text books & support objectives spec grade resources · be able to add and subtract negative...

Support Objectives Spec GradeText Books &

Resources

Be able to order positive and negative numbers given as integers, decimals and

fractions, including improper fractions N12 Collins H1 Ch 1

Be able to understand and use the symbols =,≠, >, <, ≥, ≤ N1 2 Collins F1 Ch 1, 3, 14

Be able to recall all multiplication facts to 10 × 10, and use them to derive quickly the

corresponding division facts; N2 2 Collins F2 Ch 1

Be able to add and subtract positive numbers N2 2 SMP Intermediate Ch 7

Be able to use brackets and the hierarchy of operations (not including powers); N3 3 SMP F2 Ch 6

Be able to add and subtract negative numbers N2 3 10ticks

Be able to multiply or divide any number by powers of 10; N2 2 Level 4 Packs 1, 2 & 3

Be able to multiply and divide positive and negative numbers (integers); N2 3 Level 5 Packs 1 & 5

Be able to use long multiplication & long division N2 2 Level 6 Pack 4

Understand and use place value (e.g. when working with very large or very small

numbers, and when calculating with decimals) N2 2

Be able to estimate answers and check calculations using approximation and

estimation, including answers obtained using technology N14 3

Be able to use brackets and the hierarchy of operations up to and including with powers

and roots inside the brackets, or raising brackets to powers or taking roots of bracketsN3 3

Possible Success Criteria

Given 5 digits, what are the largest or smallest answers when subtracting a two-digit

number from a three-digit number?

Use inverse operations to justify answers, e.g. 9 x 23 = 207 so 207 ÷ 9 = 23.Check answers by rounding to nearest 10, 100, or 1000 as appropriate, e.g. 29 × 31 ≈

30 × 30Given 5 digits, what is the largest even number, largest odd number, or largest or

smallest answers when subtracting a two-digit number from a three-digit number?

Outline

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Given 2.6 × 15.8 = 41.08 what is 26 × 0.158? What is 4108 ÷ 26?

Common Misconceptions

Stress the importance of knowing the multiplication tables to aid fluency.

Students may write statements such as 150 – 210 = 60.

NotesParticular emphasis should be given to the importance of students presenting their

work clearly.Formal written methods of addition, subtraction and multiplication work from right to left,

whilst formal division works from left to right.Negative numbers in real life can be modelled by interpreting scales on thermometers

using F & C.Encourage the exploration of different calculation methods.

Students should be able to write numbers in words and from words as a real-life skill.The expectation for Higher tier is that much of this work will be reinforced throughout

the course.

Support Objectives (F) Spec GradeText Books &

ResourcesBe able to Identify factors, multiples and prime numbers N4 2 Collins H1 Ch 1

Core Objectives (F/H) Collins F1 Ch 4

Be able to find the prime factor decomposition of positive integers – write as a product

using index notation;N4 3 Collins F2 Ch 2

Be able to find common factors and common multiples of two numbers; N4 2 SMP Intermediate Ch 3Be able to find the LCM and HCF of two numbers, by listing, Venn diagrams and using

prime factors – include finding LCM and HCF given the prime factorisation of two

numbers;

N5 3 SMP F1 Ch 8 & 26

Be able to solve problems using HCF and LCM, and prime numbers; N5 3 10ticksUnderstand that the prime factor decomposition of a positive integer is unique,

whichever factor pair you start with, and that every number can be written as a product

of prime factors.

N5 3 Level 5 Pack 1

Level 6 Pack 7

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Know how to test if a number up to 120 is prime.

Understand that every number can be written as a unique product of its prime factors.

Recall prime numbers up to 100.

Understand the meaning of prime factor.

Write a number as a product of its prime factors.

Use a Venn diagram to sort information.


1 is a prime number.

Particular emphasis should be made on the definition of “product” as multiplication, as

many students get confused and think it relates to addition.

Notes

Use a number square to find primes (Eratosthenes sieve).

Using a calculator to check the factors of large numbers can be useful.

Students need to be encouraged to learn squares from 2 × 2 to 15 × 15 and cubes of 2,

3, 4, 5 and 10, and corresponding square and cube roots.

Support Objectives (F) Spec Grade Collins H1 Ch 2

Be able to use diagrams to find equivalent fractions or compare fractions; N1 Collins F1 Ch 2 & 14

Be able to write fractions to describe shaded parts of diagrams; N1 SMP Intermediate Ch 11

Be able to write a fraction in its simplest form and find equivalent fractions; N8 10ticks

Know how to order fractions, by using a common denominator N1 Level 4 Pack 4

Core Objectives (F/H) Level 5 Pack 2

Be able to convert between mixed numbers and improper fractions; N1 Level 6 Pack 3

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Be able to express a given number as a fraction of another R3 Level 7-8 Pack 2

Be able to find equivalent fractions and compare the size of fractions; N1

Be able to write a fraction in its simplest form, including using it to simplify a calculation N8

Be able to find a fraction of a quantity or measurement, including within a context; R3 3

Be able to convert a fraction to a decimal to make a calculation easier; N10

Be able to convert between mixed numbers and improper fractions; N8

Be able to add, subtract, multiply and divide fractions; N2, N8 3

Be able to multiply and divide fractions, including mixed numbers and whole numbers and vice versa; N2, N8 3

Be able to add and subtract fractions, including mixed numbers; N2, N8

Understand and use unit fractions as multiplicative inverses N3, N12


Express a given number as a fraction of another, including where the fraction > 1. eg 120/100

3/5 X 15, 20 X 3/4

1/3 of 36m, 1/4 of £20

Answer the following: James delivers 56 newspapers. 3/8 of the newspapers have a

magazine. How many of the newspapers have a magazine?

Prove whether a fraction is terminating or recurring.

Convert a fraction to a decimal including where the fraction is greater than 1.


The larger the denominator, the larger the fraction.

Foundation Notes

When expressing a given number as a fraction of another, start with very simple

numbers < 1, and include some cancelling before fractions using numbers > 1.

When adding and subtracting fractions, start with same denominator, then where one

denominator is a multiple of the other (answers ≤ 1), and finally where both

denominators have to be changed (answers ≤ 1).Regular revision of fractions is essential.

Demonstrate how to the use the fraction button on the calculator.

Use real-life examples where possible.

Use long division to illustrate recurring decimals.

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Higher Notes

Ensure that you include fractions where only one of the denominators needs to be

changed, in addition to where both need to be changed for addition and subtraction.

Include multiplying and dividing integers by fractions.

Use a calculator for changing fractions into decimals and look for patterns.

Recognise that every terminating decimal has its fraction with a 2 and/or 5 as a

common factor in the denominator.

Use long division to illustrate recurring decimals.

Amounts of money should always be rounded to the nearest penny.

Encourage use of the fraction button.

Support Objectives (F) SpecText Books &

ResourcesBe able to use algebraic notation and symbols correctly A1 Collins H1 Ch 5

Know how to write an expression A1 Collins F2 Ch 4

Core Objectives (F/H) SMP Intermediate Ch 2

Know the difference between a term, expression, equation, formula and an identity; A3 SMP F1 Ch 17, 35, 37, 39

Be able to manipulate an expression by collecting like terms N3, A4 10ticks

Be able to multiply a single term over a bracket (not including surds or algebraic fractions) A4 Level 5 Pack 5

Be able to multply terms together such as 2a x 3b and 4a x 5a N3, A4 Level 6 Pack 1

Be able to simplify expressions involving brackets, i.e. expand the brackets, then add/subtract;A4

Be able to recognise factors of algebraic terms involving single brackets; A4

Be able to factorise algebraic expressions by taking out common factors. A4


Expand and simplify 3(t – 1).

Understand 6x + 4 ≠ 3(x + 2).

Argue mathematically that 2(x + 5) = 2x + 10.

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Simplify 4p – 2q2 + 1 – 3p + 5q

2.

Expand and simplify 3(t – 1) + 57.

Factorise 15x2y – 35x

2y

2.


Any poor number skills involving negatives and times tables will become evident

3(x + 4) = 3x + 4.

The convention of not writing a coefficient with a single value, i.e. x instead of 1x , may

cause confusion.

When expanding two linear expressions, poor number skills involving negatives and

times tables will become evident.

Notes

Some of this will be a reminder from Key Stage 3.

Emphasise correct use of symbolic notation, i.e. 3 × y = 3y and not y 3 and a × b =

ab .

Use lots of concrete examples when writing expressions, e.g. ‘B’ boys + ‘G’ girls.

Plenty of practice should be given and reinforce the message that making mistakes

with negatives and times tables is a different skill to that being developed.

Provide students with lots of practice.

This topic lends itself to regular reinforcement through starters in lessons.

Support Objectives (F)

G1

Understand ‘regular’ and ‘irregular’ as applied to polygons; G1

Know how to use conventional terms and notations: G1

G1

Be able to draw diagrams from written descriptions G1

G3

-points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles,

polygons, regular polygons and polygons with reflection and/or rotation symmetries

Understand the proof that the angle sum of a triangle is 180°, and derive and use the

sum of angles in a triangle;

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Be able to use the standard conventions for labelling and referring to the sides and

angles of triangles

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Be able to classify quadrilaterals by their geometric properties and distinguish between

scalene, isosceles and equilateral triangles;

G3

G3

G3

Be able to apply the properties of: G3

- angles at a point

- angles at a point on a straight line

- vertically opposite angles


Name all quadrilaterals that have a specific property.

Use geometric reasoning to answer problems giving detailed reasons.

Find the size of missing angles at a point or at a point on a straight line.

Notes

colloquial terms such as Z angles are not acceptable and should not be used


Be able to order decimals N1

Be able to add, subtract, multiply and divide positive and negative decimals N2

Be able to put digits in the correct place in a decimal calculation and use one

calculation to find the answer to anotherN2

Core Objectives (F/H)

Be able to multiply or divide by any number between 0 and 1 N2

Be able to change between terminating decimals and their corresponding fractions

(such as 3.5 and 7/2 or 0.375 or 3/8)N10

Put digits in the correct place in a decimal calculation and use one calculation to find the answer to another;N2

Extension Objectives (H)

Be able to change recurring decimals into their corresponding fractions and vice versa N10



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Be able to find missing angles in a triangle using the angle sum in a triangle AND the

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Understand and use the angle properties of quadrilaterals and the fact that the angle

sum of a quadrilateral is 360°;

Understand and use the angle properties of parallel lines and find missing angles using

the properties of corresponding and alternate angles, giving reasons;


Use mental methods for × and ÷, e.g. 5 × 0.6, 1.8 ÷ 3.

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

Given 2.6 × 15.8 = 41.08, what is 26 × 0.158? What is 4108 ÷ 26?

Calculate, e.g. 5.2 million + 4.3 million.


Significant figures and decimal place rounding are often confused.

Some students may think 35 877 = 36 to two significant figures.

Notes

Practise long multiplication and division, use mental maths problems with decimals such as 0.1, 0.001.


Include questions set in context (knowledge of terms used in household finance, for

example profit, loss, cost price, selling price, debit, credit and balance, income tax,

VAT, interest rate)


Be able to round to the nearest integer, 10, 100 or 1000 N14

Be able to round to a given number of decimal places N14

Be able to round to a given number of significant figures N14

Be able to estimate answers to one- or two-step calculations, including use of rounding

numbers and formal estimation to 1 significant figure: mainly whole numbers and then

decimals.

