CCEA GCSE Mathematics · 2019-09-26 · CCEA GCSE Mathematics Planning Framework for Templates for...

CCEA GCSE

MathematicsPlanning Framework for Templates for M1, M2, M3, M4 Units

GCSE

For first teaching from September 2017

2

Planning Framework templates for GCSE Mathematics Subject Content and the Statutory Skills and Personal Capabilities at Key Stage 4 ALL TOPICS

Accuracy and bounds Exact calculations Fractions and decimals Growth and decay Indices, Powers and Roots

Multiples and factors Number Systems Working with decimals Working with equivalences Estimations and approximations

Working with fractions Working with money Working with numbers Working with percentages Working with place value

Working with ratio Algebraic fractions Co-ordinate Geometry Equations Expressions

Expressions and formulae Graphs and gradients Indices Inequalities Sequences

The language of algebra Using graphs Working with graphs Angle properties Circle Theorems

Compound measures and units

Congruence Constructions Drawings Enlargements Mensuration problems

Perimeter, area and volume Reflections Pythagoras’ Theorem Basic Trigonometry Rotations

Shape properties Similarity Transformations Translations Trigonometry

Working with 2D shapes Working with 3D shapes Working with measures Working with scale drawings Box Plots

Cumulative frequency graphs Data collection Data interpretation Histograms Mean, median, mode,

range

Sampling Scatter graphs The handling data cycle Using statistical diagrams

Counting and listing outcomes

Experimental probability Probability and chance Probability rules Probability problems Probability tree diagrams

3

Planning Framework templates for GCSE Mathematics Subject Content and the Statutory Skills and Personal Capabilities at Key Stage 4 M1


Multiples and factors Number Systems Working with decimals Working with equivalences

Estimations and approximations




The language of algebra

Using graphs Working with graphs Angle properties CIRCLE THEOREMS



Perimeter, area and volume

Reflections Right angled triangles Pythagoras’ Theorem

Right angled triangles Basic Trigonometry

Rotations



Cumulative frequency graphs Data collection Data interpretation HISTOGRAMS Mean, median, mode,

range

Sampling Scatter graphs The handling data cycle Using statistical

diagrams Counting and listing outcomes


4

Planning Framework templates for GCSE Mathematics Subject Content and the Statutory Skills and Personal Capabilities at Key Stage 4 M1M2

Accuracy and bounds Exact calculations Fractions and decimals +Growth and decay Indices, Powers and Roots











Reflections +Right angled triangles Pythagoras’ Theorem

Right angled triangles Basic Trigonometry

Rotations



Cumulative frequency graphs Data collection Data interpretation HISTOGRAMS Mean, median, mode,

range




5

Planning Framework templates for GCSE Mathematics Subject Content and the Statutory Skills and Personal Capabilities at Key Stage 4 M1M2M3





Working with ratio +Algebraic fractions Co-ordinate Geometry Equations Expressions





Congruence Constructions Drawings Enlargements +Mensuration

problems


Reflections Right angled triangles

Pythagoras’ Theorem +Right angled triangles Basic Trigonometry

Rotations


Working with 2D shapes Working with 3D shapes Working with measures Working with scale drawings +Box Plots

+Cumulative frequency graphs Data collection Data interpretation HISTOGRAMS Mean, median, mode,

range




6

Planning Framework templates for GCSE Mathematics Subject Content and the Statutory Skills and Personal Capabilities at Key Stage M1M2M3M4

ACCURACY AND BOUNDS Exact calculations Fractions and decimals Growth and decay Indices, Powers and Roots




Working with ratio ALGEBRAIC FRACTIONS CO-ORDINATE GEOMETRY EQUATIONS EXPRESSIONS



Using graphs Working with graphs Angle properties

+CIRCLE THEOREMS Compound measures

and units

Congruence Constructions Drawings Enlargements MENSURATION PROBLEMS


Reflections Right angled triangles

Pythagoras’ Theorem Right angled triangles Basic Trigonometry

Rotations



Cumulative frequency graphs Data collection Data interpretation +HISTOGRAMS Mean, median, mode,

range

SAMPLING Scatter graphs The handling data cycle Using statistical



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Unit/option content

NUMBER

Learning outcomes or Elaboration of content

Suggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal Capabilities

ACCURACY AND BOUNDS

M1

M2

M3

M4

M5

M6

M7

M8

M1 – M4 Topic note also M5 – M8

M1 (T5)

round to a specified or appropriate degree of accuracy, number of decimal places, or 1 significant figure, including a given power of 10

M2 (T5)

round to a specified or appropriate number of significant figures

M3 (T3)

calculate the upper and lower bounds in calculations involving addition and multiplication of numbers expressed to a given degree of accuracy

M4 (T3)

calculate the upper and lower bounds in calculations involving subtraction and division of numbers expressed to a given degree of accuracy

1 Set of cards Pupils work in pairs to sort cards into number written to 1, 2 or 3 decimal places. CARDS with: 4.5, 4.52, 4.6, 4.713, 4.25, 16.4 1.27, 2.4, 0.16, 3.20, 0.024, 7.1

2 Set of cards with 4 numbers on each Work in pairs to choose the one number on each card that is written to 1 significant figure. CARDS with:

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Resources Foundation: Pages 89–95

6, 6.0, 0.61, 6.00 8.0, 0.8, 0.80, 80.0

400.0, 0.400, 400, 40.0 0.02, 0.020, 0.20, 0.0200

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Unit/option content

NUMBER



ACCURACY AND BOUNDS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)


M2 (T5)


M3 (T3)


M4 (T3)


[Students should know the content of Unit M1 before taking Unit M2.]

1 Set of cards Pupils work in pairs to sort cards into 1, 2 or 3 significant figures. CARDS with: 20, 0.6, 3.4, 0.06, 1.70, 0.9 420, 6000, 0.24, 1.35, 2.04, 0.46

2 Pupils work in pairs. Each pair given table from Q7 pg 91. They have to choose how best to round the attendances (1 or 2 or 3 significant figures). • round these accordingly • explain their reasoning

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Unit/option content

NUMBER



ACCURACY AND BOUNDS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)


M2 (T5)


M3 (T3)


M4 (T3)


[Students should know the content of Units M1 and M2 before taking Unit M3.]

1 Pupils work in pairs ordering cards into 3 groups (number that round to 48, 49 or 50) and then decide the smallest possible number and biggest possible number (to 1 decimal place) that would round to 48, 49 or 50 explaining their answer. CARDS with: 47.8, 47.51, 48.2, 48.49, 48.76, 49.3, 50.14, 50.36, 50.4999, 47.50001

2 Pupils work in pairs, one calculating the value of the minimum and maximum sum, the other the minimum and maximum product of 2 numbers each time. They then check each other’s work. CARDS with:

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Resources Higher: Pages 100–105 (only adding/multiplication questions)

18 and 24 correct to nearest integer

2.14 and 3.56 correct to 2 decimal places

6.4 and 8.2 correct to 1 decimal place

6.4 and 12 correct to 2 significant figures

10

Unit/option content

NUMBER



ACCURACY AND BOUNDS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)


M2 (T5)


M3 (T3)


M4 (T3)


[Students should know the content of Units M1, M2 and M3 before taking Unit M4.]

1 Pupils work in pairs finding the least and greatest possible answers to subtraction/division questions on cards, discussing with each other how they worked these out. CARDS with:

2 Pupils work in pairs, one calculating the value of the minimum and maximum difference, the other the minimum and maximum quotient of 2 numbers each time. They then check each other’s work. CARDS with:

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Resources Higher: Pages 100–105 (only subtraction/division questions)

10 and 7 correct to nearest integer


30 and 20 correct to nearest tens


2.14 and 1.56 correct to 2 decimal places

6.4 and 3.7 correct to 2 significant figures

0.9 and 0.4 correct to 1 significant figure

11

Unit/option content

NUMBER



FRACTIONS AND DECIMALS

M1

M2

M3

M4

M5

M6

M7

M8

M1 - M4 Topic and M5 - M8 Topic

M1 (T1)

write a simple fraction as a terminating decimal

M2 (T2)

recognise that recurring decimals are exact fractions and that some exact fractions are recurring decimals

M8 (T6)

distinguish between rational and irrational numbers

M8 (T6)

change a recurring decimal to a fraction

Provide learners with a ‘simple fractions into terminating decimals’ table. Ask learners to extract, explain and use information from the table. For example: write 1

8 as a decimal, why is 1

8 =0.125? write 5

8 as a decimal.

Fraction (simplest form)

Fraction (denominator is a power of ten)

Decimal

𝟏𝟏𝟐𝟐

𝟓𝟓𝟏𝟏𝟏𝟏 0.5

𝟏𝟏𝟒𝟒

𝟐𝟐𝟓𝟓𝟏𝟏𝟏𝟏𝟏𝟏 0.25

𝟏𝟏𝟓𝟓

𝟐𝟐𝟏𝟏𝟏𝟏 0.2

𝟏𝟏𝟖𝟖

𝟏𝟏𝟐𝟐𝟓𝟓𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 0.125

𝟏𝟏𝟏𝟏𝟏𝟏

𝟏𝟏𝟏𝟏𝟏𝟏 0.1

𝟏𝟏𝟐𝟐𝟏𝟏

𝟓𝟓𝟏𝟏𝟏𝟏𝟏𝟏 0.05

𝟏𝟏𝟐𝟐𝟓𝟓

𝟒𝟒𝟏𝟏𝟏𝟏𝟏𝟏 0.04

𝟏𝟏𝟒𝟒𝟏𝟏

𝟐𝟐𝟓𝟓𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 0.025

𝟏𝟏𝟓𝟓𝟏𝟏

𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏 0.02

𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 0.01

Ask learners to use their calculators to check the information in the table.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 2 Working with decimals Foundation CCEA GCSE Mathematics Second Edition Chapter 7 Fractions 3

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Unit/option content

NUMBER



FRACTIONS AND DECIMALS

M1

M2

M3

M4

M5

M6

M7

M8

M1 - M4 Topic and M5 - M8 Topic

M1 (T1)

write a simple fraction as a terminating decimal

M2 (T2)

recognise that recurring decimals are exact fractions and that some exact fractions are recurring decimals

M8 (T6)

distinguish between rational and irrational numbers

M8 (T6)

change a recurring decimal to a fraction

[Students should know the content of Unit M1 before taking Unit M2.] Provide learners with a ‘Fractions into recurring decimals’ table. Direct learners to noticing that the recurring decimals are all exact fractions. Ask learners if exact fractions are always recurring decimals. Discuss responses.

Fraction (simplest form)

Fraction Decimal

𝟏𝟏𝟑𝟑

𝟑𝟑𝟗𝟗

0.333333333333333

𝟏𝟏𝟔𝟔

𝟏𝟏𝟓𝟓𝟗𝟗𝟏𝟏

0.166666666666666

𝟏𝟏𝟕𝟕 𝟏𝟏𝟒𝟒𝟐𝟐𝟖𝟖𝟓𝟓𝟕𝟕

𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 0.142857142857142

𝟏𝟏𝟗𝟗 𝟏𝟏

𝟗𝟗 0.111111111111111

𝟏𝟏𝟏𝟏𝟏𝟏

𝟗𝟗𝟗𝟗𝟗𝟗

0.090909090909091

𝟏𝟏𝟏𝟏𝟐𝟐

𝟕𝟕𝟓𝟓𝟗𝟗𝟏𝟏𝟏𝟏

0.083333333333333

Ask learners to use their calculators to check the information in the table. Ask learners to extract, explain and use information from the table. For example: write 1

3 as a decimal, why is 1

3 ≈0.333? write 2

3 as a decimal.

Discuss any observations and assumptions.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 2 Working with decimals Foundation CCEA GCSE Mathematics Second Edition Chapter 7 Fractions 3

13

Unit/option content

NUMBER



GROWTH AND DECAY

M1 X

M2

M3

M4

M5 X

M6

M7

M8

M2 – M4 Topic and M6 – M8 Topic

M2 (T3)

use percentage and repeated proportional change

M3 (T3)

find the original quantity, given the result of a proportional change

M8 (T6)

set up, solve and interpret the answers in growth and decay problems, for example use the formula for compound interest


Demonstrate the solution of several growth (compound appreciation) problems by using a multiplier. In each solution model how to calculate the multiplier.

Demonstrate the solution of several decay (compound depreciation) problems by using a multiplier. In each solution model how to calculate the multiplier.

Present learners with ‘True or False?’ scenarios for example, when £220 is invested with an AER of 5% the multiplier is 1.05, true or false?

A proportional increase or decrease relationship could be presented in a multiplication triangle. The example below shows when £220 is grown by a factor of 1.05 (multiplier for a percentage increase of 5%), the result is £231.

