DAY 4
DAY 3
DAY 2
DAY 1
Revision Planner Page
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Topic
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Time
Completed
4 6 8 10 12 14 16
15 minutes 15 minutes 15 minutes 15 minutes 20 minutes 20 minutes 20 minutes
Prime Factors, HCF and LCM Fractions and Recurring Decimals Indices Standard Index Form Surds Upper and Lower Bounds Formulae and Expressions
Page 18 20 22 24 26 28
Time 15 minutes 15 minutes 15 minutes 20 minutes 20 minutes 20 minutes
Topic Brackets and Factorisation Equations 1 Equations 2 Quadratic and Cubic Equations Completing the Square Simultaneous Linear Equations
Date
Time
Completed
Page 30 32 34 36 38 40
Time 20 minutes 15 minutes 20 minutes 15 minutes 25 minutes 25 minutes
Topic Algebraic Fractions Sequences Inequalities Straight-line Graphs Curved Graphs Solving a Linear and a Non-linear Equation Simultaneously
Date
Time
Completed
Page 42 44 46 48 50 52
Time 20 minutes 20 minutes 15 minutes 20 minutes 15 minutes 20 minutes
Topic More Graphs and Other Functions Functions and Transformations Repeated Percentage Change Reverse Percentage Problems Ratio and Proportion Proportionality
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Time
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DAY 5 DAY 6 DAY 7
Page 54 56 58 60 62 64 66 68
Time 15 minutes 20 minutes 15 minutes 15 minutes 15 minutes 15 minutes 20 minutes 20 minutes
Topic Similarity and Congruency 1 Similarity and Congruency 2 Loci Angles Translations and Reflections Rotation and Enlargement Circle Theorems Pythagoras’ Theorem
Date
Time
Completed
Page 70 72 74 76 78 80
Time 20 minutes 20 minutes 25 minutes 20 minutes 20 minutes 30 minutes
Topic Trigonometry Trigonometry in 3D Sine and Cosine Rules Arc, Sector and Segment Surface Area and Volume Vectors
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Time
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Page 82 84 86 88 90 92
Time 15 minutes 20 minutes 15 minutes 15 minutes 25 minutes 20 minutes
Topic Probability Sets and Venn Diagrams Statistical Diagrams Averages Cumulative Frequency Graphs Histograms
Date
Time
Completed
94
Answers
110
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
15 Minutes
A fraction is part of a whole number. The top number is the numerator and the bottom number is the denominator.
Division ÷
There are four rules of fractions.
Before starting, write out whole or mixed numbers as improper fractions (also known as top-heavy fractions).
Addition + You need to change the fractions so that they have the same denominator. Example 5+1 9 7 = 35 + 9 63 63 = 44 63
The lowest common denominator is 63, since both 9 and 7 go into 63. Remember to add only the numerators and not the denominators.
Example
213 ÷ 127 =7÷9 3 7
Convert to top-heavy 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.
Subtraction –
Fraction Problems
You need to change the fractions so that they have the same denominator.
You may need to solve problems involving fractions.
Example 4–1 5 3 = 12 – 5 15 15 = 7 15
The lowest common denominator is 15.
Example 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? Give your answer as a fraction in its simplest form.
Remember to subtract only the numerators and not the denominators.
1+1 3 5
This is a simple addition of fractions question.
Multiplication ×
5 + 3 = 15 15
Write the fractions with a common denominator.
Before starting, write out whole or mixed numbers as improper fractions (also known as top-heavy fractions).
8 = 15
Example 2×4 7 5 = 2×4 7×5 8 = 35
8 1 – 15
The question asks for the fraction of the money that is left, so 8 from 1. subtract 15
= 7 15
The fraction is in its simplest form.
Multiply the numerators together. Multiply the denominators together.
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Changing Recurring Decimals to Fractions
SUMMARY
Recurring decimals are rational numbers because we can change them to fractions.
To add or subtract fractions, write them using the same denominator.
Examples •• 1. Change 0.15 to a fraction in its lowest terms.
To multiply fractions, multiply the numerators and multiply the denominators.
x = 0.151 515 …
Let
To divide fractions, take the reciprocal of the second fraction and multiply the fractions together.
then 100x = 15.151 515 …
When multiplying and dividing fractions, write out whole or mixed numbers as top-heavy fractions before you begin the calculation.
Multiply by 10n, where n is the length of the recurring pattern. Subtract equation from equation .
Recurring decimals are rational numbers, so they can be changed to fractions.
– 99x = 15 x = 15 99
QUESTIONS
•• x= 5 Check 5 ÷ 33 = 0.15 33 • 2. Change 0.37 into a fraction.
