Gsp training guide

RM Math Framework Edition™

The Geometer’s Sketchpad®
Training Guide

RM Math Framework Edition powered by Kaleidos:
Training Guide — The Geometer’s Sketchpad® 2

Contents
Introduction ..................................................................................................................5

Layout ...........................................................................................................................5

Tool Box buttons.....................................................................................................................6

Point Tool ..............................................................................................................................6

Circle Tool .............................................................................................................................6

Line Tool................................................................................................................................6

Text Tool................................................................................................................................7

Text boxes ..............................................................................................................................7

Custom Tool ..........................................................................................................................9

Measurement Accuracy...............................................................................................11

Drawing ......................................................................................................................13

Lines.................................................................................................................................... 13

Circles ................................................................................................................................. 13

Editing .........................................................................................................................15

Undoing .............................................................................................................................. 15

Deleting............................................................................................................................... 15

Hiding Lines or Points............................................................................................................ 15

Parallel and Perpendicular Lines ...............................................................................17

Constructing Interiors and Changing Colors............................................................................ 18

Lines.................................................................................................................................... 19

Measuring Angles ................................................................................................................. 19

To Change Labels................................................................................................................. 20


Arcs and Sectors ......................................................................................................... 21

Arcs .....................................................................................................................................21

Sectors .................................................................................................................................22

Arc/Sector Angle...................................................................................................................22

Arc Length ............................................................................................................................22

Transformations ......................................................................................................... 23

Translation: ..........................................................................................................................23

Rotation: ..............................................................................................................................23

Reflection: ............................................................................................................................24

Enlargement: ........................................................................................................................24

Graphs........................................................................................................................ 25

Grid Forms...........................................................................................................................25

Adjusting the scale on the axes...............................................................................................25

Plotting a graph ....................................................................................................................25

Plotting Derivatives ................................................................................................................26

Trigonometric Functions.........................................................................................................27

Plotting Coordinates..............................................................................................................28

Action Buttons and Tables.......................................................................................... 29

Hide and Show Buttons..........................................................................................................29

Tables..................................................................................................................................30

Animation Buttons...................................................................................................... 31

Sliders......................................................................................................................... 32

Some Simple Ideas for use in Lessons ....................................................................... 32


Introduction
The Geometer’s Sketchpad® (GSP) is a powerful geometry tool that can be used at many different
levels. The RM Math Framework Edition curriculum includes a range of activities and content pieces in
which most of the basic functions of this exciting application are utilized. You can use the Geometer’s
Sketchpad tools to create your own activities, which, in a very visual way, will help pupils learn about
and to dynamically test geometric concepts.

To load Geometer’s Sketchpad go to the Start button and click on Programs then GSP v4.05
(note if on a network, check with your administrator where GSP is located). This will open the basic
screen. Maximize this window to ensure you have the full amount of drawing space.

Layout
Across the top you have the Menu Bar

.

Down the left-hand side there is
a toolbox.

The functions of these tools are
explained below.


Tool Box buttons
Select tool (and translate)

Point tool
Circle tool
Line tool
Text tool

Custom Tool

Select Tool
This is used to select and deselect objects.
It is also used when switching from one tool to another or to stop another drawing tool being active.

Point Tool
The point tool, quite simply, allows you to draw points.

Circle Tool
This is used for drawing circles. Circles will be drawn with a center and a point on the circumference.
To draw a circle, touch where you want the center of the circle to be and drag out from there.

Line Tool
This tool has more than one option – hence the little triangle in
the bottom corner. Holding down on this triangle reveals the
full functionality

In geometry, a line is defined with infinite length. If you want to draw a line segment
between 2 points, you would use the first of the 3 line-drawing options, which is the default
option.
To draw a line segment, click on this tool, click the starting point of your line segment and drag to
where you want it to end.

The other two line tools are the ray – a half line going to infinity in one direction
and a line – a line going to infinity in both directions.


Text Tool
You can use this tool to label points, lines and circles, or to open up a text box to allow you to write
free text.

To label a point, click on the text tool then click on the point. If you do this for subsequent points, they
will be labeled in alphabetical order, with capital letters, which is mathematically correct. It is also
possible to have points labeled automatically, using the edit menu and setting the text option within
preferences (to do this for all new points. Lines are labeled in small letters. This is done by selecting
the text tool and clicking on the line.