£231

1.05 £220

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 13 Percentages 3

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Unit/option content

NUMBER



GROWTH AND DECAY

M1 X

M2

M3

M4

M5 X

M6

M7

M8


M2 (T3)

use percentage and repeated proportional change

M3 (T3)

find the original quantity, given the result of a proportional change

M8 (T6)

set up, solve and interpret the answers in growth and decay problems, for example use the formula for compound interest

[Students should know the content of Units M1 and M2 before taking Unit M3]

Present learners with a multiplication triangle that shows a percentage increase calculation. The example below shows that when an initial amount of £220 is increased by 5% the final amount is £231. ‘initial amount x multiplier = final amount’ The example also shows that ‘final amount/multiplier = initial amount’

£231

1.05 £220

Provide learners with several examples so that they can practice modelling a percentage increase by multiplying by a multiplier and a reverse percentage increase calculation by dividing by a multiplier.

Discuss alternative methods for solving this type of problem; for example, solve the problem using bar modelling.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 10 Percentages and finance

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Unit/option content

NUMBER



INDICES, POWERS AND ROOTS

M1

M2

M3

M4

M5

M6

M7

M8

M1 – M4 TOPIC and M5 – M8 TOPIC

M1 (T1)

use index notation for squares, cubes and powers of ten

M1 (T1)

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

M2 (T2)

use index notation and index laws for positive, whole number powers

M7 (T4)

use index notation and index laws for zero, positive and negative powers

M8 (T4)

use index notation and index laws for integer, fractional and negative powers

1 Pupils work in pairs linking cards showing length of the side of a square to the area of a square. CARDS with: 4 cm linked to 16 cm2, 8 cm linked to 64 cm2 9 cm linked to 81 cm2, 20 cm linked to 400 cm2

5 cm linked to 25 cm2, 7 cm linked to 49 cm2


2 As 1 but linking sides of cubes to volumes.

CARDS with: 2 cm linked to 8 cm3, 3 cm linked to 27 cm3 4 cm linked to 64 cm3, 5 cm linked to 125 cm3


3 Pupils work in pairs linking cards together.

CARDS with:

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√4 linked to 2 and -2

82 linked to 43 and 64 32 → 9 → √81

√25 → 5 → −5 1 → 12 → 13 √10003 → √100 → 10

122 linked to 144

16

Unit/option content

NUMBER



INDICES, POWERS AND ROOTS

M1

M2

M3

M4

M5

M6

M7

M8

M1 – M4 TOPIC and M5 – M8 TOPIC

M1 (T1)

use index notation for squares, cubes and powers of ten

M1 (T1)

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

M2 (T2)

use index notation and index laws for positive, whole number powers

M7 (T4)

use index notation and index laws for zero, positive and negative powers

M8 (T4)

use index notation and index laws for integer, fractional and negative powers


1 Pupils in pairs match up cards and then try to explain a quick way to multiply or divide numbers in index notation. →→ → Also include these cards

2 Pupils individually match up as many cards as possible. Also include these cards:

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Resources Foundation: Pages 43–44, Pages 261–262

22 × 23 25 34 ÷ 3 33

103 × 10

104 43 ÷ 42 4

52 × 5 53 63 ÷ 6 62

22 × 24

223 33 ÷ 3 104 × 10

43 ÷ 4

53 × 5

62 ÷ 6

→ →

→ →

→ →

23 × 2 24 25 ÷ 2 16 26 ÷ 22

32 × 33 243 37 ÷ 32 34 × 3

310 ÷ 35

104 × 102 1000000 103 × 103 (103)2 (102)3

24 × 2 24 ÷ 2 36 35 × 3 100000

(102)4

107 ÷ 10 108 ÷ 102

17

Unit/option content

NUMBER



MULTIPLES AND FACTORS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)

use the concepts and vocabulary of factor, multiple, common factor, common multiple and prime

M2 (T2)

use the concepts and vocabulary of divisor, highest common factor, least (lowest) common multiple and prime factor decomposition

M3(T2)

find the LCM and HCF of numbers written as the product of their prime factors

Provide learners with a number factors table similar to the one below:

No. No. of Factors Factors Prime Factors

1 1 1

2 2 1 2 2

3 2 1 3 3

4 3 1 2 4 2

5 2 1 5 5

6 4 1 2 3 6 2 3

7 2 1 7 7

8 4 1 2 4 8 2 Ask learners to continue the table by finding information about the factors of the numbers from 1 – 100. Direct learners to look for patterns and make observations about the information contained in their tables. For example, learners should have the opportunity to use the vocabulary of factor, multiple, common multiple and prime. Instruct learners to consider the number of factors that each number has. What can be said about numbers that: only have two factors? Or have an odd number of factors?

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 3 Types of number 1 Foundation CCEA GCSE Mathematics Second Edition Chapter 4 Types of number 2

18

Unit/option content

NUMBER




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)


M2 (T2)


M3(T2)



Provide learners with a prime factor decomposition table:

No. Factors Prime factors Prime factor decomposition

1 1 2 1 2 2 3 1 3 3 4 1 2 4 2 2 x 2 = 4 5 1 5 5 6 1 2 3 6 2 3 2 x 3 = 6 7 1 7 7 8 1 2 4 8 2 2 x 2 x 2 = 8

Using a table completed for the numbers 1-100 instruct learners to search for the HCF and the LCM of specified number pairs. Ask learners to consider how numbers with more than two factors can be built up from multiplying a combination of prime numbers together. Direct learners to observing that every whole number can be obtained by multiplying prime numbers together and that every whole number splits into primes in a unique way. Learners should practice the process of demonstrating how numbers can be written as a product of prime factors. Learners could explain their method to their partners.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 4 Types of number 2

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Unit/option content

NUMBER




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)


M2 (T2)


M3(T2)


[Students should know the content of Units M1and M2 before taking Unit M3.] Present learners with the number pairs 24 and 60, 18 and 30, and 32 and 40 and ask them to find the HCF and the LCM of the number pairs. Ask learners to place their results in a table similar to the one below:

Number 1 Number 2 Product HCF LCM Product

24 60 1440 12 120 1440

18 30 540 6 90 540

32 40 1280 8 160 1280

Discuss observations Instruct learners to write each number in the above table as a product of prime factors and place their results in a table similar to the one below:

Number 1 Number 2 Product HCF LCM Product

23x3 22x3x5 25x32 x5 22x3 23x3x5 25x32 x5

2x32 2x3x5 22 x33 x5 2x3 2x32x5 22 x33 x5

25 23x5 28x5 23 25x5 28x5

A calculator could be used. For each number pair, direct learners to compare like with like bases and powers. Working in pairs, direct learners to notice how the HCF relates to common prime factors. Discuss observations and continue the activity with further examples to consolidate how prime factor decompositions can be used to find and check HCFs and LCMs.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 1 The language of number

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Unit/option content

NUMBER



WORKING WITH DECIMALS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

add, subtract, multiply and divide decimals up to 3 decimal places

M2 (T2)

add, subtract, multiply and divide decimals of any size

1 7 pupils given 1 card each from:

4 ● 9 2 6 7 5 They are going to demonstrate practically 4.9 + 2.6 Three stand in front of class with:

4 ● 9 Two come to them with:

2 and 6 with person with ● between both 4, 9 and 2, 6 Ask the class for the answer.

Then the two with 7 and 5 replace the other number leaving 7 ● 5 in front of the class.

Teacher emphasises all 6 pupils (apart from one with ●)changed. i.e. The decimal point stays in the same place for adding.

2 Repeat for 8.1 – 2.7 = 5.4 Conclusion: The decimal point stays in the same place for subtraction.

3 Pupils pick cards and write down whether the answer is going to be smaller or bigger than the first number without using a calculator. Teacher then asks them to give a rule for knowing whether the answer will be smaller or bigger. Pupils could use calculators to check their responses.

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Resources Foundation Pages 23-30

7.5 × 0.89 8.1 ÷ 0.4

6.4 × 0.94 6 ÷ 0.21

0.742

√0.96

21.4 × 0.48 16 ÷ 1.74

24 ÷ 0.36

16.3 × 1.2

9.64 × 1.7

21

Unit/option content

NUMBER



WORKING WITH DECIMALS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

add, subtract, multiply and divide decimals up to 3 decimal places

M2 (T2)

add, subtract, multiply and divide decimals of any size


Instruct learners to solve a routine problem that requires calculating with decimals. Ask learners to explain why the method they are using works. Ask learners to compare calculator and non-calculator methods for solving the same problem. Pupils work in pairs and then individually through a set of cards. In pairs: decide whether to add, subtract, multiply or divide. Individually: work out the answer using calculator. In pairs: check each other’s answers, comment on appropriateness of answers. The cards could cover a range of topic areas for example: Growth and decay; Working with money; Working with numbers; and Working with measures.

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Resources

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Unit/option content

NUMBER



WORKING WITH EQUIVALENCES

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

understand and use equivalent fractions

M1 (T1)

understand that percentage means number of parts per hundred

M1 (T1)

use equivalences between fractions, decimals and percentages in a variety of contexts

1 Use their calculators to simplify the following fractions

2030

, 1845

, 1535

, 4872

, 120180

, 200225

, 3654

, 2763

, 2460

2 Work in pairs: Each pupil writes down 4 equivalent fractions to each of the following. They then compare their answers.

3 Work in pairs: Match up different cards that are equivalent.

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Resources Foundation: Pages 62–65. 112 - Q3 Pages 117

14

23

35

27

18

38

7

10

56

→ 25

→ 0.4 40% 34

→ 0.75 75% →

→ 0.7 35

→ 0.6 60%

→ 38

→ 0.375 7

20 → 0.35 →

70% →

37.5%

→

35%

710

23

Unit/option content

NUMBER



WORKING WITH FRACTIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

add and subtract simple fractions and simple mixed numbers

M1 (T1) (T2)

calculate a fraction of a quantity

M1 (T1) (T2)

express one quantity as a fraction of another

M2 (T2)

add, subtract, multiply and divide fractions, including mixed numbers

1 Use the grids to work out:

25

+ 12 3

4 − 1

8

23

+ 14 1

2 − 2

7

2 Pupils work in pairs in a fraction game. One picks a card at random. They choose any amount less than the value on the card and their partner has to work out the fraction of this amount compared to the money on the card (in simplest form if possible). Check using a calculator. First to get 6 right wins.

50p

20p 10p 80p 60p 30p 75p

25p

18p 36p 54p 27p 90p 84p

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Resources Foundation: Pages 72–79, Pages 66–68

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Unit/option content

NUMBER



WORKING WITH FRACTIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

add and subtract simple fractions and simple mixed numbers

M1 (T1) (T2)

calculate a fraction of a quantity

M1 (T1) (T2)

express one quantity as a fraction of another

M2 (T2)

add, subtract, multiply and divide fractions, including mixed numbers


1 No calculator Pupils work individually through a set of cards choosing the correct answer each time. Teacher then emphasises the different rules for +, −, ×, ÷

78

+ 23 =

911

2124

+ 1624

724

+ 224

34

+ 25 =

320

× 220

34

+ 52

620

23

− 15 =

115

215

− 115

1015

− 315

25

÷ 34 =

25

× 34 2

5 × 4

3

220

× 203

2 Pupils work in pairs to check what to do for 4 questions. They then work out the answer individually. They then compare their answers.

1 Find the difference between 223 and 1 3

10

2 Find the product of 212 and 33

5

3 I read 15 of the book last night and then another 2

3 tonight.

What fraction of the book have I still to read? 4 How many 3

4 m can I cut off a plank 7 m long?

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A C B

A C B

A C B

A B C

25

Unit/option content

NUMBER



WORKING WITH MONEY

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

use correct decimal notation when working with money

M1 (T1) (T2)

NEW

calculate with money and solve simple problems in the context of finance, for example profit and loss, discount, wages and salaries, bank accounts, simple interest, budgeting, debt, APR and AER

M2 (T3)

calculate with money and solve problems in a financial context for example compound interest, insurance, taxation, mortgages and investments

Use the bespoke resources created for this topic to introduce and discuss personal finance topics. Integrate promotional materials from local banks into the discussion. Use these materials to develop the mathematical techniques required to solve personal finance problems. For example, in Unit M1 interpret AER and APR as operators and calculate the interest gained after one year. Present learners with a set of problems, for example: 1(a) Increase £1700 by 3%

1(b) Anna puts £1700 into a savings account. The interest rate is 3% per annum. Work out the amount gained through simple interest after one year.

1(c) Dylan puts £1700 into a savings account. The annual equivalent rate (AER) is 3% per annum. Work out the amount of interest gained after one year.

Discuss observations.

