Radnor School District: Geometry Skills in GSP

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  Basic Constructions

Start with a blank sketch.

  1. Warm up by constructing the following figures using only the straight object tool and the "compass" tool (the circle). The constructions may not hold together all the time - methods that you use with a regular compass and straightedge don't always hold up in a "dynamic" environment. If one of your constructions fails when you move the "given" object, see if you can figure out why or when it's failing.
    1. the perpendicular bisector of a segment
    2. an equilateral triangle
    3. an angle bisector
    4. Using "construct circle by center and radius", try constructing a triangle and then duplicating it.

  2. Under File -> Document Options, choose Add Page -> Blank Page.

    Now try constructing the following figures. You may want to refer to the "Constructing Quadrilaterals" handout for a refresher.

    1. isosceles triangle
    2. equilateral triangle
    3. parallelogram
    4. rectangle
    5. rhombus
    6. square
    7. trapezoid
    When you're done constructing each figure, hide all of the unnecessary objects so that all you've got left is the vertices and edges.

  3. Under File -> Document Options, choose Add Page -> Blank Page.

    You can merge a point to another point or to a straight object as an afterthought.

    1. Construct a triangle.
    2. Rats! We wanted it to be isosceles. No problem!
    3. Construct a circle centered at one vertex and going through another vertex.
    4. Select the third vertex and the circle and choose Edit -> Merge Point to Circle.
    5. Drag that vertex and notice that it now sticks to the circle.
    6. You can "unmerge" that point later by selecting the point and choosing Edit -> Split Point from Circle.

  Labeling Objects

You can use the Text Tool to turn labels of points off and on. Once a label is on, you can use the Text Tool to change it - just double-click on the label itself (not on the object). You can also select an object and choose Edit -> Properties... From there you can click on the Label tab to change an object's label and to change whether or not the label is showing.

  Measurements and Calculations

  1. You can measure many things in a sketch, including the lengths of segments, the distances between two points points or a point and a straight object, angles, coordinates of point, and areas and perimeters of polygonal regions.

    Choose Edit -> Preferences.... Under the Units tab, note that you can change the distance and angle units, as well as the level of precision.

    1. Go back to the second page of your sketch (the page with the triangles and quadrilaterals).
    2. Select the four edges of your rectangle and choose Measure -> Length.
    3. Select three vertices of your rectangle and choose Measure -> Angle. Note that, just as in labeling an angle, the angle measured is the one at the middle vertex.
    4. Select two opposite sides of your square and choose Measure -> Slope. You might want to hide the grid (Graph -> Hide Grid).

  2. You can also calculate many different things.
    1. Choose Measure -> Calculate.
    2. Click on one of the slope measurements - it appears in the calculator. Then type / (divided by) and click on the second slope measurement. Click Okay. Drag your square around and see if the ratio changes (if it does, you haven't constructed a square!).
    3. Select another side of the square and measure its slope.
    4. Calculate the product of this slope and one of the other two. Did you get -1?

    5. Select the four vertices of the square in order (clockwise or counterclockwise) and choose Construct -> Quadrilateral Interior.
    6. With the polygon interior still selected, choose Measure -> Area.
    7. Measure the length of one side of the square and then find the area by using the Calculator to square that length. Calculate the ratio of that product to the area measurement.

  3. You can also make a table from a group of measurements.
    1. Select the measurement of the edgelength of the square and the measurement of the area of the polygon interior.
    2. Choose Graph -> Tabulate.
    3. Change the size of the square and double-click on the table. As you adjust the square now, you'll see the bottom row of data change.
    4. Double-click on the table again to make the last row "permanent" and to add another "dynamic" row.
    5. Select the table and choose Graph -> Add Table Data to see what some of the other options are.

  Making and Using Custom Tools

Custom tools are stored within sketches, and you have access to any custom tool that's in any open sketch. Exception: You always have access to any tools in any sketch stored in the Tools folder that's in the same folder as the Sketchpad application. Sketchpad checks in that folder whenever it starts up.

