Using GeoGebra for A level Further Mathematics
This GeoGebra Book provides ideas and suggestions for using GeoGebra to enhance teaching and learning A level Further Mathematics.
Table of Contents
- Complex Numbers
- Entering Complex Numbers
- Locus in the complex plane: |z-w|=k
- Complex Plane Transformation: w=1/z
- Construction: Locus of |z1-w|=|z2-w|
- Matrices
- Entering Matrices
- Matrix Transformations
- Rotation Matrix
- Invariant lines of matrix transformations
- Construction: Transformation Matrix
- Vectors
- Entering Vectors
- Vector equation of a line in 3D
- Vectors - MEI FP3 June 2010 qn 1
- Polar Curves
- Entering polar coordinates and curves
- Polar Integral
- Maclaurin Series
- Entering Maclaurin series
- Maclaurin series
- Student Tasks
- Student tasks for using GeoGebra
- Construction problems
Entering Complex Numbers
To add a complex number select the [b]Complex Number[/b] tool from the 2[sup]nd[/sup] button and click on the screen.[br][br]NB Complex Numbers will be added as z[sub]1[/sub], z[sub]2[/sub], ... a the variable z is reserved for 3D graphing.[br][br]You can also type [b]z_2=2+3i [/b]in the input bar and press enter to add the complex number [math]z_1=2+3i[/math] [br][b][br]p=z_1*z_2 [/b]will find the product. [br][b]q=z_1/z_2 [/b]will find the quotient[br][br]The command [b]ComplexRoot[f][/b] will display the complex roots of a polynomial function.
Entering Matrices
Type [b]M={{1,3},{2,4}} [/b]in the input bar and press enter to add the matrix [math]\left(\begin{matrix}1\\2\end{matrix}\begin{matrix}3\\4\end{matrix}\right)[/math].[br]Type [b]N={{0,-1},{1,0}} [/b]in the input bar and press enter to add the matrix [math]\left(\begin{matrix}0\\1\end{matrix}\begin{matrix}-1\\0\end{matrix}\right)[/math].[br][b][br]P=M*N[/b] will find the product MN.[br] [br][b]A’=M*A [/b]will find the image of a point [b]A[/b] under the transformation defined by matrix [b]M[/b].[br][br]The command [b]ApplyMatrix[M,poly1][/b] will find the image of the polygon [b]poly1[/b] under the transformation defined by matrix [b]M[/b].
Entering Vectors
Vectors can be created using the [b]Vector[/b] or [b]Vector from Point[/b] tools in the third[br]menu.[br][br][b]Vector[/b] joins two points [b]A[/b] and [b]B[/b] with the vector AB. [br][br][b]Vector from Point[/b] will translate an existing vector so that it starts from a point.[br][br]Alternatively enter a set of coordinates with a lower-case name will create the object as a position vector, e.g. [b]v=(2,3,-1)[br][br]Line[A,v] [/b]will create a a line through point [b]A[/b] in the direction [b]v[/b].[br][br]Planes can be entered directly in Cartesian form: e.g. [b]2x+3y-z=2[/b].
Entering polar coordinates and curves
Polar coordinates are entered using a semi-colon: e.g. [b](3;pi/3)[br][br][/b]The default angle measure is degrees. This can be changed in [b]Settings > Graphing [/b](cubic icon).[br][br]Polar curves can be entered directly: e.g. [b]r=3+2cos(θ)[br][/b]NB GeoGebra will plot negative values of r.[br][br]You can also use the command [b]Curve[(r;[b]θ[/b]),[b]θ[/b],start value, end value][br][/b]e.g. [b]Curve[(2 + sin(θ/2); θ), θ, 0, 4pi][/b]
Entering Maclaurin series
[list=1][br][*]Enter a function for y=f(x) such as [b]y=sin(x)[/b].[br] [br][/*][*]Add a variable: [b]n = 1[br][/b]Set the limits of the slider to: [b]1 ≤ n ≤ 15, Step 20[/b][b] [/b][/*] [*]In the input bar enter: [b]TaylorPolynomial(f, 0, n)[/b] [br] [/*][/list]
Student tasks for using GeoGebra
A set of student tasks are available at: [url=https://mei.org.uk/resource/ab80a69d-e6fd-4954-9d46-08d9d6b4f934/]https://mei.org.uk/resource/ab80a69d-e6fd-4954-9d46-08d9d6b4f934/[/url]