GCSE Foundation Maths Revision
,Grade 1 Topics: Building Strong Foundations
These fundamental topics form the bedrock of mathematical understanding. Mastering these concepts early will make
more advanced topics much easier to grasp.
Addition and Subtraction Multiplication and Division
Work from right to left, carrying or borrowing as Use column method for multiplication. For division,
needed. Always align decimal points vertically. divide, multiply, subtract, bring down.
Time Metric Conversions
Remember: 60 seconds = 1 minute, 60 minutes = 1 hour, Key prefixes: kilo (1000), centi (0.01), milli (0.001).
24 hours = 1 day. Use 24-hour format carefully. Always multiply or divide by powers of 10.
Working with Fractions Number Concepts
Writing fractions: Numerator (top) ÷ Denominator Place value: Each digit's position determines its value
(bottom) (units, tens, hundreds...)
Simplifying: Divide numerator and denominator by their Rounding: Look at the digit to the right. If ≥5, round up; if
highest common factor <5, leave unchanged
Ordering: Convert to decimals or find a common Negative numbers: Use number line. Adding a negative
denominator moves left; subtracting a negative moves right
Example: Simplify ¹²⁄₁₆ Example: Round 4.73 to 1 decimal place
HCF of 12 and 16 is 4, so ¹²⁄₁₆ = ³⁄₄ Look at 3 (second decimal): 3<5, so answer is 4.7
01 02
Order of Operations (BIDMAS) Factors, Multiples and Primes
Brackets, Indices, Division/Multiplication (left to right), Factors divide exactly into a number. Multiples are in the
Addition/Subtraction (left to right) times table. Primes have exactly two factors.
03 04
Coordinates Pictograms
(x, y) format. x tells you how far across, y tells you how far Use a key to show what each symbol represents. Count
up from the origin (0,0). symbols carefully and multiply by the key value.
Common Mistakes to Avoid
Forgetting to align decimal points when adding or subtracting
Not following BIDMAS order in calculations
Mixing up factors and multiples
Plotting coordinates in wrong order (y, x instead of x, y)
Counting pictogram symbols incorrectly or ignoring the key
,Grade 2 Topics: Developing Problem-Solving Skills
At Grade 2, you'll start applying your basic skills to solve more complex problems. These topics help you work with real-
world scenarios and interpret data effectively.
Calculation Problems Using a Calculator Systematic Listing
Read the question carefully. Identify Enter calculations carefully. Use Work methodically. Use a logical order
what calculation is needed. Show all brackets for fractions. Round final to ensure you don't miss any
your working out. answers appropriately. possibilities or repeat combinations.
Working with Proportions Algebra Basics
Fractions of an amount: Divide by the denominator, then Simplifying algebra: Collect like terms by
multiply by the numerator adding/subtracting coefficients
Example: Find ³⁄₅ of 30 Example: Simplify 3x + 2y - x + 4y
30 ÷ 5 = 6, then 6 × 3 = 18 3x - x = 2x, 2y + 4y = 6y, so answer is 2x + 6y
FDP conversions: Fraction to decimal: divide numerator Writing expressions: Use letters for unknowns. Translate
by denominator. Decimal to percentage: ×100 words into algebra
Example: Convert ²⁄₅ to percentage Example: "5 more than y" becomes y + 5
2 ÷ 5 = 0.4, then 0.4 × 100 = 40%
Function Machines Solving One-Step Equations
Input → [operation] → Output. Work forwards or Use inverse operations. Whatever you do to one side,
backwards. Reverse operations when working do to the other. Always check your answer.
backwards.
Angles Area and Perimeter
Angles on a straight line sum to 180°. Angles around a Perimeter: add all sides. Area of rectangle: length ×
point sum to 360°. Interior angles of a triangle sum to width. Area of triangle: ½ × base × height.
180°.
Probability Frequency Polygons
Probability = desired outcomes ÷ total possible Plot frequency at midpoint of each class interval.
outcomes. Always between 0 and 1 (or 0% and Join points with straight lines. Start and end at the
100%). x-axis.
Averages and Charts Stem and Leaf
Mean: sum ÷ count. Median: middle value. Mode: Stem shows tens, leaf shows units. Include a key.
most frequent. Range: highest - lowest. Bar charts Order leaves from smallest to largest. Never miss a
need labels and a scale. number.
Key Tips for Grade 2 Success
Always show your working out – you can get method marks even if your final answer is wrong. Read questions
twice to make sure you understand what's being asked. Check your answers make sense: does that probability
look too big? Does that average seem reasonable?
, Grade 3 Topics: Working with Proportional
Reasoning
Grade 3 introduces proportional reasoning, which is essential for Essential Skills
understanding how quantities relate to each other. You'll also explore
Master ratio and proportion before moving
more sophisticated algebra and geometry concepts.
to percentages. Understand that fractions,
These topics build directly on Grade 2 work but require more careful decimals, and percentages are just different
thought about how different mathematical ideas connect together. ways of expressing the same relationship.
Practice is key to becoming confident with these concepts.
