6 Mathematical skills

Exam analysis

Most tested topics

1 Arithmetic and numerical computation

A decimal digit has a value fixed by its position: moving one place left multiplies its value by 10, while moving one place right divides it by 10.

Number	Meaning
$3.2$	3 ones and 2 tenths
$0.32$	3 tenths and 2 hundredths
$0.032$	3 hundredths and 2 thousandths

Line up decimal points when adding or subtracting. For multiplication or division, calculate with the full values and check the order of magnitude; keep the unit attached to the final chemical quantity.

\text{mean rate}=\frac{1}{156,\mathrm{s}}=0.00641,\mathrm{s^{-1}}

Zeros before the first non-zero digit locate the decimal point and are not optional. Do not drop a leading zero or move the decimal point without applying the corresponding power of ten.

Standard form writes a non-zero number as $a\times10^n$ , where $1\leq |a|<10$ and $n$ is an integer.

Decimal form	Standard form	Direction
$6410$	$6.41\times10^3$	decimal moved 3 places left, so exponent is $+3$
$0.00641$	$6.41\times10^{-3}$	decimal moved 3 places right, so exponent is $-3$

(a\times10^m)(b\times10^n)=ab\times10^{m+n},\qquad \frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n}

After calculating, normalise the coefficient. For example, $12.8\times10^{-4}=1.28\times10^{-3}$ . State the unit after the complete standard-form value.

A coefficient such as 12.8 is not final standard form. A negative exponent makes a small positive magnitude; it does not make the number negative.

Ratios, fractions and percentages express relative amounts; powers and roots compact repeated multiplication and reverse it.

Tool	Reusable method	Chemistry-style example
ratio	divide by the same factor to simplify, or multiply every part by the same scale factor	$2:1$ means two reacting units for every one of the other
fraction	numerator is the selected part; denominator is the total	fraction of oxygen $=\frac{\text{oxygen amount}}{\text{total amount}}$
percentage	multiply a fraction by 100%; divide a percentage by 100% to recover the fraction	$\%\text{ yield}=\frac{\text{actual}}{\text{theoretical}}\times100$
power	$x^n$ means multiply $x$ by itself $n$ times	area scale uses a square, $x^2$
root	$\sqrt[n]{y}$ reverses raising to power $n$	if $x^2=y$ , then positive length $x=\sqrt{y}$

If a formula requires a $2:3$ ratio and the first amount is 0.40 mol, one ratio part is 0.20 mol and the second amount is $3\times0.20=0.60$ mol.

Use the same units in numerator and denominator before forming a fraction or percentage. Check whether the context permits values above 100%; percentage yield normally should not, so a value above 100% signals data or method error.

Do not add ratio parts when the task asks for one scaled amount, confuse a percentage with its decimal multiplier, or halve a power when the operation requires a root.

2 Handling data

Significant figures count meaningful digits from the first non-zero digit and communicate precision without keeping unsupported calculator digits.

Value	Significant figures	Reason
$0.00450$	3	leading zeros locate the decimal; 4, 5 and the final zero are significant
$20.06$	4	zeros between non-zero digits are significant
$2300$	ambiguous without notation	write $2.3\times10^3$ for 2 s.f. or $2.300\times10^3$ for 4 s.f.

To round to $n$ significant figures, keep the first $n$ meaningful digits, inspect the next digit, and increase the last kept digit by 1 when the next digit is 5 or more. Keep place-value zeros in the rounded result.

6.134;\text{to 2 s.f.}=6.1,\qquad 0.009876;\text{to 3 s.f.}=0.00988

Significant figures are not decimal places. Round once at the end of a multi-step calculation unless an intermediate value must be reported separately.

The arithmetic mean shares the total of a set equally across the number of values.

\text{mean}=\frac{\text{sum of all included values}}{\text{number of included values}}

For starting temperatures $17.8^{\circ}\mathrm{C}$ and $18.4^{\circ}\mathrm{C}$ , the mean is $(17.8+18.4)/2=18.1^{\circ}\mathrm{C}$ . For $6.1$ , $6.3$ and $6.0$ , the mean is $18.4/3=6.13\ldots$ , reported as $6.1$ to 2 significant figures.

State which values are included, retain units, and report at precision justified by the measurements. Exclude an anomaly only when evidence justifies that decision, then adjust both the sum and the count.

Do not divide by the number of intervals or by a memorised count. The denominator is the number of values actually included.

A bar chart compares values for separate categories; each bar's height represents the numerical value for its category.

Feature	Requirement
category axis	label each discrete category; keep bars separated
value axis	use a linear, evenly spaced scale and state the quantity and unit
bars	equal width, common baseline and accurately plotted heights
interpretation	compare heights using values or differences, not visual adjectives alone

Read a bar by tracing its top horizontally to the value scale. Compare categories with a difference, ratio or ranked statement when that answers the scientific question.

Do not join bar tops or make bars touch for separate categories. Touching intervals belong to a histogram, which is outside this Chemistry objective.