N14


Know how to use inequality notation to specify simple error intervals due to truncation

or roundingN14

Extension Objectives (H)Be able to apply and interpret limits of accuracy including upper and lower

bounds N15

Possible Success CriteriaCheck answers by rounding to nearest 10, 100, or 1000 as appropriate, e.g. 29 × 31 ≈

30 × 30

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Given 5 digits, what is the largest even number, largest odd number, or largest or

smallest answers when subtracting a two-digit number from a three-digit number?

Given 2.6 × 15.8 = 41.08 what is 26 × 0.158? What is 4108 ÷ 26?


Significant figure and decimal place rounding are often confused.

Some pupils may think 35 934 = 36 to two significant figures.

NOTES

The expectation for Higher tier is that much of this work will be reinforced throughout

the course.

Particular emphasis should be given to the importance of clear presentation of work.

Formal written methods of addition, subtraction and multiplication work from right to left,

whilst formal division works from left to right.

Encourage the exploration of different calculation methods.


Make sure students are absolutely clear about the difference between significant

figures and decimal places.

Include appropriate rounding for questions set in context

Know not to round values during intermediate steps of a calculation


Be able to convert between fractions, decimals and percentages R9

Be able to express a given number as a percentage of another number R9Be able to express one quantity as a percentage of another where the percentage is

greater than 100%R9

Be able to find a percentage of a quantity N12

Be able to find the new amount after a percentage increase or decrease N12

Be able to find a percentage of a quantity using a multiplier N12

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Be able to use a multiplier to increase or decrease by a percentage in any scenario

where percentages are usedN12

Understand how to work with percentages more than 100% R9


Be able to work out a percentage increase or decrease, including: simple interest,

income tax calculations, value of profit or loss, percentage profit or loss; N12

Be able to compare two quantities using percentages, including a range of calculations

and contexts such as those involving time or money;R9

Be able to find the original amount given the final amount after a percentage increase

or decrease (reverse percentages), including VAT; N12

Be able to use calculators for reverse percentage calculations by doing an appropriate

division; N12

Be able to use percentages in real-life situations, including percentages greater than

100%; R9

Be able to describe percentage increase/decrease with fractions, e.g. 150% increase

means times as big; N12

Understand that fractions are more accurate in calculations than rounded percentage

or decimal equivalents, and choose fractions, decimals or percentages appropriately for

calculations.

R9


What is 10%, 15%, 17.5% of £30? How do we work out 200% of something?

Be able to work out the price of a deposit, given the price of a sofa is £480 and the

deposit is 15% of the price, without a calculator.

Find fractional percentages of amounts, with and without using a calculator.

Convince me that 0.125 is 1/8

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It is not possible to have a percentage greater than 100%.

Incorrect links between fractions and decimals, such as thinking that

= 0.15, 5% = 0.5, 4% = 0.4, etc.

NOTES

When finding a percentage of a quantity or measurement, use only measurements they should know from Key Stage 3.


Use real-life examples where possible.

Emphasise the importance of being able to convert between decimals and percentages and the use of decimal multipliers to make calculations easier.

Include interpreting percentage problems using a multiplier


Be able to express the division of a quantity into a number parts as a ratio; R4

Be able to write ratios in form 1 : m or m : 1 and to describe a situation; R4,R5

Be able to write ratios in their simplest form, including three-part ratios; R4Be able to divide a given quantity into two or more parts in a given part : part or part :

whole ratio;R5

Be able to use a ratio to find one quantity when the other is known; R5

Be able to write a ratio as a fraction; N11

Be able to write a ratio as a linear function; R8Be able to identify direct proportion from a table of values, by comparing ratios of

values;R6

Be able to use a ratio to convert between measures and currencies, e.g. £1.00 = €1.36; R5

Know how to scale up recipes; R5

Be able to convert between currencies. R5 Express one quantity as a fraction of another, where the fraction is less than 1 or

greater than 1R3

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Be able to express a multiplicative relationship between two quantities as a ratio or

fractionR6

Understand and use proportion as equality of ratios R7


Be able to solve proportion problems using the unitary method; R10

Be able to recognise when values are in direct proportion by reference to the graph form; R10

Understand inverse proportion: as x increases, y decreases (inverse graphs donelater) R10

Be able to recognise when values are in direct proportion by reference to the graph form; R10

Understand direct proportion ---> relationship y = kx . R10

POSSIBLE SUCCESS CRITERIA

Write a ratio to describe a situation such as 1 blue for every 2 red, or 3 adults for every 10 children.

Recognise that two paints mixed red to yellow 5 : 4 and 20 : 16 are the same colour.

Write the solution to the following as a linear function: 3l costs £5. How much does 12l cost?

l : £ -> 3 : 5 -> (×4) -> 12 : 20.

Recognise that two paints mixed red to yellow 5 : 4 and 20 : 16 are the same colour.

If it takes 2 builders 10 days to build a wall, how long will it take 3 builders?

Scale up recipes and decide if there is enough of each ingredient.

Given two sets of data in a table, are they in direct proportion?

COMMON MISCONCEPTIONS

Students find three-part ratios difficult.

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Using a ratio to find one quantity when the other is known often results in students ‘sharing’ the known amount.

NOTES

Emphasise the importance of reading the question carefully.

Include ratios with decimals 0.2 : 1.

Converting imperial units to imperial units aren’t specifically in the programme of study,

but still useful and provide a good context for multiplicative reasoning.

It is also useful to know rough metric equivalents of pounds, feet, miles, pints and

gallons.

Include better value or best buy problems

Find out/prove whether two variables are in direct proportion by plotting the graph and

using it as a model to read off other values.

Possible link with scatter graphs.


Be able to substitute positive and negative numbers into expressions such as 3x + 4

and 2x3 and then into expressions involving brackets and powers;

A2

Be able to substitute numbers into formulae from mathematics and other subject using

simple linear formulae, e.g. l × w , v = u + at ; A2

Set up simple equations from word problems and derive simple formulae; A21

Understand the ≠ symbol (not equal), e.g. 6x + 4 ≠ 3(x + 2), and introduce identity ≡

sign; A3


Be able to solve linear equations, with integer coefficients, in which the unknown

appears on either side or on both sides of the equation; A17

Be able to solve linear equations which contain brackets, including those that have

negative signs occurring anywhere in the equation, and those with a negative solution; A17

Be able to solve linear equations in one unknown, with integer or fractional coefficients; A17

Solve simple angle problems using algebra A21

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Be able to use and substitute formulae from mathematics and other subjects, including

kinematics formulae A2

Be able to set up and solve linear equations to solve to solve a number or angle

problem; A17, A21

Be able to derive a formula and set up simple equations from word problems, then

solve these equations, interpreting the solution in the context of the problem; A17,A21

Be able to substitute positive and negative numbers into a formula, solve the resulting

equation including brackets, powers or standard form;A2


Evaluate the expressions for different values of x : 3x2 + 4 or 2x

3.

A room is 2 m longer than it is wide. If its area is 30 m2 what is its perimeter?

Use fractions when working in algebraic situations.

Substitute positive and negative numbers into formulae.


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

If a = 2 sometimes students interpret 3a as 32.

3xy and 5yx are different “types of term” and cannot be “collected” when simplifying

expressions.

The square and cube operations on a calculator may not be similar on all makes.

Not using brackets with negative numbers on a calculator.

NOTES

Use formulae from mathematics and other subjects, expressed initially in words and then using letters and symbols.

Include substitution into the kinematics formulae given on the formula sheet, i.e. v = u + at ,

v2 – u

2 = 2as , and s = ut + at

2.

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Students need to realise that not all linear equations can be solved by observation or trial and improvement, and hence the use of a formal method is important.

Students can leave their answer in fraction form where appropriate.

Students should be encouraged to use their calculator effectively by using the replay and ANS/EXE functions; reinforce the use of brackets and only rounding their final answer with trial and improvement.

Support Objectives (F) Spec Grade Text Books & Resources

Be able to use a ratio to compare a scale model to real-life object; R2 Collins F1 Ch6 & Ch10

Be able to read and construct scale drawings, drawing lines and shapes to scale;R2 SMP Intermediate Ch36

SMP F2 Ch21


Be able to estimate lengths using a scale diagram; G15

Be able to calculate bearings and solve bearings problems, including on

scaled maps, and find/mark and measure bearings G15

Be able to use and interpret maps and scale drawings, using a variety of

scales and units; R2

Understand, draw and measure bearings; G15

Be able to measure line segments and angles in geometric figures, including

interpreting maps and scale drawings and use of bearingsG15

10ticks

Level 5 Pack 3

POSSIBLE SUCCESS CRITERIA Level 6 Pack 2

Able to read and construct scale drawings.

When given the bearing of a point A from point B , can work out the

bearing of B from A .

Know that scale diagrams, including bearings and maps, are ‘similar’ to the

real-life examples.


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Correct use of a protractor may be an issue.

NOTES

Drawings should be done in pencil.Relate loci problems to real-life scenarios, including mobile phone masts

and coverage.

Construction lines should not be erased.

Support Objectives (F) Spec Text Books & Resources

Be able to draw and interpret scatter graphs; S6 Collins H1 Ex 11Understand and interpret scatter graphs in terms of the relationship

between two variables;S6 Collins F1 Ch15

Be able to draw lines of best fit by eye, understanding what these

represent;S6 SMP Intermediate Ch22

Be able to identify outliers and ignore them on scatter graphs; S6

Core Objectives (F/H)Be able to use a line of best fit, or otherwise, to predict values of a variable

given values of the other variable; S6

Be able to distinguish between positive, negative and zero correlation using

lines of best fit, and interpret correlation in terms of the problem; S6

Understand that correlation does not imply causality, and appreciate that

correlation is a measure of the strength of the association between two S6 10ticks

Know and explain an isolated point on a scatter graph; S6 Level 7-8 Pack 1

Be able to use the line of best fit make predictions; interpolate and

extrapolate apparent trends whilst knowing the dangers of so doing. S6


Justify an estimate they have made using a line of best fit.

Identify outliers and explain why they may occur.Given two sets of data in a table, model the relationship and make

predictions.

COMMON MISCONCEPTIONSLines of best fit are often forgotten, but correct answers still obtained by

sight.

Interpreting scales of different measurements and confusion between x and

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NOTESStudents need to be constantly reminded of the importance of drawing a

line of best fit.

Support with copy and complete statements, e.g. as the ___ increases, the

___ decreases.

Statistically the line of best fit should pass through the coordinate

representing the mean of the data.

Students should label the axes clearly, and use a ruler for all straight lines

and a pencil for all drawing.

Remind students that the line of best fit does not necessarily go through the

origin of the graph.


Indicate given values on a scale, including decimal value; G14 Collins F2 ch6

Interpret scales on a range of measuring instruments, mm, cm, m, ml, cl, l; G14 Collins F2 ch11

Convert between units of measure within one system, including time; G14 Collins H1 ch4.2

Make sensible estimates of a range of measures in everyday settings; G14 SMP F1 c14, 16, 20, 24

Find areas by counting squares; G16 SMP F2 ch25,33

Find the perimeter of rectangles and triangles; G17

Recall and use the formulae for the area of a triangle and rectangle; G16 10 ticks L5/P4/36-41

Core Objectives (F/H) 10 ticks L5/P5/18

Find the perimeter of parallelograms and trapezia; G17 10 ticks L5/P6/35-37

Find the perimeter of compound shapes made from two or more rectangles G17 10 ticks L6/P5/21-25

Find the area of a trapezium and recall the formula; G16 10 ticks L7-8/P6/11

Find the area of a parallelogram; G16

Calculate areas and perimeters of compound shapes made from triangles

and rectangles; G16/17

Trapezium

Solve problems where lengths have to be calculated or deduced from given informationG17 Prisms

Recall and use the formula for the volume of a cuboid G16

Use volume to solve problems; G16

Sketch and recognise nets of cuboids and prisms G17

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http://www.mymaths.co.uk/gold/areas/traparea.html

http://www.mymaths.co.uk/gold/volume/volumeMovie.html

Find surface area using rectangles and triangles; G16

Estimating by rounding measurements to 1 significant figure to check

reasonableness of answersN14


Convert between metric AREA measures. R1

Calculate area and perimeter of more complex compound shapes G16/17

Convert between metric VOLUME measures; R1

Convert between metric measures of volume and capacity, e.g. 1 ml = 1 cm3; R1

Find the surface area of other (simple) shapes with and without a diagram; G17


Find the area/perimeter of a given shape, stating the correct units.