Text boxes
To create a text box, select the text tool then double click on the page. You can type in any text from
the keyboard. You can use the bottom toolbar to alter the font type, size or color.

Another way to change the size of text is to use ALT and > to increase it, and ALT and < to decrease
it. You can also change the size via the Text palette from the Display menu.

Clicking on gives you mathematical symbols

and allows you to type fractions, square roots, etc.


For example:

The text toolbar can be automatically displayed by going to the Display menu and selecting Show
Text Palette.


Custom Tool
This is the final drawing tool found on the left hand side of the screen and can be used to produce a
copy of a GSP image or construction without having to continually reproduce it from scratch.

Example: Creating a custom tool which will draw an equilateral triangle.

Accurately construct an equilateral triangle
and then hide the construction circles.

Click on the Custom
Tool button and
choose Create New
Custom Tool

Give the Tool a name
(i.e. Equilateral Triangle)
then click on OK.

Now whenever an equilateral
triangle is needed, click on the
custom tool button and choose
Equilateral Triangle.

Then click on the screen to drag
out a triangle orienting and
sizing it to meet your
requirements.


Measurement Accuracy
It is important to set the correct accuracy for your measurements before you begin. This applies to
lengths, angles, co-ordinate points, ratios and slopes amongst others.

On the menu bar, select Edit.

Choose Preferences… at the
bottom of the list.

In the box that appears, choose
the accuracies you’d like.

Tick Apply to New Sketches if
you want to use these settings
again.


Drawing
Lines
To draw a line, click on the line (segment) tool in the toolbox and click and drag the line. This will
draw a line segment with two end points. If you want to draw closed shapes, then start the line at the
end of the previous one (e.g. over point B). Ensure that the 2 line segments are locked together; point
B should go turquoise.

If you want to move the end
points, click the Select button on
the toolbox (top tool) and click
and drag the points to their new
position.

If you want to move the whole
line, click on the body of the line
then click and drag to the new
position.

Circles
To draw a circle, click on the circle button on the toolbox, then click where you want the center of the
circle to be and drag the mouse to create the circle to whatever size you like.

If you want to change the size of
the circle, click on the Select
tool then click and drag the point
on the circumference (point P in
this diagram)

To move the whole circle,
keeping the size the same, click
the circumference (not at point P
though) and drag the circle to
where you want it to go.


Editing
Undoing
If you want to undo actions, click on the Edit menu and choose Undo (last action) from the menu. If
you need to undo the last few actions, repeat this. You can go forward again by using Redo from the
same menu.

Deleting
There are a few different ways to delete objects that you have drawn:

1. Highlight the object(s) by clicking on them, then hit the Delete button on the keyboard

2. Highlight the object(s) by clicking on them, then choose Cut from the Edit menu.

If you want to delete everything on the page, choose Select All from the Edit menu and then Cut or
Delete it all.

Hiding Lines or Points
Sometimes after you have constructed a line or point you will want to hide it, for example, if it was
only required for initial construction purposes.

To do this, select the line(s) or
point(s) you wish to hide.

Go to the Display menu and
select Hide Perpendicular
Line (or Hide Line
Segment, Hide Point, Hide
Circle, etc. – the options will
change depending on the
object you have highlighted).


Parallel and Perpendicular Lines
For some shapes or constructions you may want to construct parallel or perpendicular lines.

To draw a line parallel to another line, through a
specified point:

Draw a line segment

Draw a point

Select the point and the line

From the Construct menu, select Parallel Line

This will draw an infinite line, parallel to your
original, through the point you selected.

To construct a perpendicular line, repeat as above,
but select Perpendicular Line from the
Construct menu.

You can use parallel and perpendicular lines to help construct perfect squares and rectangles.


Constructing Interiors and Changing Colors

Draw a shape using the Line
Segment tool. (You can also use this
method if you are drawing a circle.)

To create a color fill for the shape, first
use the select tool and click on the
vertices, going round them in order.
(For a circle, select the circumference)

In this example, on the Construct
menu select Quadrilateral Interior
(this would say Triangle Interior,
Polygon Interior, Circle Interior,
etc., as appropriate).

