text books & support objectives spec grade resources · be able to add and subtract negative...
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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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Possible Success Criteria
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.
Common Misconceptions
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
Possible Success Criteria
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.
Common Misconceptions
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
Possible Success Criteria
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
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Expand and simplify 3(t – 1) + 57.
Factorise 15x2y – 35x
2y
2.
Common Misconceptions
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
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Be able to apply the properties of: G3
- angles at a point
- angles at a point on a straight line
- vertically opposite angles
Possible Success Criteria
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
Support Objectives (F)
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
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 find missing angles in a triangle using the angle sum in a triangle AND the
properties of an isosceles triangle; 4
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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;
Possible Success Criteria
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.
Common Misconceptions
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.
Amounts of money should always be rounded to the nearest penny.
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)
Support Objectives (F)
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
Core Objectives (F/H)
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?
Common Misconceptions
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.
Amounts of money should always be rounded to the nearest penny.
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
Support Objectives (F)
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
Core Objectives (F/H)
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
Possible Success Criteria
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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Common Misconceptions
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.
Amounts of money should always be rounded to the nearest penny.
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
Support Objectives (F)
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
Core Objectives (F/H)
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.
Support Objectives (F)
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
Core Objectives (F/H)
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
POSSIBLE SUCCESS CRITERIA
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.
COMMON MISCONCEPTIONS
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
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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
Core Objectives (F/H)
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.
COMMON MISCONCEPTIONS
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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
POSSIBLE SUCCESS CRITERIA
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.
Support Objectives (F)
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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Find surface area using rectangles and triangles; G16
Estimating by rounding measurements to 1 significant figure to check
reasonableness of answersN14
Extension Objectives (H)
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
Possible Success Criteria
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
Common Misconceptions
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.
Support Objectives (F) Spec GradeText Books &
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
Core Objectives (F/H) Collins F2 Ch 13
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
Extension Objectives (H)
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
Common Misconceptions
Diameter and radius are often confused, and recollection of area and circumference of
circles involves incorrect radius or diameter.
Focus on Reasoning (AO2)
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.
Focus on Problem Solving (AO3)
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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Support Objectives (F)
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
Core Objectives (F/H)
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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Extension Objectives (H)
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
Support Objectives (F)
Know which charts to use for different types of data sets; S2
Be able to produce and interpret composite bar charts; S2
Core Objectives (F/H)
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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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
Extension Objectives (H)
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
POSSIBLE SUCCESS CRITERIA
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;
Common Misconceptions
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.
Support Objectives (F)
Know how to describe in words a term-to-term sequence and identify which
terms cannot be in a sequence;A23
Core Objectives (F/H)
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
POSSIBLE SUCCESS CRITERIA
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.
Common Misconceptions
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.
Focus on Problem-solving
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.
Support Objectives (F) Spec GradeText Books &
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
Extension Objectives (H)
N5
Possible Success Criteria
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.
Support Objectives (F) SpecText Books &
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
Extension Objectives (H)
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
Possible Success Criteria
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
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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.
Focus on Reasoning (AO2)
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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
Support Objectives (F) SpecText Books &
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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Support Objectives (F) Spec GradeText Books &
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
Core Objectives (F/H) Collins F1 Ch 9
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
Extension Objectives (H)
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
Possible Success Criteria
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.
Focus on Problem Solving (AO3)
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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Support Objectives (F) Spec GradeText Books &
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
Extension Objectives (H)
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
Possible Success Criteria
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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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.
Support Objectives (F) Spec GradeText Books &
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
POSSIBLE SUCCESS CRITERIA
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
COMMON MISCONCEPTIONS
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
Focus on Problem-solving
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
POSSIBLE SUCCESS CRITERIA
Does 2, 3, 6 give a right-angled triangle?
Justify when to use Pythagoras’ Theorem and when to use trigonometry.
COMMON MISCONCEPTIONS
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.
Support Objectives (F) Spec GradeText Books &
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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POSSIBLE SUCCESS CRITERIA
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.
