7410156 IGCSE Maths TB title.indd 1
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Contents Introduction
2 5
Number Multiples of whole numbers 6 Factors of whole numbers 8 Prime numbers 10 Square numbers and cube numbers 12 Products of prime numbers 14 HCF and LCM 16
2 Fractions and percentages 2.1 2.2 H 2.3 2.4 2.5 2.6 2.7 2.8 2.9 H 2.10 H 2.11
Equivalent fractions 18 Fractions and decimals 20 Recurring decimals 22 Percentages, fractions and decimals 24 Calculating a percentage 26 Increasing or decreasing quantities by a percentage 28 Expressing one quantity as a percentage of another 30 Reverse percentage 32 Interest and depreciation 34 Compound interest problems 36 Repeated percentage change 38
3 The four rules 3.1 3.2 3.3 3.4 3.5
Order of operations Choosing the correct operation Finding a fraction of a quantity Adding and subtracting fractions Multiplying and dividing fractions
40 42 44 46 48
7.1 7.2 7.3 7.4 7.5
Ratio Speed Density and pressure Direct proportion Proportional variables
76 78 80 82 84
8 Approximation and limits of accuracy 8.1 8.2 8.3 8.4 8.5 H 8.6
Rounding whole numbers Rounding decimals Rounding to significant figures Approximation of calculations Upper and lower bounds Upper and lower bounds for calculations
86 88 90 92 94 96
9 Standard form 9.1 Standard form 9.2 Calculating with standard form H 9.3 Solving problems
98 100 102
10 Applying number and using calculators 10.1 10.2 10.3 10.4 10.5 10.6
Units of measurement Converting between metric units Reading scales Time Currency conversions Using a calculator efficiently
104 106 108 110 112 114
Algebra 11 Algebra and formulae
4 Directed numbers 4.1 4.2 4.3 4.4
Introduction to directed numbers Everyday use of directed numbers The number line Adding and subtracting directed numbers 4.5 Multiplying and dividing directed numbers
50 52 54
11.1 11.2 11.3 H 11.4
56
12 Algebraic manipulation
58
5 Squares, cubes and roots 5.1 Squares and square roots 5.2 Cubes and cube roots H 5.3 Surds
60 62 64
6 Set language and notation 6.1 Inequalities 6.2 Sets
70 72 74
7 Ratio, proportion and speed
1 Number 1.1 1.2 1.3 1.4 1.5 1.6
6.3 Venn diagrams H 6.4 More notation H 6.5 Practical problems
66 68
The language of algebra Substitution into formulae Rearranging formulae More complicated formulae
12.1 12.2 12.3 12.4 H 12.5
Simplifying expressions Expanding brackets Factorisation Expanding two brackets Multiplying more complex expressions 12.6 Quadratic factorisation
H 12.7 Factorising ax2 + bx + c
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116 118 120 122 124 126 128 130 132 134 136
Contents H 12.8 More than two brackets H 12.9 Algebraic fractions
138 140
13 Solutions of equations 13.1 13.2 H 13.3 13.4 H H H
H H
Solving linear equations 142 Setting up equations 144 More complex equations 146 Solving quadratic equations by factorisation 148 13.5 More factorisation in quadratic equations 150 13.6 Solving quadratic equations by completing the square 152 13.7 Solving quadratic equations by the quadratic formula 154 13.8 Simple simultaneous equations 156 13.9 More complex simultaneous equations 158 13.10 Linear and non-linear simultaneous equations 160
14 Graphs in practical situations 14.1 Conversion graphs 14.2 Travel graphs 14.3 Speed–time graphs
162 164 166
15 Straight line graphs 15.1 15.2 15.3 15.4 H 15.5 H 15.6 H 15.7
Using coordinates Drawing straight line graphs More straight line graphs The equation y = mx + c Finding equations Parallel and perpendicular lines Graphs and simultaneous equations
168 170 172 174 176 178
18 Indices 18.1 Using indices H 18.2 Negative indices 18.3 Multiplying and dividing with indices H 18.4 Fractional indices H 19.1 Direct proportion H 19.2 Inverse proportion
H 16.5 Graphs of sin x, cos x and tan x H 16.6 Transformations of graphs
190 192
184 186 188
17 Integer sequences 17.1 Number sequences 194 17.2 The nth term of a sequence 196 17.3 Finding the nth term of an arithmetic sequence 198 H 17.4 The sum of an arithmetic sequence 200
210 212
20 Inequalities and regions 20.1 H 20.2 20.3 20.4 H 20.5