You will then see the shape colored in
with diagonal lines over it. The shading
indicates the shape is selected. To stop
the shape being selected, click outside
it.

If you wish to change the color of the
shape, select the shape then go to the
Display menu and select Color. Pick
a color of your choice.


Lines
You can alter the color of a line and change its thickness.

Select the line.
Go to the Display menu and
choose Color to change its color.
Choose Line Width and choose
from Thin, Thick or Dashed.

Measuring Angles

Use the Line segment tool to
draw two lines to form an angle

Highlight the three points of the
angle, ensuring that the vertex is
highlighted second.

From the Measure menu
select Angle.

The angle measurement will
then appear on the screen.
Change the label if you want
using the text tool (see below).

Moving the lines or end points
of the angle will change the
displayed measurement.


To Change Labels
Click on the Text Tool.

Double click on the label.

In the box that appears, type in the text
for the new label and click OK.


Arcs and Sectors
Arcs
To construct and arc, first draw a circle. A
point will automatically be placed on the
circumference.

Using the Point Tool, place another point
on the circumference. Ensure that the point
is locked onto the circumference (i.e. when
you place the point, the circle should go
turquoise).

Using the Select tool, deselect everything
by clicking on the white screen. Highlight
the points on the circumference in a
counter-clockwise direction and then
highlight the circle itself.

Select the Construct menu followed by
Arc on a Circle.

Deselect everything and
then highlight just the circle.

Select the Display menu
and then Hide Circle.


Sectors

Having created an arc, draw in radii
to form the sector.

Highlight the arc.

On the Construct menu choose
Arc Interior, followed by Arc
Sector. The sector will now be
colored in.

Arc/Sector Angle
• Highlight the Arc.
• Select the Measure menu followed by Arc Angle.

Arc Length
• Highlight the Arc.
• Select the Measure menu followed by Arc Length.


Transformations
There are four basic transformations: translation, rotation, reflection and enlargement (or dilation as
it is called in GSP).

Translation:
On the Graph menu choose Grid Form
followed by Square Grid.

Draw a shape on the grid.

Highlight the whole shape – lines and
points.

On the Transform menu choose
Translation. Click in the circle next to
Rectangular and type in the column
vector. You’ll see a shadow of where the
shape will move to. Click Translate.

Rotation: To rotate an object, double click
the point about which you want to
rotate the object. It will ‘flash’ at
you. You can also achieve this by
single clicking the point, going to
the Transform menu and
choosing Mark Center.

Now select all the objects to be
rotated.

From the Transform menu choose
Rotate.

Type in the angle of rotation. You
will see a shadow of where the
shape will finish after the rotation.
Click OK.


Reflection:
To Reflect objects you need to define a
mirror line. To do this, either double click
the line (it will ‘flash’ at you) or select the
line and from the Transform menu select
Mark Mirror.

Now select the object(s) to be reflected and
then from the Transform menu select
Reflect.

Enlargement:

To enlarge an object, double click the
point that you wish to mark as a
center of enlargement - it will ‘flash’
at you. You can also achieve this by
single clicking the point, going to the
Transform menu and choosing
Mark Center.

Select all the objects to be enlarged.

From the Transform menu choose
Dilate.

Type in the ratio to enlarge by, then
click on Dilate. Notice the image
appears before OK is clicked.


Graphs

Grid Forms
There are three grid forms: square, rectangular and polar. To choose a grid, select the Graph menu
followed by Grid Form and then choose one of the 3 available grids.

Adjusting the scale on the axes
For Square Grids, there is a red point on the x-axis. Select this point and drag it to the left or right
to resize the x and y-axis concurrently.

For a Rectangular Grid it is possible to resize the x and y-axis independently as there is a movable
red point on each axis.

Plotting a graph
From the Graph menu choose a grid form and resize the axes as appropriate. Select the Graph
menu followed by Plot New Function, then using the keypad provided type in the relevant function
then click OK. Your graph is plotted and the function displayed.


Plotting Derivatives
When you draw a graph, the function appears on the screen, for example f(x) =x3.

To find the derivative of a function,
right-click on the function and choose
Derivative.

The derivative of the function will
appear directly below the original
function.

To plot the graph of the
derivative, right click on
the derivative formula
and choose Plot
Function.