QUICK TEST 1. Work out the following:
x = 0.3777 … 10x = 3.7777 … Multiply by 10, so that the part that is not recurring is in the units position. 100x = 37.7777 … Multiply equation by 100. Subtract equation from equation . – 90x = 34 x = 34 90 x = 17 45
Always check to see if your fraction simplifies.
a. 2 + 1 3 5
b. 2 6 – 1 7 3
c. 2 × 5 d. 3 ÷ 22 9 7 11 27 2. Change the following recurring decimals into fractions. Write each fraction in its simplest form. • • • • a. 0.7 b. 0.2 15 c. 0.35
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] •• 4 2. Prove that 0.36 is equivalent to 11 [3 marks]
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DAY 1
15 Minutes
An index is sometimes called a power.
a
The base
b
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 n
m
n–m
a ÷a =a
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 a1 = a –n
a a
1 m
= a1n m
= √ a
7 2 9 c. 3 ×103 = 310 = 3–1 3 3 9 –10 d. 7 ÷ 7 = 719
4. Evaluate: a. 33 = 3 × 3 × 3 = 27 b. 70 = 1 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.
The indices are added.
c. (4x3)2 = 16x6
QUESTIONS
Remember to square the 4 as well.
QUICK TEST 3 2
3
If in doubt, write it out: (4x ) = 4x × 4x = 16x6
3
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
c. (52)3
d. 64 3
2
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 2 4 b. 14a b = 2ab3 7ab
9x2y × 2xy3 18x3y4 c. = 6xy 6xy = 3x2y3
b. 1210 ÷ 12–3
2. Simplify the following:
2. Simplify: 4
a. 63 × 65
2.
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]
Simplify: 4 7 a. x ×15x x 4
b. 3x × 34x 2x
[2 marks] 2
[2 marks]
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DAY 1
15 Minutes
Standard index form (standard form) is useful for writing very large or very small numbers in a simpler way. When written in standard form a number will be written as:
A number between 1 and 10 1 a 10
On a Calculator To put a number written in standard form into your calculator you use the following keys:
×10x
a × 10n The value of n is the number of places the digits have to be moved to return the number to its original value.
If the number is 10 or more, n is positive.
EXP or EE
For example, (2 × 103) × (6 × 107) = 1.2 × 1011 would be keyed in as: 2
×10x
3
×
6
×10x
2
EXP
3
×
6
EXP
or
7 7
= =
If the number is less than 1, n is negative. If the number is 1 or more but less than 10, n is zero.
Examples 1. Write 2 730 000 in standard form.
Doing Calculations Examples Work out the following using a calculator. Check that you get the answers given here. 1. (6.7 × 107)3 = 3.0 × 1023 (2 s.f.)
2.73 is the number between 1 and 10 (1 2.73 10) Count how many spaces the digits have to move to restore the original number. The digits have moved 6 places to the left because it has been multiplied by 106
9 • 2. (4 × 104)2 = 4.4 (3 × 10 )
3.
(5.2 × 106) × (3 × 107) = 884.4 (1 d.p.) (4.2 × 105)2
2 .7 3 2 7 3 0 0 0 0 So, 2 730 000 = 2.73 × 106 2. Write 0.000 046 in standard form. Put the decimal point between the 4 and 6, so the number lies between 1 and 10. Move the digits five places to the right to restore the original number. The value of n is negative. So, 0.000 046 = 4.6 × 10–5
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SUMMARY
Examples On a non-calculator paper you can use indices to help work out your answers.
Numbers in standard form will be written as a × 10n.
1. (2 × 103) × (6 × 107)
1 a 10
= (2 × 6) × (103 × 107)
n is positive when the original number is 10 or more.
= 12 × 103+7
n is negative when the original number is less than 1.
= 12 × 1010 = 1.2 × 101 × 1010
n is zero when the original number is 1 or more but less than 10.
= 1.2 × 1011 2. (6 × 104) ÷ (3 × 10–2) = (6 ÷ 3) × (104 ÷ 10–2)
QUESTIONS
= 2 × 104–(–2) = 2 × 106 3. (3 × 104)2
QUICK TEST 1. Write in standard form: a. 64 000
= (3 × 104) × (3 × 104)
b. 0.000 46
= (3 × 3) × (104 × 104) = 9 × 108
2.
Work out the following. Leave in standard form. a. (3 × 104) × (4 × 106)
You also need to be able to work out more complex calculations.
b. (6 × 10–5) ÷ (3 × 10–4) 3. Work these out on a calculator: a. (4.6 × 1012) ÷ (3.2 × 10–6)
Example The mass of Saturn is 5.7 × 1026 tonnes. The mass of the Earth is 6.1 × 1021 tonnes. How many times heavier is Saturn than the Earth? Give your answer in standard form, correct to 2 significant figures.
b. (7.4 × 109)2
EXAM PRACTICE 1. a. Write 40 000 000 in standard form. [1 mark] b. Write 6 × 10–5 as an ordinary number. [1 mark]
26
5.7 × 10 = 93 442.6 6.1 × 1021 Now rewrite your answer in standard form. Saturn is 9.3 × 104 times heavier than the Earth.
2.
The mass of an atom is 2 × 10–23 grams. What is the total mass of 7 × 1016 of these atoms? Give your answer in standard form.
[3 marks]
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