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@MEIConference #MEIConf2019
#MEIConf2019
GeoGebra for BeginnersThis session will introduce some of the basic functionality
of GeoGebra. The session will give opportunities for
participants to create their own GeoGebra resources in
order to aid effective teaching and learning in their
classrooms. Examples will mainly be taken from AS Pure
Maths content, although the GeoGebra skills can be
applied to topics across any Key Stage.
No knowledge of GeoGebra is necessary.
Participants should bring their own laptop with the latest
version of GeoGebra (ideally GeoGebra Classic 5)
installed.
#MEIConf2019
GeoGebra Book
All the resources for this session can be
found in this GeoGebra Book
https://ggbm.at/znstpa3r
#MEIConf2019
About MEI Registered charity committed to improving
mathematics education
Independent UK curriculum development body
We offer continuing professional development
courses, provide specialist tuition for students
and work with employers to enhance
mathematical skills in the workplace
We also pioneer the development of innovative
teaching and learning resources
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GeoGebra for Beginners This session will introduce some of the basic functionality of GeoGebra. The session will
give opportunities for participants to create their own GeoGebra resources in order to aid
effective teaching and learning in their classrooms. Examples will mainly be taken from AS
Pure Maths content, although the GeoGebra skills can be applied to topics across any Key
Stage.
No knowledge of GeoGebra is necessary.
Participants should bring their own laptop with the latest version of GeoGebra (ideally
GeoGebra Classic 5) installed.
Some useful links
MEI GeoGebra Institute information: mei.org.uk/geogebra
Getting Started guides: mei.org.uk/geogebra#getting-started
GeoGebra book for session: https://ggbm.at/znstpa3r
GCSE/A level/Further Maths Classroom Tasks: mei.org.uk/geogebra-tasks
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Familiarisation tasks
A. Graphics View/Tools The angle in a semi-circle is a right-angle
1. Use Semi-Circle Through
Two Points to add a semi-circle to the screen with end-points A and B.
2. Use Point to add a new point on the circle, C.
3. Use the Segment tool to create line segments AB, AC and BC.
4. Use the Angle tool to measure the angle ACB.
B. Algebra View/Input bar The intersection of a line and a parabola
1. In the Input bar type: y = m x + c press enter and select Create Sliders.
2. In the Input bar type: y = x^2 and press enter.
3. Use the Intersect tool to find the points of intersection of the line and the curve.
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A basic quadratic and some useful GeoGebra tools
Construction notes and thoughts about how this could be used in your classroom
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Four prompts 1. You are given two intersecting straight lines and a point P marked on one of them.
Construct a circle that passes through P and has both lines as tangents.
2. Construct a square with two of the vertices having coordinates 3,5 and 5,0 .
In how many different ways can you do this using GeoGebra?
3. Change one aspect of the equation 2 2
6 3 16x y so that the circle lies in three
quadrants.
4. Spend a few minutes thinking of the different mathematical ways for finding the midpoint
of two points A and B. Using GeoGebra see if you can reproduce some or all of your
approaches.
Challenge: Design a task Think about a topic you are going to be teaching during the next half term. Consider how the
use of GeoGebra would enhance the teaching of an aspect of this topic.
Your task is to design a file which you will use in the teaching of your chosen topic.
It might be a file that helps you to introduce a concept, or one which illustrates an interesting
question related to the topic, or a file which helps students explore an aspect of the topic, or
a file which prompts students to think mathematically, or...
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MEI GeoGebra Tasks for GCSE Mathematics
Task 3: Algebra – Parallel lines
1. In the input bar enter: y = m x + c If prompted select Create Sliders.
2. Use the Point tool to create a point, A. NB this point should not be on the line.
3. Use the Parallel Line tool to create a line parallel to the original line through the point.
Questions for discussion
What is the relationship between parallel lines?
How could you find the equation of a line if you knew a point on the line and the equation of a line parallel to it?
Problem (Try the problem with pen and paper first then check it on your software)
Find the equation of the line that passes through the point 2,1 and is parallel to 3 2y x
.
Further Tasks
Use the Perpendicular line tool to investigate the relationship between perpendicular lines.
Investigate the distance between a pair of parallel lines.
Try changing m , changing c and
moving the point A. What do you notice – what stays the same and what changes?
If you move A to a new point can you predict what the equation will be?