Error Intervals Working with Fractions Estimation
When rounding, the actual value lies Add/subtract: common Round numbers first, then calculate.
within a range. For example, 5.4 to 1 denominator needed. Multiply: top × Useful for checking if your exact
d.p. means 5.35 ≤ x < 5.45 top, bottom × bottom. Divide: answer is sensible or for non-
multiply by reciprocal calculator questions
Ratio and Proportion
Writing and Simplifying Ratio Sharing in a Ratio
Ratio compares quantities. Simplify by dividing both parts Add ratio parts to find total parts. Divide total amount by
by their highest common factor. total parts. Multiply by each part.
Example: Simplify 12:18 Example: Share £40 in ratio 3:5
HCF of 12 and 18 is 6, so 12÷6 : 18÷6 = 2:3 3+5=8 parts. £40÷8=£5 per part. 3×£5=£15, 5×£5=£25
Direct Proportion Percentages Exchange Rates
When one quantity increases, the To find percentage: (part ÷ whole) × Multiply by the exchange rate to
other increases at the same rate. y = 100. To find percentage change: convert. Divide by the exchange rate
kx where k is the constant of (change ÷ original) × 100. to convert back. Always check
proportionality. which way round the rate is given.
Algebra and Graphs
Linear Graphs
Solving Equations y = mx + c. Plot points by
Substitution Expand brackets first if needed. substituting x-values. Join with a
Replace letters with numbers. Use Collect like terms. Use inverse straight line. Label axes and write
brackets around negative operations to isolate the variable. the equation.
numbers. Follow BIDMAS when
calculating.
Geometry and Measurement
Circles Transformations Compound Shapes
Circumference = πd or 2πr. Area = Rotation: direction, angle, centre. Split into simpler shapes.
πr². Use π ≈ 3.14 or the π button on Reflection: mirror line. Calculate each area separately.
your calculator. Always include Enlargement: scale factor, centre. Add or subtract areas as
units. Translation: vector showing appropriate. Don't forget to label.
movement.
1 2
Frequency Trees Two-Way Tables
Branch to show different categories. Write frequencies Complete the table using given information. Row totals,
on branches. Total at each level should match the column totals, and grand total must all match. Use to
starting value. find probabilities.
Common Errors in Grade 3 Topics
Forgetting to simplify ratios to their simplest form
Using the wrong formula for percentage change (using new value instead of original)
Mixing up direct and inverse proportion
Plotting graph points incorrectly or not joining with a straight line
Forgetting to square the radius when calculating circle area
Not specifying direction and centre when describing rotations
,Grade 1 Topics: Building Strong Foundations
These fundamental topics form the bedrock of mathematical understanding. Mastering these concepts early will make
more advanced topics much easier to grasp.
Addition and Subtraction Multiplication and Division
Work from right to left, carrying or borrowing as Use column method for multiplication. For division,
needed. Always align decimal points vertically. divide, multiply, subtract, bring down.
Time Metric Conversions
Remember: 60 seconds = 1 minute, 60 minutes = 1 hour, Key prefixes: kilo (1000), centi (0.01), milli (0.001).
24 hours = 1 day. Use 24-hour format carefully. Always multiply or divide by powers of 10.
Working with Fractions Number Concepts
Writing fractions: Numerator (top) ÷ Denominator Place value: Each digit's position determines its value
(bottom) (units, tens, hundreds...)
Simplifying: Divide numerator and denominator by their Rounding: Look at the digit to the right. If ≥5, round up; if
highest common factor <5, leave unchanged
Ordering: Convert to decimals or find a common Negative numbers: Use number line. Adding a negative
denominator moves left; subtracting a negative moves right
Example: Simplify ¹²⁄₁₆ Example: Round 4.73 to 1 decimal place
HCF of 12 and 16 is 4, so ¹²⁄₁₆ = ³⁄₄ Look at 3 (second decimal): 3<5, so answer is 4.7
01 02
Order of Operations (BIDMAS) Factors, Multiples and Primes
Brackets, Indices, Division/Multiplication (left to right), Factors divide exactly into a number. Multiples are in the
Addition/Subtraction (left to right) times table. Primes have exactly two factors.
03 04
Coordinates Pictograms
(x, y) format. x tells you how far across, y tells you how far Use a key to show what each symbol represents. Count
up from the origin (0,0). symbols carefully and multiply by the key value.
Common Mistakes to Avoid
Forgetting to align decimal points when adding or subtracting
Not following BIDMAS order in calculations
Mixing up factors and multiples
Plotting coordinates in wrong order (y, x instead of x, y)
Counting pictogram symbols incorrectly or ignoring the key
,Grade 2 Topics: Developing Problem-Solving Skills
At Grade 2, you'll start applying your basic skills to solve more complex problems. These topics help you work with real-
world scenarios and interpret data effectively.
Calculation Problems Using a Calculator Systematic Listing
Read the question carefully. Identify Enter calculations carefully. Use Work methodically. Use a logical order
what calculation is needed. Show all brackets for fractions. Round final to ensure you don't miss any
your working out. answers appropriately. possibilities or repeat combinations.
