DAY 4
DAY 3
DAY 2
DAY 1
Revision Planner Page 4 6 8 10 12 14 16
Time 15 minutes 20 minutes 15 minutes 15 minutes 20 minutes 20 minutes 20 minutes
Topic Prime Factors, HCF and LCM Fractions and Decimals Rounding and Estimating Indices Standard Index Form Formulae and Expressions 1 Formulae and Expressions 2
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Page 18 20 22 24 26 28
Time 15 minutes 15 minutes 20 minutes 20 minutes 20 minutes 15 minutes
Topic Brackets and Factorisation Equations 1 Equations 2 Simultaneous Linear Equations Sequences Inequalities
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Page 30 32 34 36 38
Time 20 minutes 25 minutes 20 minutes 20 minutes 20 minutes
Topic Straight-line Graphs Curved Graphs Percentages Repeated Percentage Change Reverse Percentage Problems
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Page 40 42 44 46 48 50
Time 15 minutes 20 minutes 20 minutes 15 minutes 15 minutes 15 minutes
Topic Ratio and Proportion Proportionality Measurement Interpreting Graphs Similarity 2D and 3D Shapes
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DAY 5 DAY 6 DAY 7
Page 52 54 56 58 60 62
Time 20 minutes 20 minutes 15 minutes 15 minutes 15 minutes 20 minutes
Topic Constructions Loci Angles Bearings Translations and Reflections Rotation, Enlargement and Congruency Pythagoras’ Theorem
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64
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Page 66 68
Time 20 minutes 20 minutes
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20 minutes 15 minutes 20 minutes 20 minutes
Topic Trigonometry Trigonometry and Pythagoras Problems Circles, Arcs and Sectors Surface Area and Volume 1 Surface Area and Volume 2 Vectors
70 72 74 76
Page 78 80 82 84 86 88 90
Time 15 minutes 20 minutes 20 minutes 15 minutes 15 minutes 15 minutes 20 minutes
Topic Probability Tree Diagrams Sets and Venn Diagrams Statistical Diagrams Scatter Diagrams and Time Series Averages 1 Averages 2
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92
Answers
106
Glossary
Questions marked with the symbol
should be attempted without using a calculator.
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DAY 1
15 Minutes
Prime Factors
Highest Common Factor (HCF)
Apart from prime numbers, any whole number greater than 1 can be written as a product of prime factors. This means the number is written using only prime numbers multiplied together.
The highest factor that two numbers have in common is called the HCF.
A prime number has only two factors, 1 and itself. 1 is not a prime number. The prime numbers up to 20 are:
2, 3, 5, 7, 11, 13, 17, 19
Example Find the HCF of 60 and 96. Write the numbers as products of their prime factors. 60 = 2 × 2
×3×5
96 = 2 × 2 × 2 × 2 × 2 × 3 The diagram below shows the prime factors of 60.
Ring the factors that are common. 60 = 2 × 2
60
×3×5
96 = 2 × 2 × 2 × 2 × 2 × 3 2
These give the HCF = 2 × 2 × 3
30
= 12 2
15
3
5
Divide 60 by its first prime factor, 2. Divide 30 by its first prime factor, 2. Divide 15 by its first prime factor, 3. We can now stop because the number 5 is prime. As a product of its prime factors, 60 may be written as: 60 = 2 × 2 × 3 × 5 or in index form 60 = 22 × 3 × 5
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Lowest (Least) Common Multiple (LCM)
SUMMARY
The LCM is the lowest number that is a multiple of two numbers.
Any whole number greater than 1 can be written as a product of its prime factors, apart from prime numbers themselves (1 is not prime).
Example Find the LCM of 60 and 96.
The highest factor that two numbers have in common is called the highest common factor (HCF).
Write the numbers as products of their prime factors. 60 = 2 × 2
×3×5
The lowest number that is a multiple of two numbers is called the lowest (least) common multiple (LCM).