Linear inequalities Quadratic inequalities Graphical inequalities More than one inequality More complex inequalities
214 216 218 220 222
21 Functions H H H H H
21.1 Function notation 21.2 Domain and range 21.3 Inverse functions 21.4 Composite functions 21.5 More about composite functions
224 226 228 230 232
22 Calculus H H H H
22.1 The gradient of a curve 22.2 More complex curves 22.3 Turning points 22.4 Motion of a particle
234 236 238 240
Geometry and trigonometry 23 Angle properties
16 Graphs of functions 182
206 208
19 Direct and inverse proportion
180
16.1 Quadratic graphs H 16.2 Solving equations with quadratic graphs H 16.3 Other graphs H 16.4 Estimating gradients
202 204
H H H H
23.1 Angle facts 23.2 Parallel lines 23.3 Angles in a triangle 23.4 Angles in a quadrilateral 23.5 Regular polygons 23.6 Irregular polygons 23.7 Tangents and chords 23.8 Angles in a circle 23.9 Cyclic quadrilaterals 23.10 Alternate segment theorem 23.11 Intersecting chords
242 244 246 248 250 252 254 256 258 260 262
24 Geometrical terms and relationships 24.1 24.2 24.3 24.4
Edexcel International GCSE Maths Teacher Guide
Measuring and drawing angles Bearings Congruent shapes Similar shapes
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264 266 268 270
Contents H 24.5 Areas of similar triangles H 24.6a Areas and volumes of similar shapes H 24.6b Areas and volumes of similar shapes
272
274
29 Vectors
276
H 29.1 Introduction to vectors H 29.2 Using vectors H 29.3 The magnitude of a vector
25 Geometrical constructions 25.1 Constructing shapes 25.2 Bisectors 25.3 Scale drawings
278 280 282
26 Trigonometry 26.1 26.2 26.3 26.4
H H H H H H
Pythagoras’ theorem 284 Trigonometric ratios 286 Calculating angles 288 Using sine, cosine and tangent functions 290 26.5 Which ratio to use 292 26.6 Solving problems using trigonometry 294 26.7 Angles of elevation and depression 296 26.8 Problems in three dimensions 298 26.9 Sine, cosine and tangent of obtuse angles 300 26.10a The sine rule 302 26.10b The cosine rule 304 26.11 Using sine to find the area of a triangle 306
27 Mensuration 27.1a Perimeter and area of a rectangle 308 27.1b Perimeter and area of a rectangle 310 27.2 Area of a triangle 312 27.3 Area of a parallelogram 314 27.4 Area of a trapezium 316 27.5a Circumference and area of a circle318 27.5b Circumference and area of a circle320 27.6 Surface area and volume of a cuboid 322 27.7 Volume of a prism 324 27.8 Volume and surface area of a cylinder 326 H 27.9 Arcs and sectors 328 H 27.10 Volume and surface area of a cone 330 H 27.11 Volume and surface area of a sphere 332
28 Symmetry 28.1 Lines of symmetry
28.2 Rotational symmetry 28.3 Symmetry of special two-dimensional shapes
334
336 338 340 342 344
30 Transformations 30.1 30.2 30.3 30.4 30.5 30.6
Translations Reflections Further reflections Rotations Further rotations Enlargements
346 348 350 352 354 356
Statistics and probability 31 Statistical representation 31.1 31.2 31.3 31.4 H 31.5
Frequency tables Pictograms Bar charts Pie charts Histograms
358 360 362 364 366
32 Statistical measures 32.1 32.2 32.3 32.4 32.5 32.6 32.7 H 32.8 H 32.9
The mode The median The mean The range Which average to use Using frequency tables Grouped data Measuring spread Cumulative frequency diagrams
368 370 372 374 376 378 380 382 384
33 Probability 33.1 The probability scale 33.2 Calculating probabilities 33.3 Probability that an event will not happen 33.4 Addition rule for probabilities 33.5 Probability from data 33.6 Expected frequency 33.7 Combined events H 33.8 Tree diagrams
386 388 390 392 394 396 398 400
Resource sheets 402 Answers to the examination sections from the Student Book 411
Edexcel International GCSE Maths Teacher Guide
Š HarperCollinsPublishers Ltd 2016
Welcome to the Teacher Guide for Collins Edexcel International GCSE Maths, which has been written to support the latest Edexcel International GCSE in Mathematics (Specification A) (9-1) 4MA1 syllabus. The Teacher Guide has lesson plans to accompany all the topics in the Student Book. It also includes the answers to the examination sections in the Student Book.
