Outline

# GeoGebra Apps for A-Level Pure Mathematics

## Table of Contents

### Problem solving and proof:

### Algebra indices and surds:

### The quadratic function:

- The properties of y = ax^2.
- Line of symmetry of a quadratic
- The general form of a quadratic
- The vertex of a quadratic.
- The x and y intercepts of a quadratic.
- The properties of a quadratic function
- The characteristics of a quadratic function
- All characteristics of the quadratic function.
- The discriminant
- Finding the values of k for real roots of a quadratic.
- Solving a quadratic equation by factorisation.
- Solving quadratic equations using the formula.
- Finding the equation of a quadratic from its graph
- Finding the equation of a quadratic from completing the square
- Solving a quadratic equation by factorisation

### Equations and inequalities:

### Coordinate geometry:

- The properties of the equation of a straight line.
- The gradient between two points
- y= mx + c.
- Parallel lines
- Perpendicular lines.
- The equation of a line with known point and gradient.
- The midpoint of a line joining two points P and Q
- Finding the equation of the perpendicular bisector of two points.
- Angle at the centre and circumference
- Tangent radius property of a circle
- The equation of a circle.
- The equation of a circle centre (a, b) and radius r.
- Centre of a circle through two points.
- Intersection of a line and a circle.
- The intersection of a line and a circle.
- Intersection of a curve and a circle.
- Finding values of k so that the line is a tangent to a circle.
- solving simultaneous equations with geometrical meaning
- The distance between two points

### Trigonometry:

- Deducing trig ratios for 0 and 90 degrees
- Trigonometric graphs in all four quadrants (degrees)
- Generating the graph of y=sinx in degrees.
- Generating the graph of y=cosx in degrees.
- Generating a graph of y=tanx in degrees.
- Understanding the unit circle in trigonometry in degrees
- Proof of cos(90 - x) = sin x
- Proof of sin(90 - x) = cos x
- Equation solver for sin of an angle.
- Equation solver for cos of an angle.
- Equation solver for tan of an angle.
- Pythagoras theorem introduction
- Introduction to 3-D trigonometry.
- Longest rod in a box
- Square based pyramid trigonometry
- Finding the radius of a cone.
- Graph of y=sin x/x
- The unit circle using radian measure
- Sine rule with ambiguous case
- Sine rule ambiguous case

### Polynomials:

### Graphs and transformations:

- Graphs of functions.
- Sketching a cubic function.
- Sketching the cubic function by translation and reflection.
- Sketching the cubic function by translation and reflection.
- The reciprocal function.
- Stretches of a cubic function.
- Stretches of a reciprocal function.
- Vertical translation of a graph
- Horizontal translation of a graph.
- Stretches parallel to the y-axis.
- Stretches parallel to the x-axis.
- Reflection in the x-axis.
- Reflection in the y-axis.
- Combined transformations of functions.
- Combinations of transformations of graphs of functions a stretch and a translation
- Transformations of a sine graph
- Worksheet for transformations of graphs.
- Finding maximum points using translations of functions.
- The graph of y=f(x) after a stretch parallel to the y-axis.
- The graph of y=f(x) after a stretch parallel to the x-axis.
- Stretches of functions parallel to the x-axis.
- Graphs showing proportion.
- Transformations of 1/x to obtain a bilinear function
- Inverse transformations of a function
- Max/min point on the reciprocal of a quadratic function

### Binomial theorem and counting principles:

### Differentiation:

- Gradient between two points.
- Introduction to the gradient of a curve.
- Differentiation by first principles
- To find the coordinates of where the gradient is 8
- Where the gradient of a function is the same as a line
- Where the gradient of a function is the same as a line
- The tangent and normal to a curve y = f(x)
- Tangent normal extended response.
- Increasing and decreasing function
- Showing a function always is decreasing.
- Proving a cubic function is decreasing.
- Turning points introduction.
- Using the 2nd derivative to classify stationary points.
- Using the 2nd derivative test cases of failure.
- Using the 2nd derivative to classify stationary points with a point of inflection.
- Maximum and minimum points
- A case where the 2nd derivative test does not work
- Concavity
- Maximum volume of a cylinder cut from a sphere
- The area of a rectangle enclosed in a quadratic
- The maximum volume of a cylinder inside a cone