Find areas and perimeters of shapes drawn in first quadrant;

Given dimensions of a rectangle and a pictorial representation of it when

folded, work out the dimensions of the new shape.Calculate the area and/or perimeter of shapes with different units of

measurement.

Given a 3D shape, sketch it and draw its net.Work out the length given the area of the cross-section and volume of a

cuboid


Shapes involving missing lengths of sides often result in incorrect answers.

Students very often confuse perimeter and area.

Counting squares when measuring perimeter results in incorrect answers

due to ‘double counting’.

Focus on Reasoning (AO2)

Estimating by rounding measurements to 1 significant figure to check

reasonableness of answers

Focus on Problem Solving (AO3)

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Given its dimensions, work out the volume of water needed to fill a

triangular water tank. If the tank is part-filled with a given volume of water,

find the height of the water in the tank.

Find the cross-section of a hexagonal prism given its height and the size of

one side. Given the prism’s volume, find its length.

Notes

Use questions that involve different metric measures that need converting.

Measurement is essentially a practical activity: use a range of everyday

shapes to bring reality to lessons.Ensure that students are clear about the difference between perimeter and

area.Practical examples help to clarify the concepts, i.e. floor tiles, skirting

board, etc.

Encourage students to draw a sketch where one isn’t provided.Use lots of practical examples to ensure that students can distinguish

between area, perimeter and volume. Making solids using multi-link cubes

can be useful.

Emphasise the functional elements with carpets, tiles for walls, boxes in a

larger box, etc. Best value and minimum cost can be incorporated too.Ensure that examples use different metric units of length, including

decimals.

Solve problems including examples of solids in everyday use.


ResourcesBe able to recall the definition of a circle and name and draw parts of a circle; including:

centre, radius, chord, diameter, circumference, tangent, arc, sector and segmentG9 2 Collins H1 Ex 4A


Be able to recall and use formulae for the circumference of a circle using a variety of metric measures;G14, G17 4 SMP Intermediate Ch10

Be able to recall and use formulae for the area enclosed by a circle using a variety of metric measures;G14, G17 4 SMP F2 Ch7 & 18

Be able to use π ≈ 3.142 or use the π button on a calculator; N15 4

Be able to give an answer to a question involving the circumference or area of a circle in terms of πN8 5

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Be able to calculate perimeters and areas of composite shapes made from circles and parts of circlesG16, G17 5


Be able to find radius or diameter, given area or circumference of circles in a variety of metric measures;A5, G17 5

Possible Success Criteria 10ticks

Understand that answers in terms of pi are more accurate. Level 6 Pack 5

Calculate the perimeters/areas of circles, semicircles and quarter-circles given

the radius or diameter and vice versa.Level 7-8 Pack 6


Diameter and radius are often confused, and recollection of area and circumference of

circles involves incorrect radius or diameter.


Given the diameter of a circle, find its circumference and area. Double the diameter

and find out what happens to the circumference and area. Consider what happens in

general to the circumference and area as the diameter increases.


For a wheel of given diameter, find the number of rotations for each wheel as it

completes a circuit of a circular course. Make the calculations in terms of π .Find the difference in circumference between the inner and outer lanes of a circular

running track.Given the perimeter of a semicircle or quarter-circle, find its radius.

Notes

Emphasise the need to learn the circle formulae, Don't Cha wish Circumference was Pi times d.

Ensure that students know it is more accurate to leave answers in terms of π but only

when asked to do so

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Be able to plot points in all four quadrants A8

Be able to find and use coordinates of points, for example the fourth vertex of a

rectangle given the other three verticesG11

Be able to find coordinates of a midpoint, for example on the diagonal of a rhombus A8

Be able to plot and draw graphs of y = a , x = a , y = x and y = –x , drawing and

recognising lines parallel to axes, plus y = x and y = –x ;A9

Know how to identify and interpret the gradient of a line segment; A9


Be able to recognise that equations of the form y = mx + c correspond to straight-line

graphs in the coordinate plane; A9

Understand how to identify and interpret the gradient and y-intercept of a linear graph

given by equations of the form y = mx + c ; A10

Know how to find the equation of a straight line from a graph in the form y = mx + c ; A9

Be able to plot and draw graphs of straight lines of the form y = mx + c with and a table of values; A9

Be able to sketch a graph of a linear function, using the gradient and y -intercept (i.e.

without a table of values); A9

Know how to find the equation of the line through one point with a given gradient; A9

Be able to identify and interpret gradient and y intercept from an equation ax + by = c ; A10

Know how to find the equation of a straight line from a graph in the form ax + by = c ; A9

Be able to plot and draw graphs of straight lines in the form ax + by = c ; A9

Know how to interpret and analyse information presented in a range of linear graphs:

Know how to use gradients to interpret how one variable changes in relation to another; A10

Be able to find approximate solutions to a linear equation from a graph;

Know how to identify direct proportion from a graph; R14

Be able to find the equation of a line of best fit (scatter graphs) to model the

relationship between quantities;A9

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Understand how to explore the gradients of parallel lines and lines perpendicular to each other; A9

Know how to select and use the fact that when y = mx + c is the equation of a straight

line, then the gradient of a line parallel to it will have a gradient of m and a line

perpendicular to this line will have a gradient of

A9

Be able to interpret and analyse a straight-line graph and generate equations of lines

parallel and perpendicular to the given line; A9

Notes

Students might play 'battleships' and 'Four in a row' to use cells in 2D contexts


Know which charts to use for different types of data sets; S2

Be able to produce and interpret composite bar charts; S2


Be able to produce and interpret comparative and dual bar charts; S2

Be able to produce and interpret pie charts: S2

Be able to find the mode and the frequency represented by each sector; S2Be able to compare data from pie charts that represent different-sized

samples; S2

Be able to produce and interpret frequency polygons for grouped data: S4

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Know how to from frequency polygons, read off frequency values, compare

distributions, calculate total population, mean, estimate greatest and least

possible values (and range);

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S4

Know how to produce frequency diagrams for grouped discrete data: S4

Be able to compare the mean and range of two distributions, or median or

mode as appropriate;S4

S2


Be able to produce histograms with equal and unequal class intervals with

grouped discrete data and continuous dataS3

Be able to read off frequency values, calculate total population, find

greatest and least values; S2

S3

Be able to construct and interpret time–series graphs, comment on trends; S2


Use a time–series data graph to make a prediction about a future value.

Explain why same-size sectors on pie charts with different data sets do not

represent the same number of items, but do represent the same proportion.

Make comparisons between two data sets.

Decide the most appropriate chart or table given a data set.

From a simple pie chart identify the frequency represented by

1/4 and 1/2 sections

From a simple pie chart identify the mode.

Find the angle for one item

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Know how to from frequency polygons, read off frequency values, compare

distributions, calculate total population, mean, estimate greatest and least

possible values (and range);

Be able to recognise simple patterns, characteristics relationships in bar

charts, line graphs and frequency polygons.

Be able to estimate the median from a histogram with equal class width or

any other information, such as the number of people in a given interval;


Same size sectors for different sized data sets represent the same number

rather than the same proportion.

Focus on Reasoning

1. Find the mean of the ages of a group of people whose ages are given

in months and years.

2. Find the mean, median, mode and range of a small set of numbers.

Consider the minimum information you would need to be able to identify

the numbers exactly given the averages and range.

Focus on Problem-solving

1. A set of numbers have a given mode, mean and median. Work out

what the numbers are. Try the same exercise with negative numbers.

2. For different sets of data, consider the advantages and disadvantages

of presenting the data in a variety of graphs and charts. Consider what

needs to be displayed, and how easy it will be to find the measures of

average and range from the different graphs and charts.


Know how to describe in words a term-to-term sequence and identify which

terms cannot be in a sequence;A23


Be able to generate specific terms in a sequence using the position-to-term

rule and term-to-term rule;A23

Be able to find and use (to generate terms) the n th term of an arithmetic

sequence; A25Be able to use the n th term of an arithmetic sequence to decide if a given

number is a term in the sequence, of find the first term over a certain

number; A23

Be able to identify which terms cannot be in a sequence by finding the n th

term; A24

Be able to continue geometric progression and find term to term rule,

including negative, fraction and decimal terms;A24

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Be able to distinguish between arithmetic and geometric sequences; A24

Be able to use finite/infinite and ascending/descending to describe sequences; A24

Extension Objectives (H)Be able to continue a quadratic sequence and use the n th term to generate

terms; A24

Be able to find the n th term of quadratic sequences n2, n

2 +/– b , an

2 +/– A25

Be able to recognise and use simple geometric progressions (rn where n is

an integer, and r is a rational number > 0 or a surd);

A24

Be able to solve problems involving sequences from real life situations. A24


Given a sequence, ‘Which is the 1st term greater than 50?’

What is the amount of money after x months saving the same amount or

What are the next terms in the following sequences?

1, 3, 9, … 100,50,25 2,4,8,16

Given a sequence, ‘which is the 1st term greater than 50?’

Be able to solve problems involving sequences from real-life situations, such as: 1 grain of rice on first square, 2 grains on second, 4 grains on third, etc

(geometric progression), or person saves £10 one week, £20 the next, £30

the next, etc;

What is the amount of money after x months saving the same amount,

or the height of tree that grows 6 m per year;

Compare two pocket money options, e.g. same number of £ per week

as your age from 5 until 21, or starting with £5 a week aged 5 and

increasing by 15% a year until 21.


Students struggle to relate the position of the term to “n ”.

Focus on Reasoning

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1. Find the values of the first five terms in an arithmetic sequence (e.g.

3n – 1), and plot these values on a graph with the term number on the x -

axis and the values of the terms on the y -axis. Consider why arithmetic

sequences are sometimes called linear sequences.2. For quadratic sequences with the n th term n

2 + a , find the second

differences of the first five terms. Repeat for sequences with the n th term

an2 + b , and consider how to use second differences to tell if a sequence is

quadratic or not.3. Use examples of term-to-term rules to determine which of the four

operations (+, –, ×, ÷) lead to ascending sequences, and which lead to

descending sequences.


1. For a range of geometric sequences that involve division (e.g. 200,

100, 50…), find the first term that is less than 1, and determine whether

any of the sequences have any negative terms or not.

2. Create the first few terms of a real-life geometric sequence and a

statement that a particular term will be greater or less than a given

number. Determine if the statement is true or false.3. For a knockout tournament with a given number of players, work out

the number of matches in each round, the number of rounds, and the total

number of matches.

4. Solve an equation in terms of n to find which number term in a

sequence gives the right-hand side of the equation

(e.g. 3n + 4 = 19); solve other equations with different numbers to

determine whether or not these numbers are in the sequence or not.

5. Create a pattern sequence involving two different colours of tiles, find

the n th term for each colour, and work out the largest complete tiled

pattern that can be made from given numbers of each colour of tile.