  1. Making Custom Tools: It's handy to be able to store a set of steps for making an object. Let's make some of the quadrilaterals into custom tools so that we can construct them easily in the future.
    1. Go back to your page of quadrilaterals.
    2. Select your rectangle (all four vertices and edges) and under the Custom Tool tool, select Create New Tool....
    3. Name your tool "rectangle" and click Okay.
    4. From the Custom Tool, select rectangle and click in two places - the "height" of your rectangle is determined randomly (you can change it).
    5. Make a custom tool with your parallelogram. Notice that when you use this, you need to click in three points (since there are three "given" points).
    6. You can view the script of any tool by selecting that tool and then selecting Show Script View from the Custom Tool icon.

  2. Appropriating Custom Tools from Other Sketches: Let's say you see something really slick that you like in another sketch, but you don't want to have to figure out how they did it. You can select the involved objects and make a Custom Tool to use yourself.
    1. Open the sketch Sketchpad :: Exploring Algebra 4 :: 2_Lines :: Origami.gsp. Notice that when you drag the green point (which starts at (6,8), its label is really its coordinates! The coordinates change as the point moves. This is cool, and we want to be able to do this ourselves without "knowing how".
    2. Select the point and its "moving" coordinates.
    3. Create a new tool called "Label point with coordinates".
    4. Use the point tool to add a few more points to your sketch.
    5. Use your new tool to "label" those points with their coordinates, then use the Selection Arrow to drag those points around.

  3. Managing Custom Tools: So now you've got these two tools and you want them both in the same sketch. You can easily copy tools from one sketch to another.
    1. Go back to your working sketch (the one you've made so many pages in so far).
    2. Select File -> Document Options.
    3. Next to View click Tools.
    4. Under Copy Tools choose Origami -> Label point with coordinates. The tool will appear in the Tool List at the left.
    5. (Note that you can also rename and remove tools from this page.)
    6. Click Okay, then use the Selection Arrow to construct a few points in your sketch.
    7. Select your new custom tool and label 'em!

  Managing Documents

As you've seen so far, you can create a document with multiple pages. New pages can be either blank pages or duplicates of existing pages, either within the sketch itself or within any other sketch that is currently open.

When previously you might have a folder on your computer called "Polygons", you can now just have a single Sketchpad file named "Polygons" and have separate pages within that. This also allows you to store all of your custom tools for all the regular polygons and perhaps some other quadrilaterals within that document.

In my Tools folder, I currently have three sketches. One is called Polygons, one is Centers of Triangles, and one is Utility Sketches.

  1. In my Polygons sketch, I have a square, equilateral triangle, square by center/vertex, and rhombus. I will probably add other regular polygons up to a decagon, and maybe a dodecagon, constructed from both an edge and from a center and vertex (great for inscribing polygons in circles).

  2. In my Centers of Triangle sketch, I have tools to create a centroid, circumcenter, incenter, and orthocenter. I might add a tool to construct the Fermat point of a triangle. While there are lots of other triangle centers, I don't find that I use many of them.

  3. In my Utility sketch, I have markers for a right angle, first congruent angle, second congruent angle, third congruent angle, first congruent segment, second congruent segment, third congruent segment, parallel lines, an open arrow, a closed arrow, and a fractional slope calculator (takes any slope and calcuates it in rational form - is only accurate if the two points that determine the line have integer coordinates). You can download my Utilities sketch if you'd like to use them or figure out how I made them.

  Transformations

Sketchpad is really good at transformations - much better than we could ever be on the board! Here are a few things to practice.

  1. Under File -> Document Options, choose Add Page -> Blank Page.

  2. Try all of the rigid transformations included on the handout "Rigid Transformations".

  3. To translate by a vector such as <-3,4>, you could try this:
    1. Choose Graph -> Define Coordinate System.
    2. Use your segment tool to draw a segment from the origin to anywhere (be sure the point isn't on either axis).
    3. Choose Graph -> Snap Point.
    4. Drag the free endpoint of the segment to the point (-3,4).
    5. Select the origin and then the endpoint, in that order, and choose Transform -> Mark Vector. Now anything you choose to Translate will be translated by that vector until you pick another vector. And this isn't only the vector <-3,4> - it just looks like that right now. You can change it to whatever vector you want.

There's a lot more you can do in Sketchpad, but these basics should get you going!


Shelly Berman, Annie Fetter & Suzanne Alejandre
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