10 Mathematical skills

Exam analysis

Most tested topics

1 Arithmetic and numerical computation

A decimal writes a number directly by place value. Each move one place to the left makes a digit worth ten times more; each move one place to the right makes it worth ten times less. Thus $3.47=3+4/10+7/100$ , while the zeros in $0.047$ hold the tenths place empty.

First identify the place of each digit and the size the answer should have. For addition or subtraction, align decimal points so equal place values are combined. For multiplication or division, calculate the digits and then check the decimal position against an estimate. Keep the physical unit attached to the final value.

Example: a current changes from $0.38\,\mathrm{A}$ to $1.7\,\mathrm{A}$ . The increase is $1.70-0.38=1.32\,\mathrm{A}$ ; writing $1.7$ as $1.70$ makes the place-value alignment visible but does not change its value. A unit conversion is a multiplication, not a free decimal shift: $0.025\,\mathrm{A}\times1000=25\,\mathrm{mA}$ .

A leading zero prevents misreading: write $0.6$ , not $.6$ . Moving a decimal point changes the number unless a compensating scale factor is also applied. Extra trailing zeros can communicate precision, but choosing significant figures is a separate skill; here the essential check is that every digit keeps its intended place value.

In standard form, a non-zero number is written as $a\times10^n$ , where $1\leq|a|<10$ and $n$ is an integer. A positive exponent represents repeated multiplication by 10; a negative exponent represents repeated division by 10.

$45\,000=4.5\times10^4 \qquad 0.00032=3.2\times10^{-4}$

To convert into standard form, move the decimal point until the first factor lies from 1 up to but not including 10; the number of places moved gives the exponent. When multiplying, multiply the first factors and add exponents. When dividing, divide the first factors and subtract exponents. Finally normalise the first factor if it lies outside the allowed range.

Example: light travels at $3.0\times10^8\,\mathrm{m/s}$ for $3.2\times10^7\,\mathrm{s}$ . Distance $=(3.0\times3.2)\times10^{8+7}=9.6\times10^{15}\,\mathrm{m}$ , which is approximately $10^{16}\,\mathrm{m}$ . The powers combine separately from the ordinary numbers.

The exponent belongs to 10, not to the first factor. Also, $10^{-4}$ is a small positive scale factor, not a negative number. Calculator displays such as 3.2E-4 mean $3.2\times10^{-4}$ ; they do not mean $3.2\times e-4$ .

Ratios, fractions and percentages compare quantities; powers and roots describe how a quantity is combined with itself or undo that operation. The important move is to identify the relationship before calculating, then keep numerator, denominator and scale direction consistent.

Form	Meaning and operation	Compact example
ratio $a:b$	compare $a$ with $b$ ; use a scale factor	$1:4$ means multiply a diagram length by 4
fraction $a/b$	$a$ divided by $b$ ; part of a whole	$18/60=0.30$
percentage	fraction out of 100	$0.30\times100\%=30\%$
power $x^n$	multiply $x$ by itself $n$ times	$3^2=9$
root $\sqrt[n]{x}$	undo raising to power $n$	$\sqrt{49}=7$

Example: a scale says $1\,\mathrm{cm}$ on a diagram represents $4\,\mathrm{cm}$ in the laboratory. A measured diagram distance of $6.2\,\mathrm{cm}$ therefore represents $6.2\times4=24.8\,\mathrm{cm}$ . Multiplying is correct because the real distance is four times the diagram distance; reversing the ratio would shrink it.

A ratio compares quantities in a stated order, so $1:4$ is not interchangeable with $4:1$ . A percentage is dimensionless only after like quantities and compatible units are compared. Squaring and taking a square root are inverse operations, but $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ .

An estimate replaces awkward values with nearby, easy numbers so the calculation can be completed mentally. It should preserve the operation, sign and approximate scale of the original expression, and it should be written with $\approx$ because the rounded expression is not exactly equal to the original.

