GCSE Maths Revision Notes
Common General Algebraic Errors:
(𝑎 + 𝑏)2 → 𝑎2 + 𝑏2. Writing out the bracket twice we actually find (𝑎 + 𝑏)(𝑎 + 𝑏) = 𝑎2 + 2𝑎𝑏 + 𝑏2
Similarly
2+3𝑥+2
√𝑎2 + 𝑏2 → 𝑎 + 𝑏. You can see this is not true when 𝑎 = 3, 𝑏 = 4 for example.
𝑥 → 3𝑥+2. When ‘cancelling’ fractions, we can only divide, whereas in this example we’ve incorrectly
𝑥2−4 −4
(𝑥+1)(𝑥+2) 𝑥+1
subtracted 𝑥2. If we factorised the example, it would be OK to cancel to because we have indeed
(𝑥+2)(𝑥−2) 𝑥−2
divided by 𝑥 + 2.
𝑥(𝑥 − 1) → 𝑥2 − 1. Oops!
𝑥 + 𝑎 = 𝑦 → 𝑥 + 𝑎 = 3𝑦. The 𝑎 hasn’t been multiplied by 3.
3
𝑐 − 𝑏(𝑎 − 𝑏) → 𝑐 − 𝑎𝑏 − 𝑏2. Sign error at the end.
𝑥(𝑥 + 1) − (𝑥 + 2)2 → 𝑥2 + 𝑥 − 𝑥2 + 4𝑥 + 4 . A lack of brackets when subtracting expanded expression leads to
sign errors. See (53i).
𝑎 + 3𝑥 = 𝑏 → 3𝑥 = 𝑏 + 𝑎 Sign not changed when 𝑎 moved to other side of equation.
𝑥+2𝑏 = 𝑦 → 𝑥 = 𝑦 − 2𝑏
or √𝑥 + 2𝑏 = 𝑦 → √𝑥 = 𝑦 − 2𝑏 (2𝑏 is trapped inside fraction/root so we have to
3 3
deal with the ÷ 3 and √ first)
√𝑥 = 2𝑥 → 𝑥 = 2𝑥2 When 2𝑥 is squared, you get 4𝑥2 not 2𝑥2 as 2𝑥 × 2𝑥 = 4𝑥2
1
, Number
Integers and Decimals
1
1. Understand and order Test Your Understanding: Order the following: 3 , 0.33, 0.303, 30%
integers and decimals
2. Use brackets and the BIDMAS is actually (B)(I)(DM)(AS), i.e. Division and Multiplication have the same
hierarchy of operations ‘priority’, and Addition and Subtraction have the same priority. When you have say a
(BIDMAS) mix of addition and subtraction, evaluate left-to-right. E.g. 9 − 3𝑥 + 5𝑥 simplifies to
9 + 2𝑥 NOT 9 − 8𝑥: in the latter you had done the addition first, when there was no
reason to do so.
Note that due to BIDMAS, negative numbers to a power require bracketing: −42
would produce -16 on a calculator because it does the square (‘indices’) first. You
want (−4)2. This is highly important for the 𝑏2 in the Quadratic Formula when
substituting numbers in.
3. Add, subtract, multiply and When multiplying two decimals, first multiply them as if they were whole numbers,
divide integers, negative then put the decimal point back in the result by counting the number of jumps in
numbers and decimals decimal point in the original numbers.
When subtracting negative numbers, ensure numbers are lined up in the units
column, and fill in any ‘gaps’ with 0s.
We like dividing by whole numbers. Hence if you’re dividing by a decimal, multiply
both numbers by 10 until you’re dividing by a whole number, e.g. 3.678 ÷ 0.09 →
367.8 ÷ 9
Test Your Understanding:
a. 3 − (−4)?
b. 396 × 48?
c. 5.7 − 2.89?
d. (−7.1)2?
e. 508.97 ÷ 1.1
4. Understand and use positive This just means that you understand −5 + 7 for example as starting at -5 on a number
numbers and negative integers, line and ‘moving’/translating 7 up. This won’t specifically be tested.
both as positions and
translations on a number line
5. Round whole numbers to the nearest, 10, 100, 1000, …
6. Round decimals to Be wary that any 0s after the first non-zero digit count as significant. e.g. 3.40204 to 3sf is
appropriate numbers of 3.40, NOT 3.4. Similarly it is 3.40 to 2dp, not 3.4. But 0.0020413 is 0.00204 to 3sf.