ResourcesBe able to distinguish between events which are impossible, unlikely, even chance,

likely, and certain to occur; P2 Collins H1 Ch12

Be able to mark events and/or probabilities on a probability scale of 0 to 1; P3 Collins F1 Ch 11

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Be able to write probabilities in words or fractions, decimals and percentages; P2 Collins F2 Ch 9

Core Objectives (F/H) SMP Intermediate Ch34Understand and use experimental and theoretical measures of probability, including

relative frequency to include outcomes using dice, spinners, coins, etc;P1, P3 SMP H1 Ch20

Be able to estimate the number of times an event will occur, given the probability and

the number of trials;P2 SMP F2 Ch5

Be able to find the probability of successive events, such as several throws of a single

dice; P2

Be able to list all outcomes for single events, and combined events, systematically; N5, P7 10ticks

Be able to draw sample space diagrams and use them for adding simple probabilities; P1, P7 Level 4 pack 6

Know that the sum of the probabilities of all outcomes is 1; P4 Level 5 pack 1Know 1 – p as the probability of an event not occurring where p is the probability of the

event occurring; P4 Level 6 pack 6

Be able to find a missing probability from a list or table including algebraic terms P7 Level 7-8 pack 1

Compare experimental data and theoretical probabilities; P1, P3

Compare relative frequencies from samples of different sizes P1, P3


N5


Mark events on a probability scale and use the language of probability.

If the probability of outcomes are x , 2x , 4x , 3x calculate x .

Calculate the probability of an event from a two-way table or frequency table.

Decide if a coin, spinner or game is fair.

Focus on Reasoning (AO2)Create a set of objects (e.g. pens, counters, socks) to match given probabilities for

picking items from the set.Consider the difference between theoretical and experimental outcomes for a game

with certain outcomes (e.g. spinners, dice), and how you decide whether the game is Use a pack of playing cards to consider how you calculate the probability of picking

cards that meet more than one criteria (e.g. black and a king, hearts and a 3).Consider the relative likelihood of different total scores when you roll two dice and add

their scores together. Repeat with dice with different numbers of sides.Focus on Problem Solving (AO3)Find the probabilities of picking different coloured counters from a bag. Work out how

many counters have been added so that a given new probability is correct.Spin two numbered spinners and add their scores together. Find out whether the total

scores are more likely to be above or below a given total.Use given probabilities and relative frequencies to complete a two-way table for

outcomes to an event.Use a probability table for counters in a bag to find the minimum number of counters

that must be in the bag.Use the outcomes from an experiment repeated different numbers of times, and use

the best estimate to solve a fairness or relative frequency problem.

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Be able to apply systematic listing strategies, including use of the product rule for

counting

Notes

Use this as an opportunity for practical work.

Probabilities written in fraction form should be cancelled to their simplest form.


ResourcesKnow the integer squares up to 10 x 10 and the corresponding square roots; N6 Collins H2 Ch 5

Understand the difference between positive and negative square roots; N6 SMP H1 Ch 10

Know the cubes of 1, 2, 3, 4, 5 and 10; N6 SMP Intermediate Ch 23

Be able to use index notation for squares and cubes N6 SMP F2 Ch 16

Core Objectives (F/H) SMP Higher Ch 4

Be able to recognise powers of 2, 3, 4, 5; N6

Evaluate expressions involving squares, cubes and roots N6 10ticks

Use the square, cube and power keys on a calculator; N14 Level 7-8 pack 2

Use the laws of indices to multiply and divide numbers written in index notation A1, N7


Know that any number to the power of 0 is 1 N6Be able to estimate powers and roots of any given positive number, by considering the

values it must lie between, e.g. the square root of 42 must be between 6 and 7;N6


What is the value of 25?

Prove that the square root of 45 lies between 6 and 7.

Evaluate (23 × 2

5) ÷ 2

4, 4

0

Work out the value of n in 40 = 5 × 2n.

Common MisconceptionsThe order of operations is often not applied correctly when squaring negative numbers,

and many calculators will reinforce this misconception. 10

3, for example, is interpreted as 10 × 3.


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Emphasise that were an experiment repeated it will usually lead to different outcomes,

and that increasing sample size generally leads to better estimates of probability and

population characteristics.Use problems involving ratio and percentage, similar to: A bag contains balls in the ratio

2 : 3 : 4. A ball is taken at random. Work out the probability that the ball will be … ;

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Establish a rule for how to raise a power of a product, using examples such as (2 × 4)2

Explore the difference between a negative number raised to a power and a bracketed

negative number raised to a power, e.g. –32 and (–3)

2. Create more complex Create calculations that use BIDMAS, remove the brackets, and ask students to add

brackets to the calculations to reach the given answers.Use the index law for multiplying two powers to find a rule for how to raise a power to

another power, e.g. 52 × 5

2 × 5

2, and (5

2)3.

Notes

Students need to know how to enter negative numbers into their calculator.

Use negative number and not minus number to avoid confusion with calculations


ResourcesKnow the positive and negative powers of 10 N2 Collins H2 Ch 5

Core Objectives (F/H) SMP H1 Ch 21

Be able to convert large and small numbers into standard form and vice versa; N2, N9 SMP Intermediate Ch 40

Be able to add and subtract numbers in standard form; N9 SMP F2 Ch 32

Be able to multiply and divide numbers in standard form; N9 SMP Higher Ch 12

Be able to interpret a calculator display using standard form and know how to enter

numbers in standard form.N9

10ticks

Possible Success Criteria Level 7-8 pack 2

Write 51 080 in standard form.

Write 3.74 x 10–6

as an ordinary number.

Convert a ‘near miss’ into standard form; e.g. 23 × 107.

Common MisconceptionsSome students may think that any number multiplied by a power of ten qualifies as a

number written in standard form.Focus on Problem Solving (AO3)Calculate the thickness of the paper used in different types of books, newspapers and

magazines. Discuss what this tells you about them.For models of objects of different sizes (e.g. famous landmarks, countries, the solar

system), express their scales in standard form.Use the average speed formula to estimate different calculations involving space

rockets travelling to other planets, e.g. distance to planet, time taken to reach planet, Notes

Standard form is used in science and there are lots of cross curricular opportunities.

Students need to be provided with plenty of practice in using standard form with calculators.

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ResourcesBe able to use input/output diagrams; A2 Collins H1 Ch 10

Be able to use axes and coordinates to specify points in all four quadrants in 2D; A8 Collins H2 Ex 9A

Be able to identify points with given coordinates and coordinates of a given point in all

four quadrants; A8

SMP Intermediate Ch 21D,

Ch 32B

Be able to find the coordinates of points identified by geometrical information in 2D (all

four quadrants);A8 SMP Higher Ch 6B

Be able to draw, label and scale axes; A9 SMP H1 Ch 12B


Be able to read values from straight-line graphs for real-life situations; A14 SMP F1 Ch 42B

Be able to draw straight line graphs for real-life situations, including ready reckoner

graphs, conversion graphs, fuel bills graphs, fixed charge and cost per unit; A14

Be able to draw distance–time graphs and velocity–time graphs; A14 10ticks

Be able to work out time intervals for graph scales; A14 Level 6 pack 5

Be able to interpret distance–time graphs, and calculate: the speed of individual

sections, total distance and total time; R14 Level 7-8 pack 6

Be able to interpret information presented in a range of linear and non-linear graphs; A14

Be able to interpret graphs with negative values on axes; A14

Be able to interpret gradient as the rate of change in distance–time and speed–time

graphs, graphs of containers filling and emptying, and unit price graphs.

R14


Be able to use graphs to calculate various measures (of individual sections), including:

unit price (gradient), average speed, distance, time, acceleration;A14

Be able to use graphs to calculate various measures including using enclosed areas by

counting squares or using areas of parallelograms, squares and triangles; A14


Interpret a description of a journey into a distance–time or speed–time graph.

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Calculate various measures given a graph.

Common MisconceptionsWith distance–time graphs, students struggle to understand that the perpendicular

distance from the x -axis represents distance.Focus on Reasoning (AO2)Use examples of distance–time graphs to determine the meaning of the gradient of the

line.Water runs at a constant rate into containers of different shapes. Consider the

relationships between the cross-sections of the containers and the graphs showing the

depth of water in each container over time.


Provide information on a journey that someone makes, along with a sketch of the

distance–time graph that includes a deliberate error, and identify and correct the error.

Use the corrected graph to find out further information about the journey.

Create a situation with three objects moving along given equations of lines, and

determine whether these objects can meet or not, and if so where. Try a similar

exercise to see whether one moving object makes contact with other objects at fixed

Create a linear equation for a real-life situation (e.g. salary plus commission, or taxi hire

rates), and draw and interpret the graph.

Notes

Clear presentation of axes is important.

Ensure that you include questions that include axes with negative values to represent,

for example, time before present time, temperature or depth below sea level.

Careful annotation should be encouraged: it is good practice to get the students to

check that they understand the increments on the axes.

Use standard units of measurement to draw conversion graphs.

Use various measures in distance–time and velocity–time graphs, including miles,

kilometres, seconds, and hours.

Metric-to-imperial measures are not specifically included in the programme of study, but

it is a useful skill and ideal for conversion graphs.

Emphasise that velocity has a direction.

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ResourcesUnderstand that reflections are specified by a mirror line; G7 Collins H1 Ch 8

Be able to identify correct reflections from a choice of diagrams; G7 SMP Intermediate Ch 53

Be able to colour in missing squares to complete reflections; G7 SMP Higher Ch 16

Be able to identify congruent shapes by eye; G7 SMP H1 Ch 9

Understand clockwise and anticlockwise SMP H2 Ch 2E

Support Objectives (F/H) Collins F1 Ch 8

Know how to distinguish properties that are preserved under particular transformations; G7 Collins F2 Ch 12

Be able to recognise and describe rotations – know that that they are specified by a

centre and an angle; G7 SMP F1 Ch 1

Be able to rotate 2D shapes using the origin or any other point (not necessarily on a

coordinate grid); G7 SMP F2 Ch 9, 15

Know how to identify the equation of a line of symmetry; G7 10ticks

Be able to recognise and describe reflections on a coordinate grid – know to include the

mirror line as a simple algebraic equation, x = a , y = a , y = x , y = –x and lines not

parallel to the axes;

G7 Level 4 pack 7

Be able to reflect 2D shapes using specified mirror lines including lines parallel to the

axes and also y = x and y = –x;G7 Level 5 pack 6

Be able to recognise and describe single translations using column vectors on a

coordinate grid;G7, G24 Level 6 pack 2

Know how to translate a given shape by a vector; G7, G24 Level 7-8 pack 4Understand the effect of one translation followed by another, in terms of column

vectors; G7, G24

Be able to describe and transform 2D shapes using enlargements by a positive integer

or positive fractional scale factor; G7

Know that an enlargement on a grid is specified by a centre and a scale factor; G7

Know how to identify the scale factor of an enlargement of a shape as the ratio of the

lengths of two corresponding sides; G7

Be able to enlarge a given shape using a given centre as the centre of enlargement by

counting distances from centre, and find the centre of enlargement by drawing; G7

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Know how to use congruence to show that translations, rotations and reflections

preserve length and angle, so that any figure is congruent to its image under any of

these transformations;

G8

Be able to describe and transform 2D shapes using combined rotations, reflections,

translations, or enlargements;G8


Be able to describe and transform 2D shapes using enlargements by a negative scale

factor; G7

Know how to find areas after enlargement and compare with before enlargement, to

deduce multiplicative relationship (area scale factor); given the areas of two shapes,

one an enlargement of the other, find the scale factor of the enlargement (whole

number values only);

R12

Describe the changes and invariance achieved by combinations of rotations, reflections

and translations.G8


Describe and transform a given shape by a reflection.