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MEI GeoGebra Tasks for GCSE Mathematics
Task 9: Geometry – Transformations
To construct a file showing reflection in the line y x :
1. Use Point to add four points A, B, C and D.
2. Use Polygon to create the quadrilateral ABCD.
3. In the input bar enter: y = x
4. Use Reflect about line to reflect the quadrilateral in the x-axis.
Task Create similar GeoGebra files to demonstrate the following transformations:
Reflection about: o The line y x .
o The x -axis.
o The y -axis.
Rotation about a point.
Translation by a vector.
Enlargement from a point by a scale factor.
The polygon tool can be found in the Polygon menu (More on the Graphing app). Click on A, B, C, D then A again to complete it.
The Reflect about line tool can be found in the
Transform menu (More on the Graphing app).
Click on the polygon then the line y x to reflect it.
Construct a file showing rotation by a standard
angle, such as 90°, or use a slider to vary the angle.
Construct a file showing enlargement by a standard scale factor, such as 2, or use a slider to vary the number.
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MEI GeoGebra Tasks for AS Pure
Task 5: Functions – Transformations
1. In the Input bar enter: f(x) = x2
2. In the Input bar enter: g(x) = f(x + a) + b
If prompted click to create sliders for a and b.
Questions for discussion
What transformation maps f x onto g x ?
Does this work if other functions are entered for f x ?
Problem (Try the problem with pen and paper first then check it on your software)
Show that 3 3 22 3 6 12 11x x x x .
Hence sketch the graph of 3 26 12 11y x x x .
Further Tasks
Show that 4 3 2f 8 24 32 13x x x x x can be written in the form 4
x a b
and hence find the coordinates of the minimum point on the graph of fy x .
Create sliders for c and d.
In the Input bar enter: h(x) = c*f(d*x).
What transformation maps f x onto h x ?
It is essential that this is
entered as a function f x .
Changing f x to 3f x x x
might help make it clearer.
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MEI GeoGebra Tasks for AS Pure
Task 6: Differentiation – Exploring the gradient on a curve
1. Plot a cubic function: e.g. y = x3 − x2 − 3x + 2
2. Plot the gradient function by entering Derivative(f) in the input bar.
Question for discussion
How is the shape of the gradient graph related to the shape of the original graph?
Verify your comments by trying some other functions for f x .
Problem Change your original function in GeoGebra so that its gradient function is one of the following:
Extension Task
Find the point on the function 3 2f 6 9 1x x x x where the tangent has its maximum
downwards slope. Investigate the point with maximum downward slope for other cubic functions.
Used
dx from the f x
keyboard or type f ′(x)
Bitesize GeoGebra
Tom Button Simon Clay [email protected] [email protected]
@mathstechnology @simonclay_mei
Ben Sparks Avril Steele [email protected] [email protected]
@sparksmaths
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Useful commands in GeoGebra
Using the coordinates of a point For a point A the commands x(A) and y(A) will give each coordinate as a single number.
e.g. Add a point A. Enter p=x(A) and q=y(A). Plotting y=(x-p)^2+q will give a parabola with a moveable vertex.
Sequence To enter a family of objects use: Sequence[ <Expression>, <Variable>, <Start Value>, <End Value> ] e.g. Sequence[ x²+a, a, 1, 5]
Commands The three dots on the top bar of the input keyboard or
the question mark to the bottom-right of the input bar displays the list of built-in functions.
e.g. Curve[sin(t), sin(2t),t,-pi, pi] or add a polygon p and enter Perimeter[p]
Geometry tools in GeoGebra
The GeoGebra toolbar operates as a series of drop-down lists. Click the bottom right-hand triangle of any tool to reveal a list of related tools. Select any tool to select.
e.g. Create the points (3,5) and (5,0). If these are two vertices of a square, use the geometry tools to find the other two vertices and the area of the square.
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Construction Problems
1.
Given a square with variable side construct a rectangle with the same area whose sides are in the ratio 2:1. Can you extend this to 𝑛: 1?
2.
Construct the largest circle (in-circle) in a 3-4-5 triangle. What is the radius of this circle? What about other triangles based on Pythagorean triples?
3.
Given two points A and B construct points C and D so that ABCD is a rectangle with both sides independently variable.
4.
Construct 3 circles on the vertices of a triangle such that the circles always just touch.
5. Create points A, B and C fixed to the x-axis and D fixed to the y-axis. Construct a cubic that passes through A, B, C and D.
6. Create two points A and B. Construct a cubic that has stationary points at A and B.
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Additional views in GeoGebra
Additional views can be accessed via the View menu.
Graphics 2 When the Graphics 2 panel is selected any new object will automatically be created in it.