96 = 2 × 2 × 2 × 2 × 2 × 3 60 and 96 have a common factor of 2 × 2 × 3, so it is only counted once. 60 = 2 × 2
QUESTIONS
×3×5
96 = 2 × 2 × 2 × 2 × 2 × 3 The LCM of 60 and 96 is 2×2×2×2×2×3×5
QUICK TEST 1. Write these numbers as products of their prime factors:
= 480
a. 50
b. 360
c. 16
2. Decide whether these statements are true or false: a. The HCF of 20 and 40 is 4. b. The LCM of 6 and 8 is 24. c. The HCF of 84 and 360 is 12. d. The LCM of 24 and 60 is 180.
EXAM PRACTICE 1. Find the highest common factor of 120 and 42. [3 marks] 2.
Buses to St Albans leave the bus station every 20 minutes. Buses to Hatfield leave the bus station every 14 minutes. A bus to St Albans and a bus to Hatfield both leave the bus station at 10 am. When will buses to both St Albans and Hatfield next leave the bus station at the same time? [3 marks]
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DAY 1
20 Minutes
A fraction is part of a whole number. The top number is the numerator and the bottom number is the denominator. Fractions can be cancelled by dividing the numerator and denominator by a common factor: ÷3
Addition +
12 = 4 15 5 ÷3
You need to change the fractions so that they have the same denominator. Example The lowest common denominator is 5+1 63 since both 9 and 7 go into 63. 9 7 = 35 + 9 63 63 Remember to add only the numerators = 44 and not the denominators. 63
Subtraction – You need to change the fractions so that they have the same denominator. Example 4–1 5 3 The lowest common denominator is 15. = 12 – 5 15 15 Remember to subtract only the = 7 numerators and not the denominators. 15
Division ÷ Before starting, write out whole or mixed numbers as improper fractions. Example 21 ÷ 12 3 7 =7÷9 3 7
Convert to improper fractions.
=7×7 3 9
Take the reciprocal of the second fraction and multiply both fractions.
= 49 27 = 1 22 27
Rewrite the fraction as a mixed number.
Reciprocals The reciprocal of a number x is a a x For example, the reciprocal of 47 is 74
Decimals and Fractions To change a fraction into a decimal, divide the numerator by the denominator, either by short division or by using a calculator. To change a decimal into a fraction, write the decimal as a fraction with a denominator of 10, 100, etc. (look at the last decimal place to decide) and then cancel.
Multiplication ×
Examples 2 = 2 ÷ 5 = 0.4 5
Before starting, write out whole or mixed numbers as improper fractions (also known as top-heavy fractions).
0.23 = 23 100
The last decimal place is ‘hundredths’ so the denominator is 100.
0.165 = 165 = 33 1000 200
Example 2×4 7 5 = 2×4 7×5 = 8 35
1 = 1 ÷ 8 = 0.125 8
Multiply the numerators together. Multiply the denominators together.
Decimals that never stop and have a repeating pattern are called recurring decimals. All fractions give either terminating or recurring decimals.
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SUMMARY
Examples 1 = 0.333 3333… usually written as 0.3. 3 5 = 0.454 545 45… usually written as 0.4.5. 11 4 = 0.571 428 571… usually written as 0.5. 71 428. 7
To add or subtract fractions, write them using the same denominator. To multiply fractions, multiply the numerators and multiply the denominators. To divide fractions, take the reciprocal of the second fraction and multiply the fractions together.
Fraction Problems You may need to solve problems involving fractions.
When multiplying and dividing fractions, write out whole or mixed numbers as improper fractions before you begin.
Examples 1. A school has 1400 pupils. 740 pupils are boys.
Decimals that never stop and have a repeating pattern are recurring decimals.
3 1 5 of the boys and 4 of the girls study French.
Work out the total number of pupils in the school who study French. Work out the 3 × 740 = 444 boys 5 number of boys study French who study French. 1400 – 740 = 660 are girls 1 × 660 = 165 girls 4 study French
Work out the number of girls in the school.
444 + 165 = 609 pupils study French 2. Charlotte’s take-home pay is £930. She gives her mother 13 of this and spends 15 of the £930 on going out. What fraction of the £930 is left?