Convince me the scale factor is, for example, 2.5.

Describe and transform a given shape by either a rotation or a translation.

Recognise similar shapes because they have equal corresponding angles and/or sides

scaled up in same ratio.

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

Recognise that enlargements preserve angle but not length.Understand that distances and angles are preserved under rotations, reflections and

translations so that any shape is congruent to its image. Understand that similar shapes are enlargements of each other and angles are

preserved. Common Misconceptions

The directions on a column vector often get mixed up.

Student need to understand that the ‘units of movement’ are those on the axes, and

care needs to be taken to check the scale.

Students often use the term ‘transformation’ when describing transformations instead of

the required information.

Lines parallel to the coordinate axes often get confused.

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Notes

Emphasise the need to describe the transformations fully, and if asked to describe a

‘single’ transformation they should not include two types.

Include rotations with the centre of rotation inside the shape.

Use trial and error with tracing paper to find the centre of rotation.

It is essential that the students check the increments on the coordinate grid when

translating shapes.

Emphasise the need to describe the transformations fully and if asked to describe a

‘single’ transformation they should not include two types.

Students may need reminding about how to find the equations of straight lines,

including those parallel to the axes.When reflecting shapes, the students must include mirror lines on or through original

shapes.As an extension, consider reflections with the mirror line through the shape and

enlargements with the centre of enlargement inside the shape.


ResourcesR9

Collins H1 Ch 2 2:5

onwardsSMP higher 1 Ch 8

R9SMP foundation 2 Ch10

b,c,e,f,iKnow how to find a percentage of a quantity using a multiplier;

R9

R9

Be able to use calculators for reverse percentage calculations by doing an

appropriate division; R9

Be able to use percentages in real-life situations, including percentages

greater than 100%; R9

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Be able to find the original amount given the final amount after a

percentage increase or decrease (reverse percentages), including VAT;

Be able to work out a percentage increase or decrease, including: simple

interest, income tax calculations, value of profit or loss, percentage profit or

loss;

Know how to compare two quantities using percentages, including a range

of calculations and contexts such as those involving time or money;

Be able to use a multiplier to increase or decrease by a percentage in any

scenario where percentages are used;

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Be able to describe percentage increase/decrease with fractions, e.g. 150%

increase means 2.5 times as bigR9

R9


What is 10%, 15%, 17.5% of £30?

Be able to work out the price of a deposit, given the price of a sofa is £480

and the deposit is 15% of the price, without a calculator.

Find fractional percentages of amounts, with and without using a calculator.

Convince me that 0.125 is 1/8


Incorrect links between fractions and decimals, such as thinking that

= 0.15, 5% = 0.5,

4% = 0.4, etc.

It is not possible to have a percentage greater than 100%.

Emphasise the importance of being able to convert between decimals and

percentages and the use of decimal multipliers to make calculations easier.

Focus on Reasoning

1. Create a table with annual data such as population or revenue.

Find percentage change

between years – include percentage increases and decreases


1. Two friends buy similar items in a sale. Given the sale prices and the percentage

reductions from the original prices, find which of the friends saved more

money on their item.

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Understand that fractions are more accurate in calculations than rounded

percentage or decimal equivalents, and choose fractions, decimals or

percentages appropriately for calculations.

1

5

2. The number of visitors at an event varies over a given number of

years. Given the number of

visitors in the last year, and the percentage changes year-by-year (both

increases and decreases),

find the number of visitors in the first year.

NOTES

Students should be reminded of basic percentages.

Amounts of money should always be rounded to the nearest penny, except

where successive calculations are done (i.e. compound interest, which is

covered in a later unit).

Emphasise the use of percentages in real-life situations.

Support Objectives (F/H) Spec GradeText Books &

ResourcesUnderstand, recall and use Pythagoras’ Theorem in 2D; G20

Given three sides of a triangle, be able to justify if it is right-angled or not; G6

G20

Know how to find the length of a shorter side in a right-angled triangle; G20

Know how to calculate the length of a line segment AB given pairs of points; G6

Be able to give an answer to the use of Pythagoras’ Theorem in surd form; G20


Does 2, 3, 6 give a right-angled triangle?

Justify when to use Pythagoras’ Theorem and when to use trigonometry.


Answers may be displayed on a calculator in surd form. Students forget to square root their final answer, or round their answer

prematurely.

NOTES

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Be able to alculate the length of the hypotenuse in a right-angled triangle

(including decimal lengths and a range of units);

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Students may need reminding about surds.

Drawing the squares on the three sides will help when deriving the rule.

Scale drawings are not acceptable.

Calculators need to be in degree mode.

To find in right-angled triangles the exact values of sin θ and cos θ for θ =

Use a suitable mnemonic to remember SOHCAHTOA.

Use Pythagoras’ Theorem and trigonometry together.


ResourcesKnow how to calculate mean and range, find median and mode from small data set; S4

Be able to use a spreadsheet to calculate mean and range, and find median and mode; S4

Be able to recognise the advantages and disadvantages between measures of average; S4

Be able to construct and interpret stem and leaf diagrams (including back-to-back diagrams): S4

S4

Be able to calculate the mean, mode, median and range from a frequency table (discrete data);S4

Know how to construct and interpret grouped frequency tables for continuous data: S4

Know how, for grouped data, to find the interval which contains the median and the modal class; S4

Be able to estimate the mean with grouped data; S4

S4

Use averages and range to describe a populaion S5

Support Objectives (F/H)

be able to recognise unusual data values that do not fit in an otherwise good correlation S1

understand that samples may not be representative of a population S1

understand that the size and construction of a sample will affect how representative it is S1

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Know how to find the mode, median, range, as well as the greatest and least values

from stem and leaf diagrams, and compare two distributions from stem and leaf

diagrams (mode, median, range);

Understand that the expression ‘estimate’ will be used where appropriate, when finding

the mean of grouped data using mid-interval values.

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Be able to state the median, mode, mean and range from a small data set.

Extract the averages from a stem and leaf diagram.

Estimate the mean from a table.


Students often forget the difference between continuous and discrete data. Often the ∑f × M is divided by the ∑ midpoints rather than ∑f when

estimating the mean.

NOTES

Remind students how to find the midpoint of two numbers.

Emphasise that continuous data is measured, i.e. length, weight, and

discrete data can be counted, i.e. number of shoes.


ResourcesBe able to draw sketches of 3D solids; G13

Be able to sketch and recognise basic nets of cubes and cuboids; G13 2

Be able to draw 3D shapes using isometric grids; G13 3

Core Objectives (F/H)Know how to identify planes of symmetry of 3D solids, and sketch planes of

symmetry; G12

Be able to sketch and recognise nets of cuboids and prisms; G13

Understand and draw front and side elevations and plans of shapes made

from simple solids; G13

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Given the front and side elevations and the plan of a solid, be able to draw

a sketch of the 3D solid; G13


Given a 3D shape, sketch it and draw its net.

NOTES

Encourage students to draw a sketch where one isn’t provided.

Use lots of practical examples

Scaffold drawing 3D shapes by initially using isometric paper.

Solve problems including examples of solids in everyday use

Core Objectives (F/H) Spec GradeText Books &

Resources

Be able to use the standard ruler and compass to: G2

· construct an equilateral triangle;

· bisect a given angle;

· construct a perpendicular to a given line from/at a given point;

· construct angles of 60°, 90°, 30°, 45°;

· construct a regular hexagon inside a circle, and other polygons;

· construct a perpendicular bisector of a line segment;

Be able to construct: G2

· a region bounded by a circle and an intersecting line;

· a given distance from a point and a given distance from a line;

· equal distances from two points or two line segments;

· regions which may be defined by ‘nearer to’ or ‘greater than’;Be able to find and describe regions satisfying a combination of loci,

including in 3D; G2

Know how to use constructions to solve loci problems G2

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Know that the perpendicular distance from a point to a line is the shortest

distance to the line. G2


Correct use of a protractor may be an issue.

NOTES

Drawings should be done in pencil.Relate loci problems to real-life scenarios, including mobile phone masts

and coverage.

Construction lines should not be erased.


ResourcesA22

A22

A22


A22

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Be able to solve simple linear inequalities in one variable, and

represent the solution set on a number line;

Be able to solve two linear inequalities in x , find the solution sets

and compare them to see which value of x satisfies both solve linear

inequalities in two variables algebraically;

know the conventions of an open circle on a number line for a strict

inequality and a closed circle for an included boundary

Know how to represent the solution set for inequalities using set

notation, i.e. curly brackets and ‘is an element of’ notation;

A22

know how to set up inequalities based on information given A22

Notes


ResourcesBe able to substitute positive and negative numbers into a formula A2

Know how to rearrange formulae to change the subject A5

A5

A5

A5

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Use and substitute formulae from mathematics and other subjects,

including the kinematics formulae v = u + at , v2 – u

2 = 2as , and s

= ut + at2

in graphical work the convention of a dashed line for strict

inequalities and a solid line for an included inequality will be required

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Understand that for problems identifying the solutions to two

different inequalities, show this as the intersection of the two

solution sets, i.e. solution of x ² – 3x – 10 < 0 as {x : –3 < x < 5};

Be able to change the subject of a formula involving the use of

square roots and squares;

Change the subject of a formula, including cases where the subject is

on both sides of the original formula, or involving fractions and small

1

2

A5

Change the subject of a formula such as

, where all variables are in the denominators;


ResourcesBe able to use diagrams to find equivalent fractions or compare fractions; N1

Be able to write fractions to describe shaded parts of diagrams; N1

Be able to write a fraction in its simplest form and find equivalent fractions; N8

Know how to order fractions, by using a common denominator N1

Core Objectives

Be able to convert between mixed numbers and improper fractions;

Be able to express a given number as a fraction of another R3

Be able to find equivalent fractions and compare the size of fractions; N1

Be able to write a fraction in its simplest form, including using it to simplify a calculationN8

Be able to find a fraction of a quantity or measurement, including within a context; R3

Be able to convert a fraction to a decimal to make a calculation easier; N10

Be able to convert between mixed numbers and improper fractions; N8

Be able to add, subtract, multiply and divide fractions; N2, N8

Be able to multiply and divide fractions, including mixed numbers and whole numbers and vice versa; N2, N8

Be able to add and subtract fractions, including mixed numbers; N2, N8

Understand and use unit fractions as multiplicative inverses N3, N12

Be able to convert a fraction to a recurring decimal; N10

Be able to convert a recurring decimal to a fraction; N10

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Change the subject of a formula, including cases where the subject

occurs on both sides of the formula, or where a power of the subject

By writing the denominator in terms of its prime factors, decide

whether fractions can be converted to recurring or terminating N10

1 1 1

f u v

Be able to find the reciprocal of an integer, decimal or fraction. N3


ResourcesBe able to express the division of a quantity into a number parts as a ratio;R4

Be able to express one quantity as a fraction of another R3

Be able to write ratios in form 1 : m or m : 1 and to describe a situation; R4

Be able to write ratios in their simplest form, including three-part ratios; R4

R5

Be able to use a ratio to find one quantity when the other is known; R5

Be able to write a ratio as a fraction; R6

Be able to write a ratio as a linear function; N11

R7

N13,R10

Know how to scale up recipes; R10

Be able to convert between currencies. R10

Know that 1:3 means the smaller part is one quarter of the whole R8

Understand how to represent the ratio of two quantities in direct

proportion as a linear relationship and show this graphicallyR8

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Be able to identify direct proportion from a table of values, by

comparing ratios of values;

