# Engaging students: Dilations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Andrew Sansom. His topic, from Geometry: identifying dilations.

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In recent years, Marvel Studios’ Cinematic Universe films have exposed society to dilation. One of the beloved Avengers is Ant-Man, who starred in two of his own eponymous films, as well as in Captain America: Civil War and Avengers: Endgame. Ant-man is the hero identity of one Scott Lang, an engineer trying to be a good father for his daughter. In the process, he ends up associating with Hank Pym, who had developed a technology that make it possible to shrink and enlarge objects and people. In the aforementioned films, he utilizes this ability to solve problems and combat villains.

Two particular instances where he used this ability to shrink and enlarge in meaningful ways occur in Avengers: Endgame. One such moment is when Ant-man shrinks to a smaller size than that of an insect, and crawls inside of Tony Stark’s arc-reactor. He pulls apart one wire, which causes a short, and provides a long enough distraction for his team’s escape. Later in the film, after he and a few other Avengers were buried under a collapsed building, he dilates to a gargantuan size to push aside the rubble and rescue them.

A2. How could you as a teacher create an activity or project that involves your topic?

Teachers could use this connection to Ant-man to their advantage by designing an activity where students must use geometric dilations to solve puzzles. Give the students several consecutive scenarios with diagrams and ask them to come up with a plan that Ant-man should follow to maneuver the course. In this plan, they must require at what locations Ant-man should dilate, and by what scale factor, then to where he should move to dilate again. To make this more puzzling, put another restriction on the course that it costs a certain amount of “Pym Particles” to run a distance while enlarged/shrunken or to do the shrinking in the first place. This encourages the students to minimize the dilations to reduce the cost.

Below is an (extremely rough draft) example level. Ant-man’s location is the square where his feet are. He must move right three squares. He must then dilate with a scale factor of 2, with his bottom right corner being the center of dilation. He then shrinks with a scale factor of ½ about his top left corner. He then moves right one square. He then shrinks with a scale factor of ½ about his top right corner. Then walk right 4 squares. He then expands with a scale factor of 2, shrinks with a scale factor of ½, walks right 2, expands, falls down one, then runs right.

This platformer puzzle could even be expanded into a video game of sorts maximum engagability.

B1. How can this topic be used in your students’ future courses in mathematics or science?

Dilation appears in many topics in math later than geometry. Dilation is one of the major transformations studied in Algebra 2. Studies of geometric dilation will prepare students for analyzing how scale factors will stretch or compress functions. Furthermore, comfort in geometric transformations will prepare them for advanced integration problems. If students can identify the geometry of integral, then performing transformations, including dilation, can make certain problems easier to solve. In even further math classes, including linear algebra, scaling becomes an important tool in manipulating vectors. Students should realize at that point, that dilation is a certain type of linear transformation on a set of vectors representing a shape. The concept is also critical to an intuitive understanding of what eigenvectors are.

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