Use a spreadsheet to model other scenarios relating to AER and APR.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 12 Percentages 2 Bespoke resources

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Unit/option content

NUMBER



WORKING WITH MONEY

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

use correct decimal notation when working with money

M1 (T1) (T2)

NEW

calculate with money and solve simple problems in the context of finance, for example profit and loss, discount, wages and salaries, bank accounts, simple interest, budgeting, debt, APR and AER

M2 (T3)

calculate with money and solve problems in a financial context for example compound interest, insurance, taxation, mortgages and investments


Use the bespoke resources created for this topic to introduce and discuss personal finance topics. Integrate promotional materials from local banks into the discussion. Use these materials to develop the mathematical techniques required to solve personal finance problems. Continuing on from Unit M1 when AER and APR were interpreted as operators and interest was calculated for one year, in Unit M2 calculate for example the value of an investment at the end of three years with a given AER. Set up a spreadsheet to calculate compound interest. Carry out a ‘what if’ analysis using different interest rates to compare the amount of interest earned when different interest rates are applied. For a loan use a spreadsheet to compare the amount repaid with the amount borrowed. For a savings account use a spreadsheet to compare the final amount with the amount invested. Present learners with a problem based on AER, for example: a newspaper advertisement claims that the AER on an account that pays interest monthly at a rate of 0.3% is 3.6%. Is this claim correct? Explain why you agree or disagree.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 13 Percentages 3 Bespoke resources

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Unit/option content

NUMBER



WORKING WITH NUMBERS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

use the four operations applied to positive and negative integers, including efficient written methods

M1 (T1)

order positive and negative integers, decimals and fractions

M1 use symbols =, ≠, <, >, ≤, ≥

M1 (T1)

use calculators effectively and efficiently

M1 (T5)

understand and use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals

M1 (T5)

recognise and use relationships between operations, including inverse operations

1 No calculator Give table to each pupil.

Place Belfast London Madrid Moscow Paris Dublin

Temperature (°C)

-2

4

9

-8

2

-3

They: • rewrite cities in order starting at coldest • find the difference between the coldest and warmest • research the internet to find 6 other cities, with at least 2 negative

temperatures and repeat question.

2 Pupils work in pairs. For each card they have to order the number starting with the smallest.

−4 → 0.2 → 0 →25→ 3.5 −1.7 → −1

12→ 0.485 → 0.5

−34→ −0.6 → 1

23→ 1.8 → 4 3

5→ 0.84 → 1

25→ 1.7

3 Pupils work in pairs. They have to add brackets to each card to make the equation correct.

4 + 8 × 3− 2 = 34 10 ÷ 7 − 2 × 6 = 12

12

of 18− 4 + 6 = 13 8 − 2 × 3 + 4 = 42

4

Pupils work in pairs to write an equivalent statement to the one on the card.

7.86− 5.39 = 2.47 6.48 × 1.5 = 9.72

6.72 = 44.89 9 − 1.84 = 7.16

32.0964.72

= 6.8 √72.25 = 8.5

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Resources Foundation Pages 143-147, Pages 21-23, Pages 263-265, Pages 107-111

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Unit/option content

NUMBER



WORKING WITH PERCENTAGES

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

calculate a percentage of a quantity

M1 (T2)

express one quantity as a percentage of another

M1 (T2)

calculate percentage increase/decrease

Present learners with a selection of partially shaded one hundred unit squares to help learners visualise a percentage as a proportion out of one hundred. Direct learners to express the proportion out of one hundred as simply as possible, e.g., the proportion 20

100 = 2

10 = 1

5∴ 20% = 2

10 = 1

5

Present learners with several proportions out of 100 and their simplest forms. For example: 20

100 = 1

5; 25100

= 14; 40100

= 25; and 75

100 = 3

4 . Ask learners to

cross multiply. Discuss observations. When presenting worked examples on percentages explore how problems can be solved and checked in more than one way by selecting equivalent operations. Instruct learners to solve a routine percentage problem and ask them to explain why the method they are using works. True or false? Provide learners with several statements relating to percentage calculations. Ask learners to decide on the truth or falsehood of the statements, for example, 20% of £140 = £28, true or false? 60

708 > 10%?,

30% discount on a £45 item is a saving of £13.50? and 200% of 140 = 280. Calculators can be used.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 11 Percentages 1 Foundation CCEA GCSE Mathematics Second Edition Chapter 12 Percentages 2

29

Unit/option content

NUMBER



WORKING WITH PLACE VALUE

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

understand place value and decimal places

M1 (T1)

read, write and compare decimals up to three decimal places

1 Pupils play each other a Place Value Game. They each chose a card in turn and write down the place value of

8.14

0.36

4.79

3.51

2.634

124.8

5.327

84.7

941

6802

5124

71.8

2 Pupils work in pairs rewriting the number on each card in order st

7.4, 7.041, 7.14, 7.044 8.46, 8.6, 8.04, 8.4 0.764, 0.76, 0.7, 0.767 1.94, 1.094, 1.494, 1.994

6.4, 6.24, 6.54, 6.459 4.1, 4.096, 4.05, 4.077

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Resources Foundation: Pages 4, 19-23

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Unit/option content

ALGEBRA



ALGEBRAIC FRACTIONS

M1 X

M2 X

M3

M4

M5 X

M6 X

M7

M8


M3 (T3)

add or subtract algebraic fractions, for example simplify 4a+3/10 + 6a-5/5

M3 (T3) (T4)

simplify, multiply and divide algebraic fractions with linear or quadratic numerators and denominators

M4 (T3) (T4)

add or subtract algebraic fractions with linear denominators, for example simplify 2/x+2 + 3/2x-1


1 Pupils work in pairs deciding the correct answer on each card. 3𝑎𝑎5

+ 2𝑎𝑎7

= 3𝑎𝑎35

+ 2𝑎𝑎35

2135

+ 1035

21𝑎𝑎35

+ 10𝑎𝑎35

2𝑡𝑡3

− 𝑡𝑡6 =

𝑡𝑡−3

12𝑡𝑡 − 3𝑡𝑡

4𝑡𝑡6

− 𝑡𝑡6

2𝑥𝑥

− 32𝑥𝑥

= 2𝑥𝑥

+ 3𝑥𝑥 4𝑥𝑥

𝑥𝑥 + 3𝑥𝑥

2𝑥𝑥

4+32𝑥𝑥

𝑎𝑎𝑏𝑏

− 𝑐𝑐𝑑𝑑 =

𝑎𝑎−𝑐𝑐𝑏𝑏−𝑑𝑑

𝑎𝑎𝑑𝑑−𝑐𝑐𝑏𝑏𝑑𝑑

𝑎𝑎𝑑𝑑−𝑏𝑏𝑐𝑐𝑏𝑏𝑑𝑑

2 Pupils individually choose cards and decide whether or not the expression on the card can be simplified. They then explain why they have chosen ones they think can simplify to their partner.

𝑥𝑥 + 7𝑥𝑥 − 2

3𝑥𝑥 − 6

3

𝑥𝑥2 + 𝑥𝑥𝑥𝑥

𝑥𝑥

3𝑥𝑥2 + 6𝑥𝑥𝑥𝑥

𝑥𝑥

𝑥𝑥2 + 4𝑥𝑥

4

6𝑥𝑥 − 8

3𝑥𝑥

𝑥𝑥2 − 3

𝑥𝑥 + 6𝑥𝑥2 + 6𝑥𝑥

4𝑥𝑥

2𝑥𝑥2 − 𝑥𝑥

𝑥𝑥 − 4

2𝑥𝑥 − 8

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Resources Higher: Pages 178–184

C A B

C A B

C A B

A B C

31

Unit/option content

ALGEBRA



ALGEBRAIC FRACTIONS

M1 X

M2 X

M3

M4

M5 X

M6 X

M7

M8


M3 (T3)

add or subtract algebraic fractions, for example simplify 4a+3/10 + 6a-5/5

M3 (T3) (T4)

simplify, multiply and divide algebraic fractions with linear or quadratic numerators and denominators

M4 (T3) (T4)

add or subtract algebraic fractions with linear denominators, for example simplify 2/x+2 + 3/2x-1


1

Pupils in pairs work through the following cards circling where the solution is wrong.

2𝑥𝑥 + 2

+3

2𝑥𝑥 − 1=

2(2𝑥𝑥 − 1) + 3(𝑥𝑥 + 2)(𝑥𝑥 + 2)(2𝑥𝑥 − 1) =

2𝑥𝑥 − 1 + 3𝑥𝑥 + 2(𝑥𝑥 + 2)(2𝑥𝑥 − 1)

=5𝑥𝑥 + 1

(𝑥𝑥 + 2)(2𝑥𝑥 − 1)

4

𝑡𝑡 − 2−

3𝑡𝑡𝑡𝑡

=4𝑡𝑡 + 4 − 3𝑡𝑡 + 𝑏𝑏𝑡𝑡 − 2(𝑡𝑡 + 1)

=𝑡𝑡 + 10

𝑡𝑡 − 2(𝑡𝑡 + 1)

53 − 𝑥𝑥

+2

𝑥𝑥 − 2=

5(3 − 𝑥𝑥) + 2(𝑥𝑥 − 2)(3 − 𝑥𝑥)(𝑥𝑥 − 2) =

15 − 5𝑥𝑥 + 2𝑥𝑥 − 4(3 − 𝑥𝑥)(𝑥𝑥 − 2)

=11 − 3𝑥𝑥

(3 − 𝑥𝑥)(𝑥𝑥 − 2)

3𝑞𝑞 − 2

−1

𝑞𝑞 + 4=

3(𝑞𝑞 + 4) − (𝑞𝑞 − 2)(𝑞𝑞 − 2)(𝑞𝑞 − 4) =

3𝑞𝑞 + 12 − 𝑞𝑞 − 2(𝑞𝑞 − 2)(𝑞𝑞 − 4)

=2𝑞𝑞 + 10

(𝑞𝑞 − 2)(𝑞𝑞 − 4)

7

5𝑥𝑥−

3𝑥𝑥 + 2

=7𝑥𝑥 + 14 − 15𝑥𝑥

5𝑥𝑥2 − 10𝑥𝑥=

7 − 𝑥𝑥5𝑥𝑥2 + 10𝑥𝑥

=7 − 1

5𝑥𝑥 + 10

Teacher then talks through all misunderstandings/misconceptions

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Resources Higher: Pages 191-193

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Unit/option content

ALGEBRA



CO-ORDINATE GEOMETRY

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

work with co-ordinates in all four quadrants

M2 (T2)

find the midpoint and length of a line in 2D co-ordinates

M3 (T3)

find the equation of a line through two given points or through one point with a given gradient

M3 (T4)

understand and use the gradients of parallel lines

M4 (T4)

understand the use the gradients of perpendicular lines

M8 recognise and use the equation of a circle, centre the origin and radius r

M8 find the equation of a tangent to a circle at a given point on the circle

Join the points Provide learners with co-ordinate axes with a pre-drawn horizontal line segment AB. Ask learners to see the line segment as the base of a triangle. Instruct the learners to place another point C at a specified location on the co-ordinate axes. What type of triangle has been formed? What are the co-ordinates of all 3 vertices of the triangle? Learners could continue this activity by forming other triangles and quadrilaterals. Learners should be asked to consider the properties of the shapes and to list the co-ordinates of all the vertices. Discuss observations. Alternatively learners could be asked to explain to their partners why an ordered set of three or four points on co-ordinate axes form a specified triangle or quadrilateral. This activity could be demonstrated using dynamic graphing software.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 24 Co-ordinates

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Unit/option content

ALGEBRA




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)


M2 (T2)


M3 (T3)


M3 (T4)


M4 (T4)




[Students should know the content of Unit M1 before taking Unit M2.] Finding the length of a line Measure the distance between two points A (𝑥𝑥 1, 𝑥𝑥 1) and B (𝑥𝑥 2, 𝑥𝑥 1) that both lie on a horizontal line. Calculate the distance between two points A (𝑥𝑥 1, 𝑥𝑥 1) and B (𝑥𝑥 2, 𝑥𝑥 1). Compare the two methods. Measure the distance between two points C (𝑥𝑥 1, 𝑥𝑥 1) and D (𝑥𝑥 1, 𝑥𝑥 2) that both lie on a vertical line. Calculate the distance between two points C (𝑥𝑥 1, 𝑥𝑥 1) and D (𝑥𝑥 1, 𝑥𝑥 2). Compare the two methods. Measure the distance between two points E (𝑥𝑥 1, 𝑥𝑥 1) and F (𝑥𝑥 2, 𝑥𝑥 2) in a two-dimensional plane. Calculate the distance between two points E (𝑥𝑥 1, 𝑥𝑥 1) and F (𝑥𝑥 2, 𝑥𝑥 2). Compare the two methods. Locating the midpoint of a given line … Find by practical measurement the co-ordinates of the midpoint of a line drawn on co-ordinate axes. Calculate the co-ordinates of the midpoint of a line. Compare the two methods. The line segments above could be the sides or diagonals of 2-D shapes.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 26 Midpoint and length of line segments

34

Unit/option content

ALGEBRA




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)


M2 (T2)


M3 (T3)


M3 (T4)


M4 (T4)




[Students should know the content of Units M1 and M2 before taking Unit M3.] Complete the shape Provide learners with co-ordinate axes with a pre-drawn line segment AB where A is (𝑥𝑥 1, 𝑥𝑥 1) and B is (𝑥𝑥 2, 𝑥𝑥 2). Ask learners to see the line segment as the LHS side of a parallelogram. Instruct the learners to place two other points C and D on the co-ordinate axes to complete the parallelogram. Ask learners to calculate the gradients of the four line segments that form the parallelogram using the standard formula for calculating a gradient on coordinate axes. Discuss observations. Learners could continue this activity by forming other quadrilaterals and calculating the gradients of the line segments that form the shapes. With the aid of a sketch ask learners to use the gradients of parallel lines to decide if an ordered set of points ABCD with given co-ordinates represent a parallelogram. Part of the whole Provide learners with co-ordinate axes with a pre-drawn line segment AB where A is (𝑥𝑥 1, 𝑥𝑥 1) and B is (𝑥𝑥 2, 𝑥𝑥 2). Using knowledge from the ‘Graphs and Gradients’ topic direct learners to identify that the relationship between the 𝑥𝑥 co-ordinate and the 𝑥𝑥 co-ordinate is of the form 𝑥𝑥 = 𝑚𝑚𝑥𝑥 + 𝑐𝑐. Demonstrate how to use the standard formula to find the equation of the line that the line segment is part of. Dynamic graphing software could be used to confirm the equation of the line.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 22 Straight lines and linear graphs

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Unit/option content

ALGEBRA




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)


M2 (T2)


M3 (T3)


M3 (T4)


M4 (T4)





Provide learners with a set of printouts from a dynamic graphing software package that illustrate perpendicular lines. Ask learners to find the gradients for the pairs of lines and to tabulate their results. Inform learners that there is a relationship between the gradients of perpendicular lines. Direct learners to find the products of the gradients of the perpendicular lines. Discuss observations. Represent a kite on co-ordinate axes. Ask learners to find the gradients of the line segments representing the diagonals. Ask learners to find the product of the gradients of these line segments. Discuss observations.