Use this sequence: round each value to one convenient significant figure → keep any powers of ten explicit → carry out the simple arithmetic → restore the unit → check whether the result has a sensible sign and order of size. Round enough to simplify, but not so much that a non-zero divisor becomes zero.

Example: one fusion reaction releases about $3\times10^{-12}\,\mathrm{J}$ . The number needed for $1\,\mathrm{J}$ is $1/(3\times10^{-12})=(1/3)\times10^{12}$ . Since $1/3\approx0.3$ , this is about $0.3\times10^{12}=3\times10^{11}$ reactions. The power of ten protects the scale during the mental calculation.

For a division, ask whether the denominator is smaller or larger than 1: dividing by a very small number should produce a very large result. An estimate is useful for predicting magnitude and catching calculator-entry errors; it is not permission to omit units or replace a requested accurate calculation with a rough answer.

The sine key maps an angle to a ratio; the inverse-sine key maps a valid ratio back to an angle. For this course, the angle $x$ is expressed in degrees, so check that the calculator shows DEG before entering either operation.

$x\;\xrightarrow{\sin}\;\sin x \qquad r\;\xrightarrow{\sin^{-1}}\;x$

To evaluate $\sin x$ , enter the angle and apply sin; for example, $\sin30^\circ=0.5$ . To recover an angle, apply sin⁻¹ to the ratio; for example, $\sin^{-1}(0.75)\approx48.6^\circ$ . On most calculators, inverse sine is accessed with SHIFT or 2nd followed by SIN.

A sine value must lie from $-1$ to $1$ , so an inverse-sine input outside that interval signals an earlier error. Check the output includes a degree interpretation and is plausible: for an acute angle, a ratio near 0 gives a small angle and a ratio near 1 gives an angle near $90^\circ$ .

$\sin^{-1}x$ means inverse sine, not the reciprocal $1/\sin x$ . RAD mode gives a numerically different angle, and a missing bracket can change what the calculator evaluates. Keep the complete ratio inside the inverse-sine operation before rounding the angle.

2 Handling data

Significant figures count digits from the first non-zero digit and communicate the precision justified by a value. Leading zeros locate the decimal point but are not significant; zeros between non-zero digits are significant, and trailing zeros after a decimal point can show measured precision.

To round to $n$ significant figures: find the first non-zero digit → count to the $n$ th digit → inspect the next digit → leave the $n$ th digit unchanged if the next digit is 0–4, or increase it by one if the next digit is 5–9 → replace discarded place values with zeros where needed and retain the unit.

Example: $0.004786\,\mathrm{s}$ to three significant figures is $0.00479\,\mathrm{s}$ because the significant digits begin at 4 and the next digit after 8 is 6. In standard form, the same decision is clearer: $4.786\times10^{-3}\,\mathrm{s}\approx4.79\times10^{-3}\,\mathrm{s}$ .

Significant figures are not decimal places: $0.00479$ has three significant figures but five decimal places. Do not report a calculated result with more meaningful precision than the measurements support. Keep unrounded calculator values during intermediate steps and round the final result, so repeated early rounding does not distort it.

The arithmetic mean shares the total of a set equally among its values. For experimental data, first decide which readings are valid; the divisor must be the number of readings actually included, not the number originally collected.

$\text{mean}=\frac{\text{sum of included readings}}{\text{number of included readings}}$

Inspect repeats for an anomalous value and use evidence before excluding it. Then add the valid readings with their signs, divide by their count, attach the original unit and round only the final mean appropriately. A negative reading remains negative in both the sum and the mean.

Example: count rates are 54, 58, 52, 35 and 55 counts per minute. If repeats justify treating 35 as anomalous, the included total is $54+58+52+55=219$ . The mean is $219/4=54.75$ , which is 55 counts per minute to two significant figures.

Do not remove a value simply because it changes the mean, and do not divide by five after excluding one of five readings. A mean represents the centre of the included data; it does not show their spread, and averaging cannot correct a systematic error affecting every reading.

