Scotland Β· SQASyllabus
Maths syllabus, dot point by dot point
Every dot point in the Scotland Mathssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Applications
Module overview β- How do you use differentiation to find the best possible value in a real situation and to describe how quantities change?Using differentiation to find the optimal value in optimisation problems, the greatest and least values of a function on a closed interval, rates of change, and the motion of a particle through displacement, velocity and acceleration.9 min answer β
- How do you use integration to measure area and volume and to recover a quantity from its rate of change?Using integration to find the area enclosed between a curve and a line or between two curves, the area below the x-axis, recovering displacement from velocity, and using a definite integral to evaluate an accumulated quantity in context.9 min answer β
- How do recurrence relations model step-by-step change, and when does such a sequence settle towards a limit?Recurrence relations of the form u sub n plus 1 equals a u sub n plus b, generating terms, the condition for a limit to exist, finding the limit, and interpreting a recurrence relation in a real context.9 min answer β
- How do you describe a circle algebraically, find its centre and radius, and decide how a line meets it?The equation of a circle with centre the origin and with a general centre, the general equation of a circle, finding the centre and radius, the intersection of a line and a circle, and the equation of a tangent to a circle.9 min answer β
Expressions and Functions
Module overview β- How do the laws of logarithms let you solve equations with the unknown in the power, and how do logs reveal hidden relationships in data?The laws of logarithms and their use in simplifying expressions and solving equations, the relationship between exponential and logarithmic form, and the use of logarithms to find the parameters in experimental laws of the form y equals k x to the n and y equals a b to the x.9 min answer β
- How do functions combine, invert and transform, and how do those operations move and reshape their graphs?Functions and their domain and range, composite functions, inverse functions, exponential and logarithmic graphs, and the graphs that result from translating, reflecting and stretching a known function.9 min answer β
- How do you describe a straight line algebraically, and how do gradient and angle let you decide whether lines are parallel, perpendicular or concurrent?The gradient of a line including the connection to the angle it makes with the x-axis, the equation of a line through a point with a given gradient, parallel and perpendicular lines, and the medians, altitudes and perpendicular bisectors of a triangle.9 min answer β
- How do you measure angles in radians, read exact trigonometric values, and sketch and transform trigonometric graphs?Radian measure and the conversion between degrees and radians, exact values of sine, cosine and tangent for the standard angles, the graphs of the trigonometric functions, and the transformations that change their amplitude, period and phase.9 min answer β
- How do you describe position and direction in three dimensions, and how does the scalar product reveal the angle between two vectors?Vectors in three dimensions, the magnitude and unit vector, addition and scalar multiplication, the section formula and collinearity, and the scalar product including its use to find the angle between two vectors and to test for perpendicularity.9 min answer β
Relationships and Calculus
Module overview β- How do you find the gradient of a curve at any point, and how does the derivative reveal stationary points and the shape of a graph?Differentiation of polynomial, root and reciprocal functions and of sine and cosine, the gradient of a curve and the equation of a tangent, increasing and decreasing functions, and stationary points and their nature.10 min answer β
- How do you reverse differentiation to find a function from its gradient, and how does the definite integral measure area under a curve?Integration as the reverse of differentiation, the indefinite integral of polynomial and trigonometric functions with the constant of integration, the definite integral, and the use of integration to find the area under a curve and the area between two curves.9 min answer β
- How do you complete the square, read the discriminant, and use the factor theorem to factorise and solve polynomial equations?Completing the square and the properties of the quadratic, the discriminant and the nature of the roots, the condition for a quadratic to be always positive or always negative, the factor and remainder theorems, and solving and sketching polynomials.9 min answer β
- How do the compound angle formulae let you expand and simplify trigonometric expressions, and how does the wave function combine a sine and a cosine into one?The addition (compound angle) formulae for sine and cosine, the double angle formulae, their use in proving identities and solving equations, and the wave function that expresses a sin x plus b cos x in the form k sin of x plus a.9 min answer β
- How does the chain rule let you differentiate and integrate a function nested inside another, such as a bracket raised to a power or a sine of a linear expression?Differentiating composite functions with the chain rule, including expressions of the form a function of a linear expression and sine and cosine of a linear expression, and reversing the process to integrate functions of the form (ax + b) to the n, sin(ax + b) and cos(ax + b).9 min answer β
- How do you solve trigonometric equations across a full interval, and how do the identities help you reduce them to a solvable form?Solving trigonometric equations in degrees and radians over a given interval, using the CAST diagram and the symmetry of the graphs, the trigonometric identities, and equations that reduce to a quadratic in a single trigonometric ratio.9 min answer β