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Unit/option content

ALGEBRA



EQUATIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)

set up and solve linear equations in one unknown

M2 (T2)

set up and solve linear equations in one unknown, including those with the unknown on both sides of the equation and equations of the form x/4 +3 = 7

M3 (T3)

set up and solve linear equations of the form 4a+3/10 + 6a-5/5 = 13/2

M3 (T3)

set up and solve quadratic equations using factors

M4 (T4)

set up and solve equations such as 2/x+2 + 3/2x-1 = 1

M4 (T4)

set up and solve quadratic equations using factors and the formula, where the coefficient of x2 ≠ 1 and more complex equations

1 Pupils work in pairs.

They work through the 2 cards. Each then marks up a similar question for the other to solve.

I think of a number. Call it n. I double it and then subtract 6 My answer is 10 Form and solve an equation.

I think of a number. Call it n. I add 3 to the number and then treble the answer. My final answer is 42 Form and solve an equation

Teacher emphasises the need for an equation as well as an answer each time.

2 Pupils work individually to complete each card showing every step of the Solution. Each pupil gets a mark for each correct step.

4𝑥𝑥 − 7 = 5 𝑥𝑥 = 3

8𝑥𝑥 + 5 = 21 𝑥𝑥 = 2

10 + 2𝑥𝑥 = 16 𝑥𝑥 = 3

5𝑥𝑥 − 10 = 15 𝑥𝑥 = 5

6𝑥𝑥 − 21 = 21 𝑥𝑥 = 7

9𝑥𝑥 + 4 = 40 𝑥𝑥 = 4

(both equation and answer on card with enough space for each step)

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Unit/option content

ALGEBRA



EQUATIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)


M2 (T2)


M3 (T3)


M3 (T3)


M4 (T4)


M4 (T4)



1 Pupils work in pairs. They work through the 2 cards. Each then marks up a similar question for the other to solve.

I think of a number. I double it and add 3. My answer is the same as subtracting the number from 9. Call the number n and form an equation. Hence find the value of n.

I think of a number. I multiply it by 4 and subtract 7. My answer is the same as doubling the number and adding 13. Call the number n and form an equation. Hence find the value of n.

2 Pupils work individually to complete each card showing every step of the

solution. Each pupil gets a mark for each correct step.

5𝑥𝑥 + 2 = 𝑥𝑥 − 6 𝑥𝑥 = −2

4𝑥𝑥 − 7 = 2𝑥𝑥 + 5 𝑥𝑥 = 6

6𝑥𝑥 = 2𝑥𝑥 − 8 𝑥𝑥 = −2

4𝑥𝑥 = 10 − 𝑥𝑥 𝑥𝑥 = 2

𝑥𝑥4− 2 = 6

𝑥𝑥 = 32

𝑥𝑥3

+ 5 = 2

𝑥𝑥 = −9

2𝑥𝑥3

+ 1 = 5

𝑥𝑥 = 6

3𝑥𝑥4− 2 = 3

𝑥𝑥 =203

3(2𝑥𝑥 − 1) = 4(𝑥𝑥 + 2)

𝑥𝑥 =112

(both equation and answer on card with enough space for each step)

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Resources Foundation: Pages 198-206

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Unit/option content

ALGEBRA



EQUATIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)


M2 (T2)


M3 (T3)


M3 (T3)


M4 (T4)


M4 (T4)



1 Pupils work individually to complete each card showing every step of the solution. Each pupil gets a mark for each correct step. (both equation and answer on card with enough space for each step)

2𝑥𝑥 − 1

3 +𝑥𝑥 + 6

2 = 5

𝑥𝑥 = 2

3𝑥𝑥 − 1

2 −𝑥𝑥 − 3

5 = 4

𝑥𝑥 = 3

𝑥𝑥 + 1

4 +2𝑥𝑥 − 1

3 =92

𝑥𝑥 = 5

6 − 𝑥𝑥2 −

𝑥𝑥 + 53 = 3

𝑥𝑥 = −2

2𝑥𝑥 − 3

3 +3𝑥𝑥 + 2

5 = 7

𝑥𝑥 = 6

8 − 2𝑥𝑥

7 −1 − 𝑥𝑥

2 = 0

𝑥𝑥 = −3

2 Pupils work in pairs linking 4 cards each time, showing quadratic equation, factors and solutions.

𝑥𝑥2 − 9𝑥𝑥 + 14 = 0 → (𝑥𝑥 − 7)(𝑥𝑥 − 2) → 𝑥𝑥 = 7 or 𝑥𝑥 = 2 𝑥𝑥2 − 𝑥𝑥 − 6 = 0 → (𝑥𝑥 − 3)(𝑥𝑥 + 2) → 𝑥𝑥 = 3 or 𝑥𝑥 = −2 𝑥𝑥2 + 3𝑥𝑥 = 0 → 𝑥𝑥(𝑥𝑥 + 3) → 𝑥𝑥 = 0 or 𝑥𝑥 = −3 𝑥𝑥2 + 8𝑥𝑥 + 7 = 0 → (𝑥𝑥 + 1)(𝑥𝑥 + 7) → 𝑥𝑥 = −1 or 𝑥𝑥 = −7 𝑥𝑥2 + 3𝑥𝑥 − 4 = 0 → (𝑥𝑥 + 4)(𝑥𝑥 − 1) → 𝑥𝑥 = −4 or 𝑥𝑥 = 1 𝑥𝑥2 − 9𝑥𝑥 + 20 → (𝑥𝑥 − 4)(𝑥𝑥 − 5) → 𝑥𝑥 = 4 or 𝑥𝑥 = 5

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Resources Higher: Pages 152–158, Pages 185–188 omitting any quadratic equation with 𝑥𝑥2 ≠ 1

39

Unit/option content

ALGEBRA



EQUATIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T2)


M2 (T2)


M3 (T3)


M3 (T3)


M4 (T4)


M4 (T4)



1 Pupils work individually to complete each card showing every step of the solution. Each pupil gets a mark for each correct step. (both equation and answer on card with enough space for each step)

4

𝑥𝑥 − 1 +3

𝑥𝑥 + 1 = 5

𝑥𝑥 = 2 or −35

6

𝑥𝑥 − 1 −8

𝑥𝑥 + 2 = 4

𝑥𝑥 = 2 or −72

3

2𝑥𝑥 − 3 +2

𝑥𝑥 − 2 = 3

𝑥𝑥 = 3 or 53

43𝑥𝑥 − 2 −

4𝑥𝑥 + 1 = 2

𝑥𝑥 = 1 or −83

3

𝑥𝑥 − 1 +3

𝑥𝑥 + 3 = 2

𝑥𝑥 = −2 or 3

1

𝑥𝑥 + 2 −6

2𝑥𝑥 + 3 = 1

𝑥𝑥 = −3 or −52

2 Pupils work in pairs. For each card they write down the value of 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 individually and then compare with each other. They then substitute into the quadratic formula and simplify as far as possible but not finding the square root and then compare with each other.

2𝑥𝑥2 − 𝑥𝑥 = −11 = 0

3𝑥𝑥2 + 7𝑥𝑥 + 1 = 0

4𝑥𝑥2 − 8𝑥𝑥 − 7 = 0

16𝑥𝑥 − 𝑥𝑥2 − 9 = 0

5𝑥𝑥2 − 2𝑥𝑥 − 4 = 0

8𝑥𝑥 + 𝑥𝑥2 − 5 = 0

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Resources Higher Pages: 185–198

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Unit/option content

ALGEBRA



EQUATIONS CONTINUED

M1

M2

M3

M4

M5

M6

M7

M8


M6 (T2)

use systematic trial and improvement to find approximate solutions of equations where there is no simple analytic method for solving them

M7 (T3)

set up and solve two linear simultaneous equations algebraically

M7 (T4)

set up equations and solve problems involving direct proportion, including graphical and algebraic representations

M8 (T4)

set up equations and solve problems involving indirect proportion, including graphical and algebraic representations

M8 (T4)

set up and solve two simultaneous equations, one linear and one non linear

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Resources

41

Unit/option content

ALGEBRA



EXPRESSIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

manipulate algebraic expressions by taking out common factors that are constants

M1 (T1)

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

M2 (T1)

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

M2 (T2)

manipulate algebraic expressions by taking out common factors that are terms

M3 (T3)

multiply two linear expressions

M3 (T3)

factorise quadratic expressions of the form x2+bx+c

M3 (T3)

factorise using the difference of two squares

M4 (T4)

factorise quadratic expressions of the form ax2+bx+c

1 Pupils individually sort cards into 2 sets – those that factorise and those

that don’t. They then factorise those that do.

6𝑎𝑎 + 8𝑏𝑏

2𝑣𝑣 − 3𝑤𝑤

8𝑡𝑡 + 12h

4𝑥𝑥 − 3

2𝑥𝑥 + 10

5𝑥𝑥 − 10𝑥𝑥 + 15𝑧𝑧

8𝑡𝑡 − 6𝑥𝑥 + 11𝑣𝑣

5𝑚𝑚 − 10

9𝑥𝑥 − 15

2 Pupils work in pairs and explain why the manipulation of the expressions on the cards are wrong. They then write their own notes on what you can do when collecting like terms or expanding brackets.

6𝑥𝑥 + 3𝑥𝑥 + 2𝑥𝑥 = 11𝑥𝑥3

7𝑥𝑥 + 3𝑥𝑥 − 𝑥𝑥 + 𝑥𝑥 = 8𝑥𝑥 + 4𝑥𝑥

𝑥𝑥 + 2𝑥𝑥2 + 3𝑥𝑥 = 6𝑥𝑥

4𝑎𝑎 + 3𝑏𝑏 + 2𝑎𝑎 + 5𝑏𝑏 = 6𝑎𝑎 + 8𝑏𝑏 = 14𝑎𝑎𝑏𝑏

4(𝑥𝑥 + 2) = 4𝑥𝑥 + 2

3(𝑥𝑥 − 2) = 3𝑥𝑥 + 6

3(2𝑥𝑥 + 3) = 6𝑥𝑥 + 6

5(𝑚𝑚 + 2) = 5 × 2𝑚𝑚 = 10𝑚𝑚

6(𝑥𝑥 + 2) = 6𝑥𝑥 + 12 = 18𝑥𝑥

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Resources Foundation: Pages 191–192, Pages 166–168, 171–173, 187–191

42

Unit/option content

ALGEBRA



EXPRESSIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M1 (T1)


M2 (T1)


M2 (T2)


M3 (T3)


M3 (T3)


M3 (T3)


M4 (T4)



1 Pupils individually sort cards into 2 sets – those that factorise and those that don’t. They then factorise those that do.


2𝑥𝑥2 − 𝑥𝑥𝑥𝑥

𝑥𝑥 + 𝑥𝑥2

𝑥𝑥 − 𝑥𝑥2

6𝑥𝑥 + 7𝑥𝑥

𝑥𝑥𝑥𝑥 − 𝑥𝑥𝑧𝑧

4𝑥𝑥 + 𝑥𝑥𝑧𝑧

𝑥𝑥2 + 𝑥𝑥2

6𝑥𝑥2 − 8𝑥𝑥𝑥𝑥

2 Pupils are given the 6 cards in Set A. They are then given the 12 cards in Set B and asked to match up the correct cards from B with A.