### Integration:

### Vectors:

- The addition of vectors.
- The subtraction of vectors.
- The multiplication of a vector by a scalar.
- A vector in component and magnitude-direction forms
- Position vectors in component form in 2D.
- Position vectors and scalar multiples of vectors.
- The magnitude or length of a vector in 2D and 3D
- Finding the vector joining the point P and Q and its magnitude.
- The scalar product proof
- Scalar product 3d

### Exponentials and logarithms:

### Trigonometry and circular measure

- Generating the graph of f(x) = sinx
- Generating the graph of f(x) = cosx
- Generating the graph of f(x) = tanx
- Understanding the unit circle in radians
- Small angles from the unit circle.
- Small angles measured in radians.
- Small angles measured in radians approximations using graphs.
- Modelling tides in a harbour
- Combinations of transformations for a sine graph
- The amplitude and period of sin or cos graphs
- How to change the period of a sine graph
- How to change the period of a cosine graph
- Trigonometric graphs in all four quadrants (radians)

### Sequences and series:

### Functions

- The vertical line test for a function.
- Finding maximum points using translations of functions.
- Horizontal translation of a graph.
- Horizontal translation of a graph.
- Reflection in the x-axis.
- Reflection in the y-axis.
- The graph of y=f(x) after a stretch parallel to the y-axis.
- The graph of y=f(x) after a stretch parallel to the x-axis.
- Combined transformations of functions.
- Combinations of transformations of graphs of functions a stretch and a translation
- Worksheet for transformations of graphs.
- The composite of two functions.
- The inverse of a function.
- The inverse of a quadratic function.
- The modulus function.
- The graphs of y = |f(x)| and y = f(|x|).
- Finding the gradient of the inverse function from a function.
- Domain and range of a function
- Finding the gradient of the inverse function from a function
- An odd function
- An even function

### Differentiation shape of curve, chain, product and quotient rules:

### Trigonometric functions:

### Trigonometric identities:

### Further differentiation:

- Derivative of y=a^x
- The number e
- The derivative of f(x) = lnx
- Derivative of sinx in radians
- Derivative of cosx in radians
- Derivative of tanx
- The derivative of sinx from first principles.
- The equations of tangent and normal for a parametric curve
- The parametric equation of a circle
- Projectile motion of a rocket using parametric equations.
- Tangents on an ellipse.
- Curve sketching a natural log function showing concavity

### Integration:

### Parametric equations:

- Parametric equations introduction
- Parametric equations introduction
- Paramentric equations using theta 01
- The parametric equation of a hyperbola.
- The parametric equation of a circle.
- The equations of tangent and normal for a parametric curve
- The parametric equation of a circle
- Projectile motion of a rocket using parametric equations.
- Tangents on an ellipse.
- Opposite directions on a parametric curve

### Vectors in three dimensions

- The addition of vectors.
- The subtraction of vectors.
- The multiplication of a vector by a scalar.
- 2D vectors in magnitude-direction and component forms.
- Position vectors in component form in 3D.
- The magnitude or length of a vector in 2D and 3D
- Finding the vector joining the point P and Q and its magnitude.
- A trapezium and parallel vectors.
- The scalar product proof
- Scalar product 3d
- Finding the area of a triangle in 3-Dimensions.
- Finding a vector in 3d space.
- A parallelogram in 3d space.

### Using integration to solve differential equations:

- The limiting value of a differential equation 01.
- The limiting value of a differential equation 02.
- Solving a differential equation for an electrical circuit.
- Solution of a 1st order differential equation with slope field.
- Differential equation solution of a mass being projected upwards
- 1st order differential equation by separating the variables
- 1st order trig differential equation by separating the variables
- Solution of a 1st order differential equation by separating the variables (Harder)
- Solution of a differential equation by separating the vars

### Numerical methods:

- Fixed point iteration 01
- Fixed point iteration staircase diagram.
- Fixed point iteration 02.
- Fixed point iteration cobweb diagram.
- Fixed point iteration of a trig function.
- Fixed point iteration staircase diagrams
- The Newton-Raphson method.
- Newton-Raphson method for finding roots.
- Introduction to the trapezium rule
- Trapezium rule.
- Trapezium rule for concave downwards,
- Using the trapezium rule with radians
- Using rectangles to find bounds for the area under a curve.
- Using rectangles to find bounds for the area under a curve.