More Advanced GeoGebra Projects (by Jon Ingram) · More Advanced GeoGebra Projects (by Jon Ingram)...

More Advanced GeoGebra Projects (by Jon Ingram)

There are many ways that GeoGebra can be used as more than just a graph drawing tool. You can

Use the Spreadsheet facility to graph and manipulate data;

Use Sliders to create animations;

Add Buttons, Check Boxes and Input Boxes to create interactive sheets;

Add conditions for objects to show themselves, or to dynamically change colour;

Interact with GeoGebra objects and values using GeoGebraScript;

Use the CAS facility to perform symbolic algebra;

(in version 5, and still rather flaky!) Work in 3D.

One of the best ways to explore these features is to open, use, and change worksheets which use

them, looking at the definitions of all the objects. GeoGebraTube is a collection of more than two

million GeoGebra worksheets, all freely available to download and explore:

http://tube.geogebra.org/

There are many incredible worksheets uploaded, and it’s a great resource to browse around in your

spare time.

This document links to several worksheets that I have created, which all use some more advanced

features. I have explained a little about how the worksheets were put together, and how I use them

in the classroom. Please feel free to alter and use them however you wish!

At the end of the handout is a brief introduction to using GeoGebra as a symbolic algebra system.

In this handout I refer to the following GeoGebra worksheets:

Expanding and Contracting Squares

Finding Coordinates

An Adjustable Histogram

Six Related Lines

Visualising Collisions

You can see these, and all the others I’ve uploaded, on my GeoGebraTube profile page:

http://tube.geogebra.org/joningram

I will also be writing some blog posts detailing some of the other GeoGebra features I’ve used on my

new blog:

http://jimaths.wordpress.com

http://tube.geogebra.org/

http://tube.geogebra.org/joningram

http://jimaths.wordpress.com/

Advanced Geogebra Projects

2

Expanding and Contracting Square –

http://tube.geogebra.org/material/simple/id/2041305

Description: This is a peaceful animation of an expanding and contracting square made up smaller

squares rotating around each other. I have used this several times in lessons on tiling and angles,

and it also makes a nice animation to have open at the start of the class as students are coming into

the room!

Advanced Features Used: Animated sliders, rotating by a variable.

How to: There is a hidden number slider called ‘angle’, which moves between 90 and 270 degrees.

The construction started by creating the square in the middle. All the others are rotated versions of

this one, created by the ‘rotate by angle’ button. When GeoGebra asked for an angle, I entered

angle. This means that as the ‘angle’ variable changes, the amount of rotation changes. I then

carried on constructing rotated squares to create the final pattern.

Tip: If GeoGebra worksheets don’t use scripting, then you can view all the steps in their construction

by selecting View -> Construction Protocol from the main window’s menu.

Suggestions: Create your own animated pattern using this technique, and/or try to reconstruct this

one.



3

Finding Coordinates –


Description: This worksheet has four movable points, A, B, C and D. Next to each point is a target

coordinate. When the points are dragged to the correct coordinate, a green ‘tick’ appears next to

the coordinate. When all four points are dragged to the correct location, a ‘Well done!’ message

appears. The ‘Restart’ button resets the points, and creates new targets.

Advanced Features Used: Hidden objects, buttons, scripting.

How to: As well as the four points you can see, there are four hidden points: Agoal, Bgoal,

Cgoal and Dgoal. They are moved around by a script attached to the Restart button, which you can

see by looking at its ‘Object Properties’. This is what the script tab looks like:



4

This is using GeoGebra’s inbuilt scripting language, GeoGebraScript. Every time the button is clicked,

A, B, C and D are moved to their starting locations, and Agoal, Bgoal, Cgoal, and Dgoal are moved to

random positions with various constraints. RandomBetween[a, b] creates a random whole

number between a and b (inclusive).

The text boxes are hidden using a property in their Advanced tab. In the ‘Condition to Show Object’

area of the Advanced tab, I typed

A = Agoal

Slightly confusingly, when you press return, GeoGebra changes this slightly, so it looks like this:

I only want the ‘Well Done’ message to appear when all of the points are in the correct location. In

its ‘Condition to Show Object’ box, I typed

(A = Agoal) && (B = Bgoal) && (C = Cgoal) && (D = Dgoal)

which GeoGebra also ‘helpfully’ changes to look like this:

Suggestions:

Explore different conditions. For example, x(A) and y(A) are the x- and y-coordinates of A, so setting

the condition to be x(A) = 5 would display the tick when A is on the line y=5.


5

An Adjustable Histogram -


Description: This worksheet uses the spreadsheet window to generate some data points, which are

then graphed in several different ways: as a box-and-whisker plot, and as a histogram. The end

points for the histogram’s classes are given by movable points, so you can explore what happens to

the histogram as you alter the classes.

Advanced Features Used: The spreadsheet, histograms, box-and-whisker plots, randomly generated

data.

How to: There are a lot of different interesting features to this worksheet. First, it uses the

Spreadsheet environment. This is a rather underused part of GeoGebra, which provides some really

quite nice ways to explore and visualise data. While they normally just contain numbers, each cell in

a spreadsheet can contain any GeoGebra object, including numbers, coordinates, lines, functions,

circles, text, etc. Here’s an example:

In this worksheet, each value is randomly generated from a Normal distribution with mean 10 and

standard deviation 10. If you double click on a cell, you will see the value is just

RandomNormal[10, 2]



6

There are several other random statistical functions built into GeoGebra, including

RandomBinomial.

These values are displayed in two ways. The box-and-whisker plot was generated by typing

BoxPlot[14, 0.5, A1:A50]

into the Input Line at the bottom of the GeoGebra window. This means I want a box plot centered

on y=14, with y-scale 0.5 (it goes up and down 0.5 from the starting line), and data taken from

A1:A50. Notice that you refer to data from the spreadsheet environment exactly as you would for an

Excel data range.

