a triangle

The prefix ‘centi-‘ means one hundredth, and the prefix ‘milli-‘ means one thousandth. These words are of Latin origin. The prefix ‘kilo-‘ means one thousand. This is Greek in origin. Ensure that pupils make connections with the area and volume work in Stage 6 and below, in particular the importance of the perpendicular height. NCETM: Glossary Common approaches Pupils can derive the formula for the area of a parallelogram.

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

(Given a triangle with base labelled 8 cm, height 5 cm, slope height 6 cm) Kenny thinks that the area is 40 cm2, Lenny thinks it is 20 cm2, Jenny thinks it is 240 cm2 and Benny thinks it is 24 cm2. Who do you agree with? Explain why.

NCETM: Geometry -Properties of Shapes Reasoning

Perimeter, area, surface area Square, rectangle, parallelogram, triangle, trapezium (trapezia) Polygon Square millimetre, square centimetre, square metre, square kilometre 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

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

Autumn 2d: Calculate the mean 7.18

Key concepts

Apply multiplication and division skills to interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (mean)

Support material Core material Extension material

Be able to multiple and divide integers and decimals

Understand the meaning of ‘average’ as a typicality (or location)

Calculate the mean from raw data

Make comparisons between data sets using the mean

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

Calculate the mean from a frequency table

Use comparisons of the mean to justify decisions

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 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 of 5.

The mean is X, find the missing piece of data

Average Mean Add Divide Comparison

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.

Spring 1 a: Draw, measure and name acute, obtuse and reflex angles 6.11

Key concepts

Know the relative sizes of angles and be able to use the correct mathematical vocabulary to describe them

Understand angles as turns and degrees as a measure of turn

Support material Core material Extension material

Know that angles are measured in degrees

Know that angles in a full turn total 360°, and angle in half a turn must total 180°

Estimate the size of angles

Use a protractor efficiently to measure and draw angles

Name angles and estimate their size

Recall facts about the size of acute, obtuse and reflex angles

Tessellating triangles and quadrilaterals

Suggested activities Pedagogical notes

NRICH: Square It NCETM: Activity Set B NCETM: Activity Set C KM: Maths to Infinity: Lines and Angles NRICH: Convex polygons NRICH: Outside the nonagon Learning review www.diagnosticquestions.com

The exact reason for there being 360 degrees in a full turn is unknown. There are various theories including it being an approximation of the 365 days in a year and resultant apparent movement of the sun, and the fact that it has so many factors. The SI unit for measuring angles in the radian (2π radians in a full turn). Napoleon experimented with the decimal degree, or grad (400 grads in a full turn) NCETM: Glossary Common approaches All pupils know how to use a 180° and a 360° protractor. Get students to physically stand up and turn through 90, 180 and 270 degrees turns to get the feel for angles Use forearms to show acute, right and obtuse angles

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me a right angle in this classroom. And another. And another.

Show me an angle in this classroom less (greater) than a right angle. And another. And another.

Is this a right angle? Explain your answer. NCETM: Geometry - Properties of Shapes Reasoning

Angle Degrees Right angle Acute angle Obtuse angle Reflex angle Protractor Geometry, geometrical Notation Right angle notation Arc notation for all other angles The degree symbol (°)

Some pupils may think that right angles have to look like this:

Some pupils may think that right angles have to be created from a horizontal and vertical line

Some pupils may think that all turns have to be in a clockwise direction

Spring 1 b: Find unknown angles (straight line, at a point, vertically opposite) 7.11

Key concepts

apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

Support material Core material Extension material

Know that angles are measured in degrees

Know that angles in a full turn total 360°, and angle in half a turn must total 180°

Estimate the size of angles

Use a protractor efficiently to measure and draw angles

Name acute, obtuse, reflex angles

Estimate sizes of angles

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

Identify known angle facts in more complex geometrical diagrams

Use knowledge of angles to calculate missing angles in geometrical diagrams

Know that angles in a triangles total 180°

Find missing angles in triangles

Find missing angles in isosceles triangles

Explain reasoning using vocabulary of angles

Tessellating triangles and quadrilaterals

Generalisation

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 Learning review www.diagnosticquestions.com

Get students to appreciate the straight line being half a turn by asking to stand up and turn 180 degrees clockwise, anticlockwise (end up at same position). Turn 360 degrees. Etc. NCETM: Glossary Common approaches Teachers convince pupils that the sum of the angles in a triangle is 180° by ripping the corners of triangles and fitting them together on a straight line.

