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.

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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.

 

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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.

 

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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.

 

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 Marissa Arevalo. Her topic, from Geometry: identifying dilations.

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How does this topic extend what your students should have learned in previous courses?

As teachers, we want to create connections from prior knowledge to help and assist them create a sort of base or foundation for future courses. Dilations refer to the scaling of shapes that can create similar and/or congruent shapes.

Students may not correlate this conceptual idea to scale factors of function transformations. This skill set is from Algebra I and then extended in Algebra II TEKS, which is taught after Geometry. In Algebra I, students are expected to be able to identify what occurs in a function, (i.e. a quadratic function and such). When given the parent function y=x2, if you were to change the size or steepness of the parabola you would  either multiply the function by a: y=ax2 to create a vertical stretch/compression of the function or multiply by b: y=(bx)2 to create a horizontal stretch/compression, which make a and b scale factors. By applying this knowledge, students can hopefully work to identify similar figures and proportions of shapes in relation to their sides/angles.

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In photography, prior to digital photography, we had to have photos developed in a dark room, where the only light source is in a corner of the room given by a light bulb. The darkness allows the processing of light sensitive photo material.

Equipment needed for developing photos:

Enlarger

Chemical bath

Running Water

The photo negatives are taken and enlarged through light onto a print by a specific type of transparent projector as the negatives are see-through, light projects onto the negatives which goes through the negative onto the paper on the base. The paper must eventually be developed in a chemical bath to set and hang to dry. The photos must be enlarged, which is a form of dilation by enlarging the size of the photo onto a new surface with the help of scale factors set by the type of enlarger lens on the enlarger (shown on the right). A similar concept is applied with cinema with the projection of a small film strip through a lens with a light onto a large white screen.

 

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

A really cool project that I found for a class project is called “Scale Up”. It is meant for the entire class to partake in where the teacher is to pick some  picture for the class to scale up in size in pieces. The teacher on the website chose the American Gothic picture and copied it onto an 8.5×11 in. copy paper. She then gave coordinates to each square, so as to easily give each student their own square to make in the picture. Every student was given one or two squares and together they each contributed to the bigger picture and eventually created the entire portrait out of sticky notes by either eyeballing  the approximate size of the shapes in their square or by actually scaling the actual size the lines had to be inside of the square they were assigned. This project seems like it would be fun and entertaining for the kids to do together, where they have to in the end talk with one another and discuss what it would take to dilate the photo that there were trying to make.

 

References: http://fasttimesofamiddleschoolmathteacher.blogspot.com/2014/02/scale-up-picture-class-project.html

https://en.wikipedia.org/wiki/Darkroom

http://ritter.tea.state.tx.us/rules/tac/chapter111/index.html

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 Candace Clary. Her topic, from Geometry: identifying dilations.

green lineHow could you as a teacher create an activity or project that involves your topic?

Dilations are types of transformation. One activity that I could create for my students is a matching game. I can create cards with index cards, or sheets of paper that have been cut up, that have pictures on them. Each one will be labeled and the students must classify them as dilations, why they are considered dilations, and how they were dilated. As a follow up to this activity, I could assign a topic to create their own city, or small town. They would be required to draw out their town, as well as model it using common crafts. After they do this, they will need to be able to dilate the buildings, and other such things, to make a life size city. They will not have to make the city with a model, but instead, they will need to make a blue print using their model in mind. On this blue print, they will need to inform me of the size of the dilations.

green lineHow can this topic be used in your students’ future courses in mathematics or science?

Dilations can be used in many different subjects. Dilations can be used to find sides of a triangle when learning about the triangle congruence theorems. These can be useful in algebra when finding side lengths of figures. This may not happen very often, but it is displayed in algebra. Another place that it will help, although it may not be math, it will help in math classes for architectural students, as well as help people in construction. Many science classes require science projects that work and simulate something real. Dilations can be used when making these projects because you can’t make a real river, but you can structure something that is a smaller figure to the real thing, same thing as a volcano. With architecture, dilations can help with making blue prints and can help in building these blue prints with dilations in mind. With construction, those are blue prints too. I’m not saying in order to build something you must know how to dilate something, but it will help tremendously.

 

green lineHow has this topic appeared in pop culture?

To get the students engaged in the topic, I could bring up the Disney channel movie ‘Honey I Shrunk The Kids.’ This will bring up a discussion with the kids when I ask them what the dad did with his shrink ray. Some ideas that may come up will be that he made them smaller, and then at the end of the movie he made then bigger, back to normal. But in the people were still the same people, they didn’t change, only the size did. At least I hope that is what happens in the discussion. I could then instruct the students into pretending that they had a shrink ray and ‘shrink’ some shapes, as well as other students. This activity, and their answers will be recorded on a chart that they will turn in at the end of class. They, themselves, can decide what size they want to shrink to, but they have to remember to bring the student back to normal at the end of class. I think this activity will be fun for the kids because they will never forget what a dilation is, since they have been ‘dilated’.