QUESTIONS QUICK TEST 1. Work out the following: a. 2 + 1 3 5 c. 2 × 5 9 7 2. Work out the following:
Give your answer as a fraction in its simplest form. 1+1 3 5
This is a simple addition of fractions question.
= 5 + 3 15 15
Write the fractions with a common denominator.
= 8 15 1– 8 15
You need the fraction of the money 8 from 1. that is left, so subtract 15
= 7 15
The fraction is in its simplest form.
b. 2 6 – 1 7 3 d.
3 ÷ 22 11 27
a. 2 1 + 3 1 2 5
b. 2 7 – 1 1 10 9
c. 3 1 × 2 5 15
d. 5 1 ÷ 3 4 8
EXAM PRACTICE 1. In a magazine 37 of the pages have advertisements on them. Given that 12 pages have advertisements on them, work out the number of pages in the magazine. [2 marks] 2. Rosie watches two television programmes. The first programme is 34 of an hour and the second is 2 2 3 hours long. Work out the total length of the two programmes. [3 marks] 3.
Place these fractions and decimals in order of size, smallest first: 3 8 675 [2 marks] 4 0.4 0.85 10 1000 7
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DAY 1
15 Minutes
Rounding to the Nearest 10, 100 and 1000
Significant Figures
When rounding you must always look at the digit in the next place value column. If the digit is 5 or more, round up. If the digit is 4 or less, you round down.
The first significant figure (s.f.) is the first digit that is not a zero. The 2nd, 3rd… significant figures follow on after the first significant figure. They may or may not be zeros.
For example, 2469.1 is:
The same rules apply as in decimal places.
2469 to the nearest whole number 2470 to the nearest 10 2500 to the nearest 100 2000 to the nearest 1000.
6347 = 6350 (3 s.f.)
Decimal Places
The digit is bigger than 5, so the 4 rounds to a 5.
You must fill in the end zero(s). This is often forgotten.
When rounding numbers to a given number of decimal places (d.p.), count the number of places to the right of the decimal point, then look at the next digit on the right.
Examples 1. Round 9.3156 to…
If the number is 5 or bigger, round up. If the number is 4 or smaller, the digit stays the same.
a. 3 s.f. = 9.32
The 5 has the effect of rounding the 1 to a 2.
b. 2 s.f. = 9.3
The 1 is less than 5, so the 3 does not change.
c. 1 s.f. = 9
The 3 is less than 5, so to 1 s.f. it is 9, since 9.3156 is nearer to 9 than 10.
2.3725 = 2.373 (3 d.p.) The digit is 5 so round up the 2.
The 2 rounds up to a 3.
Examples Round the following to the number of decimal places specified in the brackets. 1. 4.6931 (2 d.p.) = 4.69
2. 27.325 (2 d.p.) = 27.33
The 3 is less than 5, so the 9 does not change. The 5 has the effect of rounding the 2 to a 3.
The 7 has the 3. 149.3867 (3 d.p.) = 149.387 effect of rounding the 6 to a 7. 4. 271.74 (1 d.p.) = 271.7
2. Round 0.735 to… a. 2 s.f. = 0.74
The 5 has the effect of rounding the 3 to a 4.
b. 1 s.f. = 0.7
The 3 is less than 5, so the 7 does not change.
The 4 is less than 5, so has no effect on the 7.
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Estimating When estimating the answer to a calculation, you must round each number to 1 significant figure. 273 × 49 ≈ 300 × 50 = 500 28 30 ≈ means approximately equal to If a measurement is accurate to some given amount, then the true value lies within half a unit of that amount. The upper bound is the maximum possible value the measurement could have been. The lower bound is the minimum possible value the measurement could have been. If the weight (w) of a cat is 8.3 kg to the nearest tenth of a kilogram, then the weight would lie between 8.25 kg and 8.35 kg.
SUMMARY To round or correct to a given number of decimal places (d.p.), count that number of decimal places to the right of the decimal point. Look at the next digit on the right. If it is 5 or more you need to round up. Otherwise the digit stays the same. When measurements are given to a certain degree of accuracy: – highest possible value = upper bound – lowest possible value = lower bound For any number, the first significant figure is the first number that is not a zero. The 2nd, 3rd... significant figures follow on after the first significant figure. They may or may not be zeros.