Probability measures how likely an event is on a scale from 0 (impossible) to 1 (certain).

P(\text{event})=\frac{\text{number of favourable equally likely outcomes}}{\text{total number of equally likely outcomes}}

Form	Example for $P=0.25$
fraction	$\frac14$
decimal	$0.25$
percentage	$25\%$

For an event and its opposite, $P(\text{not event})=1-P(\text{event})$ . Repeated-event data may estimate probability as observed frequency divided by total trials.

The favourable-over-total rule requires equally likely outcomes. A probability cannot be below 0 or above 1.

A scatter diagram plots paired values for two variables so their overall association can be identified.

Pattern	Meaning
points rise from left to right	positive correlation: larger $x$ tends to occur with larger $y$
points fall from left to right	negative correlation: larger $x$ tends to occur with smaller $y$
no clear direction	no evident correlation
point far from the pattern	possible outlier or anomaly to investigate

The more tightly points cluster around a smooth trend, the stronger the association. Describe direction first, then strength if the diagram supports it; use a best-fit trend rather than joining point-to-point.

Correlation does not by itself show that one variable causes the other. A third factor or the experimental design may explain the pattern.

An order of magnitude describes the power-of-ten scale of a quantity and lets you judge whether a calculation is plausibly sized.

Quantity	Scientific notation	Scale
$0.0048$	$4.8\times10^{-3}$	about $10^{-3}$
$6200$	$6.2\times10^3$	about $10^3$
$1.7\times10^6$	already standard form	about $10^6$

For a quick estimate, round inputs to one significant figure, calculate using powers of ten, then compare the full answer with the estimate. Multiplication adds exponents and division subtracts them.

\frac{(3\times10^4)(2\times10^{-2})}{6\times10^1}\approx1\times10^1

Order of magnitude is a scale estimate, not a precise rounded answer. Keep coefficient reasoning separate from exponent arithmetic and use the estimate to catch misplaced decimal points.

3 Algebra

The symbols $<$ , $>$ , $\propto$ and $\sim$ describe different relationships and must not be used as interchangeable versions of equals.

Symbol	Read as	Meaning	Example
$a<b$	$a$ is less than $b$	$a$ has the smaller value	$pH<7$
$a>b$	$a$ is greater than $b$	$a$ has the larger value	temperature $>25^{\circ}\mathrm{C}$
$y\propto x$	$y$ is directly proportional to $x$	$y=kx$ for constant $k$ ; doubling $x$ doubles $y$	mass $\propto$ amount for a fixed substance
$a\sim b$	$a$ is approximately or of similar scale to $b$	values are close, not exactly equal	$6.41\sim6.4$

For $y\propto x$ , the ratio $y/x$ stays constant. Use $<$ and $>$ with the pointed end facing the smaller value.

A correlation does not establish proportionality, and $\sim$ does not claim exact equality. Direct proportionality must pass through the origin in a graph of $y$ against $x$ .

Changing the subject rewrites an equation so the required variable is alone on one side while the relationship remains equivalent.

Operation around target	Inverse used on both sides
$+a$	subtract $a$
$-a$	add $a$
$\times a$	divide by $a$
$\div a$	multiply by $a$
square	take the appropriate square root

c=\frac{n}{V}\quad\Longrightarrow\quad n=cV\quad\text{and}\quad V=\frac{n}{c}

Work outwards from the target variable and undo operations in reverse order. Apply every operation to the whole of both sides, then substitute the rearranged expression back into the original relationship as a check.

Do not move a term by changing its sign without performing an inverse operation on both sides. Keep brackets when an operation applies to a complete sum or difference.

Substitution replaces each symbol with its measured value only after the values have been expressed in units consistent with the equation.

Step	Action
1	write the equation and identify every symbol
2	convert values to one compatible unit system
3	substitute with brackets around negative or compound values
4	calculate while retaining meaningful intermediate digits
5	give the answer with its derived unit and appropriate precision

c=\frac{n}{V}=\frac{0.250,\mathrm{mol}}{0.500,\mathrm{dm^3}}=0.500,\mathrm{mol,dm^{-3}}

Units behave algebraically: dividing mol by $\mathrm{dm^3}$ gives $\mathrm{mol\,dm^{-3}}$ . Convert $500\,\mathrm{cm^3}$ to $0.500\,\mathrm{dm^3}$ before using a concentration equation expressed per $\mathrm{dm^3}$ .

A numerically correct substitution with incompatible units is not a valid physical result. Do not append a memorised unit without deriving it from the equation.

Solving an equation finds the value that makes both sides equal by preserving the balance at every step.

Step	Example for $3x+5=20$
simplify each side if needed	already simplified
undo addition or subtraction	$3x=15$
undo multiplication or division	$x=5$
check in the original equation	$3(5)+5=20$

Collect like terms before isolating the unknown. If the unknown appears in a denominator, multiply both sides by the denominator where it is non-zero; if brackets occur, expand them or undo their outer operation consistently.