COMMON MISCONCEPTIONS
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.
Support Objectives (F) Spec GradeText Books &
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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Ave
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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
POSSIBLE SUCCESS CRITERIA
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
COMMON MISCONCEPTIONS
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.
Core Objectives (F/H) Spec GradeText Books &
ResourcesA22
A22
A22
Extension Objectives (H)
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
Support Objectives Spec GradeText Books &
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;
Support Objectives Spec GradeText Books &
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
Support Objectives Spec GradeText Books &
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;
Support Objectives Spec GradeText Books &
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 the proof that the angle sum of a triangle is 180°, and derive and use the
sum of angles in a triangle;
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;
Support Objectives (F) Spec GradeText Books &
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;
Core Objectives (F/H)
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.
Support Objectives (F) Spec GradeText Books &
ResourcesG6, G20
G21,R12
Use the trigonometric ratios to solve 2D problems;
Be able to find angles of elevation and depression;
Core Objectives (F/H)
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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Support Objectives (F) Spec GradeText Books &
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;
Extension Objectives (H)
· 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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2n1
3n
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Support Objectives (F) Spec GradeText Books &
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.
Core Objectives (F/H)
· 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;
POSSIBLE SUCCESS CRITERIA
Solve 3x2 + 4 = 100.
Expand (x + 2)(x + 6).
Factorise x2 + 7x + 10.
Solve x2
+ 7x + 10 = 0.
Solve (x – 3)(x + 4)= 0.
COMMON MISCONCEPTIONS
x terms can sometimes be ‘collected’ with x2.
Intr
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Support Objectives (F) Spec GradeText Books &
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;
POSSIBLE SUCCESS CRITERIA
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
Core Objectives (F/H)
· Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2; N8
A24
POSSIBLE SUCCESS CRITERIA
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)
Support Objectives (F) Spec GradeText Books &
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.
Spec GradeText Books &
Resources
Extension Objectives (H)
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Understand the difference between rational and irrational numbers N8
· Rationalise the denominator involving surds;
POSSIBLE SUCCESS CRITERIA
Rationalise:
simplify (√18 + 10) +√2.
Explain the difference between rational and irrational numbers.
Support Objectives (F) Spec GradeText Books &
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;
POSSIBLE SUCCESS CRITERIA
Understand similarity as one shape being an enlargement of the other.
Understand that enlargement does not have the same effect on area and volume.
COMMON MISCONCEPTIONS
Students commonly use the same scale factor for length, area and volume.
Support Objectives (F) Spec GradeText Books &
Resources
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Co
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imil
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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;
To
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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.
POSSIBLE SUCCESS CRITERIA
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Pro
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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 )
COMMON MISCONCEPTIONS
Not using fractions or decimals when working with probability trees.
Support Objectives (F) Spec GradeText Books &
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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Arc
s &
Se
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POSSIBLE SUCCESS CRITERIA
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.
COMMON MISCONCEPTIONS
Diameter and radius are often confused and recollection which formula to use for area
and circumference of circles is often poor.
Spec GradeText Books &
Resources
N15
N16
To
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Arc
s &
Se
cto
rs
4 L
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ns
To
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Up
pe
r &
Lo
we
r B
ou
nd
s
4 L
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ns
· 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.
POSSIBLE SUCCESS CRITERIA
Round 16,000 people to the nearest 1000.
Round 1100 g to 1 significant figure.
COMMON MISCONCEPTIONS
NOTES
Encourage use a number line when introducing the concept.
Support Objectives (F) Spec GradeText Books &
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.
Core Objectives (F/H)
· Find the volume of a cylinder;
· Recall and use the formula for volume of pyramid;
· Use the formulae for volume of spheres and cones;
To
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44
Up
pe
r &
Lo
we
r B
ou
nd
s
4 L
es
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ns
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.
To
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45
Vo
lum
e
8 L
es
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ns
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.