The limits of accuracy can be written using inequalities as shown: 8.25 w 8.35 Lower bound
Upper bound
QUESTIONS QUICK TEST 1. Put a ring around the correct answer. 3724 rounded to 2 significant figures is: 3800
37
38
3700
2. Decide whether the following statements are true or false: a. 4625 rounded to 3 s.f. is 4630 b. 2.795 rounded to 1 d.p. is 2.7 c. 0.00527 rounded to 2 s.f. is 0.0053 d. 37 062 has 4 significant figures
EXAM PRACTICE 1. Work out an estimate for: 306 × 2.93 0.051
[3 marks]
2. The weight of a book is 28 grams to the nearest gram. Write down the lower bound of the weight of the book. [1 mark]
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DAY 1
15 Minutes
An index is sometimes called a power.
ab
The base
The index or power
Examples with Numbers 1. Simplify the following, leaving your answers in index notation. a. 52 × 53 = 52+3 = 55
Laws of Indices
b. 8–5 × 812 = 8–5+12 = 87
The laws of indices can be used for numbers or algebra. The base has to be the same when the laws of indices are applied.
c. (23)4 = 23×4 = 212 2. Evaluate:
Evaluate means to work out.
a. 42 = 4 × 4 = 16 b. 50 = 1
an × am = an+m an ÷ am = an–m
c. 3–2 = 12 = 1 9 3 1 2 d. 36 = √ 36 = 6 2
3 e. 8 3 = (√ 8 )2 = 22 = 4
3. Simplify the following, leaving your answers in index form.
(an)m = an×m
a. 72 × 75 = 77 b. 69 ÷ 62 = 67
a0 = 1
7 2 9 c. 3 ×103 = 310 = 3–1 3 3 d. 79 ÷ 7–10 = 719
a1 = a
4. Evaluate:
a–n = a1n
b. 70 = 1
a
1 m
m
= √ a
a. 33 = 3 × 3 × 3 = 27
1
3 c. 64 3 = √ 6 4 = 4 1
d. 81 2 = √ 8 1 = 9 1 1 e. 5–2 = 52 = 25 = 1 ( 49 ) = ( 94 ) = 81 16 5 16 –2
f.
a
n m
m
2
= ( √ a)n
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SUMMARY
Examples with Algebra 1. Simplify the following:
Make sure you know and can use all the laws of indices.
a. a4 × a–6 = a4–6 = a–2 = 12 a
A negative power is the reciprocal of the positive power.
b. 5y2 × 3y6 = 15y8
Fractional indices mean roots. The numbers are multiplied. c. (4x3)2 = 16x6
The indices are added.
QUESTIONS
Remember to square the 4 and multiply the indices.
QUICK TEST 3 2
3
3
If in doubt, write it out: (4x ) = 4x × 4x = 16x
6
1. Simplify the following, leaving your answers in index form.
d. (3x4y2)3 = 27x12y6 or 3x4y2 × 3x4y2 × 3x4y2 = 27x12y6 e. (2x)–3 = 1 3 = 1 3 (2x) 8x
7
b. 1210 ÷ 12–3
c. (52)3
d. 64 3
2
2. Simplify the following:
2. Simplify: 4
a. 63 × 65
a. 2b4 × 3b6
b. 8b–12 ÷ 4b4
c. (3b4)2
d. (5x2y3)–2
11
a. 15b ×2 3b = 45b2 = 9b9 5b 5b 2 4
b. 16a b3 = 4ab 4ab
EXAM PRACTICE 1.
3. Simplify: a. 7a2 × 3a2b = 21a4b
Evaluate: a. 50
[1 mark]
b. 7–2
[1 mark]
c.
1 1 64 3 × 144 2
[2 marks]
d.
–2 27 3
[2 marks]
2 4
b. 14a b = 2ab3 7ab 9x2y × 2xy3 18x3y4 = c. 6xy 6xy = 3x2y3
2.
Simplify: 4 7 a. x ×15x x 4
[2 marks] 2
b. 3x × 34x 2x
[2 marks]
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