The histogram was created by typing

Histogram[Boundaries, A1:A50]

into the Input Line. Here Boundaries is a list giving the boundaries of the classes. These could be

defined statically, for example by saying

Boundaries = {0, 4, 8, 12, 16, 20}

but in this worksheet I’ve been a bit more interactive, and made the class boundaries draggable. I

did this using some points on the x-axis, called P_1, P_2, up to P_6. I want to use the x-coordinates

of these points as my class boundaries, which I can do like this:

Boundaries = {x(P_0), x(P_1), x(P_2), x(P_3), x(P_4), x(P_5), x(P_6)}

If you examine the worksheet, I’ve actually been a bit more sneaky than this: the Histogram

generator doesn’t like it when your class boundaries are not in increasing order, so I used some

GeoGebraScript magic to make sure my list of points was sorted into increasing order.

Suggestions:

Explore the various options to the BoxPlot and Histogram routines, using your own or randomly

generated data.


7

Six Related Lines –


Description: Despite its simple appearance, this worksheet has lead to some really rewarding

classroom activities. When you load this worksheet, you see six moving lines. As you watch for a

while, you notice that some of the lines are related (in particular, pairs are the same colour), and

that there are some patterns to where they seem to intersect. Clicking the ‘options’ box and ‘show

equations’ reveals that the six lines are generated by the different ways of putting three numbers

into the general pattern

𝒂𝒙 + 𝒃𝒚 = 𝒄.

At a GCSE level, asking simple questions about the gradients and intercepts of these lines can be

quite rewarding. With an A-level group, this worksheet can be used as part of a lesson on proof and

conjecture generation, as there are a number of features about the related lines that you can

conjecture, investigate, and prove.

Advanced Features Used: Animated sliders, text boxes using object information, hidden checkbox

‘menus’.



8

How to:

One feature of this worksheet, that I use quite a lot in interactive worksheets for my own teaching, is

that the options are initially hidden. After creating the checkboxes to show/hide various features of

my worksheet, I then create one more checkbox called ‘Options’, which shows/hides the other

checkboxes. This makes the initial appearance of the worksheet more streamlined, which gives less

opportunity for students to get distracted when they are first looking at the animation.

I chose the animation parameters for the a, b, and c sliders so that they would slowly go out of sync

with each other, which makes the lines slowly explore the ‘parameter space’ of possible

arrangements. Specifically, I made their speeds 1, 0.9, and 0.7 (you can view and change this from

the Object Properties).

Suggestions:

About this specific worksheet:

Did you notice that when all six lines intersect, it is always at (-1, -1)? Can you prove that this is the

only possible place they can all meet?

Did you notice that lots of the intersection points are on the lines 𝑥 = 1, 𝑦 = 1 and 𝑥 = 𝑦? Perhaps

ticking the ‘show extra lines’ checkbox will help. You can even hide the lines and just look at the

intersection points.

If you just had the lines, could you work backwards and find the values of a, b, and c?

More generally:

There are lots of rich explorations generated from taking two or three numbers, and creating some

curves based on them. What happens if you graph

𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑏

𝑦 = 𝑎𝑥2 + 𝑐𝑥 + 𝑑

…

𝑦 = 𝑐𝑥2 + 𝑏𝑥 + 𝑎

for example?

You could also define a function f(x) on the Input Line, and then animate various graph

transformations.


9

Visualising Collisions -


Description: The worksheet looks fairly simple, but is quite complicated behind the scenes. It

generates a distance-time diagram showing the positions of three particles which collide with each

other after the blue particle is projected forwards to meet the others. You can vary the starting

positions, the initial velocity, the masses of the three particles, and the coefficient of restitution.

Advanced Features Used: The spreadsheet, two graphics views, buttons, multi-part functions,

GeoGebraScript.

How to: The first thing to notice about this worksheet is that there are two graphics views on screen.

Although you normally only see Graphics View 1, GeoGebra actually has two separate Graphics

Views. The second is sometimes used, as in this worksheet, as a way to display text, options, and

sliders.

All the complicated calculations going on ‘behind the scenes’ happen in the Spreadsheet View. Here

is the spreadsheet view for the above graph:



10

GeoGebra displays the values in the cells, rather than the formulae, so the easiest way to see how

these are calculated is to download the worksheet and explore it yourself. Each line is generated

from the one above it by first checking to see if there will be another collision, and, if there is, when

and where it will be, and what the new values for the velocities will be (using Conservation of Linear

Momentum and Coefficient of Restitution). The maximum time value considered is 300 seconds.

The trajectories of the three particles in the distance time graph are Polylines. This is a function

made up from lines that join specified points together. For example, if you type

Polyline[(0, 1), (2, 3), (5, 0)]

into a blank sheet, you get

Suggestions: Download the worksheet, and explore the definitions of the various points and

buttons. In particular, the button which starts the animation using the GeoGebraScript function

StartAnimation[t].


11

Using GeoGebra for Symbolic Algebra

The following is a record of a short session in GeoGebra’s CAS view, and illustrates some of the ways

in which symbolic algebra manipulation could be useful in school-level mathematics.

Integration and differentiation:

Prime factorisation:


12

Expanding and factorising:

Solving equations:

Miscellaneous:

More Advanced GeoGebra Projects (by Jon Ingram) · More Advanced GeoGebra Projects (by Jon Ingram)...

Documents

Transcript of More Advanced GeoGebra Projects (by Jon Ingram) · More Advanced GeoGebra Projects (by Jon Ingram)...