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- 1. RM Math Framework Edition™The Geometer’s Sketchpad® Training Guide
- 2. RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 2
- 3. ContentsIntroduction ..................................................................................................................5Layout ...........................................................................................................................5 Tool Box buttons.....................................................................................................................6 Point Tool ..............................................................................................................................6 Circle Tool .............................................................................................................................6 Line Tool................................................................................................................................6 Text Tool................................................................................................................................7 Text boxes ..............................................................................................................................7 Custom Tool ..........................................................................................................................9Measurement Accuracy...............................................................................................11Drawing ......................................................................................................................13 Lines.................................................................................................................................... 13 Circles ................................................................................................................................. 13Editing .........................................................................................................................15 Undoing .............................................................................................................................. 15 Deleting............................................................................................................................... 15 Hiding Lines or Points............................................................................................................ 15Parallel and Perpendicular Lines ...............................................................................17 Constructing Interiors and Changing Colors............................................................................ 18 Lines.................................................................................................................................... 19 Measuring Angles ................................................................................................................. 19 To Change Labels................................................................................................................. 20RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 3
- 4. Arcs and Sectors ......................................................................................................... 21 Arcs .....................................................................................................................................21 Sectors .................................................................................................................................22 Arc/Sector Angle...................................................................................................................22 Arc Length ............................................................................................................................22Transformations ......................................................................................................... 23 Translation: ..........................................................................................................................23 Rotation: ..............................................................................................................................23 Reflection: ............................................................................................................................24 Enlargement: ........................................................................................................................24Graphs........................................................................................................................ 25 Grid Forms...........................................................................................................................25 Adjusting the scale on the axes...............................................................................................25 Plotting a graph ....................................................................................................................25 Plotting Derivatives ................................................................................................................26 Trigonometric Functions.........................................................................................................27 Plotting Coordinates..............................................................................................................28Action Buttons and Tables.......................................................................................... 29 Hide and Show Buttons..........................................................................................................29 Tables..................................................................................................................................30Animation Buttons...................................................................................................... 31Sliders......................................................................................................................... 32Some Simple Ideas for use in Lessons ....................................................................... 32RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 4
- 5. IntroductionThe Geometer’s Sketchpad® (GSP) is a powerful geometry tool that can be used at many differentlevels. The RM Math Framework Edition curriculum includes a range of activities and content pieces inwhich most of the basic functions of this exciting application are utilized. You can use the Geometer’sSketchpad tools to create your own activities, which, in a very visual way, will help pupils learn aboutand 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 basicscreen. Maximize this window to ensure you have the full amount of drawing space.LayoutAcross the top you have the Menu Bar. Down the left-hand side there is a toolbox. The functions of these tools are explained below.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 5
- 6. Tool Box buttons Select tool (and translate) Point tool Circle tool Line tool Text tool Custom ToolSelect ToolThis 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 ToolThe point tool, quite simply, allows you to draw points.Circle ToolThis 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 ToolThis tool has more than one option – hence the little triangle inthe bottom corner. Holding down on this triangle reveals thefull 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 towhere you want it to end.The other two line tools are the ray – a half line going to infinity in one directionand a line – a line going to infinity in both directions.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 6
- 7. Text ToolYou can use this tool to label points, lines and circles, or to open up a text box to allow you to writefree text.To label a point, click on the text tool then click on the point. If you do this for subsequent points, theywill be labeled in alphabetical order, with capital letters, which is mathematically correct. It is alsopossible to have points labeled automatically, using the edit menu and setting the text option withinpreferences (to do this for all new points. Lines are labeled in small letters. This is done by selectingthe text tool and clicking on the line.Text boxesTo create a text box, select the text tool then double click on the page. You can type in any text fromthe 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 decreaseit. You can also change the size via the Text palette from the Display menu.Clicking on gives you mathematical symbolsand allows you to type fractions, square roots, etc.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 7
- 8. For example:The text toolbar can be automatically displayed by going to the Display menu and selecting ShowText Palette.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 8
