13.3 Mathematical and data-presentation skills

Learning objectives

13.3.1Numerical and algebraic calculations• Use arithmetic, decimals, fractions, percentages, ratios, reciprocals and standard form; round only final answers and use decimal places and significant figures appropriately; use positive whole-number indices, consistent units, simple equations and direct or inverse proportion.13.3.2Geometry, measurement and unit conversion• Use common geometrical terms, select suitable units and convert between units including cm³ and dm³, mg, g and kg, J and kJ, and Pa and kPa.13.3.3Graphs, charts and statistics• Draw and interpret graphs and charts; interpolate and extrapolate; determine gradients and intercepts; select suitable scales and axes; recognise direct proportionality and calculate means.13.3.4Record readings and tabulated data• Read instruments to half a smallest division where appropriate; record precision, units, repeated readings and significant figures correctly; use quantity/unit table headings and express ratios as x : y.13.3.5Plot and analyse chemical data• Plot variables on labelled axes with sensible scales; mark points accurately; draw best-fit lines or curves while treating anomalies appropriately; read values, intercepts and gradients using suitably large triangles.13.3.6Extended graph gradients• Determine the gradient of a curve by drawing a tangent.

Treat every chemistry calculation as one controlled chain: express the data in compatible forms and units, choose the relationship, rearrange if necessary, substitute, calculate, then round only the final answer.

Requirement	Reliable move
fraction, decimal, percentage	convert to the form needed; percentage = fraction × 100%
ratio	divide all terms by the same factor; preserve the order of quantities
reciprocal	$1/x$ reverses a non-zero factor or expresses an inverse relationship
standard form	write $a\times10^n$ with $1\leq a<10$ and integer $n$
equation	keep units consistent, substitute known values and use inverse operations to isolate the unknown
direct proportion	$y=kx$ ; doubling $x$ doubles $y$
inverse proportion	$y=k/x$ ; doubling $x$ halves $y$

$c=\frac{n}{V},\quad n=cV=0.250\times0.0400=0.0100\ \mathrm{mol}$

Keep extra calculator digits through intermediate steps. Report the final result with decimal places or significant figures appropriate to the given data and the measurement context.

Do not round each intermediate value or substitute quantities in inconsistent units. A numerically correct calculator display is not a complete result when a unit is required.

A unit must match the quantity and scale. Convert before substitution when an equation requires consistent units, and attach the unit to the final result.

Quantity	Exact relationship	Smaller unit → larger unit
volume	$1000\ \mathrm{cm^3}=1\ \mathrm{dm^3}$	divide by 1000
mass	$1000\ \mathrm{mg}=1\ \mathrm{g}$ ; $1000\ \mathrm{g}=1\ \mathrm{kg}$	divide by 1000 for each step
energy	$1000\ \mathrm{J}=1\ \mathrm{kJ}$	divide by 1000
pressure	$1000\ \mathrm{Pa}=1\ \mathrm{kPa}$	divide by 1000

Term	Meaning used in measurement
radius / diameter	centre-to-edge distance / full distance through the centre; diameter = 2 × radius
circumference	distance around a circle
square / rectangle	four-sided shapes used for area; a square has equal sides
diagonal	line joining opposite corners
angle / curve	amount of turn / a line whose direction changes

$40.0\ \mathrm{cm^3}=0.0400\ \mathrm{dm^3}$ ; the conversion changes the numerical value but not the physical volume.

Cubic units are already volume units: the syllabus conversion is $1000\ \mathrm{cm^3}=1\ \mathrm{dm^3}$ , not a factor of $10$ and not $10^6$ for this pair.

Choose a representation that matches the variables, then read only what its scale and range support. A graph displays a relationship; a mean combines comparable repeated values.