A

𝑥𝑥(𝑥𝑥 − 3)

𝑥𝑥(4 + 𝑥𝑥)

𝑥𝑥(𝑥𝑥 + 1)

𝑥𝑥(𝑥𝑥 + 3𝑥𝑥)

𝑥𝑥(𝑥𝑥 − 2𝑎𝑎)

𝑣𝑣(𝑣𝑣 − 2𝑤𝑤)

B

𝑥𝑥2 − 3𝑥𝑥

𝑥𝑥2 − 3

4𝑥𝑥 + 𝑥𝑥2

4𝑥𝑥 + 𝑥𝑥

𝑥𝑥2 + 1

𝑥𝑥2 + 𝑥𝑥


𝑥𝑥2 + 3𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥 − 2𝑎𝑎

𝑥𝑥𝑥𝑥 − 2𝑎𝑎𝑥𝑥

𝑣𝑣2 − 2𝑤𝑤

𝑣𝑣2 − 2𝑣𝑣𝑤𝑤

Pupils are given the 6 cards in Set C and 12 cards in D and asked to match up the correct 6 cards from D with C.

C

𝑥𝑥2 − 3𝑥𝑥

4𝑥𝑥 + 𝑥𝑥2

𝑥𝑥2 + 𝑥𝑥

𝑥𝑥2 + 3𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥 − 2𝑎𝑎𝑥𝑥

𝑣𝑣2 − 2𝑣𝑣𝑤𝑤

D


𝑥𝑥(2 − 3)

𝑥𝑥(4 + 𝑥𝑥)

𝑥𝑥(4𝑥𝑥 + 𝑥𝑥)



𝑥𝑥(𝑥𝑥 + 3𝑥𝑥)

𝑥𝑥(2𝑥𝑥 + 3𝑥𝑥)

𝑥𝑥(𝑥𝑥 − 2𝑎𝑎)


𝑣𝑣(𝑣𝑣 − 𝑤𝑤)

𝑣𝑣(𝑣𝑣 − 2𝑤𝑤)

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43

Unit/option content

ALGEBRA



EXPRESSIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M1 (T1)


M2 (T1)


M2 (T2)


M3 (T3)


M3 (T3)


M3 (T3)


M4 (T4)


[Students should know the content of Unit M1 and M2 before taking Unit M3.]

1 Pupils work in pairs. Class discussion: to multiply (𝑥𝑥 + 4) by (𝑥𝑥 + 2) This is the same as (𝑥𝑥 + 4) × 𝑥𝑥 added to (𝑥𝑥 + 4) × 2

So in pairs they do

(𝑥𝑥 + 4)(𝑥𝑥 + 2)

(𝑥𝑥 + 5)(𝑥𝑥 − 3)

(2𝑥𝑥 + 1)(3𝑥𝑥 − 2)

(3𝑥𝑥 − 2)(𝑥𝑥 + 4)

(2𝑥𝑥 − 3)(𝑥𝑥 − 4)

2 PRACTICAL ACTIVITY LEADING TO DIFFERENCE OF 2 SQUARES Each pupil cuts outs a 6 cm square. They then cut off a 2 cm corner from the square. They work out the area that is left using the compound shape.

(6 × 4 + 2 × 4 = 32) The teacher now asks them to find this area using a different method.

(6 × 6 − 2 × 2 = 36 − 4 = 32)

Pupils are asked to generalise using a × 𝑎𝑎 square cutting off b × 𝑏𝑏 square and hopefully deriving 𝑎𝑎2 − 𝑏𝑏2 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑏𝑏)from using areas

𝑎𝑎(𝑎𝑎 − 𝑏𝑏) + 𝑏𝑏(𝑎𝑎 − 𝑏𝑏) = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏)

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Resources Higher: Pages 139–143, Pages 166–173 (excluding 𝑥𝑥2 ≠ 1)

44

Unit/option content

ALGEBRA



EXPRESSIONS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M1 (T1)


M2 (T1)


M2 (T2)


M3 (T3)


M3 (T3)


M3 (T3)


M4 (T4)



1 Teacher demonstrates the following factorisation question: Does 3𝑥𝑥2 + 10𝑥𝑥 − 8 = (3𝑥𝑥 − 2)(𝑥𝑥 − 4) or (3𝑥𝑥 − 2)(𝑥𝑥 + 4) or (3𝑥𝑥 + 2)(𝑥𝑥 − 4) • Ask the class to write down the correct answer. • Ask the class to write down why A is wrong (−2 × −4 = +8) • Ask the class to write down why C is wrong. (2 × −4) = −8) but −12 + 2 = −10

• Ask the class to write down why B is correct and explain (−2 × 4 = −8) and 12 − 2 = 10

Pupils then work through the following cards choosing the correct answer and explaining why it is correct.

2𝑥𝑥2 + 11𝑥𝑥 + 5

(2𝑥𝑥 + 5)(𝑥𝑥 + 1)

(2𝑥𝑥 + 1)(𝑥𝑥 + 5)

(2𝑥𝑥 − 1)(𝑥𝑥 − 5)

3𝑥𝑥2 − 11𝑥𝑥 + 6

(3𝑥𝑥 − 2)(𝑥𝑥 + 3)

(𝑥𝑥 − 2)(3𝑥𝑥 − 3)

(3𝑥𝑥 − 2)(𝑥𝑥 − 3)

2𝑥𝑥2 + 3𝑥𝑥 − 2

(2𝑥𝑥 + 2)(𝑥𝑥 − 1)

(2𝑥𝑥 − 1)(𝑥𝑥 − 2)

(2𝑥𝑥 − 1)(𝑥𝑥 + 2)

3𝑥𝑥2 − 11𝑥𝑥 − 4

(3𝑥𝑥 − 1)(𝑥𝑥 − 4)

(3𝑥𝑥 + 1)(𝑥𝑥 − 4)

(3𝑥𝑥 − 1)(𝑥𝑥 + 4)

2𝑥𝑥2 − 3𝑥𝑥 − 10

(𝑥𝑥 + 2)(2𝑥𝑥 − 5)

(𝑥𝑥 − 2)(2𝑥𝑥 + 5)

(𝑥𝑥 − 2)(2𝑥𝑥 − 5)

6𝑥𝑥2 − 13𝑥𝑥 + 6

(2𝑥𝑥 + 3)(3𝑥𝑥 + 2)

(2𝑥𝑥 + 3)(3𝑥𝑥 − 2)

(2𝑥𝑥 − 3)(3𝑥𝑥 − 2)

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Resources Higher: Pages 172–175

A B

C

A

B C

A

B C

A

B C

A

B C

A

B C

A

B C

45

Unit/option content

ALGEBRA



EXPRESSIONS AND FORMULAE

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)

interpret simple expressions as functions with inputs and outputs

M1 (T5)

write simple formulae and expressions from real life contexts

M1 (T5)

substitute numbers into formulae (which may be expressed in words or algebraically) and expressions

M1 (T5) use standard formulae

M6 (T5)

change the subject of a simple formula

M7 (T6)

change the subject of a formula, including cases where a power or root of the subject appears and including cases where the subject appears in more than one term

1 Pupils work individually through a set of cards rewriting the information as a formula, in words or algebraically.

A plumber charges £26 for a call out fee. He then charges £32 per hour. Write the formula for the charge for 𝑥𝑥 hours. The gas company charges a fixed charge of £36 each quarter and then 40p per unit used. Write a formula for the cost of T units in £. Joe buys a car. He pays £1200 deposit. He then pays £184 per month for M months. Write a formula for the total cost. An adult train ticket costs £9 A child train ticket costs £4.50 Write a formula for the total cost of tickets for A adults and C children.

Calculators can be used.

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Resources

46

Unit/option content

ALGEBRA



GRAPHS AND GRADIENTS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

recognise and plot equations that correspond to straight line graphs in the co-ordinate plane

M2 (T2)

find and interpret gradients and intercepts of linear graphs, for example plot and interpret the graph of hiring a car at £40 per day plus a cost of 20p per mile

M3 (T3)

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

M8 interpret the gradient at a point on a curve as the instantaneous rate of change

Multiple graphs

x 1 2 3 4 5 6 7 8 9 10 y 3 6 9 12 15 18 21 24 27 30

Use the multiples of, for example 3 to demonstrate a multiples graph. The graph drawn will be of the form y is equal to a multiple of x (y=mx), in this specific case, y=3x. This activity could be developed by asking learners to plot other multiple graphs. Although ‘m’ could represent a simple multiplicative relationship, ‘m’ could also represent the unit price of a single item. Dynamic graphing software could be used. Crossed lines Ask learners to draw the lines for example, x=2 and y=4 and identify where the two lines cross. Odd one out

Learners are presented with a set of 4 graphs each drawn on co-ordinate axes and asked to identify the odd one out. For example the graphs provided could be x = 1, x = 2, y = 3 and x = 5. Learners should explain to their partners why they chose a particular graph as being the odd one out.

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Resources Foundation GCSE Mathematics Second Edition Chapter 25 Straight-line graphs and linear graphs

47

Unit/option content

ALGEBRA




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M2 (T2)


M3 (T3)




The bespoke resources created to support the revised specification contain many ideas for teaching and learning activities that relate to the topic of ‘Graphs and Gradients’. The learning outcomes for ‘Graphs and Gradients’ promote the understanding of algebraic, tabular and graphical representations of the same data. For M2, activities that for example demonstrate ‘hire costs’ can help connect gradients of linear graphs with a constant rate of change. M1 topic work can be developed by plotting graphs of the form y is equal to a multiple of x plus a constant, y = mx +c without specifically expecting learners to connect the ‘m’ with gradient and the ‘c’ with intercept.

Graphs and statements Learners are presented with a graph drawn on co-ordinate axes and asked to decide if statements describing the graph are true or false. For example, in a graph showing how the hire cost of a car (£) relates to the distance travelled (miles) in the car, is the statement the y intercept (0, 40) represents a fixed charge of £40 true or false? Do the points (10, 42) and (20, 44) represent a cost of 20p per mile? y increases by 2 when x increases by 10?

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Resources Foundation GCSE Mathematics Second Edition Chapter 25 Straight-line graphs and linear graphs

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Unit/option content

ALGEBRA




M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M2 (T2)


M3 (T3)



[Students should know the content of Unit M1 and M2 before taking Unit M3.]

For M3 focus on multiple representations to promote the understanding of algebraic, tabular and graphical representations of the same input and output values. Dynamic graphing software can be used to demonstrate key ideas throughout this topic.

Multiple representations of for example, y = 2x +3 Ask learners to consider the algebraic relationship y = 2x +3. Create an x-y (input-output) table for values of x from x = -4 to x = 4 Direct the learners to noticing that as the x value increases by 1 the y value increases by 2. Also direct learners to notice that the y value 3 occurs when x = 0. Represent the x-y table on co-ordinate axes. Ask learners to calculate the gradient of the line and state the y intercept. Work through several more examples with learners. Ask learners to make connections between the algebraic relationship, the table, the graph and the form general form y = mx +c. Discuss observations. Graphs and statements Learners are presented with a graph drawn on co-ordinate axes and asked to decide if statements describing the graph are true or false. For example, in a graph showing y = 4x – 1 learners are asked if the statement ‘the y intercept is (0, -1)’ is true or false.

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Unit/option content

ALGEBRA



THE LANGUAGE OF ALGEBRA

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

distinguish the different roles that letter symbols play in algebra, using the correct notation

M1 (T5)

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

M3 (T6)

know the difference between an equation and an identity

Pupils work in pairs. One describes what an equation is, giving an example. The other describes what an expression is, giving an example. They then agree on a joint definition of equation and expression and produce 2 more agreed examples to be discussed at whole class level.

SS and PC

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Resources

50

Unit/option content

ALGEBRA



THE LANGUAGE OF ALGEBRA

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

distinguish the different roles that letter symbols play in algebra, using the correct notation

M1 (T5)

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

M3 (T6)

know the difference between an equation and an identity


Pupils work in pairs. One describes what an equation is, giving an example. The other describes what an identity is, giving an example. They then agree on a joint definition of equation and identity and produce 2 more agreed examples to be discussed at whole class level.

SS and PC

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Resources Higher Pages 141-144

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ALGEBRA



WORKING WITH GRAPHS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)

construct and interpret linear graphs in real world contexts

M5 (T5)

plot and interpret graphs modelling real situations, for example conversion graphs, distance/time graphs and intersecting travel graphs

Select activities from the bespoke resources created to support the revised specification. These resources contain many ideas for teaching and learning activities that relate to the topic of ‘Working with graphs’. There is a focus on multiple representations and real life contexts. Practical activities include ‘the cost of carpet’ and ‘building a wall’.