- 9. Custom ToolThis is the final drawing tool found on the left hand side of the screen and can be used to produce acopy 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 9
- 10. RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 10
- 11. Measurement AccuracyIt is important to set the correct accuracy for your measurements before you begin. This applies tolengths, 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 11
- 12. RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 12
- 13. DrawingLinesTo draw a line, click on the line (segment) tool in the toolbox and click and drag the line. This willdraw a line segment with two end points. If you want to draw closed shapes, then start the line at theend of the previous one (e.g. over point B). Ensure that the 2 line segments are locked together; pointB 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.CirclesTo draw a circle, click on the circle button on the toolbox, then click where you want the center of thecircle 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 13
- 14. RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 14
- 15. EditingUndoingIf you want to undo actions, click on the Edit menu and choose Undo (last action) from the menu. Ifyou need to undo the last few actions, repeat this. You can go forward again by using Redo from thesame menu.DeletingThere 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 orDelete it all.Hiding Lines or PointsSometimes after you have constructed a line or point you will want to hide it, for example, if it wasonly 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).RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 15
- 16. RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 16
- 17. Parallel and Perpendicular LinesFor 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 17
- 18. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 18
- 19. LinesYou can alter the color of a line and change its thickness.Select the line.Go to the Display menu andchoose Color to change its color.Choose Line Width and choosefrom Thin, Thick or Dashed.Measuring AnglesUse the Line segment tool todraw two lines to form an angleHighlight the three points of theangle, ensuring that the vertex ishighlighted second.From the Measure menuselect Angle.The angle measurement willthen appear on the screen.Change the label if you wantusing the text tool (see below).Moving the lines or end pointsof the angle will change thedisplayed measurement.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 19
- 20. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 20
- 21. Arcs and SectorsArcs 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 andthen highlight just the circle.Select the Display menuand then Hide Circle.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 21
- 22. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 22
- 23. TransformationsThere are four basic transformations: translation, rotation, reflection and enlargement (or dilation asit 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 23
- 24. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 24
- 25. GraphsGrid FormsThere are three grid forms: square, rectangular and polar. To choose a grid, select the Graph menufollowed by Grid Form and then choose one of the 3 available grids.Adjusting the scale on the axesFor Square Grids, there is a red point on the x-axis. Select this point and drag it to the left or rightto 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 movablered point on each axis.Plotting a graphFrom the Graph menu choose a grid form and resize the axes as appropriate. Select the Graphmenu followed by Plot New Function, then using the keypad provided type in the relevant functionthen click OK. Your graph is plotted and the function displayed.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 25
- 26. Plotting DerivativesWhen 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 26
- 27. Trigonometric FunctionsTo plot a trigonometric function, follow the instructions above and when typing in the function, clickon the Functions button and select from the list. Remember to resize the axes appropriately beforeplotting the function.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 27
- 28. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 28
- 29. Action Buttons and TablesHide and Show ButtonsIn this example, a triangle has been drawn, all three angles have been measured, and the sum of theangles 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 29
- 30. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 30
- 31. Animation ButtonsIn this example, a circle, and then a diameter AC, have been drawn, and a point B has then beenplaced on the circumference. Point B has been joined to points A and C by line segments to formtriangle 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 clickOK.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 31
- 32. SlidersIt 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 mA 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 NewFunction from the Graph menu. When typing in the function, click on the label box ‘m’, then on thekey pad type * x + then click on the label box ‘c’ and finally click on OK. As points C and F aremoved, so the slope and intercept will change.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 32
- 33. Some Simple Ideas for use in LessonsTo draw a circle and measureits radius, circumference andarea, draw a circle using theCircle Tool from the left-handtoolbar.Deselect the circle (click theselect arrow and click anywhereoutside the circle) and then usethe Line Segment tool todraw in a radius.Select the Measure menu onthe top menu bar and chooseLength.The measurement appears inthe top left corner. You candrag this anywhere on thepage.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 33
- 34. To change the label, click thetext tool on the right-handmenu.Double click on the label.Type in your preferred labelingand 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. RM Math Framework Edition powered by Kaleidos: Training Guide — The Geometer’s Sketchpad® 34
- 35. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 35
- 36. Draw a point somewhere on theperpendicular line.Select that point, the pointwhere the radius meets thetangent, and the center of thecircle.Click the Measure option onthe menu bar and selectAngle.You can change the label box ifyou wish.You can now animate the tangent so that it moves around the circle. This will allow pupils to see thatthe 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 usingthe Motion Controller.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 36
- 37. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 37
- 38. Use the Line segment tool todraw two lines to form anangle.Highlight the three points of theangle, ensuring that the vertex ishighlighted second.From the Measure menuselect Angle.The angle measurement willthen appear on the screen.Change the label if you wantusing the text tool.Moving the lines or end pointsof the angle will change thedisplayed measurement.You could use this feature to measure angles in triangles in order to explore, for example, the sum ofthe 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 showthat the sum of the angles is still 180 degrees.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 38
- 39. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 39
- 40. To create a right triangle, draw To measure the sides, select a side anda line segment. from the Measure menu select Length. (You can change the label if you wish onceSelect one end point of the line and it’s displayed.)the line itself and from theConstruct menu choose Repeat this for all three sides.Perpendicular Line. Select the label displaying the measurementOn top of the perpendicular line, of the hypotenuse.draw a line segment from the endpoint of your initial line. From the Edit menu select ActionSelect the perpendicular line and Buttons, then select Hide/Show.from the Display menu selectHide Perpendicular Line. A Hide button is created. Clicking on this will hide your measurement. The buttonUse a line segment to create the then changes to Show Measurement.third side of your triangle. Clicking on it now will reveal theMeasure the right angle. measurement again.Trigonometry in non-right-anglesYou can select everything and copy and paste it, to save having to repeat theprocess for other triangles. You can then re-size the copied triangles to createadditional Pythagoras’ Theorem questions. You could do a similar thing usingangle measurements when studying right-angle trigonometry.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 40
- 41. 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.RM Math Framework Edition powered by Kaleidos:Training Guide — The Geometer’s Sketchpad® 41

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