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

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.

Select tool (and translate)

Point tool

Circle tool

Line tool

Text tool

Custom Tool


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.

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

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

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.

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 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.

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.

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

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.


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.

Measuring Angles

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.

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

Deselect everything and then highlight just the circle.

Select the Display menu and then Hide Circle.

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.


Sectors

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.

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.


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

Translation:

Rotation:

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.

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:

Enlargement:

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.

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.

From the Measure menu 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.

Show Angle Me asurements

D

E

F

The triangle can then be resized with the measurements hidden.

Use the Show Angle Measurements button to reveal the new measurements.

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


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.

To do this, highlight point B.

From the Edit menu select Action Buttons, followed by Animation.

Choose the direction and speed then click OK.

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


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:

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.

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


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.


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.

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.


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.

From the Measure menu 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.


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

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.

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.

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

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.


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.


Trigonometry in non-right-angles

To create a right triangle, draw a line segment.

Select one end point of the line and the line itself and from the Construct menu choose Perpendicular Line.

On top of the perpendicular line, draw a line segment from the end point of your initial line.

Select the perpendicular line and from the Display menu select Hide Perpendicular Line.

Use a line segment to create the third side of your triangle.

Measure the right angle.

To measure the sides, select a side and from the Measure menu select Length. (You can change the label if you wish once it’s displayed.)

Repeat this for all three sides.

Select the label displaying the measurement of the hypotenuse.

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

A Hide button is created. Clicking on this will hide your measurement. The button then changes to Show Measurement. Clicking on it now will reveal the measurement again.

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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