Be able to use a ratio to convert between measures and currencies,

e.g. £1.00 = €1.36;

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Be able to divide a given quantity into two or more parts in a given

part : part or part: whole ratio;


ResourcesG3

Understand ‘regular’ and ‘irregular’ as applied to polygons;

G4

Be able to explain why the angle sum of a quadrilateral is 360°;

Know how to Use the angle sums of irregular polygons;

Be able to use the sum of the exterior angles of any polygon is 360°;

Be able to use the sum of the interior angles of an n-sided polygon;

Be able to use the sum of the interior angle and the exterior angle is 180°;

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Be able to calculate and use the sums of the interior angles of polygons, use the sum of

angles in a triangle to deduce and use the angle sum in any polygon and to derive the

properties of regular polygons;

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Be able to classify quadrilaterals by their geometric properties and distinguish between

scalene, isosceles and equilateral triangles;

Be able to find missing angles in a triangle using the angle sum in a triangle AND the

properties of an isosceles triangle;

Be able to use symmetry property of an isosceles triangle to show that base angles are

equal;



Understand a proof of, and use the fact that, the exterior angle of a triangle is equal to

the sum of the interior angles at the other two vertices;

Be able to calculate the angles of regular polygons and use these to solve problems;


ResourcesBe able to write simultaneous equations to represent a situation; A19

Be able to solve simultaneous equations (linear/linear) algebraically, where: A21

· neither needs multiplying, or only one equation does;

· where both need multiplying;

· graphically;


Be able to find the exact solutions of two simultaneous equations in two unknowns;

· Use elimination or substitution to solve simultaneous equations;

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Be able to find the size of each interior angle, or the size of each exterior angle, or the

number of sides of a regular polygon, and use the sum of angles of irregular polygons;

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Be able to solve simultaneous equations representing a real-life situation, graphically

and algebraically, and interpret the solution in the context of the problem;

Solve two simultaneous inequalities algebraically and show the solution set on a

number line

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Be able to solve exactly, by elimination of an unknown, two simultaneous equations in

two unknowns:

Be able to use the side/angle properties of compound shapes made up of triangles,

lines and quadrilaterals, including solving angle and symmetry problems for shapes in

the first quadrant, more complex problems and using algebra;

Be able to use angle facts to demonstrate how shapes would ‘fit together’, and work out

interior angles of shapes in a pattern.

· linear / linear, including where both need multiplying;

· linear / quadratic;

· linear / x2 + y

2 = r

2;

· Set up and solve a pair of simultaneous equations in two variables for each of the

above scenarios, including to represent a situation;

· Interpret the solution in the context of the problem;

· Find the equation of the line through two given points.


ResourcesG6, G20

G21,R12

Use the trigonometric ratios to solve 2D problems;

Be able to find angles of elevation and depression;


Ø Compare lengths using ratio notation; make links to trigonometric ratios

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Understand, use and recall the trigonometric ratios sine, cosine and tan, and apply

them to find angles and lengths in general triangles in 2D figures;

Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the

exact value of tan θ for θ = 0°, 30°, 45° and 60°.

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Resources· Use index notation for integer powers of 10, including negative powers; N6,N7

· Recognise powers of 2, 3, 4, 5;

· Use the square, cube and power keys on a calculator and estimate powers and

roots of any given positive number, by considering the values it must lie between, e.g.

the square root of 42 must be between 6 and 7;

· Multiply and divide numbers in index form;


· Find the value of calculations using indices including positive, fractional and negative indices;

· Recall that n0 = 1 and n

–1 =

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1

n

for positive integers n as well as,

= √n and

= 3√n for any positive number n ;

· Understand that the inverse operation of raising a positive number to a power n

is raising the result of this operation to the power

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

involving multiplication and division of integer powers, fractional and negative powers,

and powers of a power;

· Solve problems using index laws;· Use brackets and the hierarchy of operations up to and including with powers

and roots inside the brackets, or raising brackets to powers or taking roots of

brackets;· Use an extended range of calculator functions, including +, –, ×, ÷, x ², √x , memory, x

y,

, brackets;

· Use calculators for all calculations: positive and negative numbers, brackets,

powers and roots, four operations.

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Resourcesbe able to multiply two brackets and simplify thee algebraiic solution

• Define a ‘quadratic’ expression;

• Multiply together two algebraic expressions with brackets; A4

• Square a linear expression, e.g. (x + 1)2;

• Factorise quadratic expressions of the form x2 + bx + c ;

• Factorise a quadratic expression using the difference of two squares;

• Solve quadratic equations by factorising;

• Solve quadratic equations that require rearranging;

• Find the roots of a quadratic function algebraically.


· Factorise quadratic expressions in the form ax2 + bx + c ;

· Solve quadratic equations by factorisation and completing the square;

· Solve quadratic equations that need rearranging;

· Set up and solve quadratic equations;

· Solve simple quadratic equations by using the quadratic formula;


Solve 3x2 + 4 = 100.

Expand (x + 2)(x + 6).

Factorise x2 + 7x + 10.

Solve x2

+ 7x + 10 = 0.

Solve (x – 3)(x + 4)= 0.


x terms can sometimes be ‘collected’ with x2.

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Resources

· Understand and use compound measures: G14

· density; N13

· pressure; R1

· speed: R11

· convert between metric speed measures;

· read values in km/h and mph from a speedometer;

· use kinematics formulae from the formulae sheet to calculate speed, acceleration;

· change d/t in m/s to a formula in km/h, i.e. d/t × (60 × 60)/1000 – with support;

· Understand and use compound measures and:

· convert between metric speed measures;

· convert between density measures;

· convert between pressure measures;


Change m/s to km/h.

Change g/cm3 to kg/m

3, kg/m

2 to g/cm

2, m/s to km/h.

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· Interpret scales on a range of measuring instruments: mg, g, kg, tonnes, km/h,

mph;

· calculate average speed, distance, time – in miles per hour as well as metric

measures;

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

speedometers and rates of change.

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Spec GradeText Books &

Resources


· Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2; N8

A24


Simplify √8.

Prove that √2 is irrational.

Multiply two brackets containing surds

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· Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 =

2√3).

Recognise and use simple geometric progressions (rn where n is an integer and r is a

surd)


Resources

Be able to show inequalities on number lines; A22

Be able to write down whole number values that satisfy an inequality;

Know how to use the correct notation to show inclusive and exclusive inequalities.


Resources


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Understand the difference between rational and irrational numbers N8

· Rationalise the denominator involving surds;


Rationalise:

simplify (√18 + 10) +√2.

Explain the difference between rational and irrational numbers.


Resources

· Solve angle problems by first proving congruence; G5

G6

G19

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· Understand and use SSS, SAS, ASA and RHS conditions to prove the

congruence of triangles using formal arguments;

· Identify shapes which are similar; including all circles or all regular polygons with

equal number of sides;

· Understand similarity of triangles and of other plane shapes, use this to make

geometric inferences, and solve angle problems using similarity;

1

3 1

1

3

· Solve angle problems in similar shapes involving 2D shapes;

· Use formal geometric proof for the similarity of two given triangles;

· Understand the effect of enlargement on angles, perimeter, area and volume of shapes and solids;

· Identify the scale factor of an enlargement of a similar shape as the ratio of

the lengths of two corresponding sides, using integer or fraction scale factors;

· Write the lengths, areas and volumes of two shapes as ratios in their simplest form;

· Find missing lengths, areas and volumes in similar 3D solids;

· Know the relationships between linear, area and volume scale factors of

mathematically similar shapes and solids;

· Use the relationship between enlargement and areas and volumes of simple shapes and solids;


Understand similarity as one shape being an enlargement of the other.

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


Students commonly use the same scale factor for length, area and volume.


Resources

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· Solve problems involving frustums of cones where you have to find missing

lengths first using similar triangles.

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

corresponding lengths of sides are not equal in size.

· Identify the scale factor of an enlargement of a shape as the ratio of the lengths

of two corresponding sides;

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

and solids;

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· Find the probability of an event happening using relative frequency;

· Estimate the number of times an event will occur, given the probability and the number of trials – for both experimental and theoretical probabilities; P2

· List all outcomes for combined events systematically; P3

· Use and draw sample space diagrams; P5

P6

P8

· Use union and intersection notation; P9

· Compare experimental data and theoretical probabilities;

· Compare relative frequencies from samples of different sizes;

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

· Understand conditional probabilities and decide if two events are independent;

· Use tree diagrams to calculate the probability of two independent events;

· Use tree diagrams to calculate the probability of two dependent events.

· Find a missing probability from a list or two-way table, including algebraic terms;

· Understand conditional probabilities and decide if two events are independent;

· Understand selection with or without replacement;

· Use a two-way table to calculate conditional probability;

· Use a tree diagram to calculate conditional probability;

· Use a Venn diagram to calculate conditional probability;

· Use: P(A or B ) = P(A ) + P(B ); P(A or B or C or …) = P(A ) + P(B ) + P(C ) + …;

P(not A ) = 1 – P(A )

· Compare experimental data and theoretical probabilities;

· Compare relative frequencies from samples of different sizes.


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Draw a Venn diagram of students studying French, German or both, and then calculate

the probability that a student studies French given that they also study German.

· Work out probabilities from Venn diagrams to represent real-life situations and

also ‘abstract’ sets of numbers/values;

· Draw a probability tree diagram based on given information, and use this to find

probability and expected number of outcome;

Solve problems such as:

· P(B ) = 0.3 and P(A or B ) = 0.7. Work out P(not A )

· P(A ) = 0.45 and P (A or B ) = 0.8. Work out P(B )


Not using fractions or decimals when working with probability trees.


Resources· Recall the definition of a circle; G18

· Identify, name and draw parts of a circle including tangent, chord and segment;

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

by a circle circumference of a circle = 2πr = πd , area of a circle = πr2;

· Find circumferences and areas enclosed by circles;

· Use π ≈ 3.142 or use the π button on a calculator;

· Give an answer to a question involving the circumference or area of a circle in terms of π ;

· Find radius or diameter, given area or perimeter of a circles;

· Find the perimeters and areas of semicircles and quarter-circles;

· Calculate perimeters and areas of composite shapes made from circles and parts of circles;

· Calculate arc lengths, angles and areas of sectors of circles;

Giving answers in terms of π

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Recall terms related to a circle.

Understand that answers in terms of pi are more accurate.

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

to one half in either direction.


Diameter and radius are often confused and recollection which formula to use for area

and circumference of circles is often poor.


Resources

N15

N16

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· Calculate the upper and lowers bounds of numbers given to varying degrees of

accuracy;

· Calculate the upper and lower bounds of an expression involving the four

operations;

· Find the upper and lower bounds in real-life situations using measurements given

to appropriate degrees of accuracy;

· Find the upper and lower bounds of calculations involving perimeters, areas and

volumes of 2D and 3D shapes;

· Calculate the upper and lower bounds of calculations, particularly when working

with measurements;

· Given a measurement/value with a 10% error interval, work out the range of

possible values.


Round 16,000 people to the nearest 1000.

Round 1100 g to 1 significant figure.


NOTES

Encourage use a number line when introducing the concept.


Resources· Recall and use the formula for the volume of a cuboid;

· Find volumes by counting cubes; R12

· Find the volume of a prism, including a triangular prism, cube and cuboid; G16

· Calculate volumes of right prisms and shapes made from cubes and cuboids; G17

· Estimate volumes etc by rounding measurements to 1 significant figure; N8

· Convert between metric volume measures;

· Convert between metric measures of volume and capacity e.g. 1ml = 1cm3.