An equation may have a restriction, such as a denominator not being zero. Do not accept a value until substitution confirms that it satisfies the original equation and its physical context.

4 Graphs

A graph and a table can represent the same relationship: each plotted point pairs one value on the horizontal axis with one value on the vertical axis.

Task	Construction	Numerical result
find $y$ for a given $x$	move vertically from $x$ to the graph, then horizontally to the $y$ -axis	read $y$ with its unit
find $x$ for a given $y$	move horizontally from $y$ to the graph, then vertically to the $x$ -axis	read $x$ with its unit
make a table	record the two coordinates of selected points	one row is one $(x,y)$ pair

Read each axis label, unit and scale before estimating. Interpolation estimates between plotted values. An intersection gives the pair of values shared by two graphs; a plateau shows that the vertical value is no longer changing as the horizontal value increases.

Keep construction lines parallel to the axes and report a precision justified by the scale. Extrapolation beyond the measured range is less secure than interpolation and should not be treated as an observed value.

A relationship of the form $y=mx+c$ is linear because equal changes in $x$ produce equal changes in $y$ ; its graph is a straight line.

y=mx+c

Term	Meaning on a graph of $y$ against $x$
$m$	gradient: the change in $y$ for each unit change in $x$
$c$	vertical-axis intercept: the value of $y$ when $x=0$
$m>0$	the line rises from left to right
$m<0$	the line falls from left to right
$m=0$	$y$ is constant, so the line is horizontal

For $y=3x+2$ , increasing $x$ by $1$ increases $y$ by $3$ , and the line crosses the $y$ -axis at $2$ . The points $(0,2)$ , $(1,5)$ and $(2,8)$ therefore lie on the same straight line.

A straight-line relationship is directly proportional only when $c=0$ , so the line passes through the origin. A straight line with a non-zero intercept is linear but not directly proportional.

Plotting two variables turns paired data into a graph so that a pattern, relationship or anomalous result can be seen.

Step	Action
1	place the independent variable on the horizontal axis and the dependent variable on the vertical axis
2	label both axes with the variable and unit
3	choose simple, even scales that use most of the grid
4	plot every coordinate accurately as a small cross
5	draw a suitable line or smooth curve of best fit through the overall trend

Continuous measurements, such as temperature and time, may take values between recorded points and usually support a line or curve. Discrete values occur as separate counts or categories and should remain distinct. A best-fit construction should balance the scatter rather than join each point in turn.

Do not force an anomalous point into the trend or omit it without evidence. Keep it plotted, identify it as a possible anomaly, and use the pattern of the remaining data when judging the best fit.

The gradient measures how quickly $y$ changes with $x$ , while the intercept is the value of $y$ where the line crosses the vertical axis.

m=rac{\Delta y}{\Delta x}=rac{y_2-y_1}{x_2-x_1},\qquad c=y-mx

Choose two widely separated points on the straight best-fit line, not necessarily measured data points. Form a large gradient triangle, calculate vertical change divided by horizontal change, then substitute one point into $c=y-mx$ or read the intercept at $x=0$ .

ext{using }(2,5) ext{ and }(8,17):\quad m=rac{17-5}{8-2}=2,\quad c=5-(2 imes2)=1,\quad y=2x+1

Gradient units are $y$ -axis units divided by $x$ -axis units. Preserve the sign: a falling line has a negative gradient. A small triangle magnifies reading error, so use much of the line where possible.

For a curved graph, the gradient changes from point to point. The gradient of a tangent at one point measures the instantaneous rate of change there.

Step	Action
1	mark the point at which the rate is required
2	draw a straight tangent that follows the curve's local direction and touches it at that point
3	choose two widely separated points on the tangent
4	calculate $\Delta y/\Delta x$ and state the derived rate unit

ext{instantaneous rate}= ext{tangent gradient}=rac{\Delta y}{\Delta x}

On a hydrogen-volume against time graph, the tangent gradient has units such as $\mathrm{cm^3\,s^{-1}}$ . A steep tangent means a faster rate; as the curve becomes less steep, the instantaneous rate decreases.

A chord joining two points on the curve gives an average rate over an interval, not the rate at one instant. Calculate the gradient from points on the tangent, not from nearby points on the curve.

6 Mathematical skills

Exam analysis

Most tested topics

1 Arithmetic and numerical computation

Use decimal numbers in chemistry calculations

Write and calculate with standard form

Use ratios, fractions, percentages, powers and roots

2 Handling data

Round results to appropriate significant figures

Calculate and interpret an arithmetic mean

Construct and interpret a bar chart

Use simple probability

Identify patterns in a scatter diagram

Estimate with orders of magnitude

3 Algebra

Read inequalities, proportionality and approximation symbols

Change the subject of an equation

Substitute values with consistent units

Solve a simple algebraic equation

4 Graphs

Translate between graphs and numerical values

Recognise a linear relationship

Plot two variables from data

Determine the gradient and intercept of a line

Use a tangent to find an instantaneous rate