decimal places or significant Note that 3.9853 to 1dp is 4.0.
figures
Test Your Understanding: Write the following to the indicated number of significant
figures or decimal places:
a. 24703 to 2sf
b. 15.0849 to 1dp
c. 25.969403 to 3sf
d. 495.18473 to 3dp
7. Multiply and divide by any You should appreciate that multiplying by a number between 0 and 1 makes it smaller,
number between 0 and 1 and dividing by it makes it bigger. You should recognise that ÷ 0.5 is the same as × 2 and
÷ 0.25 the same as × 4 and so on (see Fractions): this is particularly useful in estimation
(see below)
8. Check their calculations by For estimation questions, the general rule of thumb is to round each number to 1
rounding, eg 29 31 30 30 significant figure, unless it is close to some other nice number (such as 0.26 → 0.25
because it’s a quarter), e.g:
4.98 × 31 5 × 30
≈ = 300
0.49 0.5
Test Your Understanding: Estimate the following.
79×6.89
a. 0.52
4.03×4.94
b.
0.24
9. Check answers to a division Not tested as such.
sum using multiplication eg use
inverse operations
10. Multiply and divide whole numbers by a given multiple of 10
11. Put digits in the correct place in a decimal number
2
,Fractions
45
12. Find equivalent fractions Test Your Understanding: Put 54 in its simplest form.
and write a fraction in its
simplest form.
13. Compare the sizes of The strategy is to find a common denominator for all fractions, so that we can just easily
5 3 20 21 3
fractions compare the numerators, e.g. 7 and 4 can be converted to 28 and 28, thus 4 is larger.
Sometimes the ordering will be obvious by thinking of the fractions on a number line.
4 5
Test Your Understanding: Which of 5 and 7 is bigger?
3
14. Find fractions of an amount Test Your Understanding: Find 7 of 35
15. Convert between mixed For mixed number to improper fractions, to get the new numerator times the whole part
numbers and improper with the denominator and add the numerator. The denominator stays the same. e.g
2 17
fractions 35 → 5
For improper fractions to mixed numbers, see how many times the denominator goes into
39 4
the numerator and find the remainder also. 7 → 5 7
16. Add and subtract fractions The ‘foolproof’ way is to cross-multiply: multiply the two denominators, then times each
3 1 12+7 19
numerator by the other fraction’s denominator and add. e.g. 7 + 4 = 28 = 28.
1 4 3 4 7
However, sometimes you only need to change one of the fractions, e.g. 3 + 9 → 9 + 9 = 9
Test Your Understanding:
4 3
a. 9 + 5
4 1
b. 17 − 2
1 4
c. 5 4 + 4 5
17. Multiply and divide If any whole numbers, put over 1: 4 → 4. If any mixed numbers, convert to improper
1
fractions including mixed fractions first. When multiplying, just multiply numerators and denominators separately.
numbers. When dividing, ‘flip’ (reciprocate) the second fraction and instead multiply.
Test Your Understanding:
3 4
a. 5
×9
4
b. 25 × 3
1 1
c. 14 ÷25
1
d. 4÷15
Factors, Multiples, Primes, Roots, Powers
18. Identify factors, multiples Ensure you don’t confuse factors and multiples. Factors of 6: 1, 2, 3, 6. Multiples of 6: 6, 12,
and prime numbers 18, 24, …
19. Find the prime factor Use a prime factor tree. Don’t forget to write × between each prime factor. Try to collect the
decomposition of positive same prime factors together using power notation. e.g. 120 = 2 × 2 × 2 × 3 × 5, but it
integers would be better to write 23 × 3 × 5
Test Your Understanding: Express 240 as the product of its prime factors.
20. Find the common factors See below.
and common multiples of two
numbers
21. Find the Highest Common There are two ways you could find the Lowest Common Multiple:
Factor (HCF) and the Least Write out multiples of the larger number until you see a multiple of the smaller
Common Multiple (LCM) of two number. e.g. for 60 and 54, write out multiples of 60: 60, 120, 180, 240, 300, 360,
numbers 420, 480, 540. And 540 is the first multiple of 54 in this list so is the LCM.
