Scottish National 4 (Nat 4) Mathematics (National 4) Syllabus

What is the formula for the area of a triangle?

$A = \dfrac{1}{2} \times b \times h$, where $b$ is the base and $h$ is the perpendicular height.

Solve the linear equation $3x + 5 = 20$.

Subtract $5$: $3x = 15$. Divide by $3$: $x = 5$.

Solve the equation $2(x - 3) = 10$.

Expand: $2x - 6 = 10$. Add $6$: $2x = 16$. Divide by $2$: $x = 8$.

Solve the inequality $4x - 1 < 11$.

Add $1$: $4x < 12$. Divide by $4$: $x < 3$.

What special rule applies when solving an inequality by multiplying or dividing by a negative number?

You must reverse (flip) the inequality sign. For example, $-2x < 6$ becomes $x > -3$.

State Pythagoras' theorem.

In a right-angled triangle, $a^{2} + b^{2} = c^{2}$, where $c$ is the hypotenuse (the side opposite the right angle) and $a$, $b$ are the two shorter sides.

A right-angled triangle has shorter sides $3$ cm and $4$ cm. Find the hypotenuse.

$c = \sqrt{3^{2} + 4^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$ cm.

How do you use Pythagoras to find a shorter side when the hypotenuse is known?

Subtract: $a = \sqrt{c^{2} - b^{2}}$. Rearrange $a^{2} + b^{2} = c^{2}$ to isolate the unknown shorter side, then take the square root.

What is the sum of the angles in a triangle and in a quadrilateral?

The angles in a triangle add to $180^{\circ}$; the angles in a quadrilateral add to $360^{\circ}$.

Define complementary and supplementary angles.

Complementary angles add to $90^{\circ}$; supplementary angles add to $180^{\circ}$ (angles on a straight line).

What are the properties of angles formed when two straight lines cross (vertically opposite angles)?

Vertically opposite angles are equal. Angles around a point add to $360^{\circ}$.

What is the difference between an equilateral, isosceles and scalene triangle?

Equilateral: all three sides and angles equal ($60^{\circ}$ each). Isosceles: two equal sides and two equal base angles. Scalene: all sides and all angles different.

What is the place value of the digit $7$ in the number $4\,275.6$?

The $7$ is in the tens column, so it represents $70$.

How do you round $3.467$ to two decimal places?

Look at the third decimal digit ($7$); since it is $5$ or more, round up: $3.467 \approx 3.47$.

State the correct order of operations (BODMAS/BIDMAS).

Brackets, Of/Indices (powers), Division and Multiplication (left to right), then Addition and Subtraction (left to right).

Evaluate using order of operations: $2 + 3 \times 4$.

Multiplication first: $3 \times 4 = 12$, then $2 + 12 = 14$.

How do you find a fraction of a quantity, e.g. $\dfrac{3}{4}$ of $\pounds20$?

Divide by the denominator then multiply by the numerator: $20 \div 4 = 5$, then $5 \times 3 = \pounds15$.

How do you convert a percentage to a decimal and to a fraction?

Divide by $100$. For example, $25\% = 0.25 = \dfrac{25}{100} = \dfrac{1}{4}$.

How do you find a percentage of an amount, e.g. $15\%$ of $\pounds80$?

Convert to a decimal and multiply: $0.15 \times 80 = \pounds12$.

How do you simplify a ratio such as $12 : 18$?

Divide both parts by their highest common factor ($6$): $12 : 18 = 2 : 3$.

How do you share an amount in a given ratio, e.g. share $\pounds40$ in the ratio $3 : 5$?

Add the parts: $3 + 5 = 8$. One part $= 40 \div 8 = \pounds5$. So the shares are $3 \times 5 = \pounds15$ and $5 \times 5 = \pounds25$.

How do you calculate gross pay and net pay?

Gross pay is total earnings before deductions. Net pay is what remains after deductions (such as income tax and National Insurance) are subtracted from gross pay.

How do you calculate simple interest?

$I = \dfrac{P \times R \times T}{100}$, where $P$ is the principal, $R$ the annual interest rate (%) and $T$ the time in years.

How do you find the best value (best buy) between two products?

Work out the cost per unit (e.g. price per gram or per item) for each, then compare; the lower unit cost is the better value.

How many minutes are in $2$ hours $15$ minutes, and how is $90$ minutes written in hours and minutes?

$2$ h $15$ min $= 135$ minutes. $90$ minutes $= 1$ hour $30$ minutes (note time is base-60, not decimal).

Convert $3.5$ km to metres and $250$ cm to metres.

$3.5$ km $= 3.5 \times 1000 = 3500$ m. $250$ cm $= 250 \div 100 = 2.5$ m.

What is the formula linking distance, speed and time?

$\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}$, so Distance $=$ Speed $\times$ Time and Time $= \dfrac{\text{Distance}}{\text{Speed}}$.

What is a tally chart / frequency table used for?

To record and organise how often each value or category occurs in a data set, counting in groups of five tally marks to give the frequency of each item.

What is the difference between a bar chart and a line graph?

A bar chart uses separate bars to compare amounts across categories; a line graph joins plotted points to show a trend or change over time (continuous data).

How do you read information from a pie chart?

Each sector represents a fraction of the whole (the full circle is $360^{\circ}$ or $100\%$); the size of each sector shows that category's proportion of the total.

How do you calculate the mean of a set of numbers?

Add up all the values and divide by how many values there are: $\text{mean} = \dfrac{\text{sum of values}}{\text{number of values}}$.

Define the median and the mode of a data set.

The median is the middle value when the data is arranged in order. The mode is the value that occurs most often.

How do you calculate the range of a data set, and what does it measure?

$\text{Range} = \text{highest value} - \text{lowest value}$. It measures the spread of the data.

Find the median of: $3, 7, 4, 9, 5$.

Order them: $3, 4, 5, 7, 9$. The middle value is $5$, so the median is $5$.

What is the probability scale, and what do $0$ and $1$ mean on it?

Probability ranges from $0$ to $1$. A probability of $0$ means the event is impossible; a probability of $1$ means it is certain.

State the formula for the probability of an event.

$P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$.

What is the probability of rolling a $4$ on a fair six-sided dice?

There is $1$ favourable outcome out of $6$, so $P(4) = \dfrac{1}{6}$.

How do you find the probability that an event does NOT happen?

Subtract the probability it does happen from $1$: $P(\text{not } A) = 1 - P(A)$.

When interpreting data or a graph, what does it mean to draw a valid conclusion?

It means making a statement that is genuinely supported by the data shown — identifying trends, comparisons or patterns — rather than assuming information the data does not actually provide.