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 Straight line 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?

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

Some pupils may make conceptual mistakes when adding and subtracting mentally. For example, they may see that one of two angles on a straight line is 127° and quickly respond that the other angle must be 63°.

Some pupils when presented with several angles at different points along a straight line will think that all the angles on that line should add up to 180 degrees.

Spring 1 c: Properties of triangles and quadrilaterals 7.4, 7.5

Key concepts

apply the properties of angles in a triangle and in a quadrilateral

Support material Core material Extension material

Know that angles on a straight line sum to 180 degrees

Know that angles around a point sum to 360 degrees

Be able to add and subtract integers and decimals

Know that angles in a triangles total 180°

Find missing angles in triangles

Find missing angles in isosceles triangles

Know that angles in a quadrilateral add up to 360 degrees

Find missing angles in quadrilaterals

Explain reasoning using vocabulary of angles

Tessellating triangles and quadrilaterals

Generalisation

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 Learning review www.diagnosticquestions.com

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 in Stage 8 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! NCETM: Glossary Common approaches Teachers convince pupils that the sum of the angles in a triangle is 180° by ripping the corners of triangles and fitting them together on a straight line. Teachers convince students that ANY quadrilateral sums to 360 by asking students to split a variety of quadrilaterals into triangles – always two triangles possible from vertices! Additionally, rip corners of quadrilaterals and see how they always stick together into a ‘full circle’

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.

Angle Degrees Right angle Acute angle Obtuse angle Reflex angle Protractor Vertically opposite Geometry, geometrical 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 make conceptual mistakes when adding and subtracting mentally. For example, they may see that one of two angles on a straight line is 127° and quickly respond that the other angle must be 63°.

a b 40°

a b c

Spring 2a: Equivalent fractions, mixed numbers and improper 7.7

Key concepts

express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1

recognise in equivalence of fractions, greater than and less than 1, in top heavy and mixed number form

Support material Core material Extension material

Understand the concept of a fraction as a proportion

Understand the meaning of the numerator and denominator

Understand and use improper fractions

Understand that the denominator is a number of EQUAL sized parts

Write one quantity as a fraction of another where the fraction is less than 1

Write one quantity as a fraction of another where the fraction is greater than 1

Recognise equivalent fractions from diagrams

Complete diagrams to show equivalent fractions

Create diagrams to show equivalent fractions

Write a fraction in its lowest terms by cancelling common factors

Convert between mixed numbers and improper fractions

Identify equivalent fractions

Egyptian fractions

Suggested activities Pedagogical notes

NRICH: Fraction Match NRICH: Matching Fractions NCETM: Activity F - Comparing Fractions KM: Crazy cancelling, silly simplifying NRICH: Rod fractions Learning review KM: 7M3 BAM Task KM: 3M8 BAM Task, 3M9 BAM Task NCETM: NC Assessment Materials (Teaching and Assessing Mastery)

Describe 1/3 as ‘there are three equal parts and I take one’, and 3/4 as ‘there are four equal parts and I take three’. Be alert to pupils reinforcing misconceptions through language such as ‘the bigger half’. To explore the equivalency of fractions make several copies of a diagram with half shaded. Show that splitting these diagrams with varying numbers of lines does not alter the fraction of the shape that is shaded. NCETM: Teaching fractions NCETM: The Bar Model NCETM: Glossary NRICH: Teaching fractions with understanding NCETM: Teaching fractions NCETM: Departmental workshop: Fractions NCETM: Glossary Common approaches Use visual aids here: bar models, emphasizing equal parts represented in diagrams

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show this fraction as part of a square / rectangle / number line / …

Which you would prefer, ½ of a cake, 2/3 of a cake or 2/4 of a cake?

Convince me that 1

2=

2

4

Show me a picture of 1

5. And another. And another.

NCETM: Fractions Reasoning

Fraction Improper fraction Proper fraction Vulgar fraction Improper fraction Proportion Notation Diagonal fraction bar / horizontal fraction bar

A fraction can be visualised as divisions of a shape (especially a circle) but some pupils may not recognise that these divisions must be equal in size, or that they can be divisions of any shape.

