A stretcher is a rubber band that has been cut and then tied, creating a knot in the middle. A stretcher also has 2 dots, one on each side of its knot and equidistant from the knot-in this case, the dots are 2 inches from the knot. (See Figure 1.) In the questions that follow, you will use a stretcher to create a new version of a triangle. The new version of the triangle is called a dilation of the original triangle. Stretchers are tools that can help you create dilations of any figure .
1. Follow the directions below to create a dilated version of ABC on Grid 1(page 5).
a. Hold your stretcher so one of the dots is on point P.
b. Pull the other end of the stretcher so the knot is on point A of the original triangle. (Check to make sure the dot is still directly on point P.) Now, make a mark on the grid where the second dot of the stretcher is located. Label this new point D. (See Figure 2.)
c. Repeat the process you just completed, but this time move the stretcher so the knot is on point B of the original triangle. Use the second dot of the stretcher to help you locate point E of the new triangle.
d. Repeat this process one more time, placing the knot on point C to locate the position of point F of the new triangle.
e. Use a straightedge to connect the 3 vertices (D, E, and F) of your new triangle.
2. Compare the original triangle (ABC) to the dilated version of the triangle (DEF).
a. How are DEF and ABC alike? (List 2 or 3 mathematical ways in which they are alike.)
b. How are DEF and ABC different? (List 2 or 3 mathematical differences.)
c. On Grid 1, draw line segments to represent the distances from the knot to each dot for the 3 positions of the stretcher. For example, draw a segment that connects points P and B, and then another that connects points B and E. Do this for all 3 vertices of the triangle.
d. Compare segment PA's length to that of segment AD. Do the same for segment PB and segment BE. Now do the same for segment PC and segment CF. What do you notice about the relationship between the lengths?
3. On Grid 1, draw a new point 4 grid spaces below the original point P and label it P2. Without using the stretcher, predict what a dilated version of ABC will
look like and where it will land on the grid when you use P2 instead of P.
a. Use a different-color pencil or marker and draw your predicted triangle on the grid (but don't use the stretcher!).
b. How did you decide where to put the new dilated triangle and what it should look like? Describe what you thought about as you made your prediction.
c. Check your prediction. Use P2 and your stretcher to create a new dilated triangle based on ABC. Use a different-color pencil or marker to draw this triangle. Was your prediction correct? Explain.
4. Take a look at the Geometer's Sketchpad applet "Dilations" available in your facilitator's Fostering Geometric Thinking Toolkit DVD.l In this file, DEF is the dilated version of ABC using point P as the center of dilation. Move the point P around on the grid and explore what happens to the dilated triangle DEF.
a. What do you notice?
b. What questions come to mind?
5. Return to Grid 1 and draw a third point P, 3 grid spaces to the right of P. Label this new point P3. Without using the applet or your stretcher, construct a new dilated triangle based on P3 and ABC. Use a different-color pencil or marker to draw this triangle. How did you decide where the new triangle should be on the grid?
6. Maria made the dilated triangle shown on Grid 2 (page 6) using a stretcher like yours. Someone erased her original triangle. Without using a stretcher, reconstruct her original triangle on Grid 2. Describe how you figured out what her original triangle would look like and where it would be located.
7. Here is a drawing of the kind of stretcher you've been using .
You probably figured out that this stretcher doubles the lengths of the sides of triangles like ABC. So, we could call it a 2-stretcher. Suppose you wanted to make a 3-stretcher-that is, a stretcher that triples the lengths of the sides of ABC. Where would you put the right dot for a 3-stretcher? Explain.
Where would you put the right dot for a 1/2-stretcher? Explain.