# 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.

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.

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.

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.

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

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

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