Spring 2b: Compare and order fractions and decimals 7.2

Key concepts

order positive and negative integers, decimals and fractions

use the symbols =, ≠, <, >, ≤, ≥

Support material Core material Extension material

Understand that negative numbers are numbers less than zero

Order a set of decimals with a mixed number of decimal places (up to a maximum of three)

Order fractions where the denominators are the same

Know how to simplify a fraction by cancelling common factors

Identify a common denominator that can be used to order a set of fractions

Order fractions where the denominators are not multiples of each other

Order a set of numbers including a mixture of fractions, decimals and negative numbers

Use inequality symbols to compare numbers

Make correct use of the symbols = and ≠

Recurring decimal to fractions

Fractions of tangrams

Suggested activities Pedagogical notes

KM: Decimal ordering cards 2 KM: Maths to Infinity: Fractions, decimals and percentages KM: Maths to Infinity: Directed numbers NRICH: Greater than or less than? NCETM: Activity A – visualising fractions along a line Learning review www.diagnosticquestions.com

Pupil must use language correctly to avoid reinforcing misconceptions: for example, 0.45 should never be read as ‘zero point forty-five’; 5 > 3 should be read as ‘five is greater than 3’, not ‘5 is bigger than 3’. Ensure that pupils read information carefully and check whether the required order is smallest first or greatest first. The equals sign was designed by Robert Recorde in 1557 who also introduced the plus (+) and minus (-) symbols. NCETM: Glossary Common approaches Teachers use the language ‘negative number’ to avoid future confusion with calculation that can result by using ‘minus number’ Every classroom has a negative number washing line on the wall

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Jenny writes down 0.400 > 0.58. Kenny writes down 0.400 < 0.58. Who do you agree with? Explain your answer.

Find a fraction which is greater than 3/5 and less than 7/8. And another. And another …

Convince me that -15 < -3

Show me a decimal and fraction equivalent pair. And another. And another.

Jenny is counting in tenths ‘…. 2.7, 2.8, 2.9, 2.10, 2.11 …’. Do you agree with Jenny? Explain your answer.

NCETM: Fractions Reasoning

Integer Numerator Denominator Place value Notation The ‘equals’ sign: = The ‘not equal’ sign: ≠ The inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (more than or equal to)

Some pupils may believe that 0.400 is greater than 0.58

Some pupils may believe that -6 is greater than -3. For this reason ensure pupils avoid saying ‘bigger than’

Pupils may believe, incorrectly, that: - A fraction with a larger denominator is a larger fraction - A fraction with a larger numerator is a larger fraction - A fraction involving larger numbers is a larger fraction

Spring 2c: Fraction of a quantity (Stage 3 & 4)

Key concepts

solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number

Support material Core material Extension material

Recognise and use tenths

Divide one digit numbers by 10

Divide numbers

Calculate a unit fraction of an amount when the answer is a whole number

Calculate a non-unit fraction of an amount when the answer is a whole number

Fractions of tangrams

Shape block challenges

Suggested activities Pedagogical notes

KM: Decimal ordering cards 2 KM: Maths to Infinity: Fractions, decimals and percentages KM: Maths to Infinity: Directed numbers NRICH: Greater than or less than? Learning review www.diagnosticquestions.com

NCETM: Teaching fractions NCETM: Fractions videos NCETM: The Bar Model NCETM: Glossary Common approaches Teachers use the language ‘improper fraction’ and not ‘top-heavy fraction’ When finding fractions of an amount pupils work denominators of at least 2 to 10

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Jenny writes down 0.400 > 0.58. Kenny writes down 0.400 < 0.58. Who do you agree with? Explain your answer.

Find a fraction which is greater than 3/5 and less than 7/8. And another. And another …

Convince me that -15 < -3

Place value Tenth, hundredth Decimal Divide Fraction Unit fraction, non-unit fraction Improper fraction Numerator, denominator Notation Decimal point t, h notation for tenths, hundredths Horizontal bar for fractions Diagonal bar for fractions

Some pupils may think that you find the non-unit fraction of an amount by dividing by the denominator (as with unit fractions) and then dividing by the numerator. They do not make the connection that ¾ = 3 x ¼.

