## Mathematics

#### Learner Journey Summary - Higher

##### Reasoning skills

- The learner will use mathematical reasoning skills (these can be used in combination or separately)
- Interpreting a situation where mathematics can be used and identifying a strategy
- Explaining a solution and, where appropriate, relating it to context

##### Algebraic and trigonometric skills

The learner will use algebraic and trigonometric skills and apply them in context.

###### Manipulating algebraic expressions

- Factorising a cubic or quartic polynomial expression.
- Simplifying a numerical expression using the laws of logarithms and exponents.
- Manipulating trigonometric expressions..
- Application of the addition or double angle formulae.
- Application of trigonometric identities.
- Identifying and sketching related functions.
- Identifying a function from a graph, or sketching a function after a transformation.
- Sketching the inverse of a logarithmic or an exponential function.
- Completing the square in a quadratic expression where the coefficient of 2 x is non-unitary.
- Determining composite and inverse functions.
- Determining a composite function which can be trigonometric, logarithmic, exponential or algebraic. functions - knowledge and use of the terms domain and range is expected.

###### Solving algebraic equations

- Solving a cubic or quartic polynomial equation.
- Given the nature of the roots of an equation, use the discriminant to find an unknown.
- Solve quadratic inequalities.
- Solving logarithmic and exponential equations.
- Using the laws of logarithms and exponents.
- Use a straight line graph to confirm relationships.
- Model mathematically situations involving the logarithmic or exponential function.
- Finding the coordinates of the point(s) of the intersection of a straight line and a curve or of two curves.
- Solving trigonometric equations.
- Solving trigonometric equations in degrees or radians including those involving the wave function or trigonometric formulae or identities, in a given interval.

##### Geometric skills

The learner will use geometric skills and apply them in context.

- Determining vector connections.
- Determining the resultant of vector pathways in three dimensions.
- Working with collinearity.
- Determining the coordinates of an internal division point of a line.
- Working with vectors.
- Evaluate a scalar product given suitable information and determine the angle between two vectors.
- Apply properties of the scalar product.
- Using and finding unit vectors including i, j, k as a basis.

##### Calculus skills

The learner will use calculus skills and apply them in context.

- Differentiating functions.
- Differentiating an algebraic function which is, or can be simplified to, an expression in powers of x.
- Differentiating k x sin , k x cos.
- Differentiating a composite function using the chain rule Using differentiation to investigate the nature and properties of functions.
- Determining the equation of a tangent to a curve at a given point by differentiation.
- Determining where a function is strictly increasing/decreasing.
- Sketching the graph of an algebraic function by determining stationary points and their nature as well as intersections with the axes and behaviour of f x for large positive and negative values of x.
- Integrating functions.
- Integrating an algebraic function which is, or can be, simplified to an expression of powers of x.
- Solving differential equations.
- Using integration to calculate definite integrals.
- Calculating definite integrals of functions with limits which are integers, radians, surds or fractions.
- Applying differential calculus.
- Determining the optimal solution for a given problem.
- Determining the greatest/least values of a function on a closed interval.
- Solving problems using rate of change.
- Applying integral calculus.
- Finding the area between a curve and the x-axis.
- Finding the area between a straight line and a curve or two curves.
- Determine and use a function from a given rate of change and initial conditions.

##### Algebraic and geometric skills

The learner will use algebraic and geometric skills and apply them in context.

- Applying algebraic skills to rectilinear shapes.
- Finding the equation of a line parallel to and a line perpendicular to a given line.
- Using properties of medians, altitudes and perpendicular bisectors in problems involving the equation of a line and intersection of lines.
- Determine whether or not two lines are perpendicular.
- Applying algebraic skills to circles and graphs.
- Determining and using the equation of a circle.
- Using properties of tangency in the solution of a problem.
- Determining the intersection of circles or a line and a circle.
- Modelling situations using sequences.
- Determining a recurrence relation from given information and using it to calculate a required term.
- Finding and interpreting the limit of a sequence, where it exists.