Find the prime factorisation of each number. Then see which things ‘wins’ in each
factor. E.g. 60 = 22 × 3 × 5 and 54 = 2 × 33. Of the 2s the 22 ‘wins’ over the 2 so
we use 22. The 33 ‘wins’ over the 3. And the ‘5’ beats nothing. So the LCM is
22 × 33 × 5 = 540.
To find the HCF:
Write out the factors of each number, and look for the highest number in both lists.
Alternatively you can again use the prime factorisation of each number, but this
time see which factor ‘loses’ (where ‘nothing’ loses against anything). So in
60 = 22 × 3 × 5 and 54 = 2 × 33, the 2 loses, the 3 loses and ‘nothing’ loses
against 5, so the HCF is 2 × 3 = 6. The previous method is faster however.
3
, More Notes: LCM questions often come in an applied setting. e.g. Two buses come every 5
minutes and 7 minutes respectively: if they both come now, when will they next both come
at the same time? (i.e. in 35 minutes time)
Test Your Understanding:
a. Find the Lowest Common Multiple of 120 and 96.
b. Find the Highest Common Factor of 48 and 60?
c. Two train services both arrive at 9am. The first comes every 20 minutes. The second
comes every 25 minutes. What is the time when the two train services both come at
the same time?
d. Cookies come in packs of 12 and chocolate bars in packs of 9. I want to have the
same number of cookies and chocolate bars. What’s the smallest number of packs
of cookies can I buy?
22. Recall integer squares from This is just saying you should know these of-by-heart. Of course, in exam, you could work out
2 × 2 to 15 × 15 and the 142 say by long multiplication, but it’s highly useful to memorise these if you haven’t
corresponding square roots. already.
Recall the cubes of 2, 3, 4, 5 and
10.
23. Use standard form, When the power of 10 is negative (i.e. we have a small number), the number of 0s on the
expressed in conventional front (including the one before the decimal place) is the number in the index. So 0.00147 =
notation. Be able to write very 1.47 × 10−3 as there’s 3 leading 0s. But if ever in doubt count the number of times the
large and very small numbers decimal place has to move until you get to a number between 1 and 9. 9̇.
presented in a context in
standard form. Convert When converting a number not in standard form, if you divide the first part by 10, you need
between ordinary and standard to multiply the second part by 10 (i.e. by increasing the power by 1) to cancel out the effect.
form representations. Convert When adding or subtracting two numbers in standard form, it’s often easiest to just convert
between ordinary and standard the numbers into normal numbers, then add/subtract, then convert back to standard form.
form representations. Interpret
a calculator display using Example: (3 × 107) × (4 × 102) = 12 × 109 = 1.2 × 1010
standard form (2 × 105) ÷ (8 × 109) = 0.25 × 10−4 = 2.5 × 10−5
(1.2 × 105) + (3 × 104) = 120000 + 30000
= 150000 = 1.5 × 105
Test Your Understanding:
a. Express 367 000 in standard form.
b. Express 0.00048 in standard form.
c. Express 2.67 × 10−2 as an ordinary number.
d. Express 5.2 × 103 as an ordinary number.
e. Calculate (5.2 × 103) × (2 × 107) giving your answer in standard form.
f. Calculate (2.4 × 108) ÷ (4 × 105) giving your answer in standard form.
g. Calculate (4 × 105) + (3 × 104)
24. Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer
powers, and powers of a power
(Covered later)
Fractions, Decimals and Percentages
25. Understand that a e.g. 34% = 34
100
percentage is a fraction in
hundredths
26. Convert between fractions, To convert (non-recurring) decimals to fractions, just put over 10/100/1000 depending on
decimals and percentages how many digits are after the decimal point, then simplify. e.g. 0.56 = 56 = 14 and
100 25
0.035 = 35 = 7
1000 200
Test Your Understanding:
a. Convert 0.84 to a fraction in its simplest form.
3
b. Express 8 as a percentage.
27. Convert between recurring To convert from a fraction to recurring decimal, just use long division. You will need to add
decimals and exact fractions as “.00000” to the number you’re dividing to get extra digits for which to put your remainders
well as understanding the proof on. Make sure you put the decimal point in the same place on the result, and stop once you
see the repetition in digits.
