Where's The Math? Analyzing our Kaleidoscope Images
Lesson 16 of 16
Objective: SWBAT understand the relationship between mirror angles and images through observation and graphing.
Yesterday's Kaleidoscope lesson was a fun, hands-on activity. In today's lesson we will study some of the mathematics behind reflections. I plan to keep the same pairs of students that worked on the kaleidoscopes together yesterday. I encourage them to maintain the same collaborative spirit, since we are still working on the same math problems. I'll say something like, "Today we'll complete our Kaleidoscopes by measuring angles, making tables of data, graphing the data and discussing the patterns that we observe." Then, I will use the ten slides in Transformations APK to re-visit some of what we have learned about transformations and symmetry in this unit. As I lead the presentation, I will call on students at random to read and answer the questions on each slide. Along the way we will discuss all of the ideas and concepts that students raise or ask about (see my Revisiting Past Concepts reflection).
- A set of hinged mirrors: (see image hinge) larger size mirrors may be better for observing multiple images.
- A protractor
- a non-symmetric image (you can cut out the image in the worksheet)
- a small object (I often use a key)
- Graph paper
- One copy of the Lesson Data Sheet per group
- One copy of activity instructions per group
I begin today's activity by handing out a set of activity instructions and a Lesson Data Sheet to each student. I let the students know that they can begin working immediately. I encourage them to take their time, help each other out, and discuss what they see using their knowledge of mathematics.
I know that multiple reflections can be difficult to visualize so I plan on walking through groups as they work and watch patiently as they fiddle with their mirrors, measure angles, and draw their images. I also watch for those students who have forgotten how to use their protractors to make sure they complete their data table correctly. I added a coordinate plane with a scales for angle degrees and number of reflections to save time.
I expect that when they get to Question 11, many students may struggle to determine a formula relating the angle formed by the mirrors and the number of images. If a group struggles too long, I may help out and wisely lead them to the formula:
#images = (360/mirrors angle) - 1
Once students are done and before proceeding to the lesson closure, I like to "milk" the content a bit by asking questions like the following:
- Explain why you see one image when standing in front of a mirror using the formula you derived above. (360/180 -1 = 1)
- What kind of graph did you expect to see? What type of graph resulted? (many students expect to see a line graph)
- Does the graph ever intersect the x axis? y axis? Explain your reasoning.
- Which images are rotations of the pre-image? Indicate the degree of the rotation and the direction (clockwise or counter)
- Can we connect the points on the graph? Explain answer.
See samples of student work below, including pictures of reflections taken as students completed their data sheets:
Once students have finished their Data Sheets, I randomly call on volunteers to share their answers.
As the images of our 3 mirror case is projected on the board, we discuss what we see. I ask a student to describe the images. (Students should indicate that there are three 60 degree angles and therefore 6 images, including the pre-image are seen about each vertex. Students should also indicate which reflections are rotations of the pre image)
A student asked to place color paper clips inside our 3 mirror setup, and we took a picture. Here it is:
This array of images is what we see in our kaleidscopes. Here is a video made looking into the 3 mirror setup: 3 mirror vid
- If mirror strips are available, I ask students to describe the images when an object is placed in between parallel mirrors. Students should see an infinite number of images: parallel mirrors
- I then ask that one of the mirrors be slightly turned at an angle. Students should see that the line of images curves parallel mirrors curved
For extra credit, I sometimes ask my students to find how concave and convex mirrors would reflect objects placed in front of them. Ask that they explain the images using terms like "angle of incidence" and "angle of reflection".