Mathematical requirements

Learning objectives

—M.1-Arithmetic, standard form and significant figuresUse decimal and standard form notation, calculators, means, powers, roots, trigonometric functions, justified significant figures and approximations to check calculated magnitudes.—M.2-Algebra, models, units and dimensional consistencyChange the subject of equations, solve simple and simultaneous equations, use the quadratic formula, substitute quantities with consistent units, check dimensional consistency, use percentages and interpret mathematical symbols.—M.3-Geometry, trigonometry and vectorsUse areas, volumes, Pythagoras, triangle similarity, trigonometric relationships, vector addition and resolution into perpendicular components in physical contexts.—M.4-Graphs, gradients, intercepts and areasSelect variables and scales, determine gradients/intercepts/intersections, draw best-fit or curved trend lines, recognise common graph forms, use tangents to curves and use area under a curve where physically meaningful.—M.5-A Level radians, exponentials and logarithmsUse radians, small-angle approximations, exponentials and logarithms, recognise logarithmic forms of products, quotients, powers and exponentials, and use logarithmic plots to test exponential or power-law variation.

$(a×10ᵐ)(b×10ⁿ)=ab×10ᵐ⁺ⁿ (a×10ᵐ)/(b×10ⁿ)=(a/b)×10ᵐ⁻ⁿ normalise so 1≤|a|<10$

Convert prefixes before calculation: G=10⁹, M=10⁶, k=10³, m=10⁻³, μ=10⁻⁶, n=10⁻⁹. For squared/cubed quantities, apply the power to the conversion factor too.

Keep unrounded calculator values through intermediate steps. Round only the final answer, usually to the significant figures justified by the least precise measured input or by the question instruction.

Do an order-of-magnitude check: round inputs to one convenient significant figure, calculate the approximate power of ten, and compare sign, scale and unit with the calculator result.

Enter brackets explicitly for numerators, denominators, roots and powers; verify degree/radian mode before trigonometry. For a mean, add all values and divide by the number of values.

Significant figures describe justified precision, not the number of decimal places. A trailing zero after a decimal may be significant; leading zeros are not.

Write the model. 2. Rearrange for the target symbol. 3. Convert every quantity to consistent units. 4. Substitute with brackets. 5. Calculate. 6. Check dimensions, sign and physical scale.

Undo operations in reverse order and apply the same operation to both sides. For simultaneous equations eliminate one variable; for a quadratic use x=(−b±√(b²−4ac))/(2a), then reject branches that violate the physical conditions.

$[v]=L T⁻¹, [a]=L T⁻², [F]=M L T⁻² An equation can be valid only if every added term and both sides have identical dimensions.$

Keep unit factors with values and derive the result unit from the algebra. A dimensionally consistent equation may still have a wrong numerical constant or wrong physical model, so dimensional agreement is necessary but not sufficient.

$percentage change=((new−original)/original)×100% percentage difference=|A−B|/reference×100%$

Do not mix cm with m, μF with F or minutes with seconds inside one substitution. Rearranging symbolically first makes powers, roots and reciprocal relationships visible and reduces calculator errors.

Sketch the geometry, label lengths/angles and vector directions, choose positive perpendicular axes, then project each vector onto those axes before adding components.

$If θ is measured from the +x direction: Vₓ=V cosθ, Vᵧ=V sinθ Rₓ=ΣVₓ, Rᵧ=ΣVᵧ R=√(Rₓ²+Rᵧ²), tanφ=Rᵧ/Rₓ$

$a²+b²=c² sinθ=opposite/hypotenuse cosθ=adjacent/hypotenuse tanθ=opposite/adjacent$

Use similarity when corresponding side ratios are equal. Common geometry: circle area πr², sphere area 4πr², sphere volume 4πr³/3, cylinder volume πr²L. Convert all lengths before squaring or cubing.

A negative component means opposite to the chosen positive axis, not a negative magnitude. Use inverse trigonometry and the signs/quadrant of both components to state the final direction.

Cosine is not always horizontal and sine is not always vertical: cosine gives the component adjacent to the labelled angle. Add scalar magnitudes only when vectors are collinear in the same signed axis.

Identify x and y quantities with units, use most of both axes with simple scales, plot accurately, then choose a straight best-fit line or smooth curve that represents the trend rather than joining points dot-to-dot.

$gradient=Δy/Δx units=(y-axis unit)/(x-axis unit) Use two well-separated points on the best-fit line, not necessarily measured data points.$

For an instantaneous rate on a curve, draw a tangent at the required point and calculate its gradient using a large triangle. An intersection is where two plotted quantities satisfy both relationships; an intercept is the model value when the other axis variable is zero.

$area under y against x ≈ Σ y Δx units=(y-axis unit)(x-axis unit) It represents a physical quantity only when the governing relation makes that product meaningful.$

A straight line indicates y=mx+c; proportionality requires a straight line through the origin. Curvature may signal powers, reciprocals, exponentials, changing rate or a limited model—use physics and any requested transformation to distinguish them.

A visually straight trend is not automatically direct proportionality. Always test the intercept within uncertainty and attach compound units to gradient and area.

$θ(rad)=θ(°)π/180 s=rθ, arc length s for |θ|≪1 rad: sinθ≈tanθ≈θ and cosθ≈1−θ²/2$

$ln(ab)=ln a+ln b ln(a/b)=ln a−ln b ln(aⁿ)=n ln a ln(eˣ)=x$

$y=axⁿ ⇒ ln(y/y₀)=n ln(x/x₀)+constant plot ln(y/y₀) against ln(x/x₀): gradient=n$

$y=aeᵏˣ ⇒ ln(y/y₀)=kx+constant plot ln(y/y₀) against x: gradient=k$

The intercept determines the scale factor in the chosen reference-unit system. Gradient units are dimensionless for a log-log power plot and inverse x-units for ln(y/y₀) against x. Use natural or base-10 logs consistently with the model.

Small-angle approximations require radians. A logarithm must act on a dimensionless ratio such as y/y₀; writing ln(3 m) without a reference unit is not physically complete.

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—Key quantities, symbols and unitsRead

Mathematical requirements

Learning objectives

Keep powers of ten and precision under control from input to final answer

Rearrange symbols first, then substitute SI values and audit dimensions

Let the diagram and chosen axes decide the trigonometry and vector signs

Read every graph feature as a physical ratio, value or accumulated quantity

Use radians for angle physics and logarithms to expose model parameters

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