## Mathematics

#### Learner Journey Summary - Higher

##### Reasoning skills
1. The learner will use mathematical reasoning skills (these can be used in combination or separately)
2. Interpreting a situation where mathematics can be used and identifying a strategy
3. 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
1. Factorising a cubic or quartic polynomial expression.
2. Simplifying a numerical expression using the laws of logarithms and exponents.
3. Manipulating trigonometric expressions..
4. Application of the addition or double angle formulae.
5. Application of trigonometric identities.
6. Identifying and sketching related functions.
7. Identifying a function from a graph, or sketching a function after a transformation.
8. Sketching the inverse of a logarithmic or an exponential function.
9. Completing the square in a quadratic expression where the coefficient of 2 x is non-unitary.
10. Determining composite and inverse functions.
11. 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
1. Solving a cubic or quartic polynomial equation.
2. Given the nature of the roots of an equation, use the discriminant to find an unknown.
4. Solving logarithmic and exponential equations.
5. Using the laws of logarithms and exponents.
6. Use a straight line graph to confirm relationships.
7. Model mathematically situations involving the logarithmic or exponential function.
8. Finding the coordinates of the point(s) of the intersection of a straight line and a curve or of two curves.
9. Solving trigonometric equations.
10. 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.

1. Determining vector connections.
2. Determining the resultant of vector pathways in three dimensions.
3. Working with collinearity.
4. Determining the coordinates of an internal division point of a line.
5. Working with vectors.
6. Evaluate a scalar product given suitable information and determine the angle between two vectors.
7. Apply properties of the scalar product.
8. 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.

1. Differentiating functions.
2. Differentiating an algebraic function which is, or can be simplified to, an expression in powers of x.
3. Differentiating k x sin , k x cos.
4. Differentiating a composite function using the chain rule Using differentiation to investigate the nature and properties of functions.
5. Determining the equation of a tangent to a curve at a given point by differentiation.
6. Determining where a function is strictly increasing/decreasing.
7. 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.
8. Integrating functions.
9. Integrating an algebraic function which is, or can be, simplified to an expression of powers of x.
10. Solving differential equations.
11. Using integration to calculate definite integrals.
12. Calculating definite integrals of functions with limits which are integers, radians, surds or fractions.
13. Applying differential calculus.
14. Determining the optimal solution for a given problem.
15. Determining the greatest/least values of a function on a closed interval.
16. Solving problems using rate of change.
17. Applying integral calculus.
18. Finding the area between a curve and the x-axis.
19. Finding the area between a straight line and a curve or two curves.
20. 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.

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