· Find the volume of a cylinder;

· Recall and use the formula for volume of pyramid;

· Use the formulae for volume of spheres and cones;

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Students readily accept the rounding for lower bounds, but take some convincing in

relation to upper bounds.

Students should use ‘half a unit above’ and ‘half a unit below’ to find upper and lower

bounds.

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Work out the upper and lower bounds of a formula where all terms are given to 1

decimal place.

Be able to justify that measurements to the nearest whole unit may be inaccurate by up

to one half in either direction.

· Solve problems involving more complex shapes and solids, including segments

of circles and frustums of cones;

· Giving answers in terms of π ;

· Form equations involving more complex shapes and solve these equations.


Justify whether a certain number of small boxes fit inside a larger box.

Calculate the volume of a triangular prism with correct units.



Volume often gets confused with area.

NOTES

Discuss the correct use of units.

Drawings should be done in pencil.

Consider ‘how many small boxes fit in a larger box’-type questions.

Ensure students can recall area and circumference formulae accuratley (songs)

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Practical examples should be used to enable students to understand the difference

between perimeter, area and volume. (F)

Ensure that students know it is more accurate to leave answers in terms of π but only

when asked to do so.

· Find the volumes of compound solids constructed from cubes, cuboids, cones,

pyramids, spheres, hemispheres, cylinders;

Calculate the perimeters/areas of circles, semicircles and quarter-circles given the

radius or diameter and vice versa.Given two solids with the same volume and the dimensions of one, write and solve an

equation in terms of π to find the dimensions of the other, e.g. a sphere is melted down

to make ball bearings of a given radius, how many will it make?


ResourcesCore Objectives (F/H)

· Use statistics found in all graphs/charts in this unit to describe a population; S3

· Know the appropriate uses of cumulative frequency diagrams; S4

· Construct and interpret cumulative frequency tables;

· Construct and interpret cumulative frequency graphs/diagrams and from the graph:

· estimate frequency greater/less than a given value;

· find the median and quartile values and interquartile range;


Construct cumulative frequency graphs and box plots from frequency tables.


NOTES

Ensure that axes are clearly labelled.

As an extension, use the formula for identifying an outlier, (i.e. if data point is below

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· Produce box plots from raw data and when given quartiles, median and identify

any outliers;

Students often confuse the methods involved with cumulative frequency, estimating the

mean and histograms when dealing with data tables.

As a way to introduce measures of spread, it may be useful to find mode, median,

range and interquartile range from stem and leaf diagrams (including back-to-back) to

compare two data sets.

LQ – 1.5 × IQR or above UQ + 1.5 × IQR, it is an outlier). Get them to identify outliers in

the data, and give bounds for data.

· Compare the mean and range of two distributions, or median and interquartile

range, as appropriate;

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

draw conclusions;

Compare two data sets and justify their comparisons based on measures extracted

from their diagrams where appropriate in terms of the context of the data.


Resources

Support Objectives (F/H) S4

· Know the appropriate uses of histograms;

· Construct and interpret histograms from class intervals with unequal width;

· Use and understand frequency density;

· From histograms:

· complete a grouped frequency table;

· understand and define frequency density;

· Estimate the mean from a histogram;


Construct histograms from frequency tables.


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Estimate the median from a histogram with unequal class widths or any other

information from a histogram, such as the number of people in a given interval.

LQ – 1.5 × IQR or above UQ + 1.5 × IQR, it is an outlier). Get them to identify outliers in

the data, and give bounds for data.

Students often confuse the methods involved with cumulative frequency, estimating the

mean and histograms when dealing with data tables.

Labelling axes incorrectly in terms of the scales, and also using ‘Frequency’ instead of

‘Frequency Density’ or ‘Cumulative Frequency’.


Resources· Find the volume and surface area of a cylinder;

G17


· Recall and use the formula for volume of pyramid;

· Find the surface area of a pyramid;

· Use the formulae for volume and surface area of spheres and cones;

· Giving answers in terms of π ;

· Form equations involving more complex shapes and solve these equations.




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· Solve problems involving more complex shapes and solids, including segments

of circles and frustums of cones;

· Find the surface area and volumes of compound solids constructed from cubes,

cuboids, cones, pyramids, spheres, hemispheres, cylinders;



Calculate the perimeters/areas of circles, semicircles and quarter-circles given the

radius or diameter and vice versa.Given two solids with the same volume and the dimensions of one, write and solve an

equation in terms of π to find the dimensions of the other, e.g. a sphere is melted down

to make ball bearings of a given radius, how many will it make?


Resources

Be able to express the division of a quantity into a number parts as a ratio;N11, N12, N13, R3, R4, R5, R6, R7, R8, R10

Be able to write ratios in form 1 : m or m : 1 and to describe a situation;

Be able to write ratios in their simplest form, including three-part ratios;

Be able to use a ratio to find one quantity when the other is known;

Be able to write a ratio as a fraction;

Be able to write a ratio as a linear function;

Know how to scale up recipes;

Be able to convert between currencies.

· Solve word problems involving direct and indirect proportion;

· Work out which product is the better buy;

· Scale up recipes;

· Solve proportion problems using the unitary method;

· Recognise when values are in direct proportion by reference to the graph form;

· Understand inverse proportion: as x increases, y decreases (inverse graphs done in later unit);

· Recognise when values are in direct proportion by reference to the graph form;

· Understand direct proportion ---> relationship y = kx .

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Be able to identify direct proportion from a table of values, by comparing ratios of

values;

Be able to use a ratio to convert between measures and currencies, e.g. £1.00 = €1.36;

Be able to divide a given quantity into two or more parts in a given part : part or part :

whole ratio;


Resources

· Solve simple equations; A17

A18

A21

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

· Rearrange simple equations;

· Substitute into a formula, and solve the resulting equation;

· Find an approximate solution to a linear equation using a graph;

· Solve simple angle problems using algebra.

• Solve quadratic equations by factorising;

• Solve quadratic equations that require rearranging;

• Find the roots of a quadratic function algebraically.

· Factorise quadratic expressions in the form ax2 + bx + c;

· Solve quadratic equations by factorisation and completing the square;

· Solve quadratic equations that need rearranging;

· Set up and solve quadratic equations;

· Solve simple quadratic equations by using the quadratic formula;

· Find an approximate solution to a quadratic equation using a graph;

· Be able to identify from a graph if a quadratic equation has any real roots;

· Find approximate solutions to quadratic equations using a graph;

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

on either side or on both sides of the equation;

· Solve linear equations which contain brackets, including those that have negative

signs occurring anywhere in the equation, and those with a negative solution;

· Sketch a graph of a quadratic function, by factorising, identifying roots and

y -intercept, turning point;

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Solve: x + 5 = 12

Solve: x – 6 = 3

Solve:

= 5

Solve and sketch : 2x – 5 =19

Solve 3x2 + 4 = 100.

Expand (x + 2)(x + 6).

Factorise x2 + 7x + 10.

Solve x2

+ 7x + 10 = 0.

Solve and sketch (x – 3)(x + 4)= 0.

Solve 3x2 + 4 = 100.


Rules of adding and subtracting negatives.

Inverse operations can be misapplied.

NOTES

Emphasise good use of notation.

If students are using calculators for the quadratic formula, they can come to rely on

them and miss the fact that some solutions can be left in surd form.

Know when to solve a quadratic equation using the quadratic formula, and when it is

suitable to leave your answer in surd form.

· Sketch a graph of a quadratic function and a linear function, identifying

intersection points;

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2

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Students can leave their answer in fraction form where appropriate.

Clear presentation of working out is essential.

Link with graphical representations.


Resources


Ø Use the form y = mx + c to identify parallel lines A9Ø Find the equation of the line through two given points, or through one point with a

given gradientA10

Ø Identify and interpret gradients and intercepts of linear functions graphically and

algebraicallyA14

Ø Plot and interpret graphs (including reciprocal graphs) and graphs of non-standard

functions in real contexts, to find approximate solutions to problems such as simple

kinematics problems involving distance, speed and acceleration including problems

requiring a graphical solution

A17

Ø Solve linear equations in one unknown algebraically including those with the

unknown on both sides of the equation


Ø Use the form y = mx + c to identify perpendicular lines A9

Ø Plot and interpret exponential graphs A14

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Students need to realise that not all linear equations can be solved by observation or

trial and improvement, and hence the use of a formal method is important.

Reinforce the fact that some problems may produce one inappropriate solution which

can be ignored.

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Resources

recognise, sketch and interpret graphs of linear functions, quadratic functions A12

A14


· Recognise, sketch and interpret graphs of simple cubic functions;

recognise, sketch and interpret graphs of the reciprocal function y = 1/x with x ≠ 0



Select and use the correct mathematical techniques to draw linear, quadratic, cubic and reciprocal graphs.

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recognise, sketch and interpret graphs of exponential functions y = k^x for

positive values of k

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Identify a variety of functions by the shape of the graph.


Students struggle with the concept of solutions and what they represent in concrete terms.

NOTES

Use lots of practical examples to help model the quadratic function, e.g. draw a graph

to model the trajectory of a projectile and predict when/where it will land.

Ensure axes are labelled and pencils used for drawing.

Graphical calculations or appropriate ICT will allow students to see the impact of

changing variables within a function.


Resources

Ø Solve geometrical problems on co-ordinate axes G11

G7

G8


Ø Find the surface area of pyramids and composite solids G17

Ø Calculate surface area of spheres, cones and composite solids

Ø Calculate the volume of spheres, pyramids, cones and composite solids

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ØIdentify, describe and construct congruent and similar shapes, including on co-

ordinate axes, by considering rotation, reflection, translation and enlargement

Ø Describe the changes and invariance achieved by combinations of rotations,

reflections and translations

Ø Calculate arc lengths, angles and areas of sectors of circles G18


G7


Resources

A4,A5,A6,A7

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ØIdentify, describe and construct congruent and similar shapes, including on co-

ordinate axes, by considering rotation, reflection, translation and enlargement (including

fractional and negative scale factors)

simplify and manipulate algebraic expressions [S] (including those involving surds [U]

and algebraic fractions [B]) by:

• collecting like terms;

• multiplying a single term over a bracket;

• taking out common factors; [S]

• expanding products of two [U] or more [B] binomials;

• factorising quadratic expressions of the form x^2 + bx + c, including the difference of

two squares [U]; factorising quadratic expressions of the form ax^2 + bx + c [B]

• simplifying expressions involving sums, products and powers, including the laws of

indices [S]

Ø Understand and use standard mathematical formulae

Ø Rearrange formulae to change the subject

Ø Know the difference between an equation and an identityØ Argue mathematically to show algebraic expressions are equivalent, and use algebra

to support and construct arguments and proofs Ø Where appropriate, interpret simple expressions as functions with inputs and outputs

Ø Interpret the reverse process as the ‘inverse function’

Ø Interpret the succession of two functions as a ‘composite function’



G20

Know the exact values of sin q and cos q for q = 00, 30

0, 45

0, 60

0 and 90

0 G21

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simplify and manipulate algebraic expressions [S] (including those involving surds [U]

and algebraic fractions [B]) by:

• collecting like terms;

• multiplying a single term over a bracket;

• taking out common factors; [S]

• expanding products of two [U] or more [B] binomials;

• factorising quadratic expressions of the form x^2 + bx + c, including the difference of

two squares [U]; factorising quadratic expressions of the form ax^2 + bx + c [B]

• simplifying expressions involving sums, products and powers, including the laws of

indices [S]T

op

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know the formulae for: Pythagoras’ theorem a^2 + b^2 = c^2, and the trigonometric

ratios, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ =

opposite/adjacent; apply them to find angles and lengths in right-angled triangles [U]

and, where possible, general triangles [B] in two- [U] and three- [B] dimensional figures

[U]

Know the exact value of tan q for 00, 30

0, 45

0, 60

0 G21

G6

Compare lengths using ratio notation; make links to trigonometric ratios R12


Resources


· Simplify algebraic fractions; A4

· Multiply and divide algebraic fractions;

Solve quadratic equations arising from algebraic fraction equations;

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Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to

conjecture and derive results about angles and sides, including Pythagoras Theorem,

and use known results to obtain simple proofs

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Resources


R16

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Set up, solve and interpret the answers in growth and decay problems, including

compound interest and depreciation


NOTES

Financial contexts could include percentage or growth rate.


