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Super 8s: GCSE Higher (Calculator)
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Foundational skills
Rounding to significant figures
\[ 0.003847 \rightarrow 0.00385 \text{ (3 s.f.)} \]
Round a number to a given number of significant figures.
Standard form calculations
\[ (3.2 \times 10^4) \times (2 \times 10^{-2}) \]
Multiply or divide numbers in standard form.
Percentage calculations
Percentage of an amount
\[ 17.5\% \text{ of } £840 \]
Find a percentage of an amount using a calculator.
Percentage change
\[ £450 \rightarrow £520 = \text{?}\% \]
Calculate the percentage increase or decrease.
Reverse percentage
\[ \text{After 20% off: } £64 \rightarrow \text{?} \]
Find the original amount before a percentage change.
Compound interest
\[ £2000 \text{ at } 3.5\% \text{ for 4 years} \]
Calculate compound interest over several years.
Depreciation
\[ £15000 \text{ car, } 18\% \text{ per year} \]
Calculate depreciation over time.
Ratio and proportion
Sharing in a ratio
\[ £720 \text{ in ratio } 3:5:4 \]
Divide an amount in a given ratio.
Combining ratios
\[ a:b = 2:3, \; b:c = 4:5 \]
Combine two ratios with a common term.
Direct proportion
\[ y \propto x, \; y = 12 \text{ when } x = 4 \]
Find unknown values using direct proportion.
Inverse proportion
\[ y \propto \frac{1}{x}, \; y = 6 \text{ when } x = 4 \]
Find unknown values using inverse proportion.
Scale factors for area and volume
\[ k = 3 \Rightarrow \text{Vol} \times 27 \]
Use linear, area and volume scale factors.
Algebra – equations
Solving linear equations
\[ 3(2x + 5) = 4x – 7 \]
Solve equations with brackets and unknowns on both sides.
Rearranging formulae
\[ V = \frac{4}{3}\pi r^3 \; \text{(make } r \text{ subject)} \]
Change the subject of a formula.
Solving quadratics by factorising
\[ x^2 – 5x + 6 = 0 \]
Solve a quadratic equation by factorising.
Solving quadratics using the formula
\[ 2x^2 + 5x – 7 = 0 \]
Use the quadratic formula to solve equations.
Simultaneous equations (linear)
\[ 3x + 2y = 12, \; 5x – y = 7 \]
Solve two linear equations.
Simultaneous equations (non-linear)
\[ y = x^2, \; y = 2x + 3 \]
Solve when one equation is quadratic.
Trigonometry
SOHCAHTOA
\[ \sin\theta = \frac{\text{opp}}{\text{hyp}}, \; \cos\theta = \frac{\text{adj}}{\text{hyp}} \]
Find missing sides or angles using sin, cos, tan.
Pythagoras’ theorem
\[ a^2 + b^2 = c^2 \]
Find missing sides in right-angled triangles.
Area using ½ab sin C
\[ \text{Area} = \frac{1}{2}ab\sin C \]
Find area of a triangle given two sides and included angle.
Sine rule
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Find missing sides or angles using the sine rule.
Cosine rule
\[ a^2 = b^2 + c^2 – 2bc\cos A \]
Find missing sides or angles using the cosine rule.
Arc length and sector area
\[ \frac{\theta}{360} \times \pi r^2 \]
Calculate arc lengths and areas of sectors.
Geometry – area and volume
Volume of prisms
\[ V = \text{Area} \times \text{length} \]
Calculate the volume of various prisms.
Volume of pyramids and cones
\[ V = \frac{1}{3} \times \text{base area} \times h \]
Calculate volumes using ⅓ base × height.
Volume of spheres
\[ V = \frac{4}{3}\pi r^3 \]
Calculate volumes using ⁴⁄₃πr³.
Surface area
\[ SA = 2\pi r^2 + 2\pi rh \]
Calculate surface areas of 3D shapes.
Statistics
Mean from a frequency table
\[ \bar{x} = \frac{\Sigma fx}{\Sigma f} \]
Calculate the mean from grouped or ungrouped data.
Cumulative frequency and median
\[ \text{Median at } \frac{n}{2} \]
Read median/quartiles from a cumulative frequency diagram.
Histograms with unequal class widths
\[ \text{Frequency density} = \frac{f}{w} \]
Read or draw histograms with varying class widths.
Probability
Tree diagrams for combined events
\[ P(A \text{ and } B) = P(A) \times P(B|A) \]
Calculate probabilities using tree diagrams.
Conditional probability
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Use conditional probability to find overall probabilities.
Independent and mutually exclusive events
\[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
Use rules for independent and mutually exclusive events.
Graphs and functions
Distance-time graphs
\[ \text{Speed} = \text{gradient} \]
Read and interpret distance-time graphs.
Equation of a line through two points
\[ y = mx + c \text{ from } (x_1, y_1), (x_2, y_2) \]
Find the equation of a line given two points.
Special cases
Upper and lower bounds
\[ 3.5 \text{ cm (1 d.p.)} \Rightarrow 3.45 \leq x < 3.55 \]
Calculate using upper and lower bounds.
Iteration
\[ x_{n+1} = \sqrt[3]{5 + 2x_n} \]
Use an iteration formula to find approximate solutions.
Vector arithmetic
\[ 2\mathbf{a} – 3\mathbf{b} \]
Add, subtract and multiply vectors by scalars.
Simplifying algebraic fractions
\[ \frac{x^2 – 9}{x^2 + 5x + 6} \]
Factorise and simplify algebraic fractions.
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