A bar chart compares a numerical value across distinct categories. The category variable is discontinuous—for example, material or planet—so each bar belongs to a named group rather than to every value along a continuous scale.

Choose a linear numerical scale that uses most of the plotting area. Label the category axis, label the numerical axis with its quantity and unit, and draw every bar to the correct height. Keep bar widths consistent; gaps usually make separate categories clear. The orientation and category order do not change the data if the labels remain unambiguous.

Read a bar by tracing its top to the numerical scale, then compare or combine only the categories named. Example: if three non-renewable categories contribute 27.5%, 35% and 19.5%, their total is $27.5+35+19.5=82\%$ . The calculation should use the bar heights, not an estimate based only on appearance.

Do not use a bar chart merely because the dependent variable is numerical; the deciding feature is that the horizontal groups are categories. Continuous paired measurements usually need a line or scatter graph. A non-linear axis distorts comparisons, and missing units make bar heights ambiguous.

Frequency is the number of observations in a value or class interval. A frequency table performs the essential grouping first; a frequency diagram or histogram then makes that distribution visible without changing the underlying counts.

Representation	Horizontal structure	Vertical information	Key construction rule
frequency table	values or class intervals	written counts	intervals must not overlap or leave intended values unclassified
frequency diagram	discrete values or classes	frequency	label values/classes and use a linear scale
histogram	continuous class intervals	frequency for equal widths; frequency density for unequal widths	bars touch and bar area represents frequency

$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}$

Example: a class from 10 up to 20 has frequency 12, so its width is 10 and its frequency density is $12/10=1.2$ . A class from 20 up to 25 with frequency 9 has density $9/5=1.8$ ; its narrower but taller bar still has area proportional to 9.

A histogram is not a bar chart with the gaps removed. Its horizontal axis is continuous and bar width carries numerical meaning. If all class widths are equal, frequency heights preserve the same comparison; with unequal widths, plotting raw frequency as height gives misleading areas.

Probability is a number from 0 to 1: 0 means impossible and 1 means certain. For equally likely outcomes, theoretical probability is the number of favourable outcomes divided by the total number of possible outcomes.

$P(A)=\frac{\text{favourable outcomes}}{\text{total equally likely outcomes}} \qquad P(\text{not }A)=1-P(A)$

When probability is estimated from observations, use relative frequency: number of times the outcome occurs divided by the number of trials. Example: if a detector records the chosen event in 30 of 120 repeated trials, the experimental probability is $30/120=0.25$ and the probability of no such event is $1-0.25=0.75$ .

Check that the result lies between 0 and 1; multiplying by 100 converts it to a percentage. A larger number of trials usually gives a more stable experimental estimate, but it does not force the observed proportion to equal the theoretical value exactly.

The favourable-over-total shortcut requires outcomes to be equally likely. Probability describes long-run likelihood, not a guarantee about the next single trial. Do not add probabilities unless the events being combined cannot occur together, and do not confuse a count with a probability until it has been divided by the total.

A scatter diagram plots paired measurements of two variables, one point per pair. Its purpose is to reveal whether the variables tend to change together; the overall cloud of points matters more than the route from one point to the next.

Pattern as the horizontal variable increases	Interpretation
points tend upward	positive correlation
points tend downward	negative correlation
no consistent upward or downward tendency	no clear correlation
points lie close to a trend	stronger correlation than a widely scattered cloud

Label both axes with quantities and units, choose linear scales and plot every pair accurately. Identify any anomalous point, then use a line or curve of best fit to represent the central trend with points reasonably balanced around it. Estimate within the measured range by reading from that trend, not by joining dots.

Correlation does not prove that one variable causes the other; a third factor or coincidence may explain the pattern. An anomaly should be checked rather than automatically deleted, and extrapolation beyond the measured range is less secure because the relationship may change there.