Learners could interpret graphs generated by a dynamic mathematics software package.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 25 Straight-line graphs and linear graphs Bespoke resources

52

Unit/option content

GEOMETRY AND MEASURES



ANGLE PROPERTIES CIRLCE THEOREMS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

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

M1 (T2)

understand and use alternate and corresponding angles on parallel lines

M4 (T4)

understand and use circle theorems

M5 (T1)

use the sum of angles in a triangle, for example to deduce the angle sum in any polygon

M6 (T2)

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

Tiling patterns Provide learners with examples of tiling patterns that have been formed by lines that connect or intersect a set of horizontal parallel lines. Examples could be taken from photographs, tiling magazines or sketching grids for creating tiling patterns. Explain that there are many angle rules that apply to lines that connect or intersect parallel lines. Initially ask learners to measure the angles in a tiling pattern that is made up of identical parallelograms. What do they notice? Is it necessary to measure all the angles? How many angle values are needed in order to know the sizes of all the other angles in the tiling pattern? Discuss observations. Provide learners with an information sheet that includes vocabulary such as: angles at a point; angles on a straight line; vertically opposite angles; alternate angles; and corresponding angles. Encourage learners to use mathematical vocabulary to explain the results they have noticed. Continue this activity by using other tiling patterns, fragments of tiling patterns or perhaps images of the shadows cast by posts and railings with some given angles and directing learners to use mathematical vocabulary to explain how they calculated and checked missing marked angles in the source material.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 33 Lines and angles Foundation CCEA GCSE Mathematics Second Edition Chapter 37 Parallel lines

53

Unit/option content




ANGLE PROPERTIES CIRLCE THEOREMS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

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

M1 (T2)

understand and use alternate and corresponding angles on parallel lines

M4 (T4)

understand and use circle theorems

M5 (T1)

use the sum of angles in a triangle, for example to deduce the angle sum in any polygon

M6 (T2)

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

[Students should know the content of Units M1, M2 and M3 before taking Unit M4.] In particular students should be familiar with the vocabulary: arc, centre, chord, circumference, diameter, radius, sector, segment and tangent. Investigation (Circle Theorems) Provide learners with drawings of circles with a marked centre point or Concentric Circle Graph Paper from for example, MathSphere. http://www.mathsphere.co.uk/resources/MathSphereFreeGraphPaper.htm Explain that there are many angle rules that apply to lines and shapes connected to circles. To start investigating these suggest that learners investigate angles in the same segment standing on the same chord. Ask learners to draw a chord AB joining two points on the circumference of the circle. This chord will split the circle into two segments. In the major segment ask learners to mark some points on the circumference of the circle. These points could be labelled as P, Q, R, S. Draw an angle APB. Measure the angle APB. Continue this process and measure the angles AQB, ARB, ASB. What is noticed? Discuss observations. What if the chord AB is the diameter of the circle? What if the angles are in the minor segment? What can be said about angles in the same segment and standing on the same chord? Continue to investigate other angle rules. Dynamic graphing software could be used to support this activity. As a collaborative activity learners could create a circle theorems poster.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 33 Angles in circles

http://www.mathsphere.co.uk/resources/MathSphereFreeGraphPaper.htm

54

Unit/option content




COMPOUND MEASURES AND UNITS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)

use compound measures and units such as speed, heart beats per minute and miles per gallon

M2 (T5)

use compound measures and units such as density and kg/m3

M3 NEW

use compound measures and units such as pressure and N/m2

(pressure is force divided by area)

Odd one out Present learners with a set of four statements that relate to the average speed of a journey. Three equivalent statements could be provided, for example: a car travelled 40 miles in one hour; a car travelled 10 miles in 15 minutes, a car travelled 20 miles in half an hour; and a car travelled 12 miles in 20 minutes. In pairs, learners explain their choice of the odd one out. Learners could use a graphical representation of an hour long journey to explain the odd one out. Distance, Speed, Time triangles Demonstrate how to use a distance, speed, time triangle to read the relationship between distance, speed and time in three different ways and to state the appropriate units for measuring a particular speed.

D

S T

miles

mph hr

Learners could calculate the average speeds of journeys by selecting information about distances and times from an online maps application.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 51 Compound measures

55

Unit/option content





M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)


M2 (T5)


M3 NEW



[Students should know the content of Unit M1 before taking Unit M2.] Explain keywords In pairs ask learners to explain the difference between a simple measure and a compound measure to their partner. Recall standard compound measures and appropriate standard units Ask learners to write a definition of density and include an appropriate unit in their response. Mass, density, volume triangles Demonstrate how to use a mass, density, volume triangle to read the relationship between mass, density and volume in three different ways and to state the appropriate units for measuring a particular density.

m

d V

g

g/cm3 cm3

Ask learners to calculate the density of a small block of wood and to suggest the most appropriate unit to measure the density of the small block.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 51 Compound measures

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Unit/option content





M1

M2

M3

M4

M5

M6

M7

M8


M1 (T5)


M2 (T5)


M3 NEW




Explain keywords In pairs ask learners to explain the difference between force and pressure. Recall standard compound measures and appropriate standard units Ask learners to write a definition of pressure and include an appropriate unit in their response. Force, Pressure, Area triangles Demonstrate how to use a force, pressure, area triangle to read the relationship between force, pressure and area in three different ways and to state the appropriate units for measuring pressure.

F

P A

N

N/m2 m2

Ask learners to apply their knowledge about pressure to explain why some public buildings still display signs saying ‘No stiletto heels’.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 40 Compound measures

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Unit/option content




DRAWINGS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

draw diagrams from a written description

M1 (T1)

measure line segments and angles in geometric figures

M5 (T1)

draw triangles and other 2D shapes using a ruler and a protractor

As a paired activity ask learners to take turns to describe a provided picture or screen shot of a geometric shape to their partner. Learners could be given a working with shapes notation and vocabulary list to help them with this task. Then ask learners to draw a diagram of a shape from a written description. For example the properties of a parallelogram could be listed and learners could be asked to draw and name the shape. Learners must check that each other’s drawings match the description given.

Provide learners with a range of designs and patterns that contain geometric figures. Ask the learners to select a shape from the design or pattern. Ask the learners to label and measure the line segments and angles of their chosen shape.

Provide learners with a mixture of drawings of different types of triangle. Ask learners to: label each triangle; measure the length of each side; measure all the interior angles; and record their results. Direct learners to noticing that the longest side is opposite the largest angle. What can be said about the angles if all the sides have different lengths? What can be said about the angles if all the sides have the same length? Discuss observations.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 33 Lines and angles

58

Unit/option content




MENSURATION PROBLEMS

M1 X

M2 X

M3

M4

M5 X

M6 X

M7

M8


M3 (T6)

solve mensuration problems that involve arc length and area of a sector and surface area and volume of a cylinder, cone or sphere

M4 (T6)

solve more complex mensuration problems, for example frustums


1 Pupils are given the following formulae: Cylinder Curved surface area = 2𝜋𝜋𝜋𝜋h Sphere Curved surface area = 4𝜋𝜋𝜋𝜋2 Volume = 4

3𝜋𝜋𝜋𝜋3

Cone Curved surface area = 𝜋𝜋𝜋𝜋𝜋𝜋 The following solids are put out:

A Cylinder with no top

B Cylinder with no top or bottom

C Hemisphere

D Cone The pupils have to work out the formulae to find: • total area of A • total area of B • volume of C • total area of C • total area of D

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Resources Higher: Pages 422-425, 411-416, 419-Q10 Page 421

59

Unit/option content




MENSURATION PROBLEMS

M1 X

M2 X

M3

M4

M5 X

M6 X

M7

M8


M3 (T6)

solve mensuration problems that involve arc length and area of a sector and surface area and volume of a cylinder, cone or sphere

M4 (T6)

solve more complex mensuration problems, for example frustums


1 Pupils make up a cone from heavy paper. They then cut off the top of the cone to get a frustum.

2 Using their frustum they discuss in parts: • how to find the volume of the frustum; • how to find the curved surface area of the frustum.

3 Teacher gives each pupil the frustum drawn below and formula sheet In pairs they discuss: Volume • What measurement must you find to be able to work out the volume of

the frustum? • How can you work out this measurement? They then work out the volume. Curved Surface Area • What measurements must you find to be able to work out the curved

surface area? • How can you work out these measurements? They then use calculators to find the curved surface area.

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Resources Higher: Pages 417-419, 421-422

r = 4.5cm

60

Unit/option content




PERIMETER

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T5)

calculate perimeters of triangles and rectangles and simple compound shapes made from triangles and rectangles

M1 (T2)

calculate circumferences of circles

M2 (T1) (T2)

calculate perimeters of kite, parallelogram, rhombus and trapezium

M2 (T3)

calculate perimeters of composite shapes

1 The teacher gives each pupil cut out circles, string and a ruler.

They each measure the diameter (d) of their circle and, using the string, the circumference (c). They work out 𝑐𝑐

𝑑𝑑 using a calculator.

All results for d, c, 𝑐𝑐

𝑑𝑑 for the class are recorded on a spreadsheet.

Each pupil is then asked to write down a comment on these results.

2 Each pupil is given cm squared graph paper. They are asked to draw rectangles with perimeter (i) 10 cm (ii) 14 cm (iii) 16 cm (iv) 20 cm The teacher then displays all the different rectangles drawn for each of the 4 perimeters. Pupils are asked how they did this activity, how they decided on the length and breadth each time.

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Resources Foundation: Pages 417-420, 459-463, 474-478 (perimeter only)

61

Unit/option content




PERIMETER

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T5)

calculate perimeters of triangles and rectangles and simple compound shapes made from triangles and rectangles

M1 (T2)

calculate circumferences of circles

M2 (T1) (T2)

calculate perimeters of kite, parallelogram, rhombus and trapezium

M2 (T3)

calculate perimeters of composite shapes

[Students should know the content of Unit M1 before taking Unit M2.] As a paired activity provide pupils with drawings of a variety of composite shapes. Direct learners to discuss and explain how best to find the perimeter of each composite shape. They then work out the perimeters. Follow up with class discussion looking at the different methods used each time.

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Resources Foundation: Pages 464-465

62

Unit/option content




AREA

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T5)

calculate areas of triangles and rectangles and simple compound shapes made from triangles and rectangles

M1 (T2)

calculate areas of circles

M1 (T2) (T3)

calculate surface area of cubes and cuboids

M2 (T2) (T3)

calculate areas of kite, parallelogram, rhombus and trapezium

M2 (T3)

calculate areas of composite shapes

1 The teacher gives each pupil a cut out cardboard circle divided equally into

24 equal sectors. The pupil cuts out the 24 sectors and rearranges them to form an approximate rectangle. They then work in groups to see if they can find a formula for finding the area of the rectangle and hence the area of the circle. Teacher hints if necessary: • Width of the rectangle? r • Length of the rectangle? 1

2 c

• What is 12

c? 𝜋𝜋𝜋𝜋 • Area of rectangle? 𝜋𝜋𝜋𝜋 × 𝜋𝜋 = 𝜋𝜋𝜋𝜋2 • Area of circle? 𝜋𝜋𝜋𝜋2

2 Provide pupils with cards with drawings of: triangles; squares; rectangles; and circles. The dimensions of the shapes are also marked on the drawings. For example, ask pupils to arrange shapes in groups of equal areas.

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Resources Foundation: Pgs 420–426, 465–469. 430–435

63

Unit/option content




AREA

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) (T5)

calculate areas of triangles and rectangles and simple compound shapes made from triangles and rectangles

M1 (T2)

calculate areas of circles

M1 (T2) (T3)

calculate surface area of cubes and cuboids

M2 (T2) (T3)

calculate areas of kite, parallelogram, rhombus and trapezium

M2 (T3)

calculate areas of composite shapes

[Students should know the content of Unit M1 before taking Unit M2.] As a paired activity provide pupils with drawings of a variety of composite shapes. Direct learners to discuss and explain how best to find the area of each composite shape. They then work out the areas. Follow up with class discussion looking at the different methods used each time.

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Resources Foundation: Pgs 478–482, 469–470, 474–478 (area only)

64

Unit/option content




VOLUME

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

calculate volumes of cubes and cuboids

M2 (T2)

calculate volumes of right prisms

1 Pupils given a set of different sized cubes/cuboids and asked to put them in

order of volume. They then measure the length, breadth and height for each solid and calculate the volume. They compare these with their estimation.

2 Pupils work through a series of cards showing volumes and 2 dimensions. They have to work out the missing dimension.

Cuboid volume 60 cm3 length 6 cm breadth 4 cm

Cuboid volume 24 cm3 breadth 3 cm height 4 cm

Cuboid volume 26 cm3 length 5 cm height 2 cm

Cuboid volume 8 cm3 length 2.5 cm breadth 2cm

Cuboid volume 64 cm3 length 8 cm height 5 cm

Cuboid volume 120 cm3 breadth 10 cm height 5 cm

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65

Unit/option content




VOLUME

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

calculate volumes of cubes and cuboids

M2 (T2)

calculate volumes of right prisms


1 Pupils are given a set of different sized/shaped prisms. They estimate the volumes of them and put them in order. They then take appropriate measurements to calculate the volume of each.

2 Pupils match up cards giving volume, cross sectional area and length of prisms.

8 cm → 24 cm2 → 192 cm3

10 cm → 40 cm2 → 400 cm3

1.6 cm → 20 cm2 → 32 cm3

15 cm → 60 cm2 → 900 cm3

16 cm → 80 cm2 → 1280 cm3

5.4 cm → 30 cm2 → 162 cm3

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66

Present learners with a list of Pythagorean triplets and ask them to show that the square of one of the numbers is equal to the sum of the squares of the other two numbers.

