GCSE Maths Foundation & Higher Help Book

This is your complete guide to mastering GCSE maths, covering both Foundation and Higher tier topics. The book works for all major exam boards including Edexcel, AQA, and OCR.

You'll notice some topics are in italics - these are Higher tier only content, so Foundation students can skip them. The shaded formulas are the ones you absolutely must memorise because they won't be given in your exam formula booklet.

Each topic links to video tutorials and worksheets, making it dead easy to get extra practice when you need it. Whether you're aiming for a grade 4 or pushing for that grade 9, this resource has got your back.

Top Tip: Focus on the non-shaded formulas first - these will be provided in your exam, so understanding how to use them is more important than memorising them.

Core Mathematical Concepts

Growth and decay problems are everywhere - from your savings account to a car losing value. The key is finding the right multiplier. For growth, use a multiplier greater than 1 (like 1.05 for 5% growth). For decay, use less than 1 (like 0.92 for 8% decrease).

Exponential graphs follow the pattern y = aˣ and create smooth curves. When a > 1, you get growth curves shooting upwards. When 0 < a < 1, you get decay curves falling down. These graphs model real situations like population growth or radioactive decay.

Compound interest beats simple interest every time because you earn "interest on interest". Simple interest only calculates on your original amount, whilst compound interest grows your money faster by including previous interest in the calculation.

VAT is just a 20% tax added to goods and services. Multiply by 1.2 to add VAT, or divide by 1.2 to find the pre-VAT price. Dead simple once you remember which way round to do it!

Money Saver: Understanding compound interest helps you see why starting to save early makes such a massive difference to your future wealth.

Working with Surds and Decimals

Surds are those "exact" square roots like √2 that never end as decimals. Don't convert them to decimals - leave them as surds for exact answers. To simplify, find the largest square number that divides into your surd, then split it up.

When multiplying and dividing surds, remember these golden rules: √a × √a = a, and √a × √b = √ab. You can only add surds that are "like" terms - think of √5 + 2√5 = 3√5, just like algebra.

Rationalising the denominator means getting rid of surds from the bottom of fractions. For simple cases, multiply top and bottom by the surd. For expressions like 5 + √3, multiply by the opposite sign version to create a difference of two squares.

Converting recurring decimals to fractions uses a clever algebraic trick. Set x equal to your decimal, multiply by powers of 10 until the pattern repeats, then subtract to solve for x. The dots show you exactly which digits repeat.

Exam Gold: Always simplify your surds completely - examiners love to see √12 written as 2√3, not left as √12.

Ratio, Proportion and Rates

Ratios work exactly like fractions - just divide by common factors to simplify. When sharing amounts, add up all the parts first, then work out what one part is worth by dividing the total amount by the number of parts.

Best buys are found by calculating the unit cost - divide the price by the quantity. The item with the lowest cost per unit wins. Don't round too early or you might get the wrong answer!

Direct proportion means as one thing increases, the other increases too $y = kx$. Inverse proportion means as one increases, the other decreases $y = k/x$. The constant k stays the same in both cases.

Exchange rates are straightforward multiplication and division. If £1 = $1.6, multiply pounds by 1.6 to get dollars, or divide dollars by 1.6 to get pounds. Always check which direction you're converting!

Real Life: Understanding proportion helps you adjust recipes, work out exchange rates on holiday, and spot the best deals when shopping.

Advanced Algebraic Techniques

Difference of two squares follows the pattern a² - b² = $a + b$$a - b$. When you expand it back out, the middle terms always cancel perfectly, leaving just the first and last terms.

Completing the square transforms expressions like x² + bx + c into the form $x + p$² + q. Take half the coefficient of x, square it, then adjust accordingly. This reveals the minimum/maximum point of quadratic graphs.

When the coefficient of x² isn't 1, factor it out first before completing the square. Work inside brackets first, then multiply back through. This technique becomes crucial for solving quadratic equations that won't factorise.

The completed square form tells you loads about a quadratic function - the turning point coordinates, the axis of symmetry, and whether it has a maximum or minimum. It's like having X-ray vision for quadratic graphs!

Graph Insight: Completing the square instantly shows you the vertex of a parabola - the (p, q) values in $x + p$² + q give you the turning point coordinates.