Spring 2d: Add, subtract multiply and divide fractions 7.12

Key concepts

apply the four operations, including formal written methods, to simple fractions (proper and improper), and mixed numbers

Support material Core material Extension material

Add and subtract unit fractions

Multiply a fraction by an integer

Divide a fraction by an integer

Multiply a proper fraction by a proper fraction

Apply addition to proper fractions, improper fractions and mixed numbers

Apply subtraction to proper fractions, improper fractions and mixed numbers

Multiply proper and improper fractions

Multiply mixed numbers

Divide a proper fraction by a proper fraction

Apply division to improper fractions and mixed numbers

Eqyptian fractions

Problem 3

Fractions of tangrams

Shape block challenges

Apply to area and perimeter problems where side lengths and/or

answers are fractions

Suggested activities Pedagogical notes

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 Learning review KM: 7M4 BAM Task, 7M5 BAM Task

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.

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? Explain.

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

Summer 1a: BIDMAS, substitution and expressions 7.6

Key concepts

understand and use the concepts and vocabulary of expressions, equations, formulae and terms

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

simplify and manipulate algebraic expressions by collecting like terms and multiplying a single term over a bracket

where appropriate, interpret simple expressions as functions with inputs and outputs

substitute numerical values into formulae and expressions

use conventional notation for priority of operations, including brackets

Support material Core material Extension material

Use symbols (including letters) to represent missing numbers

Substitute numbers into worded formulae

Substitute numbers into simple algebraic formulae

Know the order of operations

Know the meaning of expression, term, formula, equation, function

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

Know how to multiply a (positive) single term over a bracket (the distributive law)

Substitute positive numbers into expressions and formulae

Given a function, establish outputs from given inputs

Given a function, establish inputs from given outputs

Use a mapping diagram (function machine) to represent a function

Use an expression to represent a function

Create expressions to represent worded situations

Use the order of operations correctly in algebraic situations

Four fours

Algebraic mean

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 KM: Maths to Infinity: Brackets NRICH: Your number is … NRICH: Crossed ends NRICH: Number pyramids and More number pyramids Learning review KM: 7M7 BAM Task, 7M8 BAM Task, 7M9 BAM Task

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 All pupils are expected to learn about the connection between mapping diagrams and formulae (to represent functions) in preparation for future representations of functions graphically.

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

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.

Some pupils may think that 3(g+4) = 3g+4

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

Summer 1b: Sequences (term-to-term, not nth term) 7.9

Key concepts

generate terms of a sequence from a term-to-term rule

identify term-to-term rules for sequences

Support material Core material Extension material

Know the basic rules of algebra

Know the vocabulary of sequences

Find the next term in a linear sequence

Find a missing term in a linear sequence

Generate a linear sequence from its description

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

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

Find the term-to-term rule for a sequence

Describe a number sequence

Solve problems involving the term-to-term rule for a sequence

Solve problems involving the term-to-term rule for a non-numerical sequence

The hand shake problem

The leaping frogs problem

Suggested activities Pedagogical notes

KM: Maths to Infinity: Sequences NRICH: Shifting times tables NRICH: Odds and evens and more evens Learning review www.diagnosticquestions.com

‘Term-to-term rule’ is the only new vocabulary for this unit. Position-to-term rule, and the use of the nth term, are not developed until later stages. NRICH: Go forth and generalise NCETM: Algebra Common approaches All students are taught to describe the term-to-term rule for both numerical and non-numerical sequences

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

Show me a (non-)linear sequence. And another. And another.

What’s the same, what’s different: 2, 5, 8, 11, 14, … and 4, 7, 10, 13, 16, …?

Create a (non-linear/linear) sequence with a 3rd term of ‘7’

Always/ Sometimes /Never: The 10th term of is double the 5th term of the (linear) sequence

Kenny thinks that the 20th term of the sequence 5, 9, 13, 17, 21, … will be 105. Do you agree with Kenny? Explain your answer.