4
Common General Algebraic Errors:
(𝑎 + 𝑏)2 → 𝑎2 + 𝑏2. Writing out the bracket twice we actually find (𝑎 + 𝑏)(𝑎 + 𝑏) = 𝑎2 + 2𝑎𝑏 + 𝑏2
Similarly
2+3𝑥+2
√𝑎2 + 𝑏2 → 𝑎 + 𝑏. You can see this is not true when 𝑎 = 3, 𝑏 = 4 for example.
𝑥 → 3𝑥+2. When ‘cancelling’ fractions, we can only divide, whereas in this example we’ve incorrectly
𝑥2−4 −4
(𝑥+1)(𝑥+2) 𝑥+1
subtracted 𝑥2. If we factorised the example, it would be OK to cancel to because we have indeed
(𝑥+2)(𝑥−2) 𝑥−2
divided by 𝑥 + 2.
𝑥(𝑥 − 1) → 𝑥2 − 1. Oops!
𝑥 + 𝑎 = 𝑦 → 𝑥 + 𝑎 = 3𝑦. The 𝑎 hasn’t been multiplied by 3.
3
𝑐 − 𝑏(𝑎 − 𝑏) → 𝑐 − 𝑎𝑏 − 𝑏2. Sign error at the end.
𝑥(𝑥 + 1) − (𝑥 + 2)2 → 𝑥2 + 𝑥 − 𝑥2 + 4𝑥 + 4 . A lack of brackets when subtracting expanded expression leads to
sign errors. See (53i).
𝑎 + 3𝑥 = 𝑏 → 3𝑥 = 𝑏 + 𝑎 Sign not changed when 𝑎 moved to other side of equation.
𝑥+2𝑏 = 𝑦 → 𝑥 = 𝑦 − 2𝑏
or √𝑥 + 2𝑏 = 𝑦 → √𝑥 = 𝑦 − 2𝑏 (2𝑏 is trapped inside fraction/root so we have to
3 3
deal with the ÷ 3 and √ first)
√𝑥 = 2𝑥 → 𝑥 = 2𝑥2 When 2𝑥 is squared, you get 4𝑥2 not 2𝑥2 as 2𝑥 × 2𝑥 = 4𝑥2
1
, Number
Integers and Decimals
1
1. Understand and order Test Your Understanding: Order the following: 3 , 0.33, 0.303, 30%
integers and decimals
2. Use brackets and the BIDMAS is actually (B)(I)(DM)(AS), i.e. Division and Multiplication have the same
hierarchy of operations ‘priority’, and Addition and Subtraction have the same priority. When you have say a
(BIDMAS) mix of addition and subtraction, evaluate left-to-right. E.g. 9 − 3𝑥 + 5𝑥 simplifies to
9 + 2𝑥 NOT 9 − 8𝑥: in the latter you had done the addition first, when there was no
reason to do so.
Note that due to BIDMAS, negative numbers to a power require bracketing: −42
would produce -16 on a calculator because it does the square (‘indices’) first. You
want (−4)2. This is highly important for the 𝑏2 in the Quadratic Formula when
substituting numbers in.
3. Add, subtract, multiply and When multiplying two decimals, first multiply them as if they were whole numbers,
divide integers, negative then put the decimal point back in the result by counting the number of jumps in
numbers and decimals decimal point in the original numbers.
When subtracting negative numbers, ensure numbers are lined up in the units
column, and fill in any ‘gaps’ with 0s.
We like dividing by whole numbers. Hence if you’re dividing by a decimal, multiply
both numbers by 10 until you’re dividing by a whole number, e.g. 3.678 ÷ 0.09 →
367.8 ÷ 9
Test Your Understanding:
a. 3 − (−4)?
b. 396 × 48?
c. 5.7 − 2.89?
d. (−7.1)2?
e. 508.97 ÷ 1.1
4. Understand and use positive This just means that you understand −5 + 7 for example as starting at -5 on a number
numbers and negative integers, line and ‘moving’/translating 7 up. This won’t specifically be tested.
both as positions and
translations on a number line
5. Round whole numbers to the nearest, 10, 100, 1000, …
6. Round decimals to Be wary that any 0s after the first non-zero digit count as significant. e.g. 3.40204 to 3sf is
appropriate numbers of 3.40, NOT 3.4. Similarly it is 3.40 to 2dp, not 3.4. But 0.0020413 is 0.00204 to 3sf.