Trigonometric Functions
To plot a trigonometric function, follow the instructions above and when typing in the function, click
on the Functions button and select from the list. Remember to resize the axes appropriately before
plotting the function.


Plotting Coordinates

On the Graph menu choose an
appropriate grid form.

Again from the Graph menu,
select Plot Points.

Type in the coordinates of the point
required and then click Plot.

To label the point with its
coordinates, highlight the
point.

select Coordinates.


Action Buttons and Tables
Hide and Show Buttons
In this example, a triangle has been drawn, all three angles have been measured, and the sum of the
angles has been calculated using the calculation facility in the Measure menu.

To create a Hide/Show button for
the measurements and the sum,
highlight all three measurements and
the sum of the angles.

From the Edit menu select Action
Buttons, followed by Hide/Show.

The measurements can now be
hidden by clicking on:

Hide Angle Measurements

D The triangle can then be
resized with the
E measurements hidden.
Show A ngle Me asurements
Use the Show Angle
Measurements button to
reveal the new
F
measurements.


Tables

To tabulate the results from
the above example,
highlight the 3 angle
measures and their sum.

From the Graph menu
select Tabulate.

Resize the triangle to adjust
the angle sizes.

To capture the new
measurements in the table,
simply double click on the
table and it will update.


Animation Buttons
In this example, a circle, and then a diameter AC, have been drawn, and a point B has then been
placed on the circumference. Point B has been joined to points A and C by line segments to form
triangle ABC. Angle B has then been measured.

We can animate point B so that it moves
around the circumference.

To do this, highlight point B.

From the Edit menu select Action Buttons,
followed by Animation.
Choose the direction and speed then click
OK.


Sliders
It is useful to construct a slider for controlling a numeric value. For example, when exploring straight-
line graphs (y = mx + c) it may be useful to create a slider to vary the slope and the y- intercept.

To set up a slider for the slope m:

Select a rectangular grid from the Graph menu
Draw an infinite horizontal line
Label the two points on the line A and B
Put a point somewhere between A and B. Label it C
Highlight points A, B and C in that order
From the Measure menu select Ratio
Select the text tool and double click on the ratio measurement. Label it m
Draw a line segment between A and C
Click on the infinite line and point B and hide them
By moving point C to the left and right this now changes the value of m

A slider could also be created for c, the y-intercept value.

To use this ratio as the value of m when plotting the graph of y = mx+c, choose Plot New
Function from the Graph menu. When typing in the function, click on the label box ‘m’, then on the
key pad type * x + then click on the label box ‘c’ and finally click on OK. As points C and F are
moved, so the slope and intercept will change.


Some Simple Ideas for use in Lessons

To draw a circle and measure
its radius, circumference and
area, draw a circle using the
Circle Tool from the left-hand
toolbar.

Deselect the circle (click the
select arrow and click anywhere
outside the circle) and then use
the Line Segment tool to
draw in a radius.

Select the Measure menu on
the top menu bar and choose
Length.

The measurement appears in
the top left corner. You can
drag this anywhere on the
page.


To change the label, click the
text tool on the right-hand
menu.

Double click on the label.

Type in your preferred labeling
and click OK.

You can also alter the size of the text or
its color.

Click on the text tool and double click
anywhere on the page. This brings up
the text toolbar at the bottom of the
page. You can also do this by going to
the Display menu and select Show
Text Palette.

Select the text box that has the radius
measurement. Choose a text size and
color using the options on the text
toolbar. You can also choose a font
and make it bold or italic.


To measure the circumference
of the circle, highlight the
circumference and ensure that
nothing else is highlighted (e.g.
the radius measurement box).

Click Measure on the menu
bar and choose
Circumference.

Similarly, you could choose
Area to measure the circle’s
area.

If you now drag the point on the
circumference to change the
size of the circle, the
measurements will
automatically change.

For tangents to circles,
draw a circle and draw in
a radius (don’t draw the
radius to the point that
appears on the
circumference though).

Highlight the radius and
the point where the radius
meets the circumference.

Click on the Construct
menu and select
Perpendicular Line.


Draw a point somewhere on the
perpendicular line.

Select that point, the point
where the radius meets the
tangent, and the center of the
circle.

Click the Measure option on
the menu bar and select
Angle.

