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GCSE Maths Revision Summary

Jun 18, 2025

Overview

This lecture provides a focused revision of the main topics in the CA GCSE Maths CM4 course, covering algebra, geometry, statistics, and number. It emphasizes key methods, worked examples, and exam strategies to ensure familiarity with all required topics.

Algebra: Quadratics & Algebraic Fractions

  • Factorize quadratics using either inspection or the split-in-the-middle-term technique:
    • For ax² + bx + c, set up two brackets and find pairs of factors for a and c that combine to give b.
    • Example: Factorize 5x² + 13x + 6 by inspection or by splitting the middle term.
  • The difference of two squares: a² - b² = (a + b)(a - b).
    • Example: 20x² - 125y² = 5(4x² - 25y²) = 5(2x + 5y)(2x - 5y).
  • Solve quadratic equations by factorizing and setting each bracket to zero.
    • Example: 3x² - 10x - 48 = 0 → factorize, then solve for x.
  • Combine algebraic fractions by finding a common denominator, expanding, and simplifying.
    • Example: Express 2/(3x+1) + 1/(x-1) as a single fraction.
  • Solve equations with algebraic fractions by combining into a single fraction, clearing denominators, and rearranging into quadratic form.
  • Use the quadratic formula for equations that cannot be factorized:
    • ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
    • Identify a, b, and c from the equation and substitute into the formula.
    • Example: Solve 2x² - x - 9 = 0 using the quadratic formula.
  • Apply quadratic methods to word problems by forming equations from context, then solving for the required values.

Algebra: Perpendicular Lines

  • Perpendicular lines have gradients that multiply to -1.
    • To find the gradient of a perpendicular line, use the negative reciprocal: if the original gradient is m, the perpendicular gradient is -1/m.
  • To find the equation of a perpendicular line:
    • Use the new gradient and substitute a given point to solve for the y-intercept.
    • Example: Find the equation of a line perpendicular to y = -4x + 7 passing through (8, 3).

Geometry: Circle Theorems & Shape

  • The angle in a semicircle is always 90°.
  • Radii from the center to the circumference create isosceles triangles within the circle.
  • The angle at the circumference is half the angle at the center for the same arc.
  • Angles in the same segment from a common chord are equal.
  • Opposite angles in a cyclic quadrilateral (all vertices on the circle) sum to 180°.
  • The radius and tangent meet at 90°.
  • Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
  • Two tangents from a point to a circle are equal in length.
  • A radius that bisects a chord meets the chord at 90°.

Geometry: Volume & Surface Area

  • Volume of a cone or pyramid: ( \frac{1}{3} ) × area of base × height.
    • Example: Volume of a cone with radius r and height h is ( \frac{1}{3} \pi r^2 h ).
  • To find the volume of a frustum:
    • Calculate the volume of the original (larger) cone or pyramid.
    • Calculate the volume of the smaller (chopped off) cone or pyramid.
    • Subtract the smaller volume from the larger to get the frustum’s volume.
  • Surface area of a cone:
    • Curved surface area: ( \pi r l ) (where l is the slant height).
    • Add the area of the base: ( \pi r^2 ).
    • For a frustum, subtract the surface area of the removed top and account for both circular ends.

Statistics: Stratified Sampling & Histograms

  • Stratified sampling ensures the sample reflects the proportions of groups in the population:
    • Use: ( \frac{\text{number in category}}{\text{total}} \times \text{sample size} ).
    • Example: If there are 120 Year 7s out of 300 students and a sample of 15, sample 6 Year 7s.
  • Frequency density for histograms: ( \text{frequency} / \text{class width} ).
    • Draw histograms using frequency density as the height of each bar.
  • To find frequency from a histogram: frequency = frequency density × class width.
  • Mean estimate from grouped data:
    • Multiply the midpoint of each class by its frequency, sum these, and divide by the total frequency.
  • Median estimate:
    • Use cumulative frequency to locate the median position.
    • Alternatively, use linear interpolation:
      Lower bound + (number into category / number in category) × class width.

Number: Applying Bounds

  • For values rounded to a certain decimal place, the upper bound is half a unit above, and the lower bound is half a unit below the given value.
    • Example: 8.4 (1 d.p.) has bounds 8.35 (lower) and 8.45 (upper).
  • Greatest possible range: subtract the smallest lower bound from the largest upper bound.
  • Greatest possible speed: divide the largest possible distance (upper bound) by the smallest possible time (lower bound).
  • For functions involving bounds, to maximize or minimize the result, use the largest/smallest possible values for numerators and denominators as appropriate.

Key Terms & Definitions

  • Quadratic Formula: Formula to solve ax² + bx + c = 0.
  • Frequency Density: Frequency divided by class width in histograms.
  • Cyclic Quadrilateral: A quadrilateral with all vertices on the circle.
  • Stratified Sample: A sample that maintains the group proportions of the population.
  • Negative Reciprocal: For gradient m, the negative reciprocal is -1/m.

Action Items / Next Steps

  • Print and use the revision checklist provided in the course materials.
  • Complete the Ultimate CM4 Revision Question Booklet for targeted practice.
  • Watch the detailed video lessons for any topics you find challenging.
  • Practice regularly using five-a-day exercises and revision cards for retention.
  • Use the QR code resources to check answers and access further explanations.

Full transcript