Radnor School District: Algebra Skills in GSP

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  Measuring the Coordinates of Points, "Snap Points"

Start with a blank sketch.

  1. Use the point tool to construct three points in your sketch. Using the selection arrow, select all three of them and measure their coordinates. Move them each around the sketch to see how the coordinates change.

  2. Select Graph -> Snap Points. Notice that nothing happens until you start to drag a point, at which time it jumps to a spot with integer coordinates.

  3. If you want a point at a specific spot and want it to stay there, use Graph -> Plot Points... This lets you put in the x- and y-coordinate for the point (or for multiple points). Notice that once you plot points in this way, you can't move them (try it!).

    Example: Susan the Ant runs 3 yards a minute. Tom the Ant runs 1 yard a minute. Susan gives him a 10 yard lead. After how many minutes will Susan catch Tom? How far will they have run?

    We know that when they start, Tom will be at 10 yards. If the x-axes is time in minutes and the y-axis is distance in yards, he starts at (0,10). Susan starts at (0,0). After 3 minutes, Tom will have run 3 more yards, so he's now at the point (3,13). Susan will have run 9 yards, so she's now at (3,9). Plot three of those points on your graph (the origin is already there, and you will probably need to move the origin to the lower lefthand corner and move the unit point closer to the origin so that you can see all of the points). Then construct the line that represents each ant, and construct their point of intersection. Then measure the coordinates of that point.

  4. Measurement Preferences: You can change the preferences so that the coordinates and such are shown in units, tenth, hundredths, etc. Choose Edit -> Preferences and make sure you're on the Units tab.

  Plotting Functions

non-dynamic :: editing functions :: parameters :: using sliders :: second-degree equations :: inverse functions :: rectangular grids :: polar grids :: as (x,y) relationships :: intersections of functions

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

  1. Non-Dynamic: select Graph -> Plot New Function. Type in 3x + 5 and hit Okay. You can enter the x directly from the keyboard or choose the x from the provided keypad. Notice that the function is written in the form f(x).

  2. Editing Functions: Use the Selection Arrow to double-click on the function. You can now change it in any way you like. For example, select the 5 and change it to a -1. (Notice that it gives you "+ -1", which you may or may not want.)

  3. Using Parameters: Instead of typing in the exact function you want, you can define it using parameters.
    1. Select Graph -> New Parameter... Name it m and give it the value 3 for now.
    2. Create a second parameter named b and give it the value -1.
    3. Edit your function by double-clicking on it.
    4. Select the 3 and click on the m measurement in your sketch. Now your function read f(x) = m x + -1. Replace the -1 with the b in the same way. Notice that your function hasn't changed because you're using the same numbers you used before.

    5. Double-click on the m parameter, change it to -2, and select Okay. Now your function has changed.

  4. Using Sliders: Change the coefficients for a function by dynamically dragging something instead of changing one parameter at a time can be done using "sliders". While you could make your own sliders, they also come in a tool with Sketchpad.
    1. Open the sketch Sketchpad::Samples::Custom Tools::Sliders.gsp.
    2. Go back to the sketch you've been working on. The tools in the Sliders.gsp document will be available to you since that document is open.
    3. Under the Custom Tool tool, choose Sliders -> Basic Horizontal.
    4. Click in two places in your sketch - this will make two sliders. (Each slider requires you to define a single point as the "givens".)
    5. Using your Selection Arrow, move the sliders and measurements so that they are easy to keep track of. Use the Text Tool to relabel the measurements m and b and also relabel the adjustable points m and b. (I often hide the "origin" point of the slider so that students don't try to grab it - these are the red points in the picture below.

    6. Edit your function again, this time changing m and b to the measurements of the sliders (instead of the parameters).

    You can use as many sliders as you need - keep the Sliders sketch open in order to have access to them.

  5. Second-Degree Equations: You can edit your function to include just about anything you want. To include exponents, use the ^ key in the calculator or type it from your keyboard.
    1. Edit your function so that it's f(x) = ax2 + bx.