Unit/option content




RIGHT ANGLED TRIANGLES PYTHAGORAS’ THEOREM BASIC TRIGONOMETRY

M1 X

M2

M3

M4

M5 X

M6

M7

M8


M2 (T2) (T3)

use Pythagoras’ theorem in 2D problems

M3 (T3)

understand and use the trigonometric ratios of sine, cosine and tangent to solve 2D problems, including those involving angles of elevation and depression

[Students should know the content of Unit M1 before taking Unit M2.] Introduce this topic as a History of Mathematics topic. Discuss with the class how the Pythagoreans made many contributions to mathematics. For this topic focus on square numbers, Pythagorean triplets and Pythagoras’ Theorem. Ask learners to research Pythagorean triplets and Pythagoras’ Theorem. Also ask learners to find a definition for the word hypotenuse. Discuss findings. Present learners with a set of drawings of rectangles labelled ABCD with dimensions given in the table below and ask them to measure the length of the diagonals of each of the rectangles. Direct learners to notice a connection between the Pythagorean triplets and the diagonal lengths. Discuss observations. Ask learners to use their observations to predict and then measure the length of the diagonal AC which is the shortest distance (hypotenuse) between vertices A and C in a rectangle that has dimensions of 7cm and 24 cm. Discuss findings and formalise activity with further examples.

Side 1 Side 2 Diagonal (Side 1)2 (Side 2)2 (Diagonal)2

3 4 5 12 8 15

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 42 Pythagoras’ theorem

a b c a2 b2 c2

3 4 5 5 12 13 8 15 17

Note: a, b and c are lengths. a2, b2 and c2 are areas.

67

Unit/option content




RIGHT ANGLED TRIANGLES PYTHAGORAS’ THEOREM BASIC TRIGONOMETRY

M1 X

M2

M3

M4

M5 X

M6

M7

M8


M2 (T2) (T3)

use Pythagoras’ theorem in 2D problems

M3 (T3)

understand and use the trigonometric ratios of sine, cosine and tangent to solve 2D problems, including those involving angles of elevation and depression

[Students should know the content of Units M1 and M2 before taking Unit M3.] Opposite, Sine, Hypotenuse; Adjacent, Cosine, Hypotenuse; Opposite, Tangent, Adjacent triangles Demonstrate how to label the sides of a right-angled triangle as opposite, adjacent and hypotenuse. Discuss the relationships sine, cosine and tangent. Demonstrate how to use an Opposite, Sine, Hypotenuse triangle to read the relationship between opposite, sine and hypotenuse in three different ways. Demonstrate how to use an Adjacent, Cosine, Hypotenuse triangle to read the relationship between adjacent, cosine and hypotenuse in three different ways. Ask learners to explain to their partners how to use an Opposite, Tangent, Adjacent triangle to read the relationship between opposite, tangent and adjacent, in three different ways. Solve a mixture of right-angled triangle problems by using these relationships. What if the hypotenuse is equal to 1? What if the tangent is 1 Scientific calculators should be used, however, learners may benefit in the initial stages of work from using tables.

O

A

O

S H C H T A

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 35 Pythagoras and trigonometry

68

Unit/option content




SHAPE PROPERTIES

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

apply the properties and definitions of triangles including, right-angled, scalene, isosceles and equilateral

M1 (T1)

apply the properties and definitions of quadrilaterals, including square, rectangle, parallelogram, trapezium, triangles, kite and rhombus

M1 (T1)

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

M3 (T1)

Identify and apply circle definitions and properties, including tangent, arc, sector and segment.

Use dynamic mathematics software to demonstrate the properties and definitions of triangles, quadrilaterals and circles.

As a collaborative activity learners could create A3 size posters to display the properties and definitions of equilateral, isosceles, right-angled and scalene triangles. Learners could also create similar posters for quadrilaterals and for circles. In considering circles learners could research the number π and use different approximations to find the circumference and area of a circle with, for example, a diameter of 10m.

Provide learners with a set of images and statements that could be used for a matching activity to consolidate the properties of quadrilaterals.

Present learners with a variety of perimeter, area and volume problems that require the use and application of the properties and definitions of shapes in order to solve them. For example, if the widest distance across a circular pool is 10m, what is the area of the pool?

The School of Athens is a very famous Raphael painting. The painting features some of the greatest Greek mathematicians. Research the mathematicians that appear in the painting and what they discovered.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 34 Shapes Foundation CCEA GCSE Mathematics Second Edition Chapter 35 Angles in triangles and quadrilaterals

69

Unit/option content




SHAPE PROPERTIES

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

apply the properties and definitions of triangles including, right-angled, scalene, isosceles and equilateral

M1 (T1)

apply the properties and definitions of quadrilaterals, including square, rectangle, parallelogram, trapezium, triangles, kite and rhombus

M1 (T1)

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

M3 (T1)

Identify and apply circle definitions and properties, including tangent, arc, sector and segment.

[Students should know the content of Units M1 and M2 before taking Unit M3.] Provide learners with drawings of circles with a marked centre point or Concentric Circle Graph Paper from, for example, MathSphere.

http://www.mathsphere.co.uk/resources/MathSphereFreeGraphPaper.htm Ask learners to identify, mark or draw the following: arc, centre, chord, circumference, diameter, radius, sector, segment and tangent on a circle. On a unit circle ask learners to identify an arc length of π. As a collaborative activity learners could create a mini-poster to illustrate the language of circles. The poster could also include a reference to π and perimeter and area formulae for whole circles and for fractions of circles. Provide learners with a set of measurement problems that require the use and application of circle definitions and properties. For example two points A and B are placed on the circumference of a unit circle. What is the distance between them? When is the distance between the points equal to the length of the radius? Discuss any assumptions and observations.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 33 Angles in circles


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Unit/option content




WORKING WITH 2D SHAPES

M1

M2

M3

M4

M5

M6

M7

M8


M1

use conventional terms and notations such as points, lines, vertices, edges, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotational symmetries

M1 (T1)

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

Provide learners with a wide variety of images of points, lines, angles and 2D shapes and ask learners to match pre-written statements to the images. The set of statements should include all the key mathematical vocabulary relating to this topic. The images could be diagrams or screen shots from an online dynamic graphing software package. As a collaborative activity learners could write a “Working with 2D Shapes” dictionary which could include definitions and images or photographs of everyday items. Learners should be encouraged to include as much mathematical vocabulary as possible when writing descriptions of the images or photographs. This dictionary could be used as reference material when learning about other 2D Geometry and Measure topics. Alternatively learners could create a scrapbook or posters about 2D shapes. ICT could be used to support this activity.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 33 Lines and angles Foundation CCEA GCSE Mathematics Second Edition Chapter 34 Shapes Foundation CCEA GCSE Mathematics Second Edition Chapter 45 Symmetry and transformations

71

Unit/option content




WORKING WITH 3D SHAPES

M1

M2

M3

M4

M5

M6

M7

M8


M1

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

M1 (T1)

draw and interpret 2D representations of 3D shapes, for example nets, plans and elevations

Provide learners with a selection of images of 3D shapes and ask learners to match pre-written statements to the images. The set of statements should include all the key mathematical vocabulary relating to this topic. The images could be photographs of everyday objects, diagrams, or screen shots from an online dynamic mathematics software package. Demonstrate to learners how to draw blocks on isometric paper. http://www.mathsphere.co.uk/resources/MathSphereFreeGraphPaper.htm Learners should be encouraged to draw the cross-section of a block first as this approach will help with more complicated drawings. Demonstrate to learners how to identify the plan, front elevation and a side elevation for each block. Together draw the plan, the front elevation and a side elevation for a particular block on standard graph paper. Then demonstrate how to make an accurate drawing of the net of the block. What is the surface area of the block? What is the volume of the block? Discuss any observations. This 3D to 2D drawing activity could be continued by using other prisms and irregular arrangements of blocks of cubes. True or false? Provide learners with several statements relating to an image of a net. Ask learners to decide if the statements are either true or false.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 34 Shapes


72

Unit/option content




WORKING WITH MEASURES

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

understand and use metric units of measurement

M1 (T1)

make sensible estimates of a range of measures

M1 (T1)

convert metric measurements from one unit to another

M1 (T1) (T2)

solve problems involving length, area, volume/capacity, mass, time and temperature

M5 (T1)

interpret scales on a range of measuring instruments and recognise the continuous nature of measurement and the approximate nature of measurement

M5 (T5)

know and use imperial measures still in use and their approximate metric equivalents

1 Pupils fill in a sheet estimating the following:

length of a pen

width of their desk

height of the classroom door

weight of an apple

capacity of a glass of water

area of their book

They then measure these and compare with their estimates.

2 Pupils in pairs match up the following cards.

4.8 cm → 480 cm → 4800 mm

52 kg →52000 g

36 cm → 360 mm

2𝜋𝜋 → 2000 𝑚𝑚𝜋𝜋

0.94 km → 940 m → 94000 cm

1.2 kg → 1200g

480 mm

5200 g

3.6 mm

200 ml

20 ml

94 m

940000 cm

120 g

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Resources Foundation: Pages 45, 47, 355-368, 369-381

73

Unit/option content

HANDLING DATA



BOX PLOTS

M1 X

M2 X

M3

M4

M5 X

M6 X

M7

M8


M3 (T3)

calculate quartiles and inter-quartile range from ungrouped data and understand their uses

M3 (T3)

display information using box plots

[Students should know the content of Units M1 and M2 before taking Unit M3.] Provide learners with a data set of approximately 50 values. For example, learners are provided with data values from a statistical investigation about estimating skills. Ask learners to complete the following statement: 50% of the estimates are below ________. Ask learners to consider the lower 50% of the estimates. What is the median of these values? What is the median of the upper 50% of the estimates? Discuss observations. Define keywords Ask learners to define the first quartile (Q1). Ask learners to define the third quartile (Q3). Compare and contrast Ask learners to compare and contrast the range and the interquartile range. Direct learners to state advantages and disadvantages. Five number summaries Provide learners with a set of five number summaries and ask them to draw box plots to represent the information. Provide learners with screen shots each displaying either one or two box plots. The box plots should all have a context. Ask learners to read and interpret the box plots. For example, which box plot shows the least amount of spread? Why is this important?

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 45 Cumulative frequency curves and box plots

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Unit/option content

HANDLING DATA



CUMULATIVE FREQUENCY GRAPHS

M1 X

M2 X

M3

M4

M5 X

M6 X

M7

M8


M3 (T3)

construct and interpret cumulative frequency tables and the cumulative frequency curve

M3 (T3)

estimate the median, quartiles and interquartile range

[Students should know the content of Units M1 and M2 before taking Unit M3.] Provide learners with a set of cumulative frequency data tables. Demonstrate to learners how to extract information from the tables. For example, the grouped frequency table below shows the estimates of a group of people who were asked to estimate the size of an angle (400). How many people were in the group? How many estimates were less than 400? Which class contains the median estimate? Discuss responses and address any misconceptions. Ask the learners if the estimates are close to the true value?

Est. Angle (x) Frequency (f) Cum. Freq.

20 ≤ x < 25 3 3 25 ≤ x < 30 2 5 30 ≤ x < 35 6 11 35 ≤ x < 40 5 16 40 ≤ x < 45 7 23 45 ≤ x < 50 5 28 50 ≤ x < 55 1 29 55 ≤ x < 60 0 29

Graphs and Statements An graph is provided and several statements which relate to it. Learners have to decide which statements are true and which statements are false. For example, learners could be given a copy of a cumulative frequency diagram and asked to decide which statement out of a set of statements correctly identifies the median and the interquartile range. This could be a paired activity.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 45 Cumulative frequency curves and box plots

75

objects per cm

Unit/option content

HANDLING DATA



HISTOGRAMS

M1 X

M2 X

M3 X

M4

M5 X

M6 X

M7 X

M8

M4 Topic note also M8

M4 (T4)

construct and interpret histograms for grouped continuous data with unequal class intervals

[Students should know the content of Units M1, M2 and M3 before taking Unit M4.] Frequency density triangle Demonstrate how to use a frequency, frequency density, class width triangle to read the relationship between frequency, frequency density and class width in three different ways and to state appropriate units for measuring frequency density, for example on a histogram that shows the length of different objects in cm the frequency density units will be objects per cm. Ask learners to explain the difference between frequency and frequency density.

f

fd c

No of objects

cm

Histograms and Statements A histogram generated by a dynamic graphing software package is provided and several statements which relate to it. Learners have to decide which statements are true and which statements are false. For example learners could be given a set of statements about: the frequency of each class; the type of data represented; the mean of the distribution; or the median of the distribution. Are the statements true or false? Learners should justify responses. Discuss any misconceptions.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 46 Histograms and sampling

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Unit/option content

HANDLING DATA



MEAN, MODE, MEDIAN AND RANGE

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

find mean, median, mode and range for ungrouped data and understand their uses

M1 (T2)

calculate mean from an ungrouped frequency table and identify the mode and median

M2 (T2)

estimate mean from a grouped frequency distribution

M2 (T2)

identify the modal class and the median class from a grouped frequency distribution

M3 (T3)


M3 (T3)


Define keywords

From memory ask learners to write a definition of a keyword related to this topic. For example write a definition for the keyword ‘range’.

