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

Mathematics (National 4) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (National 4) in Scottish National 4 (Nat 4), not a summary of it.

1. 1

   Expressions and Formulae

   4 topics

   • Working with surds and indices
     • Simplifying powers
     • Scientific notation
   • Algebraic expressions
     • Expanding brackets
     • Collecting like terms and factorising
   • Gradient of a straight line
     • Calculating gradient from coordinates
   • Area and volume of solids
     • Composite shapes
     • Prisms and cylinders
2. 2

   Relationships

   4 topics

   • Solving linear equations and inequalities
     • Equations with brackets
     • Equations with variables on both sides
   • Linear graphs and equations of a line
     • Plotting y = mx + c
     • Interpreting graphs
   • Pythagoras' theorem
     • Finding the hypotenuse
     • Applications in real contexts
   • Properties of shapes and angles
     • Angles in circles
     • Symmetry
3. 3

   Numeracy

   4 topics

   • Calculations with whole numbers and decimals
     • Order of operations
     • Rounding and estimation
   • Fractions, percentages and ratio
     • Percentage increase and decrease
     • Sharing in a ratio
   • Money and financial calculations
     • Wages, budgeting and best deals
     • Currency conversion
   • Measurement and time
     • Distance, speed and time
     • Reading scales and timetables
4. 4

   Statistics and Data Handling

   4 topics

   • Collecting and representing data
     • Tables, charts and graphs
     • Pie charts and frequency tables
   • Summarising data
     • Mean, median, mode and range
   • Probability of events
     • Expressing likelihood as a fraction
   • Interpreting and drawing conclusions
     • Comparing data sets

Mathematics (National 4) flashcards for Scottish National 4 (Nat 4)

25 of 59 cards from the Mathematics (National 4) deck — real questions with worked answers.

1. What is a surd?

   A surd is an irrational root that cannot be simplified to a whole number, e.g. $\sqrt{2}$ or $\sqrt{3}$. It is a root left in exact form rather than written as a rounded decimal.

2. State the multiplication rule for surds.

   $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. For example, $\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6$.

3. State the division rule for surds.

   $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$. For example, $\dfrac{\sqrt{20}}{\sqrt{5}} = \sqrt{4} = 2$.

4. How do you simplify a surd such as $\sqrt{72}$?

   Factor out the largest perfect square: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}$.

5. State the law of indices for multiplication of powers with the same base.

   $a^{m} \times a^{n} = a^{m+n}$ — add the indices.

6. State the law of indices for division of powers with the same base.

   $a^{m} \div a^{n} = a^{m-n}$ — subtract the indices.

7. What does a power of zero equal, and what does a negative index mean?

   Any non-zero number to the power zero equals $1$: $a^{0} = 1$. A negative index means a reciprocal: $a^{-n} = \dfrac{1}{a^{n}}$.

8. Expand the brackets: $3(2x + 5)$.

   Multiply each term inside by $3$: $3(2x+5) = 6x + 15$.

9. Simplify by collecting like terms: $5a + 3b - 2a + 4b$.

   $5a - 2a = 3a$ and $3b + 4b = 7b$, so the result is $3a + 7b$.

10. Factorise fully: $6x + 9$.

    Take out the highest common factor $3$: $6x + 9 = 3(2x + 3)$.

11. What is the formula for the gradient of a straight line through two points?

    $m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}$, i.e. the change in $y$ divided by the change in $x$ (rise over run).

12. Find the gradient of the line through $(1, 2)$ and $(4, 11)$.

    $m = \dfrac{11 - 2}{4 - 1} = \dfrac{9}{3} = 3$.

13. What kind of gradient does a line sloping up to the right have, versus down to the right?

    A line going up from left to right has a positive gradient; a line going down from left to right has a negative gradient. A horizontal line has gradient $0$.

14. What is the equation of a straight line in gradient–intercept form?

    $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept (where the line crosses the $y$-axis).

15. In the line $y = 4x - 7$, what are the gradient and the $y$-intercept?

    The gradient is $m = 4$ and the $y$-intercept is $c = -7$, so the line crosses the $y$-axis at $(0, -7)$.

16. What is the equation of a horizontal line and a vertical line?

    A horizontal line has equation $y = c$ (constant $y$); a vertical line has equation $x = a$ (constant $x$).

17. State the formula for the volume of a cuboid.

    $V = l \times b \times h$ (length times breadth times height).

18. State the formula for the volume of a cylinder.

    $V = \pi r^{2} h$, where $r$ is the base radius and $h$ is the height.

19. State the formula for the volume of a prism.

    $V = A \times h$, where $A$ is the cross-sectional area and $h$ is the length (height) of the prism.

20. What is the formula for the area of a circle and the circumference?

    Area $= \pi r^{2}$ and circumference $= 2\pi r$ (or $\pi d$), where $r$ is the radius and $d$ the diameter.

21. 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.

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

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

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

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

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

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

25. 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 $x > -3$.

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