im.g.5.3.2.Pyramid Mobile

Author:mark vasicek, Anthony OR 柯志明, Mathieu Blossier

Dilate rectangles

This activity is modified for online learning. If we were in the classroom, we would build models of pyramids in 3-D using the vertex as the point of dilation, and layers that were scaled sections (fractions) of the base. The teacher will give a demonstration of each of the applets before you are given time to explore them.

After watching the demonstration, try this: 1. Move point A to make AB=4. ABCD will be a 4 x 4 square. 2. Move point E to rest directly above point A. 3. Move the plane to change the size of the cross section, square GHIJ.

Question 1

Each layer of the pyramid is a dilation. The point of dilation is the vertex, point E. When the scale factor is 1, the dilation (cross section GHIJ, or polygon 2) is the same as the base (ABCD or polygon 1). With the base length AB = 4, the scale factor will be a ratio: k = HG/AB. Make a table with the following values: AB = 4 Area poly1 = 16 k HG Area poly2 .25 ____ _____ .5 ____ _____ .75 ____ _____ 1 ____ _____

Question 2

Repeat this using a different base length (such as AB = 6, or AB = 8 or AB = 10). Again, record AB = ___, Area poly1 = ___, and make the table.

Question 3

Staying in 3 dimensions, what would a dilation by the scale factor k = 0 look like? Where would it be located in the pyramid?

Question 4

How was the area affected with changes to the cross section? Specifically, was the area of the dilated rectangle also changed by a factor of k?

Question 5

Is dilating a square using a factor of 0.9, then dilating the image using a scale factor of 0.9 the same as dilating the original square using a factor of 0.8? Explain and show your reasoning.

This activity was inspired by illustrative mathematics, geometry, unit 5, lesson 3, https://im.kendallhunt.com/HS/teachers/2/5/3/preparation.html Licensed under the Creative Commons Attribution 4.0 license, https://creativecommons.org/licenses/by/4.0/ Modifications were made by Mark E. Vasicek to accommodate online instruction.