An order of magnitude is the nearest power of ten describing a value's scale. Write the value as $a\times10^n$ with $1\leq a<10$ ; compare $a$ with $\sqrt{10}\approx3.16$ to decide which neighbouring power of ten is closer.

$a<3.16:\;\text{order}=10^n \qquad a\geq3.16:\;\text{order}=10^{n+1}$

Example: $7.2\times10^{-4}$ is closer to $10^{-3}$ than to $10^{-4}$ , so its order of magnitude is $10^{-3}$ . For a calculation, first estimate the standard form: $(2\times10^3)(4\times10^{-6})=8\times10^{-3}$ , whose order of magnitude is $10^{-2}$ .

For products, add powers of ten; for quotients, subtract them; then normalise the coefficient before choosing the nearest power. Use the result to compare scales or check a detailed calculation. Values whose orders differ by three powers of ten differ in scale by about a factor of 1000.

Order of magnitude is not the exponent copied from unexamined standard form and is not the same as rounding to one significant figure. For example, $7\times10^4$ to one significant figure remains $7\times10^4$ , but its nearest order of magnitude is $10^5$ .

3 Algebra

An algebraic symbol states a precise relationship between quantities. Read the complete statement from left to right and keep each symbol's meaning separate from equality.

Symbol	Meaning	Example and reading
$<$	less than	$8\,\mathrm{m}<12\,\mathrm{m}$ : 8 m is less than 12 m
$>$	greater than	$5\,\mathrm{s}>3\,\mathrm{s}$ : 5 s is greater than 3 s
$\propto$	proportional to	$F\propto a$ : for fixed mass, $F/a$ is constant
$\sim$	approximately	$9.8\,\mathrm{N/kg}\sim10\,\mathrm{N/kg}$

A proportionality becomes an equation only after a constant is introduced: $y\propto x$ means $y=kx$ for constant $k$ . If $x$ doubles under the same conditions, $y$ doubles. An inverse relationship must be written explicitly, such as $y\propto1/x$ .

Do not read $\propto$ as “equals”; the constant and its unit may be essential. The symbol $\sim$ reports an approximation, not exact equality. For inequalities, reversing the order reverses the symbol: $3<7$ and $7>3$ describe the same comparison.

The subject is the single symbol isolated on one side of an equation. Changing the subject rewrites the same relationship in an equivalent form; every operation used to isolate the target must preserve equality.

Choose the required subject, then undo operations in reverse order. Apply the same operation to both sides, treat a numerator or bracket as one complete expression, and simplify only after the target is isolated. A useful check is to rearrange the result back to the original form.

$v^2=u^2+2as \quad\Rightarrow\quad v^2-u^2=2as \quad\Rightarrow\quad s=\frac{v^2-u^2}{2a}$

In the example, subtracting $u^2$ from both sides removes the added term; dividing both sides by $2a$ then leaves $s$ . The entire difference $v^2-u^2$ remains in the numerator. No numerical values are needed because the job is to produce a reusable symbolic formula.

“Move it across and change the sign” hides the balancing operation and often loses brackets or factors. Addition is undone by subtraction, multiplication by division, squaring by a square root when appropriate. Never divide by an expression that could be zero without recognising that condition.

Substitution replaces each symbol in a physical equation with its measured value. The numerical calculation is valid only when the units are compatible with the equation, so convert first and keep the value and unit together throughout the reasoning.

Write the equation → identify every symbol and required unit → convert prefixes or time units → insert values with brackets where signs or powers could be ambiguous → calculate → state the derived unit and appropriate final precision. Showing the substitution makes both the arithmetic and unit choices checkable.

$E=Pt,\quad P=60\,\mathrm{W},\quad t=2.0\,\mathrm{min}=120\,\mathrm{s}$

Because $1\,\mathrm{W}=1\,\mathrm{J/s}$ , use seconds: $E=(60\,\mathrm{J/s})(120\,\mathrm{s})=7200\,\mathrm{J}=7.2\times10^3\,\mathrm{J}$ . The seconds cancel, leaving joules. Substituting 2.0 directly would incorrectly treat minutes as seconds and make the result sixty times too small.