Lesson plans Each topic in the Student Book (1.1, 1.2, 1.3, etc.) is supported by a double-page lesson plan, and each one follows the same format making it easy to use and prepare lessons. The following sections of the plan help in preparing lessons:
Specification references show which parts of the course are delivered in the lesson.
Key words highlight important mathematical vocabulary.
Prior knowledge highlights the underpinning maths that students need for the lesson.
Learning objectives indicate clearly what the students should master during the lesson and are a useful way of measuring its success. Cross references to the relevant Student Book pages make it easy to integrate the book into class teaching. Common mistakes and remediation are pinpointed so they can be quickly recognised and rectified. Useful tips help students remember key concepts easily.
These are followed by suggestions for the structure of the lesson: Starter ideas involve the whole class and give you ideas on how to capture attention and interest of students.
Main lesson activities help you lead students into exercise questions. Plenaries offer guidance on how to round off the three-part lessons.
Resource sheets mentioned in some of the lesson plans are towards the back of this book. You can request all of the lesson plans and resource sheets in Word format so that they can be edited to suit the needs of individual classes or departmental schemes of work by emailing education@harpercollins.co.uk with ‘Edexcel International GCSE Maths Teacher Pack request’ in the subject line.
Scheme of Work Download a 2-year scheme of work that gives one possible teaching order in an editable spreadsheet format from www.collins.co.uk.
Edexcel International GCSE Maths Teacher Guide © HarperCollinsPublishers Ltd 2016
1.1
Multiples of whole numbers
Specification references Statement
Foundation
1.1G 1.1H
Use the terms odd and even numbers, multiples. Identify common multiples.
Collins references
Student Book page 8
Learning objectives
Find multiples of whole numbers. Recognise multiples of numbers.
Key words
multiple, common multiple, even, odd
Prior knowledge Students will need to know the multiplication tables to 10 × 10.
Common mistakes and remediation A common mistake is to fail to interpret the functional aspect of the topic correctly, e.g. if there are 10 people getting in taxis each holding four people, students may give answers of two and a half taxis required or two taxis with people left behind. Explain to students that they need to think carefully about their answer.
Useful tips Encourage students to remember some simple multiplication rules: multiples of 2 always end in 0, 2, 4, 6, or 8; multiples of 3 have digits that add up to a multiple of 3; multiples of 5 always end in 0 or 5; multiples of 9 have digits that add up to a multiple of 9; and multiples of 10 always end in 0.
Starter
Ask students, around the class, to take turns to increase an amount by a given number, e.g. the first person may say, “Six,” the next, “Twelve,” the next, “Eighteen,” and so on. Continue until someone makes a mistake. Go beyond the 10 × 10 multiplication tables for more able students.
Move on to counting down, e.g. start at 60 and count down in sixes.
Draw ‘multipillars’ of different tables. These also make good displays of the tables.
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Edexcel International GCSE Maths Teacher Guide
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Lesson 1.1 Multiples of whole numbers
Main lesson activity
Remind students that multiples are the answers that appear in multiplication or times tables.
