Teacher Information: Transformations - Dilation

This transformation, Dilation, has a different character than the other four Transformation manipulatives. Translation, Reflection, and Rotation are all called "Isometries" because they keep the same ("iso-") distances and angles and shape. Each can be explored independently, and then the Composition manipulative allows two of the isometries to be performed sequentially. A dilation moves every point in the plane away from, or toward, a given Center point by some fixed scaling factor. Dilation retains shape (the images are similar to the original) but changes each length by the same scale factor, so that Dilation is not included as an isometry.

When the manipulative opens, there is a figure (constructed from Pattern Block pieces), a black dot showing the Center of Dilation, and a smaller copy of the full-size original figure, lying on the line from the Center through the middle of the original. Students should freely explore the objects on-screen to observe how changes affect relative sizes and placements.

Moving the Center of Dilation (black dot) doesn’t affect the original figure, but the image (the copy) remains on the line from the Center. A dilation moves every point in the plane toward (or away from) the Center. Intuitively, everything "shrinks" toward the Center or "blows up" away from the Center.

Moving the original figure, either dragging or rotating, leaves the Center unchanged, but the image copies every move of the original. Notice what happens to lines that are parallel. That is, find a segment in the original and a parallel segment in the copy. Rotate the original. Do the lines remain parallel after the rotation? What happens if the student tries to move the image figure?

Now change the scale factor with the slider at the bottom. When the scale factor is 0.50, the image should be exactly half as large as the original. What happens with a scale factor of 1? To compare sizes, add coordinate axes (box at lower left), and by moving the original we can see that the sides of each colored triangle are 1 unit. Move the scale slider to 2 and move things around to see how long the sides of the image triangles are. What happens if the original figure is dragged to the garbage can at the lower right? Should we be surprised?

More experienced students can be encouraged to do some computational work. The following activitiess are intended just to suggest avenues of exploration. Clear the workspace and move the Center of Dilation to the origin. Click one of the Pattern Block buttons to put a block in the workspace and drag it into place so that one corner has integer coordinates on the x-axis. Then set the scale factor slider to 0.5 and observe the coordinates of the corresponding corner of the image. What if the scale factor is 0.33? Can we place an object so that the line from the Center has slope 1? If one corner of an original is located at the point (3,3), what should the scale factor be to have the corresponding image corner at the point (2,2)? Make predictions and then check. Now, without changing any setting or object on-screen, if we were to add another object and put a corner at the point (-6,3), where should the corresponding corner be located? Check.