Alternatively you can define which panel objects are visible in using: Settings/Object Properties > Advanced > Location
Spreadsheet The spreadsheet uses the common form of cell references (letters for columns and
numbers for rows).
3D Graphics Planes in the form a x + b y + c z = d and functions z = f(x,y) can be entered directly.
CAS CAS can be used to factorise, expand, substitute, solve, differentiate and integrate
expressions and equations.
Probability calculator Various different distributions can be viewed graphically and numerically.
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Using Graphics 2
Quadratic inequalities
Adding the function and a point and the tangent 1 2 3
In the input bar type y=x^2+bx+c and press enter. If prompted select Create Sliders. Enable the second graphics panel: View > Graphics 2 In the input bar type x^2+bx+c<0 and press enter.
Exploring gradients on curves
Adding the function and a point and the tangent 1 2 3 4
In the input bar type y=sin(x) and press enter. Use the Point tool to add a point, A, on the curve. Use the Tangent tool to find the tangent to the curve at the point A. Use the Slope tool to measure the gradient of the tangent.
Tracing the gradient function in Graphics 2
5 6 7 8
In the input bar type x_1=x(A) and press enter. Enable Graphics 2: View/Perspectives > Graphics 2. In the input bar type (x_1,m) and press enter. Right-click point B and select Trace On.
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MEI GeoGebra Tasks for Further Maths: Pure
Task 8 – Vectors: Intersection of three planes
1. Enable the 3D view: View > 3D
2. In the input bar enter the equations of three planes: x − 2y + 4z = 4 x + y − z = 2 x + 3y + z = 6
3. Use the Intersect two surfaces tool to find the lines of intersection of two of the pairs of planes.
4. Use the Intersect tool to find the point of intersection of these lines.
Question for discussion
How can you use a matrix to find the point of intersection?
Problem (Try just using the inverse matrix function on your calculator then check it on your
software) Find the point of intersection of the planes:
5 2 5
5 2 11
2 3
x y z
x y z
x y z
Further Tasks
Investigate how many ways three planes can be orientated so that they do not have a unique point of intersection.
For each of these cases determine whether there are no points of intersection or infinitely many.
7th menu in Classic or
Basic Tools on the app.
2nd menu in Classic or
Points Tools on the app.
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GeoGebra: How to create a vector equation of a line in 3D
GeoGebra Classic 3D view or GeoGebra 3D Graphing Calculator
Creating the line based on points A and B
1 2 3 4 5
In the Input bar enter: O=(0,0,0) In the Input bar enter: A=(-1,1,2) In the Input bar enter: B=(2,2,3) Use the Vector tool to create the vectors OA and AB. Rename these vectors OA and AB. Use the Line tool (3rd menu) to create the line through A and B.
Creating a dynamic point P
6 7 8 9
In the input bar enter: λ=0.5 (In GeoGebra Classic 5 enable the slider). In the Input bar enter: P=A+λ×AB Use the Vector tool to create the vectors OP and AP. Rename these vectors as OP and AP. (GeoGebra Classic Only) In the Graphics view add a Text box. Switch the LateX formula on and enter OP = OA + λ AB. OP, OA, λ and AB should be selected from the objects menu.
Examples of using GeoGebra for 3D Vector Geometry: www.geogebra.org/m/GTPCFBVW
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Using GeoGebra for Statistics
GeoGebra Classic 5 is recommended for statistics work. It can be downloaded for free from www.geogebra.org/download
Cleaning, grouping and sorting data It is easier to group, clean or sort data in Excel before importing into GeoGebra.
Importing data To see the Spreadsheet select: View > Spreadsheet view. Data can be copied into the spreadsheet view directly from Excel.
One-variable analysis In Spreadsheet view highlight one column of data and select One Variable Analysis from the 2nd set of tools.
This displays 1-variable statistics, such as mean and standard deviation, and offers diagrams, such as boxplots and histograms.
Multi-variable analysis Different sets of data can be compared on the same screen.
In Spreadsheet view highlight multiple columns of data and select Multiple Variable Analysis from the 2nd set of tools.
Two-variable regression analysis In Spreadsheet view highlight two columns of data and select Two Variable Analysis from the 2nd set of tools:
A linear regression line can be added. To plot these points in the Graphics View right-click and select Copy to Graphics View.
Using Excel and GeoGebra for the Large Data sets in A level Mathematics: There is a collection of help videos at: mei.org.uk/large-data-sets In addition, large data set lesson plans for each specification can be found in Integral: integralmaths.org/