For the 10 times table, the answer always ends in 0, numbers in the five times table always end in either 0 or 5, numbers in the two times table are always even. The digits in multiples of three always add up to a multiple of three, the digits in multiples of nine always add up to a multiple of nine. Multiples of six are always even numbers and the digits add up to a multiple of three. Multiples of four are even when divided by 2.
Make sure that students know how to use a calculator to find multiples; e.g. for multiples of 5 press 5 then = then +5 then repeatedly press =, and the multiples will be displayed.
Now display this table of numbers and, as students give answers, write the answers on the board.
Ask students to give you the multiples of 2. (34, 48, 102, 470, 630, 876, 1000)
Repeat for multiples of 4 and 9.
Encourage students to look at the numbers in various multiplication tables and try to identify patterns or rules.
Ask students to give you the multiples of 3. (48, 102, 123, 501, 630, 876) Ask students to pick out the multiples of 6 from the lists above. (48, 102, 630, 876) Ask students to give you the multiples of 5. (55, 470, 630, 1000) Ask students to give you the multiples of 10. (470, 630, 1000). Point out they could use the multiples of 2 and 5. More able students could identify the multiple of 18 as multiples of both 2 and 9 (630). They can then do Exercise 1A.
Plenary
Put the number 392 on the board. Ask the students which of the numbers, from one to ten, this number is a multiple of. Clearly two is one answer. Half of 392 is 196 so four is also a multiple, but what about other numbers? Three and nine are not answers as 3 + 9 + 2 = 14 is not a multiple of three. Also five and ten are not answers as the number does not end in 0 or 5.
Use calculators to test whether it is a multiple of seven (it is). Repeat with further examples, such as 630 or 720. Use smaller numbers within the 10 × 10 multiplication table for less able students, e.g. 48, 63 and 72.
Edexcel International GCSE Maths Teacher Guide
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7
1.2
Factors of whole numbers
Specification references Statement
Foundation
1.1G 1.1H
Use the term factor. Identify common factors.
Collins references
Student Book pages 9 – 10
Learning objectives
Identify the factors of a number. Identify common factors.
Key words
factor, common factor, factor pair
Prior knowledge Students will need to know the multiplication tables to 10 × 10.
Common mistakes and remediation Students often miss out one and the number itself. Remind them always to begin with one. Many students will mix up the terms multiple and factor. Relate the word multiple to multiplications.
Useful tips Remind students that factors always include 1 and the number itself.
Starter
Ask students to draw rectangles, each with an area of 12 cm2. Look at the different possibilities and list the lengths of the sides. Repeat with other areas.
Show students how to draw factor bugs and let them complete their own bugs for different numbers. Check that students have always included the number itself and 1 as factors.
Now ask students to find the number between 20 and 30 that has eight factors. (24 → 1, 2, 3, 4, 6, 8, 12 and 24)
For more able students, ask further questions, e.g. “How many factors has 36?” (9 → 1, 2, 3, 4, 6, 9, 12, 18, 36)
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Edexcel International GCSE Maths Teacher Guide
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Lesson 1.2 Factors of whole numbers
Main lesson activity
Make sure that students realise that factors always come in pairs, so when they have found one factor they can find the other. The exception to this is when they are finding factors of square numbers, where one pair is the same number repeated.
Draw out the fact that every number has 1 and itself as a factor pair.
Work through Example 1.
Introduce the terms factor pair and common factor. Explain to students that, in order not to miss any factors, it is sensible to start with 1 and work upwards, obtaining the factor pairs. Using this method, when they have tried all the numbers up to half the number itself, students will have found all the factors. Ask students to make a table as shown. 1 factor
3 factors
4 factors
5 factors
6 factors
Under each column, enter numbers from 1 to 20, e.g. 10 has four factors so is entered in that column. 1 factor 1
2 factors
2 factors 2, 3, 5, 7, 11, 13, 17, 19
3 factors 4, 9
4 factors 6, 8, 10, 14, 15
5 factors 16
6 factors 12, 18, 20
Students can now do Exercise 1B. When using a calculator to find factors, encourage less able students particularly to work systematically.
