Scottish National 5 (Nat 5) Mathematics Syllabus

Mathematics syllabus: all 5 chapters

Expand any chapter to see its topics and sub-topics.

1. 1

   Numerical and Algebraic Skills

   4 topics

   • Surds and Indices
     • Simplifying surds and rationalising denominators
     • Laws of indices including negative and fractional powers
     • Scientific notation
   • Algebraic Expressions
     • Expanding brackets and collecting like terms
     • Factorising: common factor, difference of two squares, trinomials
     • Completing the square
   • Algebraic Fractions
     • Simplifying algebraic fractions
     • Adding, subtracting, multiplying and dividing fractions
   • Percentages and Proportion
     • Appreciation, depreciation and compound interest
     • Direct and inverse proportion
2. 2

   Equations, Graphs and Functions

   4 topics

   • Straight Line
     • Gradient and the equation y = mx + c
     • Finding the equation of a line through two points
     • Parallel lines and intercepts
   • Quadratic Functions and Graphs
     • Sketching parabolas and identifying turning points
     • Solving quadratics by factorising, formula and graph
     • The discriminant and nature of roots
   • Simultaneous Equations
     • Solving graphically
     • Solving algebraically by substitution and elimination
   • Functional Notation
     • Evaluating f(x)
3. 3

   Geometry and Measurement

   5 topics

   • Pythagoras and Its Converse
     • Finding lengths in 2D and 3D
   • Arcs and Sectors
     • Arc length and sector area
   • Volume of Solids
     • Spheres, cones, pyramids and composite solids
   • Angle Properties of Circles
     • Tangent and radius, angles in semicircles, symmetry
   • Gradient and Similarity
     • Linear scale factor for length, area and volume
4. 4

   Trigonometry

   4 topics

   • Right-Angled Triangles
     • SOH CAH TOA for sides and angles
   • The Sine and Cosine Rules
     • Finding sides and angles in non-right triangles
     • Area of a triangle using (1/2)ab sin C
   • Trigonometric Graphs and Equations
     • Period, amplitude and transformations of sin, cos, tan
     • Solving trigonometric equations
   • Trigonometric Identities
     • Using tan x = sin x / cos x and sin squared plus cos squared = 1
5. 5

   Statistics and Applications

   3 topics

   • Comparing Data Sets
     • Quartiles, interquartile range and semi-interquartile range
     • Standard deviation
   • Scattergraphs and Best Fit
     • Line of best fit and its equation
   • Vectors
     • Components, magnitude and addition of vectors in 2D and 3D

Scottish National 5 (Nat 5) Mathematics flashcards — 50 questions with answers

1. Simplify $\sqrt{50}$ into its simplest surd form.

   $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

2. How do you rationalise the denominator of $\frac{1}{\sqrt{a}}$?

   Multiply numerator and denominator by $\sqrt{a}$: $\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$

3. State the index laws for multiplication, division and powers of powers.

   $a^{m} \times a^{n} = a^{m+n}$, $\quad a^{m} \div a^{n} = a^{m-n}$, $\quad (a^{m})^{n} = a^{mn}$

4. What do the negative and fractional index laws state?

   $a^{-n} = \frac{1}{a^{n}}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}$ (with $a^{0}=1$)

5. Expand and simplify $(x+3)(x-5)$.

   $x^{2} - 5x + 3x - 15 = x^{2} - 2x - 15$

6. Factorise the difference of two squares $a^{2} - b^{2}$.

   $a^{2} - b^{2} = (a+b)(a-b)$

7. Factorise $x^{2} + 7x + 12$.

   $(x+3)(x+4)$, since $3 \times 4 = 12$ and $3 + 4 = 7$

8. Complete the square for $x^{2} + 6x + 1$.

   $(x+3)^{2} - 9 + 1 = (x+3)^{2} - 8$

9. How do you add or subtract algebraic fractions such as $\frac{a}{b} + \frac{c}{d}$?

   Use a common denominator: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

10. Simplify $\frac{x^{2}-9}{x+3}$.

    $\frac{(x+3)(x-3)}{x+3} = x - 3$ (for $x \neq -3$)

11. How do you find the multiplier for an $r\%$ increase and an $r\%$ decrease?

    Increase multiplier $= 1 + \frac{r}{100}$; decrease multiplier $= 1 - \frac{r}{100}$

12. State the compound interest / appreciation formula over $n$ years.

    $A = P\left(1 + \frac{r}{100}\right)^{n}$, where $P$ is the principal and $r$ the percentage rate

13. A value is $\pounds 240$ after a $20\%$ increase. How do you find the original value?

    Divide by the multiplier: $\frac{240}{1.2} = \pounds 200$

14. What is direct proportion and its equation?

    $y$ varies directly as $x$ means $y = kx$ for constant $k$; as one doubles, so does the other

15. What is the equation of a straight line in gradient-intercept form?

    $y = mx + c$, where $m$ is the gradient and $c$ the $y$-intercept

16. State the formula for the gradient of a line through $(x_{1},y_{1})$ and $(x_{2},y_{2})$.

    $m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

17. How do you write the equation of a line given gradient $m$ and a point $(a,b)$?

    $y - b = m(x - a)$

18. For the parabola $y = (x-p)^{2} + q$, where is the turning point?

    The turning point is at $(p, q)$; it is a minimum (the axis of symmetry is $x = p$)

19. State the quadratic formula for $ax^{2} + bx + c = 0$.

    $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

20. What does the discriminant $b^{2} - 4ac$ tell you about the roots?

    $>0$: two distinct real roots; $=0$: one repeated real root; $<0$: no real roots

Scottish National 5 (Nat 5) Mathematics FAQ