Working with Proportions Algebra Basics
Fractions of an amount: Divide by the denominator, then Simplifying algebra: Collect like terms by
multiply by the numerator adding/subtracting coefficients
Example: Find ³⁄₅ of 30 Example: Simplify 3x + 2y - x + 4y
30 ÷ 5 = 6, then 6 × 3 = 18 3x - x = 2x, 2y + 4y = 6y, so answer is 2x + 6y
FDP conversions: Fraction to decimal: divide numerator Writing expressions: Use letters for unknowns. Translate
by denominator. Decimal to percentage: ×100 words into algebra
Example: Convert ²⁄₅ to percentage Example: "5 more than y" becomes y + 5
2 ÷ 5 = 0.4, then 0.4 × 100 = 40%
Function Machines Solving One-Step Equations
Input → [operation] → Output. Work forwards or Use inverse operations. Whatever you do to one side,
backwards. Reverse operations when working do to the other. Always check your answer.
backwards.
Angles Area and Perimeter
Angles on a straight line sum to 180°. Angles around a Perimeter: add all sides. Area of rectangle: length ×
point sum to 360°. Interior angles of a triangle sum to width. Area of triangle: ½ × base × height.
180°.
Probability Frequency Polygons
Probability = desired outcomes ÷ total possible Plot frequency at midpoint of each class interval.
outcomes. Always between 0 and 1 (or 0% and Join points with straight lines. Start and end at the
100%). x-axis.
Averages and Charts Stem and Leaf
Mean: sum ÷ count. Median: middle value. Mode: Stem shows tens, leaf shows units. Include a key.
most frequent. Range: highest - lowest. Bar charts Order leaves from smallest to largest. Never miss a
need labels and a scale. number.
Key Tips for Grade 2 Success
Always show your working out – you can get method marks even if your final answer is wrong. Read questions
twice to make sure you understand what's being asked. Check your answers make sense: does that probability
look too big? Does that average seem reasonable?
, Grade 3 Topics: Working with Proportional
Reasoning
Grade 3 introduces proportional reasoning, which is essential for Essential Skills
understanding how quantities relate to each other. You'll also explore
Master ratio and proportion before moving
more sophisticated algebra and geometry concepts.
to percentages. Understand that fractions,
These topics build directly on Grade 2 work but require more careful decimals, and percentages are just different
thought about how different mathematical ideas connect together. ways of expressing the same relationship.
Practice is key to becoming confident with these concepts.
Error Intervals Working with Fractions Estimation
When rounding, the actual value lies Add/subtract: common Round numbers first, then calculate.
within a range. For example, 5.4 to 1 denominator needed. Multiply: top × Useful for checking if your exact
d.p. means 5.35 ≤ x < 5.45 top, bottom × bottom. Divide: answer is sensible or for non-
multiply by reciprocal calculator questions
Ratio and Proportion
Writing and Simplifying Ratio Sharing in a Ratio
Ratio compares quantities. Simplify by dividing both parts Add ratio parts to find total parts. Divide total amount by
by their highest common factor. total parts. Multiply by each part.
Example: Simplify 12:18 Example: Share £40 in ratio 3:5
HCF of 12 and 18 is 6, so 12÷6 : 18÷6 = 2:3 3+5=8 parts. £40÷8=£5 per part. 3×£5=£15, 5×£5=£25
Direct Proportion Percentages Exchange Rates
When one quantity increases, the To find percentage: (part ÷ whole) × Multiply by the exchange rate to
other increases at the same rate. y = 100. To find percentage change: convert. Divide by the exchange rate
kx where k is the constant of (change ÷ original) × 100. to convert back. Always check
proportionality. which way round the rate is given.
Algebra and Graphs
Linear Graphs
Solving Equations y = mx + c. Plot points by
Substitution Expand brackets first if needed. substituting x-values. Join with a
Replace letters with numbers. Use Collect like terms. Use inverse straight line. Label axes and write
brackets around negative operations to isolate the variable. the equation.
numbers. Follow BIDMAS when
calculating.
Geometry and Measurement
Circles Transformations Compound Shapes
Circumference = πd or 2πr. Area = Rotation: direction, angle, centre. Split into simpler shapes.
πr². Use π ≈ 3.14 or the π button on Reflection: mirror line. Calculate each area separately.
your calculator. Always include Enlargement: scale factor, centre. Add or subtract areas as
units. Translation: vector showing appropriate. Don't forget to label.
movement.
1 2
Frequency Trees Two-Way Tables
Branch to show different categories. Write frequencies Complete the table using given information. Row totals,
on branches. Total at each level should match the column totals, and grand total must all match. Use to
starting value. find probabilities.
Common Errors in Grade 3 Topics
Forgetting to simplify ratios to their simplest form
Using the wrong formula for percentage change (using new value instead of original)
Mixing up direct and inverse proportion
Plotting graph points incorrectly or not joining with a straight line
Forgetting to square the radius when calculating circle area
Not specifying direction and centre when describing rotations