decimal places or significant Note that 3.9853 to 1dp is 4.0.
figures
Test Your Understanding: Write the following to the indicated number of significant
figures or decimal places:
a. 24703 to 2sf
b. 15.0849 to 1dp
c. 25.969403 to 3sf
d. 495.18473 to 3dp
7. Multiply and divide by any You should appreciate that multiplying by a number between 0 and 1 makes it smaller,
number between 0 and 1 and dividing by it makes it bigger. You should recognise that ÷ 0.5 is the same as × 2 and
÷ 0.25 the same as × 4 and so on (see Fractions): this is particularly useful in estimation
(see below)
8. Check their calculations by For estimation questions, the general rule of thumb is to round each number to 1
rounding, eg 29 31 30 30 significant figure, unless it is close to some other nice number (such as 0.26 → 0.25
because it’s a quarter), e.g:
4.98 × 31 5 × 30
≈ = 300
0.49 0.5
Test Your Understanding: Estimate the following.
79×6.89
a. 0.52
4.03×4.94
b.
0.24
9. Check answers to a division Not tested as such.
sum using multiplication eg use
inverse operations
10. Multiply and divide whole numbers by a given multiple of 10
11. Put digits in the correct place in a decimal number
2
,Fractions
45
12. Find equivalent fractions Test Your Understanding: Put 54 in its simplest form.
and write a fraction in its
simplest form.
13. Compare the sizes of The strategy is to find a common denominator for all fractions, so that we can just easily
5 3 20 21 3
fractions compare the numerators, e.g. 7 and 4 can be converted to 28 and 28, thus 4 is larger.
Sometimes the ordering will be obvious by thinking of the fractions on a number line.
4 5
Test Your Understanding: Which of 5 and 7 is bigger?
3
14. Find fractions of an amount Test Your Understanding: Find 7 of 35
15. Convert between mixed For mixed number to improper fractions, to get the new numerator times the whole part
numbers and improper with the denominator and add the numerator. The denominator stays the same. e.g
2 17
fractions 35 → 5
For improper fractions to mixed numbers, see how many times the denominator goes into
39 4
the numerator and find the remainder also. 7 → 5 7
16. Add and subtract fractions The ‘foolproof’ way is to cross-multiply: multiply the two denominators, then times each
3 1 12+7 19
numerator by the other fraction’s denominator and add. e.g. 7 + 4 = 28 = 28.
1 4 3 4 7
However, sometimes you only need to change one of the fractions, e.g. 3 + 9 → 9 + 9 = 9
Test Your Understanding:
4 3
a. 9 + 5
4 1
b. 17 − 2
1 4
c. 5 4 + 4 5
17. Multiply and divide If any whole numbers, put over 1: 4 → 4. If any mixed numbers, convert to improper
1
fractions including mixed fractions first. When multiplying, just multiply numerators and denominators separately.
numbers. When dividing, ‘flip’ (reciprocate) the second fraction and instead multiply.
Test Your Understanding:
3 4
a. 5
×9
4
b. 25 × 3
1 1
c. 14 ÷25
1
d. 4÷15
Factors, Multiples, Primes, Roots, Powers
18. Identify factors, multiples Ensure you don’t confuse factors and multiples. Factors of 6: 1, 2, 3, 6. Multiples of 6: 6, 12,
and prime numbers 18, 24, …
19. Find the prime factor Use a prime factor tree. Don’t forget to write × between each prime factor. Try to collect the
decomposition of positive same prime factors together using power notation. e.g. 120 = 2 × 2 × 2 × 3 × 5, but it
integers would be better to write 23 × 3 × 5
Test Your Understanding: Express 240 as the product of its prime factors.
20. Find the common factors See below.
and common multiples of two
numbers
21. Find the Highest Common There are two ways you could find the Lowest Common Multiple:
Factor (HCF) and the Least Write out multiples of the larger number until you see a multiple of the smaller
Common Multiple (LCM) of two number. e.g. for 60 and 54, write out multiples of 60: 60, 120, 180, 240, 300, 360,
numbers 420, 480, 540. And 540 is the first multiple of 54 in this list so is the LCM.