You can change the label box if
you wish.

You can now animate the tangent so that it moves around the circle. This will allow pupils to see that
the angle doesn’t change. To do this:

• Highlight the point where the radius meets the tangent (ensure nothing else is highlighted).

• On the Display menu select Animate Point.

The tangent now moves round the circle. You can pause, slow down, or speed up the animation using
the Motion Controller.


To construct circle sectors, draw
a circle and draw in two radii to
form a sector.

Draw a point on the arc within
the sector.

Select the 3 points in the sector,
in order.

On the Construct menu
choose Arc Through 3
Points.

Highlight the arc by double
clicking on it.

you can choose to measure
the Arc Angle or Arc
Length.

To color the sector, highlight
the arc and from the
Construct menu choose Arc
Interior then Arc Sector.

If you want to change the color
of the sector, click on it and
from the Display menu choose
Color.


Use the Line segment tool to
draw two lines to form an
angle.

Highlight the three points of the
angle, ensuring that the vertex is
highlighted second.

select Angle.

The angle measurement will
then appear on the screen.
Change the label if you want
using the text tool.

Moving the lines or end points
of the angle will change the
displayed measurement.

You could use this feature to measure angles in triangles in order to explore, for example, the sum of
the angles in a triangle.
Measure each of the angles in the
triangle.

From the Measure menu select
Calculate.

Click on the measurement of one of
the angles and on the calculator,
click +. Click on the next angle
measurement and click +. Click
on the third angle measurement
and click OK.

The sum of the angles appears on
screen.

You can now drag the triangle to create different-sized angles and repeat the above process to show
that the sum of the angles is still 180 degrees.


Open a Square Grid from the
Graph menu.

From the Graph menu select
Snap Points.

Use line segment tool to draw a
triangle in the top right quadrant.
Highlight the vertices and from the
Construct menu select Triangle
Interior.

Select the vertices and from the Measure menu select Coordinates. The vertices are
now labeled with the coordinates shown.

Double click on the y-axis to set this up as your mirror line (it will ‘flash’ at you when you
do this).

Select the triangle and from the Transform menu choose Reflect.

Your triangle will be reflected. You can change its color if you wish, using the Display
menu.

Set up a diagram with a triangle, as
described above.

Double click the Origin to set the
origin as the center of rotation.

Select the triangle, then from the
Transform menu select Rotate.

Choose the angle you wish to rotate
by. Note: rotations are counter-
clockwise. Choose a negative angle
to create a clockwise rotation.

Click Rotate.

Your rotated triangle is plotted on the
diagram.


To create a right triangle, draw To measure the sides, select a side and
a line segment. from the Measure menu select Length.
(You can change the label if you wish once
Select one end point of the line and it’s displayed.)
the line itself and from the
Construct menu choose Repeat this for all three sides.
Perpendicular Line.
Select the label displaying the measurement
On top of the perpendicular line, of the hypotenuse.
draw a line segment from the end
point of your initial line. From the Edit menu select Action
Select the perpendicular line and Buttons, then select Hide/Show.
from the Display menu select
Hide Perpendicular Line. A Hide button is created. Clicking on this
will hide your measurement. The button
Use a line segment to create the then changes to Show Measurement.
third side of your triangle. Clicking on it now will reveal the
Measure the right angle. measurement again.
Trigonometry in non-right-angles

You can select everything and copy and paste it, to save having to repeat the
process for other triangles. You can then re-size the copied triangles to create
additional Pythagoras’ Theorem questions. You could do a similar thing using
angle measurements when studying right-angle trigonometry.


Geometer’s Sketchpad is a very effective tool for teaching the Sine Rule, Cosine Rule and
the Area of a Triangle.

For example, for the Sine Rule, draw a triangle using line segments.

Measure two angles – select the 3 points of the angle and choose Angles from the Measure
menu.

Measure one side by selecting the side and choosing Length from the Measure menu.

You can measure the length of the side you wish to calculate, then hide its length by selecting
the measurement and from the Edit menu choose Action Buttons, then Hide/Show.
Clicking the Hide button that is then created will make the measurement disappear. Click the
Show button to make it reappear.

The triangle can be re-sized to create a different problem.

Examples for the Cosine Rule can be created in a similar way.


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