VIEWS: 1 PAGES: 11 POSTED ON: 11/6/2012 Public Domain
The Geometer’s Sketchpad DGS software provides a range of tools for you to construct geometric objects from a range of `primitive’ objects (points, segments, lines, circles etc.) using both `classical’ constructions (midpoint, perpendicular, parallel etc.) and transformations (reflect, rotate, translate etc.) Once drawn, measurements can be taken from objects (length, angle, area etc.). The `dynamic’ aspect comes from the ability to drag defining objects, such as points, round the screen to deform the resulting shape – and hence to be able to look for aspects that always stay the same (`invariants’). The dragging is usually done with some analogue control device such as a mouse, touch pad, tracker-ball or stylus. Dynamic geometry software is ideally suited for use with an interactive whiteboard where the dragging can be done directly on the board. The two most common DGS packages for schools are Cabri Geometry and the Geometer’s Sketchpad . MathsNet maintains a page detailing these packages at www.mathsnet.net/geometry/ and also has many examples of interactive geometry web-pages. In this tutorial you will learn some of the basics in using Sketchpad. Cabri and Sketchpad have much more in common than they have differences. The most significant difference is in the order in which you perform constructions. In French you would say “le train bleu” which translates in English to “the blue train”. In Cabri you specify first the construction to use, and then select the objects which define it. In Sketchpad you first define the objects to use and then select the construction to use. We will now start by drawing the perpendicular bisector of the segment between two points. Use the links on the left to learn more about the Geometer's Sketchpad. In Sketchpad the toolbar is on the left of the screen. We need to use the second icon, the Point Tool, in order to create points. 1 Click two different places on the screen to create the two points. The most recent one will be highlighted. Now use the first tool, the Selection Arrow, and click on the first point in order to highlight that as well. Now we have two points selected. Above the screen are the menus. Pull down the Construct Menu. You will see that most items are `greyed-out’, but that Segment is in black. Click this to create the segment. Since the segment is the most recent object, it is highlighted. So just pull down the Construct Menu, and this time you will find that Midpoint is in black. Click this to create the midpoint. In order to draw the perpendicular bisector we need to select both the midpoint, through which it passes, and the segment, to which it is perpendicular. Then open up the ‘Construct’ menu and click ‘Perpendicular Line’ to complete this task. 2 Now the figure has been created the perpendicular bisector is highlighted. Just click the mouse anywhere on a blank area of the screen to turn off all highlights. Then you can drag either of the original two points and check that your construction always holds good. Now we will do some prettying up. First to label the points use the 5th tool icon: the Text Tool. Move the cursor over any of the points and click to insert a label. Since the ends of the segment were the first two points to be created it is not surprising that Sketchpad allocates A and B as their labels. When you click the midpoint it produces C as the label. To change this, double-click the label. In the dialog box you can change the label C to M, say. If you click on the `Style’ button you can also change the font, colour, style and/or size of the label. 3 For example you can select Arial as the font, Italic as the style, 24 point as the size and magenta as the colour. When you have finished labelling, you can also drag the labels into more convenient places. In order to change the style of the segment, first select the Selection Arrow Tool, and then use if to highlight the segment. Pull down the ‘Display’ menu and select ‘Line Width’. This then offers three choices – select ‘Thick’. Similarly you can select the perpendicular bisector and change its style to thick. With the bisector still highlighted you can also use the ‘Edit’ menu to select ‘Colour’ and change it to red, say. Now we will look at some of the measurement tools. First use the ‘Point’ tool to construct a point on the bisector, and then use the ‘Text’ tool to label it C. Use the ‘Selection Arrow’ tool to check that C can only slide on the bisector. 4 With points A and C highlighted use the ‘Construct’ menu to create the segment AC, and repeat for BC. Adjust the style and colour of these new sides of the isosceles triangle ABC. Now highlight all three of the vertices A, B and C. Use the Construct Menu - the last option there is, to create a Triangle Interior. This is the way Sketchpad defines polygons. Since the interior is the last object to be created, is it highlighted. Now open up the ‘Measure’ menu and select Area. This will find the area of the triangle ABC. Check that if you drag C then the area changes. In order to measure one of the base angles, like ABC, first click any blank part of the screen to make sure nothing else is selected. Then click in turn on points A, B and C. Use the ‘Measure’ menu, and this time select ‘Angle’. 