  6. Swapping x and y: From the Function editor you can choose functions of the form y = f(x) or x = f(y).
    1. Choose Graph -> Plot New Function and select "x = f(y)" from the Equation menu at the bottom of the window.
    2. Enter ay2 + by and hit okay.
    3. This gives you the inverse of the function that you graphed in the previous step, and both functions change as you move a and b.

      Here's a way to show that the graph of an inverse function is really just a reflection of the function across the line y = x:

      1. Use the point tool to construct a point on the function plot.
      2. Plot some point on the line y = x, like (2, 2); highlight that point and the origin, and use Construct -> Line to construct the line y = x.
      3. Mark the line y = x as a mirror; reflect the point on the function plot.
      4. Select the function plot, the point on the plot and the reflection, and Construct -> Locus. Use Display -> Line Width to make this thin or dashed, and Display -> Color to color it red.
      5. If you make the plot of the function from step 6.b display as thick and yellow, it will be obvious that these are equivalent ways to generate the inverse of a function.

  7. Square and Rectangular Grid Forms: Choose Graph -> Plot New Function. Enter the function f(x) = -1.5x^4 + 5x^2 + 2x. If you really want to "zoom in" on the origin without losing the "top" of the graph, you're stuck if you're using a "square" grid. But changing to a rectangular grid allows you to inspect more of the function around the origin without losing anything. Choose Graph -> Grid Form -> Rectangular Grid and notice that you now have a draggable point at (0,1), in addition to the one you had before at (1,0). These two points are independent, so you can zoom in vertically and out horizontally until you've got the view of the function that you want.

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

  1. Polar Graphs: In addition to square and rectangular grids, you can work in a polar grid.
    1. Choose Graph -> Grid Form -> Polar Grid.
    2. Use the horizontal slider tool to make two sliders labeled a and b.
    3. Choose Graph -> Plot New Function. Notice that the function now reads f(θ) instead of f(x).
    4. Plot the function f(θ) = a * sin(b * θ). If you aren't already using radians to measure angles, Sketchpad asks if you want to switch to radians (slick!).
    5. Drag the sliders for a and b to see how they affect the function.
    6. Edit your function to be anything you like.

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

  1. "Functions" as (x,y) Relationships: These aren't really functions in the formal sense of what you've been doing, but you can construct loci of (x,y) relationships. We'll look at the relationships between the radius of a circle and the circle's area and circumference.
    1. Construct a ray, and construct a circle centered at the endpoint of the ray and going through a point on the ray that is NOT the control point of the ray.
    2. Select the circle and measure its radius, circumference, and area.
    3. Select the measurements for the radius and the circumference in that order and choose Graph -> Plot as (x,y). (If you can't see the new point that was constructed in the first quadrant, move the origin down to the lower lefthand corner of the page. If you still can't see it, move the unit point at (1,0) closer to the origin.
    4. Drag the control point of the circle and see how this new point moves.
    5. We'd like to see all of the possible locations for this point, so let's construct a locus. Choose the point you graphed and the control point of the circle and choose Construct -> Locus.
    6. Repeat these steps for the radius and the area - select the measurements for radius and area, choose Graph -> Plot as (x,y), select the new point and the control point of the circle, and choose Construct -> Locus.
    7. You might want to change the grid to rectangular so that you can focus on the graph more clearer. Then again, you might not if you want to "see" the slope of the line (which is "meaningful") and the shape of the half-parabola.

  2. Intersections of Functions: You can't actually plot a point at the intersection of two functions. You can, however, plot a point at the intersection of two lines which may be "functions". See the example above in Measuring the Coordinates of Point.

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

  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: As you saw earlier in using the slider tool, it's handy to be able to store a set of steps for making an object. Let's say, for example, that you want a tool that will give you the equation and slope of any line that you click on.
    1. Use the Line Tool to construct a line. Measure its equation.
    2. Use the Selection Arrow to select the line again and measure its slope.
    3. Select the line, its equation, and its slope and from the Custom Tool Icon, choose Create New Tool....
    4. Name the tool "measure slope and equation" and click Okay.
    5. Use the Line Tool to construct a new line on the graph.
    6. From the Custom Tool Icon, select "measure slope and equation" and click once on your new line. Voila!

  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 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!

Shelly Berman, Annie Fetter & Suzanne Alejandre
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