Units are not decorations added after a unitless calculation. Do not mix centimetres with metres, minutes with seconds, or prefixed and base units unless the equation is written for that combination. Retain full calculator precision during the calculation and round the final result rather than each intermediate value.

Solving an equation finds the value of an unknown that makes both sides equal. Preserve the balance by performing the same inverse operation on both sides until the unknown is isolated.

Simplify each side if needed, undo addition or subtraction, then undo multiplication or division. Keep signs and brackets visible. When the unknown represents a physical quantity, attach the unit after the algebra and check that the value is sensible for the given quantities.

$12=2+5t \quad\Rightarrow\quad 10=5t \quad\Rightarrow\quad t=2$

The first step subtracts 2 from both sides; the second divides both sides by 5. If the equation describes time in seconds, the solution is $t=2\,\mathrm{s}$ . Verify it by substitution: $2+5(2)=12$ , so the found value satisfies the original equation.

Changing the subject produces a symbolic formula; solving uses the given numbers to find a particular unknown value. Do not change a sign merely because a term appears on the other side—state the inverse operation. A solution that fails when substituted back reveals an arithmetic or algebraic error.

4 Graphs

A graph and a numerical table can represent the same paired measurements. Each plotted point has an $x$ -coordinate and a $y$ -coordinate, so translating between the forms means preserving each pair, its scale and its units.

Direction	Reliable method
Table $\to$ graph	Read one row as $(x,y)$ , locate $x$ horizontally, then $y$ vertically, and mark their intersection.
Graph $\to$ number	Start at the required axis value, move to the plotted point or curve, then project to the other axis and read its scale.

If a temperature–time graph passes through $(40\,\mathrm{s},32\,^{\circ}\mathrm{C})$ , the matching numerical statement is: at $40\,\mathrm{s}$ , the temperature is $32\,^{\circ}\mathrm{C}$ . A value read between labelled divisions must use the size of each small division, not merely the nearest printed label.

A value read from a curve is normally an estimate. Interpolation uses a value inside the measured range; extending the trend beyond the data is extrapolation and is less secure. Do not swap the coordinates or drop their units when translating.

The equation $y=mx+c$ represents a straight-line relationship. Its gradient $m$ is constant: every equal increase in $x$ produces the same change in $y$ . The intercept $c$ is the value of $y$ when $x=0$ .

$\Delta y=m\,\Delta x,\qquad y(0)=c$

For $y=2.5x+4$ , increasing $x$ by 3 increases $y$ by $2.5\times3=7.5$ , wherever the interval begins. The line crosses the $y$ -axis at 4. In a physical graph, $m$ carries the units of the $y$ -quantity divided by the units of the $x$ -quantity, while $c$ carries the units of $y$ .

A positive gradient rises from left to right, a negative gradient falls, and zero gradient gives a horizontal line. Changing $c$ shifts the line vertically without changing its gradient.

A straight line is not automatically direct proportion. Direct proportion has the special form $y=mx$ , so $c=0$ and the line passes through the origin. A non-zero intercept still gives a linear relationship, but not direct proportionality.

A two-variable graph places every data pair at one coordinate. Usually the independent or chosen variable goes on the $x$ -axis and the measured response on the $y$ -axis; the labels must name each quantity and its unit.

Choose simple, continuous linear scales that cover the whole data range and use more than half the available grid in both directions. Mark equal numerical steps at equal distances, plot each coordinate to the nearest half-small-square where the grid permits, and check every point against the table before adding any line or curve.

For wire length in mm and force in N, a row $(15,0.92)$ is plotted by moving to $15\,\mathrm{mm}$ on the horizontal axis and $0.92\,\mathrm{N}$ on the vertical axis. Discrete values occur as separate allowed values; continuous measurements can take values between recorded readings. Both still require correctly scaled axes.