Plenary
Give students a number, e.g. 48. Ask them to write it as a multiplication, e.g. 6 × 8. Carry on breaking down the separate values as multiplications (do not allow 1×), e.g. 2 × 3 × 8, then 2 × 3 × 2 × 4, then 2 × 3 × 2 × 2 × 2. Ask students how they can use 2 × 3 × 2 × 2 × 2 to obtain factors, e.g. 2 × 2 × 2 × 2 = 16 so the factor pair is 3 and 16. Repeat with 20 (2 × 2 × 5), 60 (2 × 3 × 2 × 5), 100 (2 × 2 × 5 × 5), etc. Finally, write the product 1 × 2 × 3 × 5 × 7 = 210 on the board and ask students to give all the factors. (1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210).
Edexcel International GCSE Maths Teacher Guide
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9
1.3
Prime numbers
Specification references Statement
Foundation
1.1G
Use the term prime number.
Collins references
Student Book page 10
Learning objectives
Identify prime numbers.
Key words
prime number
Prior knowledge Students will need to know the multiplication tables to 10 × 10 and the prime numbers up to 50.
Common mistakes and remediation Sometimes students confuse prime numbers with odd numbers. They frequently also include 1 and forget 2. Regular practice at reciting the first few prime numbers should help to avoid these errors.
Useful tips Stress that 1 is not defined as a prime number as it only has one factor.
Starter
Ask students to give numbers that have only two factors. Explain that any number with exactly two factors, itself and 1, is called a prime number. Tell students that there are 25 prime numbers under 100.
Main lesson activity
Tell students that they are now going to do an activity that will result in them finding all the prime numbers less than 100.
Ask students to draw a 100 number square as shown, and to put a cross through the number 1.
Draw a circle around the number 2 and then cross out all the multiples of 2. Draw a circle around the next number that hasn’t been crossed out (In this case 3) and cross out all the multiples of that number.
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Edexcel International GCSE Maths Teacher Guide
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Lesson 1.3 Prime numbers
Repeat the process until all the numbers in the table are either circled or crossed out. The numbers circled are the prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97).
More able students can go beyond 100 to see how many primes they can find. (Primes up to 200: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191,193, 197, 199).
Explain that to show that a number is not prime, all students need to do is find a factor that is not the number itself (1 is a factor of all numbers). For example 432 is not prime as it is even. The only even prime number is 2.
Less able students may need to be reminded of the rules for divisibility by 2, 3 and 5. Students can now complete Exercise 1C.
Plenary
Check students’ understanding by playing ‘True and false’. Say a number and ask students to put their thumbs up if it is prime and down if it is not prime.
Ask students to use whiteboards to write down a factor of the number if it is not prime; e.g. for 49, the student has to display 7. Gradually increase the numbers to make the plenary more challenging for the students.
Edexcel International GCSE Maths Teacher Guide
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11
1.4
Square numbers and cube numbers
Specification references Statement
Foundation
1.4A
Identify square numbers and cube numbers. Calculate squares and cubes.
1.4B
Collins references
Student Book pages 11 – 13
Learning objectives
Identify square numbers and cube numbers. Use a calculator to find the square and the cube of a number.
Key words
square, square number, cube, cube number
Prior knowledge Students will need to know the multiplication tables to 10 × 10.
Common mistakes and remediation Students may multiply the number by two instead of squaring it. Encourage students to write out the calculation in full to avoid this error, e.g. 3.22 = 3.2 × 3.2 = 10.24. When using a calculator less able students often type –32 and then do not realise that –9 is an incorrect answer. Show students that they need to type (–3)2.
Useful tips It may be helpful to display a chart of the squares, up to 152: 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81, 102 = 100, 112 = 121, 122 = 144, 132 = 169, 142 = 196, 152 = 225. Many of the questions in these two exercises will have alternative methods of solution. Discuss these with the students so that they can select their preferred method.
