Using the Geometers Sketchpad at key stage three
I found out about the Geometers Sketchpad about 4 years ago when I was looking for
useful software in mathematics teaching. There were so many complimentary reviews
written about Dynamic Geometry that I decided it was definitely worth a look. I could not
find any local schools that used it but the Leicester City numeracy advisor, Sue Welford,
put me in touch with a teacher in Hawaii, Cathi Sanders. Cathi sent me some files she had
written for the GSP. I managed to get hold of a demo version of the GSP and then use
Cathi’s files to teach myself how to use the software.
The next step was to persuade my colleagues at the school where I was then teaching. I
had the demo version put on to the school’s network and hosted an after school session in
order to let the GSP strut its stuff. I managed to impress our vice principal who decided
the school could cough up nearly £500 to pay for a site licence. It was pretty expensive as
an investment in mathematics software but, I believe, it is well worth it. Since then and a
move of school later, I have learnt more of how valuable a resource Dynamic Geometry
can be. In my present school we only have 10 computers in the mathematics department
and so we bought a 10 user licence which was a bit cheaper at just under £300. Our ICT
technician just allocated the GSP to the maths stations.
I use the Geometers Sketchpad in two ways. One way is to set up demonstration files in a
Read Only folder which the students can access and use to investigate geometrical
properties such as the angle sum of a triangle. Alternatively I can hook up one computer
to a large monitor and demonstrate to the whole class. The other way I use the GSP with
my classes is to let them set up their own sketches, either to design a symmetrical pattern
or to investigate some aspect of geometry. For example having worked with the tri-angle
sum sketch I get the students to draw a quadrilateral, get the software to measure its
angles and the angle sum. When they drag corners of their quadrilateral they find that the
sum stays 360 degrees for convex quadrilaterals. However this does not hold true for
quadrilaterals which have reflex angles. This is because the GSP always measures angles
up to 180 degrees. With a few carefully chosen questions the students can be helped to
see this. It is one of the cons of using this software.
The students have a lot of fun using the GSP to create designs using reflection and
rotational symmetry. They use the line tool to draw one part of the design then use the
Transform menu to reflect or rotate it. I have seen some wonderful results such as the
symmetrical pineapple and the ‘cat eating a mouse’ that was rotated through 120 degrees.
I also use the GSP to help the students to practise drawing images under transformations.
I have set up some files where I have drawn an object, used the software to transform it
then hidden the object. The students have to use the line tool to draw the image then use a
Show button on the screen to bring up the correct image so they can check their solution.
I have created some files on Geometers Sketchpad Version 4 for use with year 8 students
(aged 12-13 years). The particular part of the curriculum is Geometical reasoning:lines,
angles and shapes.The key objectives are:
 Identify alternate angles and corresponding angles;
 Understand a proof that;
The sum of the angles of a triangle is 180 degrees and of a quadrilateral is 360
degrees
(National Numeracy Strategy Framework for teaching Mathematics, Year 8 programme
of study)
In year 7 the students have learnt the sum of angles on a straight line, sum of angles in a
triangle and that opposite angles are equal. They have also learnt how to measure angles
to the nearest degree and how to label lines using 3 letters.
I begin this topic by asking the class to think of letters of the alphabet made of parallel
lines. When several suggestions have been drawn on the board I tell the students that we
will be looking more closely at F and Z. The students draw some different versions of F
and Z in their books: squat letters, tall letters and the parallel lines slanted to various
degrees. For each drawing the students measure and record the angles in their diagrams.
We then discuss whether they found any equal angles, bearing in mind that we cannot
measure with complete accuracy. We record our findings as rules and then look for Fs
and Zs in a pair of parallel lines and a transversal.
2 parallel lines & transversal
I have written the files F angles.gsp and alternate angles.gsp as demonstrations of
corresponding and alternate angles. I have kept the diagrams looking like the letters of the
alphabet. The students can open these files and use the arrow tool to drag and distort the
diagrams while observing that alternate angles are always equal, corresponding angles are
always equal and allied angles sum to 180 degrees. The beauty of the software is that it
allows the students to observe many examples of the rules whereas they can only observe
a small number of examples at the desk. It also serves as another (more fun?) media to
reinforce the concepts of alternate and corresponding angles.
The file tri-anglesum.gsp shows a triangle with a line going through the top corner which
is parallel to the base. The students can drag any point and distort the diagram. The lines
will remain parallel and the same pairs of angles will always be equal. Alternate angles
can be used to show that since the angle sum of a straight line is 180 degrees the angle
sum of a triangle is 180 degrees.
The file quadrilateral.gsp shows a quadrilateral and the sum of its angles. The students
find that while ABCD is convex the angle sum is recorded as 360 degrees. When ABCD
is concave the angle sum changes. Why is this? We can see that the angle sum should be
360 degrees because a quadrilateral can be divided into 2 triangles. The students will
notice that the software records the reflex angles as smaller than 180 degrees. This shows
a disadvantage of the software as it does not measure reflex angles and so spoils the
calculation.
These files can be used on individual computer work stations with the students working
singly or in pairs. Alternatively the teacher can use one computer with a projector or
interactive whiteboard so that the whole class can discuss the demonstrations. This gives
the teacher the opportunity to guide the students to notice where the equal angles are
found in the diagram and which angles are being added. Overall using dynamic geometry
software gives students and teacher the opportunity to visualise the geometrical rules in
action. The students can also use the software to set up these demonstrations themselves
which would be helpful as practice for setting up their own files later.
One of my year 9 students had finished an exercise on enlargement and so I set him an
investigation task from the text book. He had to enlarge an object to create the first image
then enlarge the first image using the reciprocal scale factor but a different centre of
enlargement to create the second image. When this was done it looked as if a translation
would map the object on to the second image. Andrew tried this a couple of times and it
seemed to be the case. During the next lesson when the class were using the GSP to
produce enlarged designs on the computer, Andrew had the idea of checking his
hypothesis using the GSP. He set up the enlargement using a given point as the centre of
enlargement and enlarged the object 2 times. He then enlarged the image to half its size
using a different centre of enlargement. Again it appeared that to go from the object to the
second image would just take one translation. The beauty of the software is that Andrew
was then able to drag the centres of enlargement around and show that the two
enlargements still amounted to a translation for many more different cases than he would
have time to draw. Of course this is not a proof but it is a powerful demonstration of the
hypothesis. Dynamic Geometry can provide a demonstration of a geometrical property in
an infinity of cases.