POSSIBLE SUCCESS CRITERIA
Justify whether a certain number of small boxes fit inside a larger box.
Calculate the volume of a triangular prism with correct units.
Understand that answers in terms of pi are more accurate.
COMMON MISCONCEPTIONS
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)
To
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Vo
lum
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8 L
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ns
Diameter and radius are often confused, and recollection of area and circumference of
circles involves incorrect radius or diameter.
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?
Spec GradeText Books &
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;
POSSIBLE SUCCESS CRITERIA
Construct cumulative frequency graphs and box plots from frequency tables.
COMMON MISCONCEPTIONS
NOTES
Ensure that axes are clearly labelled.
As an extension, use the formula for identifying an outlier, (i.e. if data point is below
To
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Cu
mu
lati
ve
Fre
qu
en
cy
an
d B
ox
Plo
ts
4 L
es
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ns
· 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.
Spec GradeText Books &
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;
POSSIBLE SUCCESS CRITERIA
Construct histograms from frequency tables.
COMMON MISCONCEPTIONS
To
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mu
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ve
Fre
qu
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Plo
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To
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His
tog
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s
4 L
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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’.
Support Objectives (F) Spec GradeText Books &
Resources· Find the volume and surface area of a cylinder;
G17
Core Objectives (F/H)
· 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.
POSSIBLE SUCCESS CRITERIA
Understand that answers in terms of pi are more accurate.
COMMON MISCONCEPTIONS
To
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His
tog
ram
s
4 L
es
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ns
To
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48
Co
ne
s &
Sp
here
s
4 L
es
so
ns
· 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;
Diameter and radius are often confused, and recollection of area and circumference of
circles involves incorrect radius or diameter.
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?
Support Objectives (F) Spec GradeText Books &
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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Rati
o &
Pro
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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;
Support Objectives (F) Spec GradeText Books &
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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Rati
o &
Pro
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n R
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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;
To
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POSSIBLE SUCCESS CRITERIA
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.
COMMON MISCONCEPTIONS
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;
To
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eir
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ns
2
x
Students can leave their answer in fraction form where appropriate.
Clear presentation of working out is essential.
Link with graphical representations.
Support Objectives (F) Spec GradeText Books &
Resources
Core Objectives (F/H)
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
Extension Objectives (H)
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.
To
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Support Objectives (F) Spec GradeText Books &
Resources
recognise, sketch and interpret graphs of linear functions, quadratic functions A12
A14
Core Objectives (F/H)
· Recognise, sketch and interpret graphs of simple cubic functions;
recognise, sketch and interpret graphs of the reciprocal function y = 1/x with x ≠ 0
Extension Objectives (H)
POSSIBLE SUCCESS CRITERIA
Select and use the correct mathematical techniques to draw linear, quadratic, cubic and reciprocal graphs.
To
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recognise, sketch and interpret graphs of exponential functions y = k^x for
positive values of k
To
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Sk
etc
hin
g G
rap
hs
4 L
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ns
Identify a variety of functions by the shape of the graph.
COMMON MISCONCEPTIONS
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.
Support Objectives (F) Spec GradeText Books &
Resources
Ø Solve geometrical problems on co-ordinate axes G11
G7
G8
Core Objectives (F/H)
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
To
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etc
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Ge
om
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ap
& r
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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
Extension Objectives (H)
G7
Support Objectives (F) Spec GradeText Books &
Resources
A4,A5,A6,A7
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To
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Fu
rth
er
Qu
ad
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& R
ea
rra
ng
ing
Fo
rmu
lae
& I
de
nti
tie
s
8 L
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ns
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’
Spec GradeText Books &
ResourcesCore Objectives (F/H)
G20
Know the exact values of sin q and cos q for q = 00, 30
0, 45
0, 60
0 and 90
0 G21
To
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Fu
rth
er
Qu
ad
rati
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& R
ea
rra
ng
ing
Fo
rmu
lae
& I
de
nti
tie
s
8 L
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ns
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
ic 5
5
Tri
go
no
me
try
re
ca
p &
ex
ten
sio
n
8 L
es
so
ns
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
Spec GradeText Books &
Resources
Extension Objectives (H)
· Simplify algebraic fractions; A4
· Multiply and divide algebraic fractions;
Solve quadratic equations arising from algebraic fraction equations;
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Tri
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ap
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xte
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ion
8 L
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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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Spec GradeText Books &
Resources
Core Objectives (F/H)
R16
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Fra
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Gro
wth
& D
ec
ay
4 L
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Set up, solve and interpret the answers in growth and decay problems, including
compound interest and depreciation
Extension Objectives (H)
NOTES
Financial contexts could include percentage or growth rate.