Resources

· Find the equation of a tangent to a circle at a given point, by: A16

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Amounts of money should be rounded to the nearest penny, but emphasise the

importance of not rounding until the end of the calculation if doing in stages.

Set up, solve and interpret the answers in growth and decay problems and work

with general iterative processes

· Work out the multiplier for repeated proportional change as a single decimal

number;

· Represent repeated proportional change using a multiplier raised to a power, use

this to solve problems involving compound interest and depreciation;

Include fractional percentages of amounts with compound interest and encourage use

of single multipliers.

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· finding the gradient of the radius that meets the circle at that point (circles all

centre the origin);

· finding the gradient of the tangent perpendicular to it;

· using the given point;

· Recognise and construct the graph of a circle using x2 + y

2 = r

2 for radius r

centred at the origin of coordinates.


Justify the relationship between the gradient of a tangent and the radius.

Produce an equation of a line given a gradient and a coordinate.


NOTES

Extension Objectives (H) Spec GradeText Books &

Resources

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Find the gradient of a radius of a circle drawn on a coordinate grid and relate this to the

gradient of the tangent.

Students find it difficult working with negative reciprocals of fractions and negative

fractions.

Work with positive gradients of radii initially and review reciprocals prior to starting this

topic.

It is useful to start this topic through visual proofs, working out the gradient of the radius

and the tangent, before discussing the relationship.

· Prove and use the facts that: G10

· the angle in a semicircle is a right angle;

· the perpendicular from the centre of a circle to a chord bisects the chord;

· angles in the same segment are equal;

· alternate segment theorem;

· opposite angles of a cyclic quadrilateral sum to 180°;

· Find and give reasons for missing angles on diagrams using:

· circle theorems;

· isosceles triangles (radius properties) in circles;

· the fact that the angle between a tangent and radius is 90°;

· the fact that tangents from an external point are equal in length.


Justify clearly missing angles on diagrams using the various circle theorems.


Much of the confusion arises from mixing up the diameter and the radius.

NOTES

Reasoning needs to be carefully constructed and correct notation should be used throughout.

Students should label any diagrams clearly, as this will assist them; particular emphasis

should be made on labelling any radii in the first instance.

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· the angle subtended by an arc at the centre of a circle is twice the angle

subtended at any point on the circumference;

· Understand and use the fact that the tangent at any point on a circle is

perpendicular to the radius at that point;


Resources

Ø Solve linear equations in one unknown algebraically including those with the

unknown on both sides of the equation A17

Ø Find approximate solutions using a graph A18Ø Solve quadratic equations (including those that require rearrangement)

algebraically by factorising, by completing the square and by using the quadratic A12

Ø Find approximate solutions using a graph

Ø Recognise, sketch and interpret graphs of linear and quadratic functionsØ Identify and interpret roots, intercepts and turning points of quadratic functions

graphically; deduce roots algebraically and turning points by completing the square A11

Ø Translate simple situations or procedures into algebraic expressions or formulae;

derive an equation and the solve the equation and interpret the solution A21

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ResourcesExtension Objectives (H)

G22

Know and apply Area = G23

ab sin C to calculate the area, sides or angles of any triangle.

· Use the sine and cosine rules to solve 3D problems.


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


Not using the correct rule, or attempting to use ‘normal trig’ in non-right-angled triangles.

NOTES

Ensure that finding angles with ‘normal trig’ is refreshed prior to this topic.

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The cosine rule is used when we have SAS and used to find the side opposite the

‘included’ angle or when we have SSS to find an angle.

Students may find it useful to be reminded of simple geometrical facts, i.e. the shortest

side is always opposite the shortest angle in a triangle.

The sine and cosine rules and general formula for the area of a triangle are not given

on the formulae sheet.

· Know the sine and cosine rules, and use to solve 2D problems (including

involving bearings).

Justify when to use the cosine rule, sine rule, Pythagoras’ Theorem or normal

trigonometric ratios to solve problems.

When finding angles students will be unable to rearrange the cosine rule or fail to find

the inverse of cos θ .

1

2


Resources· Recognise and interpret graphs showing direct and indirect proportion. R10

R13

R14



Know the symbol for ‘is proportional to’.


Direct and inverse proportion can get mixed up.

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In multi-step questions emphasise the importance of not rounding prematurely and

using exact values where appropriate.

Whilst 3D coordinates are not included in the programme of study, they provide a visual

introduction to trigonometry in 3D.

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· Identify direct proportion from a table of values, by comparing ratios of values, for

x squared and x cubed relationships.

· Write statements of proportionality for quantities proportional to the square, cube

or other power of another quantity.

· Set up and use equations to solve word and other problems involving direct

proportion involving square and cubic proportionality.

· Use y = kx to solve direct proportion problems, including questions where

students find k , and then use k to find another value.

· Solve problems involving inverse proportion using graphs by plotting and reading

values from graphs.

· Solve problems involving inverse proportionality, including problems where y is

inversely proportional to the square of x .

· Set up and use equations to solve word and other problems involving direct

proportion or inverse proportion.

Understand that when two quantities are in direct proportion, the ratio between them

remains constant.

NOTES


Resources

· Show inequalities on number lines;

· Write down whole number values that satisfy an inequality; A22

· Solve simple linear inequalities in one variable, and represent the solution set on a number line;

· Solve two linear inequalities in x , find the solution sets and compare them to see

which value of x satisfies both solve linear inequalities in two variables algebraically;

Represent the solution set for inequalities using set notation, i.e. curly brackets and ‘is an element of’ notation;· for problems identifying the solutions to two different inequalities, show this as

the intersection of the two solution sets, i.e. solution of x ² – 3x – 10 < 0 as {x : –3 < x <

5};· Use the correct notation to show inclusive and exclusive inequalities.



Be able to state the solution set of x ² – 3x – 10 < 0 as

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Consider using science contexts for problems involving inverse proportionality, e.g.

volume of gas inversely proportional to the pressure or frequency is inversely

proportional to wavelength.

· Solve quadratic inequalities in one variable, by factorising and sketching

the graph to find critical values;

{x: x < -3} {x : x > 5}.


NOTES

Link to units 2 and 9a, where quadratics and simultaneous equations were solved.

Students can leave their answers in fractional form where appropriate.

Set notation is a new topic.



G25

· Find the length of a vector using Pythagoras’ Theorem.

· Calculate the resultant of two vectors.

· Solve geometric problems in 2D where vectors are divided in a given ratio.



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When solving inequalities students often state their final answer as a number quantity,

and exclude the inequality or change it to =.

Emphasise the importance of leaving their answer as an inequality (and not changing it

to =).

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· Produced geometrical proofs to prove points are collinear and vectors/lines are

parallel.

· Understand that 2a is parallel to a and twice its length, and that a is parallel to –a

in the opposite direction.

· Represent vectors, combinations of vectors and scalar multiples in the plane

pictorially.

· Calculate the sum of two vectors, the difference of two vectors and a scalar

multiple of a vector using column vectors (including algebraic terms).

· Understand and use vector notation, including column notation, and understand

and interpret vectors as displacement in the plane with an associated direction.

Add and subtract vectors algebraically and use column vectors.

Solve geometric problems and produce proofs.


NOTES


ResourcesSupport Objectives (F)

recognise, sketch and interpret graphs of linear functions, quadratic

Core Objectives (F/H) A12

recognise, sketch and interpret graphs of simple cubic functions, the

reciprocal function y = 1/x with x ≠ 0


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recognise, sketch and interpret graphs of exponential functions y =

k^x for positive values of k, and the trigonometric functions (with

arguments in degrees) y = sin x, y = cos x and y = tan x for angles

Students find it difficult to understand that parallel vectors are equal as they are in

different locations in the plane.

Students find manipulation of column vectors relatively easy compared to pictorial and

algebraic manipulation methods – encourage them to draw any vectors they calculate

on the picture.

Geometry of a hexagon provides a good source of parallel, reverse and multiples of

vectors.

3D vectors or i, j and k notation can be introduced and further extension work can be

found in GCE Mechanics 1 textbooks.

Remind students to underline vectors or use an arrow above them, or they will be

regarded as just lengths.

Extend geometric proofs by showing that the medians of a triangle intersect at a single

point.


Resources

Extension Objectives (H) A13

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· Interpret and analyse transformations of graphs of functions

and write the functions algebraically, e.g. write the equation of f(x )

+ a , or f(x – a ):

· apply to the graph of y = f(x ) the transformations y = –f(x ),

y = f(–x ), y = –f(–x ) for linear, quadratic, cubic functions;

apply to the graph of y = f(x ) the transformations y = f(x ) + a , y = f(ax ), y = f(x + a ),

y = a f(x ) for linear, quadratic, cubic functions;


The effects of transforming functions is often confused.


ResourcesExtension Objectives (H)

· Use iteration with simple converging sequences. A20

find approximate solutions to equations numerically using iteration

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· apply to the graph of y = f(x ) the transformations y = –f(x ),

y = f(–x ), y = –f(–x ) for linear, quadratic, cubic functions;

NOTES

The extent of algebraic iteration required needs to be confirmed.


Resources


interpret the gradient of a straight line graph as a rate of change;

recognise and interpret graphs that illustrate direct and inverse

proportion [U]

R14


R15

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· Interpret the gradient of linear or non-linear graphs, and

estimate the gradient of a quadratic or non-linear graph at a given

point by sketching the tangent and finding its gradient;

· for a non-linear distance–time graph, estimate the speed at

one point in time, from the tangent, and the average speed over

several seconds by finding the gradient of the chord;

· Interpret the gradient of non-linear graph in curved

distance–time and velocity–time graphs:

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· Interpret the gradient of a linear or non-linear graph in financial contexts;

· Interpret the rate of change of graphs of containers filling and emptying;

· Interpret the rate of change of unit price in price graphs.

NOTES


ResourcesSupport Objectives (F/H)

· Estimate area under a quadratic graph by dividing it into trapezia;A15

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· for a non-linear distance–time graph, estimate the speed at

one point in time, from the tangent, and the average speed over

several seconds by finding the gradient of the chord;

When interpreting rates of change of unit price in price graphs, a steeper graph means

larger unit price.

· for a non-linear velocity–time graph, estimate the acceleration

at one point in time, from the tangent, and the average acceleration

over several seconds by finding the gradient of the chord;

When interpreting rates of change with graphs of containers filling and emptying, a

steeper gradient means a faster rate of change.


Explain why you cannot find the area under a reciprocal or tan graph.

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Emphasise the importance of being able to convert between decimals and percentages and the use of decimal multipliers to make calculations easier.

Students need to realise that not all linear equations can be solved by observation or trial and improvement, and hence the use of a formal method is important.

Students should be encouraged to use their calculator effectively by using the replay and ANS/EXE functions; reinforce the use of brackets and only rounding their final answer with trial and improvement.

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