5 As a final exercise, construct the perpendicular bisector of BC. Click the spot where this cuts CM and label the new point O. This is the circumcentre of ABC. Highlight O and C and use the Construct menu to draw the circle by ‘Centre and Point’. Use the Measure menu to find its Circumference. Now we have a rather cluttered picture. Before we look at some transformations we will hide objects like the measurements, the circumcircle etc. Just click each of the objects you wish to hide and use the ‘Display’ menu to select ‘Hide Objects’. In order to reflect ABC in side AB we first have to define AB as the mirror line. Highlight the segment AB and use the ‘Transform’ menu to select ‘Mark Mirror’. 6 Now select the objects to reflect – these are the triangle interior, the segments AC, BC and the point B. Then use the ‘Transform’ menu to ‘Reflect’ the highlighted objects. Tidy up the picture using the ‘Text’ tool to create a new label, and the ‘Display’ Menu’s ‘Colour’ tool to shade the image triangle ABC’. Check what happens as you drag C on the bisector. In order to rotate an object we can first specify the angle and the centre. Click in turn on C’. A and C. Use the ‘Transform’ menu to select ‘Mark Angle’. Similarly click A and use the ‘Transform’ menu to select ‘Mark Centre’. In order to rotate the whole picture it is easiest to select all the objects by a process called `dragging a marquee’ around the rhombus. With everything selected you can then use the ‘Transform’ menu to rotate the object around the given centre through the given angle. The dialog box for Rotation also lets you type in a fixed angle for the rotation if required. In order to define a translation you need to define a vector by clicking in order on two points which define the translation vector, and then using the ‘Transform’ menu to ‘Define Vector’. 7 In order to start a new drawing, use the ‘File’ menu and select ‘New Sketch’. We will now explore some of the functions to do with co-ordinates. Use the ‘Graph’ menu and select ‘Show Grid’. This will choose an origin in the centre of the screen, axes parallel to the edges of the screen and a unit of 1 cm. Remember that 1 cm. only refers to a printed page. Viewing the display on a small monitor or on a large whiteboard will preserve ratios but not lengths. Use the ‘Point’ tool to put a point on the x-axis and another on the y-axis. Label the points as shown. With X and Y selected use the ‘Construct’ menu and select ‘Line’. Highlight X as we shall now measure its x-coordinate. 8 Use the ‘Measure’ menu and select ‘Abcissa’ to find the x- coordinate of X. Similarly find the y-coordinate of Y. Then highlight the line XY. Now you can use the ‘Measure’ menu to find the equation of the line XY. Slide X and Y about to see their effect on the equation. Is the constant in the equation always the same as the y-coordinate of Y? It should be. Now select the points Y and O in order and use the ‘Construct’ menu to create the Circle By Centre and Point. Change the style and colour of the circle. Highlight it and use the Measure menu to find its equation. How does this change as you slide Y? 9 Use the ‘File’ menu to save your work. Also to print it out. If you want to include a diagram you have constructed in Sketchpad in a Word document then just highlight the objects you want to export. Use the ‘Edit’ menu to select ‘Copy’. Go into your document and select ‘Paste’. Now use the ‘File’ menu for a new sketch, and the ‘Graph’ menu to ‘Show Grid’. Use the ‘Graph’ menu to ‘Plot Points’, and enter zero and two as the x- and y-coordinates of a new point. Plot it and close the dialog box. Label the origin O, the unit point U and the new point as A. We will now create a regular polygon. First we will enter the number of sides using the Graph Menu and selecting New Parameter. Name the parameter `n’ and give it a value of 5, with no units. 10 With the parameter highlighted use the ‘Measure’ menu and select ‘Calculate’. Use the on screen calculator to enter 360, then from the ‘Units’ button select `degrees’. Enter a divide sign, and then from the ‘Variables’ button select the parameter `n’. Finally use the ‘Calculate’ button to display the result. Now you can define this resulting angle as the angle of rotation, and the point O as the centre of rotation. Use the transformation menu’s rotate selection to rotate A around O through 72 degrees. Complete the pentagon and calculate its area. You can change the size of any text, such as measurement, by highlighting it and using the ‘Display’ menu to change the ‘Text’ style. This completes our short tour through some of the geometric features of Sketchpad. 11