Task	Mathematical tool	Interpretation
show how one continuous variable changes with another	line graph or scatter graph with best-fit line/curve	direction, shape, intercept and values within the plotted range
compare separate categories	suitable chart	compare bar lengths or sector proportions using the labelled scale/key
estimate between measured points	interpolation	estimate within the evidence range
extend beyond measured points	extrapolation	prediction with greater uncertainty
rate of change on a straight line	gradient $=\Delta y/\Delta x$	include units from the two axes
starting value	intercept	value where the line crosses an axis
summarise repeats	mean $=\Sigma x/n$	use only values selected by the stated repeat/anomaly rule

Direct proportionality appears as a straight line through the origin: $y=kx$ . A straight line with a non-zero intercept is linear but not directly proportional.

Interpolation and extrapolation are not equally secure. Do not claim direct proportion merely because the graph is straight, and do not average quantities that are not comparable repeats.

The recorded value must reflect what the instrument can resolve. Read an analogue scale to the nearest half of the smallest division where required and preserve appropriate precision in the table.

Feature	Required convention
analogue reading	interpolate to the nearest half-smallest division where appropriate
measured value	decimal places and significant figures reflect the instrument's precision
calculated value	use the same number of significant figures as the least precise raw datum used in that calculation
table heading	quantity or symbol followed by solidus and unit, e.g. `time / s`
table body	numbers only; do not repeat units in each cell
repeated readings	record each reading before calculating or selecting a representative value
ratio	write in the order requested as $x:y$

Values measured with the same instrument in one table column should normally show consistent decimal places, including trailing zeros that communicate the instrument's resolution.

Do not add unjustified digits to a coarse reading or remove meaningful trailing zeros. Significant figures describe precision; they do not make an inaccurate method accurate.

A graph must make the data relationship readable without distorting it: variables, units, scale, plotted points and the best-fit relationship all carry meaning.

Step	Construction rule
axes	independent variable on $x$ and dependent variable on $y$ , unless instructed otherwise
labels	transfer quantity and unit from the table heading, e.g. `time / s`
scale	use sensible 1, 2 or 5-based intervals and more than half the grid in both directions
points	mark small crosses or encircled dots to half a smallest square
best fit	draw one thin, smooth line or curve with roughly even scatter on both sides; ignore a clearly anomalous point when fitting
readings	interpolate/extrapolate and read values or intercepts to half a smallest square
straight-line gradient	use $\Delta y/\Delta x$ from a marked triangle whose hypotenuse spans at least half the best-fit line

Gradient units are $y$ -axis units divided by $x$ -axis units. Use two points on the best-fit line or its large gradient triangle, not necessarily two experimental points.

Do not join plotted points dot-to-dot unless instructed. A best-fit line need not pass through the origin or every point, and an anomaly remains plotted even when ignored for the fit.

For Extended candidates, the gradient of a curve at one point is the gradient of the tangent there. The tangent is a straight line with the same local direction as the curve.

Step	Action
1	locate the required point on the smooth curve
2	draw a straight tangent that touches the curve at that point and follows its local direction, with roughly balanced separation on either side
3	choose two well-separated points on the tangent, not on the curve
4	draw a large gradient triangle and read the coordinate differences from the axes
5	calculate $\Delta y/\Delta x$ and attach $y$ -unit per $x$ -unit

$\text{instantaneous gradient}=\frac{y_2-y_1}{x_2-x_1}$

A steeper tangent has a larger gradient magnitude. A horizontal tangent has gradient zero; a downward tangent gives a negative gradient with the usual axis directions.

Do not calculate between two points on the curve: that gives an average gradient over an interval. The tangent must be drawn at the specified point before the triangle is chosen.

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—Cambridge IGCSE Chemistry (0620)Read

13.3 Mathematical and data-presentation skills

Learning objectives

Control the numerical chain in chemistry calculations

Choose and convert measurement units

Choose and interpret graphs, charts and statistics

Record readings and tables with matched precision

Construct and analyse a chemical-data graph

Find the instantaneous gradient of a curve

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