Starter
Put the sequence 1, 4, 9, 16, 25 on the board. Ask the students if they can give the next two terms (36, 49). Ask them how the pattern is building up. They may say that it goes up by 3, 5, 7, 9, . . . but make sure they eventually spot that it is 1 × 1, 2 × 2, 3 × 3, etc. Carry on the sequence as far as possible without using a calculator. Now try the ‘Brainwashing’ game. This involves asking the students a simple multiplication but when they give the answer they also have to say the answer to an agreed square number. For example, if the agreed brainwash is ‘14 squared is equal to 196’ ask, “What is 8 × 8?” The student has to reply, “8 × 8 = 64 and 14 squared is equal to 196.” Ask the next student, “What is 6 × 6?” and this student has to reply, “6 × 6 = 36 and 14 squared is equal to 196.” Students will soon remember the answer to 14 squared.
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Edexcel International GCSE Maths Teacher Guide
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Lesson 1.4 Square numbers and cube numbers
Main lesson activity
When any integer is multiplied by itself, the result is a square number.
Write on the board: 52 = 5 × 5 = 25
Square numbers form square patterns when they are drawn as arrays of dots. The short way of writing ‘squared’ is to put a small 2 after and above a number. For example, 12 squared is written as 122. 502 = 50 × 50 = 2500 5002 = 500 × 500 = 250 000
Ask students to give the answer to 50002 (5000 × 5000 = 25 000 000)
Ask students to describe the pattern. Then ask for other square numbers and continue the same pattern. For example, ask for 82, 802, 8002. Cover all square numbers up to 15 × 15.
Less able students could start by just looking at the first 10 square numbers, while more able students could go beyond 15 × 15.
Now show students the calculator key for working out square numbers:
Show students the calculator key for working out cube numbers.
Introduce the term ‘cube number’. Work through the text at the bottom of page 11 of the Student Book. Students can now do Exercise 1D. Exercise 1E is a summary of the exercises covered so far: multiples, factors, prime numbers and square numbers. It differentiates by asking questions covering more than one topic. Less able students may need to be reminded of the basic facts before attempting the exercise. More able students should not need such help.
Plenary
Put the following on the board: 1 = 1
Ask if they can fill in the missing square number for: 1 + 3 + ... + 17 + 19 = ? (100 as the value is ((19 + 1) ÷ 2)2.
1+3=4
1+3+5=9
Ask students if they can continue the pattern, e.g.: 1 + 3 + 5 + 7 = 16 Ask students to describe what is on each side of the equals sign. (Sum of consecutive odd numbers, square numbers.)
Now ask students to give the answers to, for example, 8 × 8 followed by 7 × 9 (64 and 63).
Ask students if they can see the pattern.
Repeat for similar calculations, e.g. 6 × 6 followed by 5 × 7 (36 and 35) or 15 × 15 followed by 14 × 16 (225 and 224).
Edexcel International GCSE Maths Teacher Guide
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13
1.5
Products of prime numbers
Specification references Statement
Core
1.1H 1.4D
Identify prime factors. Express integers as the product of powers of prime factors.
Collins references
Student Book pages 14 – 15
Learning objectives
Identify prime factors.
Key words
prime factor, product
Prior knowledge Students should understand what a factor is. Students should know what a prime number is.
Common mistakes and remediation A common mistake, when students are using the division method, is to leave out the last factor when rewriting the answer in index notation. Encourage students to check their answer by multiplying the factors together and checking the answer is the original number.
Useful tips It may be helpful to display the important facts. A factor is a number that divides into another number without a remainder. A prime number has only two factors: itself and 1.
Starter
Students work in pairs, with two whiteboards. They write ‘Prime’ on one whiteboard.
Repeat for a variety of numbers.
Say a number. Students write the number on the plain whiteboard and list all its factors beneath it. If the number is prime, they hold up both whiteboards. Otherwise, they hold up the whiteboard with the number and its factors. For example, if the number is 5, they write this on the whiteboard, with ‘1, 5’ below it, and hold it up alongside the ‘Prime’ board. If the number is 6, they write this on the whiteboard with ‘1, 2, 3, 6’ below it.
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Edexcel International GCSE Maths Teacher Guide
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