A line of best fit represents the overall trend and need not pass through every point; a dot-to-dot join instead treats every fluctuation as meaningful. Do not use an irregular scale, omit units, or force a line through the origin unless the evidence or physical model requires it.

The gradient of a straight graph measures how much the vertical quantity changes per unit change in the horizontal quantity. The intercept is where the line crosses an axis; in $y=mx+c$ , $c$ is specifically the $y$ -intercept at $x=0$ .

$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

Choose two well-separated points on the straight line, not automatically two experimental points. Draw a large right-angled gradient triangle, read both coordinate pairs from the axis scales, subtract in the same direction, and divide vertical change by horizontal change. State the sign and the compound unit.

If a velocity–time line passes through $(2\,\mathrm{s},5\,\mathrm{m\,s^{-1}})$ and $(8\,\mathrm{s},17\,\mathrm{m\,s^{-1}})$ , then $m=(17-5)/(8-2)=2\,\mathrm{m\,s^{-2}}$ . Extending or reading the line at $t=0$ gives the intercept $c=1\,\mathrm{m\,s^{-1}}$ .

The triangle dimensions are changes, not the coordinate values themselves. Reversing only one subtraction gives the wrong sign; reversing both leaves the gradient unchanged. A larger triangle usually reduces the percentage reading uncertainty.

A curve has a changing gradient, so its rate of change depends on position. A tangent drawn at one point has the same local direction as the curve there; the tangent's gradient estimates the instantaneous rate at that point.

Mark the point of interest and place a straight line so that it just follows the curve's direction there, with the curve lying evenly around the line locally. Extend the tangent, choose two well-separated points on the tangent, then calculate $\Delta y/\Delta x$ using the axis scales and units.

$\text{instantaneous rate at }x_0\approx\frac{y_2-y_1}{x_2-x_1}\quad\text{using two points on the tangent at }x_0$

On a distance–time graph, the tangent gradient has units $\mathrm{m/s}$ and estimates speed at that instant. A steeper positive tangent means a larger positive rate; a horizontal tangent gives zero rate; a downward tangent gives a negative rate under the chosen sign convention.

A chord joining two points on the curve gives an average rate over an interval, not the instantaneous rate at one point. Calculate with points on the tangent—even if they are not data points—and do not assume a tangent must touch the curve only once globally.

The area between a curve and the $x$ -axis combines the two plotted quantities. Its physical meaning and unit come from multiplying the vertical-axis quantity by the horizontal-axis quantity; the context decides what that product represents.

$\text{area unit}=(\text{$y$-axis unit})(\text{$x$-axis unit})$

For a velocity–time graph, $(\mathrm{m/s})(\mathrm{s})=\mathrm{m}$ , so area represents displacement. A constant velocity of $6\,\mathrm{m/s}$ for $4\,\mathrm{s}$ gives a rectangular area $6\times4=24\,\mathrm{m}$ . A triangular or trapezium section can be calculated geometrically.

For an irregular curve, find the value represented by one small grid square, count complete squares, combine partial squares into approximate wholes, and multiply the total by the value per square. Using smaller squares or sensible trapezia can improve the estimate while keeping the curve boundary visible.

The shaded shape has no automatic physical label: determine it from the axis product. Area below the $x$ -axis is negative in a signed total, so on a velocity–time graph it represents displacement in the opposite direction; total distance would require adding the magnitudes of the separate areas.

5 Geometry and trigonometry

An angle measures the turn from one line or direction to another. The unit is the degree, written $^{\circ}$ , and the vertex is the point where the two arms meet. A numerical angle is meaningful only when both arms are identified.

$\text{full turn}=360^{\circ},\qquad \text{half turn}=180^{\circ},\qquad \text{right angle}=90^{\circ}$

To measure an angle, place the protractor centre on the vertex, align its zero line with the stated reference arm, and read the scale where the second arm crosses. Choose the scale that starts at zero on the aligned arm and report the value in degrees.