Pattern Sequence Linear Term Term-to-term rule Ascending Descending

When describing a number sequence some students may not appreciate the fact that the starting number is required as well as a term-to-term rule

Some pupils may think that all sequences are ascending

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

Summer 2a: Construct and interpret statistical diagrams including pie charts 7.17

Key concepts

interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data and know their appropriate use

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 and know their appropriate use

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

Choose appropriate graphs or charts to represent data

Construct and interpret vertical line charts

Converting and comparing between different representations

Suggested activities Pedagogical notes

KM: Maths to Infinity: Averages, Charts and Tables NRICH: Picturing the World NRICH: Charting Success Learning review www.diagnosticquestions.com

Pupils may previously have constructed pie charts when the total of frequencies is a factor of 360. More complex cases can now be introduced. Much of the content of this unit has been covered previously in different stages. This is an opportunity to bring together the full range of skills encountered up to this point, and to develop a more refined understanding of usage and vocabulary. William Playfair, a Scottish engineer and economist, introduced the bar chart and line graph in 1786. He also introduced the pie chart in 1801. NCETM: Glossary Common approaches Pie charts are constructed by calculating the angle for each section by dividing 360 by the total frequency and not using percentages. The angle for the first section is measured from a vertical radius. Subsequent sections are measured using the boundary line of the previous section.

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.

Always / Sometimes / Never: Bar charts are vertical

Always / Sometimes / Never: Bar charts, pie charts, pictograms and vertical line charts can be used to represent any data

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? Explain your answer.

Data, Categorical data, Discrete data Pictogram, Symbol, Key Frequency Table, Frequency table Tally Bar chart Time graph, Time series Bar-line graph, Vertical line chart Scale, Graph, Axis, axes Line graph Pie chart Sector Angle Maximum, minimum

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

Summer 2b: Convert between fractions, decimals and percentages 7.7, 8.6

Key concepts

interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data and know their appropriate use

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

Support material Core material Extension material

Know the integer and decimal place value columns

Find a fraction of an amount

Find and recognise equivalent fractions

Write one quantity as a fraction of another

Understand that a percentage means ‘number of parts per hundred’

Write a percentage as a fraction

Write a quantity as a percentage 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

Applications of percentages

Suggested activities Pedagogical notes

KM: Crazy cancelling, silly simplifying NRICH: Rod fractions 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: 7M3 BAM Task 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 All pupils are made aware that ‘per cent’ is derived from Latin and means ‘out of one hundred’

Reasoning opportunities and probing questions Mathematical language Possible misconceptions

What is the same and what is different: 1/10 and 10% … 1/5 and 20%?

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 ?

Percentage Proportion Fraction Mixed number Top-heavy fraction Percentage Decimal Proportion Terminating Recurring Simplify, Cancel

Pupils may not make the connection that a percentage is a different way of describing a proportion

Pupils may think that it is not possible to have a percentage greater than 100%

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%.

Summer 2c: Percentage of a quantity, plus find the whole given the part and the percentage 7.12

Key concepts

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

solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison

compare two quantities using percentages

solve problems involving percentage change, including percentage increase/decrease

Support material Core material Extension material

Convert between mixed numbers and improper fractions

Find equivalent fractions

Add and subtract fractions when one denominator is a multiple of the other

Multiply a proper fraction by a whole number

Use the formal written method of short multiplication

Know the effect of multiplying and dividing by 10 and 100

Know percentage equivalents of 1/2, 1/4, 3/4, 1/5, 2/5, 4/5

Find 10% of a quantity

Use non-calculator methods to find a percentage of an amount

Use decimal or fraction equivalents to find a percentage of an amount where appropriate

Solve problems involving the use of percentages to make comparisons

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

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

Applications of percentages

Problem 12

Suggested activities Pedagogical notes

KM: Stick on the Maths: Percentage increases and decreases KM: Maths to Infinity: FDPRP KM: Percentage methods 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 Learning review KM: 7M4 BAM Task, 7M5 BAM Task

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. 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

Lenny says ‘20% of £60 is £3 because 60 ÷20 = 3’. Do you agree?

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.

NCETM: Fractions Reasoning NCETM: Ratio and Proportion Reasoning

Percentage Percent, percentage Multiplier Increase, decrease Notation Mixed number notation

Some pupils may think that as you divide by 10 to find 10%, you divided by 15 to find 15%, divide by 20 to find 20%, etc.

Some pupils may think the multiplier for, say, a 20% decrease is 0.2 rather than 0.8

Some pupils may think that percentage change = actual change ÷ new amount