Find the prime factorisation of each number. Then see which things ‘wins’ in each
factor. E.g. 60 = 22 × 3 × 5 and 54 = 2 × 33. Of the 2s the 22 ‘wins’ over the 2 so
we use 22. The 33 ‘wins’ over the 3. And the ‘5’ beats nothing. So the LCM is
22 × 33 × 5 = 540.
To find the HCF:
Write out the factors of each number, and look for the highest number in both lists.
Alternatively you can again use the prime factorisation of each number, but this
time see which factor ‘loses’ (where ‘nothing’ loses against anything). So in
60 = 22 × 3 × 5 and 54 = 2 × 33, the 2 loses, the 3 loses and ‘nothing’ loses
against 5, so the HCF is 2 × 3 = 6. The previous method is faster however.
3
, More Notes: LCM questions often come in an applied setting. e.g. Two buses come every 5
minutes and 7 minutes respectively: if they both come now, when will they next both come
at the same time? (i.e. in 35 minutes time)
Test Your Understanding:
a. Find the Lowest Common Multiple of 120 and 96.
b. Find the Highest Common Factor of 48 and 60?
c. Two train services both arrive at 9am. The first comes every 20 minutes. The second
comes every 25 minutes. What is the time when the two train services both come at
the same time?
d. Cookies come in packs of 12 and chocolate bars in packs of 9. I want to have the
same number of cookies and chocolate bars. What’s the smallest number of packs
of cookies can I buy?
22. Recall integer squares from This is just saying you should know these of-by-heart. Of course, in exam, you could work out
2 × 2 to 15 × 15 and the 142 say by long multiplication, but it’s highly useful to memorise these if you haven’t
corresponding square roots. already.
Recall the cubes of 2, 3, 4, 5 and
10.
23. Use standard form, When the power of 10 is negative (i.e. we have a small number), the number of 0s on the
expressed in conventional front (including the one before the decimal place) is the number in the index. So 0.00147 =
notation. Be able to write very 1.47 × 10−3 as there’s 3 leading 0s. But if ever in doubt count the number of times the
large and very small numbers decimal place has to move until you get to a number between 1 and 9. 9̇.
presented in a context in
standard form. Convert When converting a number not in standard form, if you divide the first part by 10, you need
between ordinary and standard to multiply the second part by 10 (i.e. by increasing the power by 1) to cancel out the effect.
form representations. Convert When adding or subtracting two numbers in standard form, it’s often easiest to just convert
between ordinary and standard the numbers into normal numbers, then add/subtract, then convert back to standard form.
form representations. Interpret
a calculator display using Example: (3 × 107) × (4 × 102) = 12 × 109 = 1.2 × 1010
standard form (2 × 105) ÷ (8 × 109) = 0.25 × 10−4 = 2.5 × 10−5
(1.2 × 105) + (3 × 104) = 120000 + 30000
= 150000 = 1.5 × 105
Test Your Understanding:
a. Express 367 000 in standard form.
b. Express 0.00048 in standard form.
c. Express 2.67 × 10−2 as an ordinary number.
d. Express 5.2 × 103 as an ordinary number.
e. Calculate (5.2 × 103) × (2 × 107) giving your answer in standard form.
f. Calculate (2.4 × 108) ÷ (4 × 105) giving your answer in standard form.
g. Calculate (4 × 105) + (3 × 104)
24. Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer
powers, and powers of a power
(Covered later)
Fractions, Decimals and Percentages
25. Understand that a e.g. 34% = 34
100
percentage is a fraction in
hundredths
26. Convert between fractions, To convert (non-recurring) decimals to fractions, just put over 10/100/1000 depending on
decimals and percentages how many digits are after the decimal point, then simplify. e.g. 0.56 = 56 = 14 and
100 25
0.035 = 35 = 7
1000 200
Test Your Understanding:
a. Convert 0.84 to a fraction in its simplest form.
3
b. Express 8 as a percentage.
27. Convert between recurring To convert from a fraction to recurring decimal, just use long division. You will need to add
decimals and exact fractions as “.00000” to the number you’re dividing to get extra digits for which to put your remainders
well as understanding the proof on. Make sure you put the decimal point in the same place on the result, and stop once you
see the repetition in digits.
4