Explain keywords

In pairs use a small data set to explain the difference between the keywords ‘mean’ and ‘median’ to your partner.

The Handling Data cycle (Data Analysis)

Analyse the data collected as part of a statistical investigation about estimation skills.

Learners could for example, analyse ‘How many sweets are there in the jar?’ guesses, or the responses from a survey about estimating the length of a line and the size of an angle.

ICT

Use a spreadsheet to carry out a ‘what would happen if’ analysis, for example, by asking the question ‘what would happen to the mean if one of the pieces of data had a different value?’.

Use a spreadsheet to model the calculation of a mean from an ungrouped frequency table.

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Resources Foundation GCSE Mathematics Second Edition Chapter 53 Statistical averages and spread

77

Unit/option content

HANDLING DATA



MEAN, MODE, MEDIAN AND RANGE

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1)

find mean, median, mode and range for ungrouped data and understand their uses

M1 (T2)

calculate mean from an ungrouped frequency table and identify the mode and median

M2 (T2)

estimate mean from a grouped frequency distribution

M2 (T2)

identify the modal class and the median class from a grouped frequency distribution

M3 (T3)


M3 (T3)


[Students should know the content of Unit M1 before taking Unit M2.] Compare and Contrast Calculate the mean for a data set that contains, for example, 50 pieces of data. Represent the data in a grouped frequency table. Estimate the mean of the data contained in the grouped frequency table. Compare and contrast the results and the methods. An example of an appropriate set of data could be the responses that 50 people give when they are asked to estimate the length of a line and the size of an angle. The learners should collect the data used and interpret the results of their calculations. For example, on average are the estimates close to the correct measurement? Are the length estimates better than the angle estimates? Worked examples and repeated practice Present several worked examples on, for example, estimating the range and mean and identifying the median and modal class/es from a grouped frequency table. Set pupils a series of similar problems to solve using the knowledge and understanding modelled in the worked examples. Ask the pupils to reflect on how their solutions were improving as they worked through the set of problems.

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Resources Foundation GCSE Mathematics Second Edition Chapter 53 Statistical averages and spread

78

Unit/option content

HANDLING DATA



SAMPLING

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

understand what is meant by a sample and a population

M1 (T2)

understand simple random sampling and the effect of sample size on the reliability of conclusions

M1 (T2)

identify possible sources of bias

M3 (T2)

infer properties of populations or distributions from a sample and know the limitations of doing so

M4 (T4)

understand and use stratified sampling techniques

Define keywords From memory ask learners to write a definition of a keyword related to this topic area. For example write a definition for the keyword ‘sample’. Write a definition for the keyword ‘simple random sample’. Explain keywords Ask learners to explain the effect of sample size on the reliability of conclusions to their partner. Learners should refer to at least one collected data set.

The Handling Data cycle (planning – sample selection)

Comment on the selection of a sample of people who took part in a statistical investigation about estimation skills. For example, from a population size of 75 (48 females and 27 males), a sample size of 50 is selected (25 females, 25 males). Is the sample representative?

Investigate a real world problem using the statistical problem solving process. For example: A school would like to find out on average how far people travel to attend an open evening at the school. Define the population and give advice on selecting a representative sample.

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Resources Foundation GCSE Mathematics Second Edition Chapter 52 Data and questionnaires

79

Unit/option content

HANDLING DATA



SAMPLING

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M1 (T2)


M1 (T2)


M3 (T2)


M4 (T4)


[Students should know the content of Units M1 and M2 before taking Unit M3.] Define keywords From memory ask learners to write a definition of a keyword related to this topic area. For example write a definition for the keyword ‘population’. Write a definition for the keyword ‘infer’. Explain keywords Ask learners to explain the difference between the keywords deduce and infer. Learners should refer to population data and sample data .

The handling data cycle

Present summary data from a statistical investigation to the learners and ask them to comment on the results. For example, as part of a statistical investigation a sample of 25 females and 25 males were asked to estimate the size of an angle (500). If for example, the summary data indicates that the males in the sample were better at estimating the size of the angle does this finding suggest that in general males are better at estimating angles than females? Discuss opinions. A context for carrying out this investigation could be estimating a pie chart proportion or the size of a cut piece of cake or pizza.

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Resources (Higher CCEA GCSE Mathematics Second Edition Chapter 46 Histograms and sampling)

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Unit/option content

HANDLING DATA



SAMPLING

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)


M1 (T2)


M1 (T2)


M3 (T2)


M4 (T4)


[Students should know the content of Units M1, M2 and M3 before taking Unit M4.] Define keywords From memory ask learners to write a definition of a keyword related to this topic area. For example write a definition for the keyword ‘stratification’. Write a definition for the keyword ‘sampling fraction’. Sampling fraction triangle Demonstrate how to use a sampling fraction triangle to read the relationship between sample size, sampling fraction and population size in three different ways.

s

f P

Provide learners with a data table showing for example, the numbers of males and females employed in different departments within an organisation. A 10% representative is required. Ask learners to calculate the number of men and women that should be selected for the sample. Discuss any assumptions and difficulties. Display results of calculations.

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Resources Higher CCEA GCSE Mathematics Second Edition Chapter 46 Histograms and sampling

s sample size

f sampling fraction

P Population size

81

Unit/option content

HANDLING DATA



SCATTER GRAPHS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

plot and interpret scatter diagrams and recognise correlation

M2 (T2)

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

M2 (T2)

draw conclusion from scatter diagrams

M2 (T2)

use terms such as positive correlation, negative correlation and little or no correlation

M2 (T2)

interpolate and extrapolate from data and know the dangers of doing so

M2 (T2) identify outliers

M2 (T2)

appreciate that correlation does not imply causality

One variable at a time Learners record the height of 10 people in their class and display the results on a horizontal number line. This type of representation could be referred to as a number line plot.

The learners then record the arm span of the same 10 people and plot each person’s height and arm span on coordinate axes. Each point plotted (height, arm span) represents the height and arm span of one of the 10 people. This type of representation should be referred to as a scatter diagram.

The scatter chart tool on a spreadsheet could also be used for this activity.

Ask learners to interpret the scatter diagram. Is there an association between height and arm span? If so, describe the association between the two variables. Discuss responses and formalise activity by introducing the vocabulary of correlation.

Embed this practical activity into a statistical investigation. An appropriate question to investigate could be for example, does arm span depend on height? Learners could collect, present and analyse data in order to draw a conclusion about a possible relationship between height and arm span.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 55 Statistical diagrams 2

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Unit/option content

HANDLING DATA



SCATTER DIAGRAMS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T2)

plot and interpret scatter diagrams and recognise correlation

M2 (T2)

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

M2 (T2)

draw conclusion from scatter diagrams

M2 (T2)

use terms such as positive correlation, negative correlation and little or no correlation

M2 (T2)

interpolate and extrapolate from data and know the dangers of doing so

M2 (T2) identify outliers

M2 (T2)

appreciate that correlation does not imply causality

[Students should know the content of Unit M1 before taking Unit M2.] Graphs and Statements An graph is provided and several statements which relate to it. Learners have to decide which statements are true and which statements are false. For example learners could be given a screen shot of a scatter diagram generated by a dynamic mathematics software package and asked to decide which statement out of a set of statements correctly identifies the type of correlation displayed in the scatter diagram. Learners could also be asked to decide if a point is an outlier.

A graph is provided and learners have to interpret the graph. For example a scatter diagram with a line of best fit drawn is given to learners. The learners have to use the line of best fit to estimate a value slightly beyond the data given in the scatter diagram. Learners should justify their answer to their partner or group. Responses should be monitored by the teacher.

Learners could be given a scatter diagram that displays the results of an investigation about height and arm span and asked if the correlation suggests causality.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 55 Statistical diagrams 2

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Unit/option content

HANDLING DATA



THE HANDLING DATA CYCLE DATA COLLECTION (DATA ANALYSIS) DATA INTERPRETATION

M1 M2 M3 M4 M5 M6 M7 M8


M1 (T1)

understand and use the handling data cycle to solve problems

M1 (T2)

design an experiment or survey to test hypotheses

M1 (T1)

design data collection sheets, distinguishing between different types of data

M1 (T1)

look at data to find patterns and exceptions

M1 (T2)

compare distributions and make inferences

Define keywords From memory ask learners to write a definition of a keyword related to these topic areas. For example write a definition for the keyword ‘hypothesis’. Write a definition for the keyword ‘quantitative data’. Explain keywords Ask learners to explain the difference between discrete and continuous data to their partner. Learners should refer to several data sets . The Handling Data cycle (Pose, Collect, Analysis, Interpret)

Ask learners to outline the stages of the handling data cycle and explain how this cycle could be used to investigate a real world problem, for example, a school would like to find out on average how far people travel to attend an open evening at the school. A modelling diagram could be used to structure responses.

Learners could investigate the hypothesis ‘Females are more accurate in their estimates of length and angle than males’. This activity should include collecting data, analysing results and interpreting findings. Learners could be provided with a keywords list covering all aspects of the handling data cycle and directed to select appropriately from this vocabulary in a report about their findings. Why are the findings useful?

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Resources

84

Unit/option content

HANDLING DATA



USING STATISTICAL DIAGRAMS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) NEW

sort, classify and tabulate qualitative (categorical) data and discrete or continuous quantitative data, including the use of 2 circle Venn diagrams to sort data

M1 (T1)

extract data from printed tables and lists

M1 (T1)

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

M1 (T1) (T2) NEW

construct and interpret a wide range of graphs and diagrams including frequency tables and diagrams, pictograms, bar charts, pie charts, line graphs, frequency trees and flow charts, and draw conclusions, recognising that graphs maybe misleading

M2 NEW

use 3 circle Venn diagrams to sort data

Bespoke resources The bespoke resources created to support the revised specification contain many ideas for teaching and learning activities that relate to the topics of ‘frequency tree diagrams’ and ‘Venn diagrams’. There is a focus on multiple representations and real life contexts. Examples are also given to show how frequency tree diagrams lead into: counting and listing outcomes and probability tree diagrams. Questions are set in practical and mathematical contexts.

Graphs and Statements

A graph is provided and several statements which relate to it. Learners have to decide which statements best describe the information presented in the graph. For example, a bar chart is given to learners along with several True/False statements that relate to the chart. Learners have to decide the truth or falsehood of the statements given.

Images and statements

An image is provided and several statements which relate to it. Learners have to decide which statements are true and which statements are false. For example learners could be given a completed frequency tree diagram or 2 circle Venn diagram to interpret.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 52 Data and questionnaires Foundation CCEA GCSE Mathematics Second Edition Practice Book Chapter 52 Data and questionnaires Foundation CCEA GCSE Mathematics Second Edition Chapter 54 Statistical diagrams 1 Foundation CCEA GCSE Mathematics Second Edition Chapter 55 Statistical diagrams 2

85

Unit/option content

HANDLING DATA



USING STATISTICAL DIAGRAMS

M1

M2

M3

M4

M5

M6

M7

M8


M1 (T1) NEW

sort, classify and tabulate qualitative (categorical) data and discrete or continuous quantitative data, including the use of 2 circle Venn diagrams to sort data

M1 (T1)

extract data from printed tables and lists

M1 (T1)

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

M1 (T1) (T2) NEW

construct and interpret a wide range of graphs and diagrams including frequency tables and diagrams, pictograms, bar charts, pie charts, line graphs, frequency trees and flow charts, and draw conclusions, recognising that graphs maybe misleading

M2 NEW

use 3 circle Venn diagrams to sort data

[Students should know the content of Unit M1 before taking Unit M2.] Bespoke resources

The bespoke resources created to support the revised specification contain many ideas for teaching and learning activities that relate to the topic of ‘Venn diagrams’. Examples for M2 build on the knowledge developed in M1.

Demonstrate to learners how to solve the following problem: 300 commuters were asked what kind of transport they used to get to work in the morning. 5 people said they used train, car and bus. 9 people used a train and a car, 30 used a train and a bus, 7 used a car and a bus. Altogether 61 used a train, 98 used a car and 147 used a bus. How many people used neither a train, car nor bus to get to work?

Learners could also use Venn Diagrams to group a list of numbers into sets. For example, group 4, 6, 8, 9, 16, 25, 27, 36, 49 and 64 into sets for even, square and cube numbers.

Images and Statements

An image is provided and several statements which relate to it. In pairs learners have to decide which statements are true and which statements are false. For example learners could be given a 3 circle Venn diagram displaying information about tea, coffee and juice sales to interpret.

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Resources Foundation CCEA GCSE Mathematics Second Edition Chapter 52 Data and questionnaires

CCEA GCSE Mathematics · 2019-09-26 · CCEA GCSE Mathematics Planning Framework for Templates for...

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