Core Objectives (F/H) Spec GradeText Books &
Resources
· Find the equation of a tangent to a circle at a given point, by: A16
To
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Gro
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& D
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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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irc
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ns
· 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.
POSSIBLE SUCCESS CRITERIA
Justify the relationship between the gradient of a tangent and the radius.
Produce an equation of a line given a gradient and a coordinate.
COMMON MISCONCEPTIONS
NOTES
Extension Objectives (H) Spec GradeText Books &
Resources
4 L
es
so
ns
To
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Eq
uati
on
of
a C
irc
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To
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Cir
cle
Th
eo
rem
s
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.
POSSIBLE SUCCESS CRITERIA
Justify clearly missing angles on diagrams using the various circle theorems.
COMMON MISCONCEPTIONS
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.
4 L
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Cir
cle
Th
eo
rem
s
· 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;
Core Objectives (F/H) Spec GradeText Books &
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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Cir
cle
Th
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s
Spec GradeText Books &
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.
POSSIBLE SUCCESS CRITERIA
Find the area of a segment of a circle given the radius and length of the chord.
COMMON MISCONCEPTIONS
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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Sin
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sin
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ule
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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
Support Objectives (F) Spec GradeText Books &
Resources· Recognise and interpret graphs showing direct and indirect proportion. R10
R13
R14
Core Objectives (F/H)
POSSIBLE SUCCESS CRITERIA
Know the symbol for ‘is proportional to’.
COMMON MISCONCEPTIONS
Direct and inverse proportion can get mixed up.
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Sin
e &
Co
sin
e R
ule
4 L
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ns
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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Dir
ec
t &
In
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Pro
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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
Support Objectives (F) Spec GradeText Books &
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.
Extension Objectives (H)
POSSIBLE SUCCESS CRITERIA
Be able to state the solution set of x ² – 3x – 10 < 0 as
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Ine
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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}.
COMMON MISCONCEPTIONS
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.
Spec GradeText Books &
ResourcesCore Objectives (F/H)
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.
Extension Objectives (H)
POSSIBLE SUCCESS CRITERIA
To
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Ine
qu
ali
tie
s
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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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Ve
cto
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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.
COMMON MISCONCEPTIONS
NOTES
Spec GradeText Books &
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
Extension Objectives (H)
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cto
rs
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Sk
etc
hin
g G
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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.
Spec GradeText Books &
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;
COMMON MISCONCEPTIONS
The effects of transforming functions is often confused.
Spec GradeText Books &
ResourcesExtension Objectives (H)
· Use iteration with simple converging sequences. A20
find approximate solutions to equations numerically using iteration
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Tra
ns
form
ing
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Nu
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Me
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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.
Spec GradeText Books &
Resources
Core Objectives (F/H)
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
Extension Objectives (H)
R15
To
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die
nt
& R
ate
of
Ch
an
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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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Nu
me
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Me
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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
Spec GradeText Books &
ResourcesSupport Objectives (F/H)
· Estimate area under a quadratic graph by dividing it into trapezia;A15
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r a
Cu
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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.
POSSIBLE SUCCESS CRITERIA
Explain why you cannot find the area under a reciprocal or tan graph.
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& A
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un
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r a
Cu
rve
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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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