In ray physics, the angle of incidence is measured between the incoming ray and the normal—the line perpendicular to the surface at the point of incidence. If the ray makes $11^{\circ}$ with the surface, it makes $90^{\circ}-11^{\circ}=79^{\circ}$ with the normal.

Do not measure from whichever nearby line looks convenient. Measuring from the surface instead of the normal gives the complementary angle, and reading the wrong protractor scale can produce a second incorrect value.

A three-dimensional object has length, width and height, but a two-dimensional representation can display only two spatial directions at once. Read the labels, view direction and scale to decide which real dimensions and relationships the drawing preserves.

Representation	What it can show directly
plan view	length and width as seen from above
front or side elevation	one horizontal dimension and height
perspective sketch	the object's overall 3D form, but apparent lengths may be distorted
scale diagram	measurable lengths linked to real lengths by a stated factor

If a scale diagram states $1\,\mathrm{cm}$ on the page represents $10\,\mathrm{cm}$ in the laboratory, a measured height of $3.1\,\mathrm{cm}$ represents $3.1\times10=31\,\mathrm{cm}$ . Apply the scale factor to a length measured in the same direction, then attach the real unit.

Match corresponding corners, edges and faces across views; a feature hidden in one view may be visible in another. A cross-section describes a slice through the object rather than its outside silhouette.

A labelled physics sketch is not automatically drawn to scale. Never infer a real length, angle or proportion from appearance alone unless dimensions or a scale are supplied; perspective can make equal or parallel edges look unequal or convergent.

Area measures a two-dimensional region, surface area totals the areas of an object's outside faces, and volume measures the three-dimensional space it occupies. The requested quantity determines both the formula and the power of the unit.

Shape or object	Calculation	Unit type
rectangle	$A=lw$	square units
triangle	$A=\frac12 bh$	square units
cube, side $a$	$S=6a^2$	square units
cube, side $a$	$V=a^3$	cubic units

Use a triangle's perpendicular height, not necessarily a sloping side. Convert every length to the same unit before multiplying. For a cube, one face is a square of area $a^2$ and there are six equal faces; volume multiplies three perpendicular dimensions.

A cube with side $4\,\mathrm{cm}$ has surface area $6(4^2)=96\,\mathrm{cm^2}$ and volume $4^3=64\,\mathrm{cm^3}$ . The numbers differ because surface area counts six faces, while volume fills the interior. An irregular contact area on $2\,\mathrm{cm}$ grid squares can be estimated by counting complete and combined partial squares, then multiplying by $4\,\mathrm{cm^2}$ per square.

Do not report area in cm or volume in $\mathrm{cm^2}$ . Squaring or cubing the numerical length also squares or cubes its unit; converting after the calculation without applying that power changes the physical quantity.

10 Mathematical skills

Exam analysis

Most tested topics

1 Arithmetic and numerical computation

Use decimal place value without losing the scale

Standard form keeps very large and small values manageable

Choose the numerical relationship that matches the physics

Estimate with controlled rounding, not a guess

Use sine and inverse sine in degree mode

2 Handling data

Use significant figures to match the evidence

Find a mean from the valid readings

Use a bar chart for categories

Organise frequencies before choosing the diagram

Probability measures how likely an outcome is

Read a scatter diagram as a pattern of pairs

Use powers of ten to judge order of magnitude

3 Algebra

Choose the symbol that states the relationship

Change the subject while keeping the equation balanced

Substitute values only after the units agree

Solve an equation by undoing operations in order

4 Graphs

Move between a graph and its numbers

Read the straight-line model $y=mx+c$

Plot two measured variables accurately

Find gradient and intercept from a straight line

Use a tangent for an instantaneous rate

Give physical meaning to area under a graph

5 Geometry and trigonometry

Measure an angle from the correct reference line

Read a three-dimensional object from a two-dimensional view

Choose area, surface area or volume—and keep the units