Formulas and Equations

Writing formulas from word problems requires identifying the variables (use letters) and constants (fixed numbers). Look for key phrases like "per item" (multiplication) and "call-out charge" (addition).

Substituting into formulas means swapping letters for numbers carefully. Watch the order of operations - square first, then multiply. Remember that subtracting a negative means adding, and squaring always gives a positive result.

Rearranging equations uses the same balance method as solving equations. Do the opposite operation to both sides until your desired letter is alone. It doesn't matter if your subject ends up on the right or left side.

Solving linear equations requires getting all the unknowns on one side and numbers on the other. When unknowns appear on both sides, collect like terms first. For equations with fractions, multiply through by the lowest common multiple to clear them.

Success Strategy: Always check your answer by substituting it back into the original equation - it's the best way to catch silly mistakes.

Simultaneous and Quadratic Equations

Simultaneous equations need two equations to find two unknowns. Make the coefficients of x (or y) the same, then add or subtract the equations to eliminate one variable. Solve for the remaining variable, then substitute back.

Quadratic equations come in different forms requiring different techniques. For ax² = b, just square root both sides (remember positive and negative solutions). For ax² + bx = 0, factor out x and set each factor to zero.

Factoring quadratics when a = 1 follows familiar patterns. Once factored into two brackets, set each bracket equal to zero to find your solutions. Always check both answers make sense in context.

When a ≠ 1, factoring becomes trickier but follows the same principles. Sometimes solutions might not be realistic (like negative lengths), so always check your answers make sense for the problem.

Problem-Solving: Quadratic equations often have two solutions, but real-world problems might only accept one - negative time or length usually doesn't make sense!

Straight Line Graphs

Gradient measures how steep a line is. Count up/down and left/right between two points, then divide vertical change by horizontal change. Positive gradients slope upwards, negative ones slope downwards.

Finding equations of straight lines needs the gradient and one point. Substitute into y = mx + c to find c, then write the complete equation. You can pick either point when you have two - the answer will be the same.

Parallel lines have identical gradients but different y-intercepts. Perpendicular lines have gradients that multiply to give -1. If one line has gradient m, the perpendicular line has gradient -1/m.

The equation y = mx + c tells you everything about a straight line. The m value shows the steepness and direction, whilst c shows where it crosses the y-axis. Master this form and linear graphs become simple.

Visual Tip: Steep gradients mean big changes in y for small changes in x. A gradient of 5 means the line goes up 5 units for every 1 unit across.

Advanced Graph Types

Graph recognition helps you identify different function types instantly. Linear graphs are straight lines, quadratic graphs are smooth curves (parabolas), cubic graphs have up to three x-intercepts, and reciprocal graphs have two separate branches.

Quadratic graphs create U-shaped or inverted-U curves called parabolas. When a > 0, you get a U-shape with a minimum point. When a < 0, you get an upside-down U with a maximum point.

Sketching quadratics from equations requires finding key features: x-intercepts $set y = 0$, y-intercept $set x = 0$, and turning point. Factored form makes finding roots easy, whilst completed square form reveals the vertex.

The turning point of a quadratic is crucial for sketching. In completed square form $x + p$² + q, the turning point is at $-p, q$. This point is either the highest or lowest point on the entire curve.

Sketch Success: You don't need to plot every point perfectly - focus on the key features like intercepts and turning points, then draw a smooth curve through them.

Cubic Graphs and Functions

Cubic graphs create sweeping S-shaped curves that can cross the x-axis up to three times. When the coefficient of x³ is positive, the curve enters bottom-left and exits top-right. When negative, it's the opposite.

Sketching cubic graphs from factored form y = $x - p$$x - q$$x - r$ is straightforward. The curve crosses the x-axis at x = p, q, and r. Find the y-intercept by setting x = 0 and multiplying the constants.

Repeated roots create special behaviour where the curve touches the x-axis but doesn't cross it. For example, y = $x - 3$$x + 5$² touches at x = -5 but crosses at x = 3.

The shape of cubic graphs depends entirely on whether the leading coefficient is positive or negative. Once you know this and the x-intercepts, you can sketch the entire curve confidently.

Curve Confidence: Cubic graphs always have that characteristic S-shape. Practice recognising the entry and exit quadrants based on the sign of the x³ coefficient.