The first is from a preservice secondary teacher named Jill Beauchamp. She is active in coaching cheer, and in our local Dutch culture. (And it's almost tulip time.) I'm pretty sure she's a licensed wooden shoe dancer.

On an assignment that gave a choice of follow up options after playing with Pierre Van Hiele's mosaic puzzle in class (from “Begin with Play,” by Pierre van Hiele, Teaching Children Mathematics, Feb 1999), Jill chose to make a activity based on my triangle puzzle. And she was willing to share it! I like how she really captured Van Hiele's idea of beginning with play, and uses the puzzles to get at the triangle properties. She makes the most of what I was designing the puzzle to do, have one triangle of each type.

The assignment:

Teaching Math –Mosaic Making

Choose one or more of the following to do for this:

Document your work, and be sure to include a reflection.

- Analyze Van Hiele’s mosaic. What geometric properties of the pieces permit all the combinations we saw in class?
- Create your own mosaic puzzle and document your design process.
- Create a new lesson using PvH’s mosaic or my 7 triangle mosaic at http://mathhombre.blogspot.com/2010/12/triangle-puzzle.html

*Schema*: I decided to take a look at your 7 triangle mosaic – nice work! This would be difficult for me to create on the computer, so I am very impressed. When first thinking about a lesson in regards to the mosaic, I could only consider it being a fun puzzle. With our exposure in class to different workshops regarding the original mosaic, I began to think about the properties each triangle in your mosaic possessed. You have:

- -Two right triangles
- -One isosceles triangles
- -One right isosceles triangle
- -One equilateral triangle
- -Two scalene triangles (One acute and one obtuse)

*Focus*: With my class, I would want to explore why these triangles fit together the way they do. Assuming the students have not yet learned about triangles, this could be used as an introduction. Let’s say I have this class for 60 min. Here’s how my day would go.

**Lesson: Properties of Triangles**

*I*

*ntroduction*: (5 min) Talk about puzzles!

- What kinds of puzzles do students like to do?
- What makes a puzzle puzzling?
- What are some mathematical properties of puzzles?

*Introduce Mosaic*

*Mosaic Play and Record*: (15 min) Allow students to play with the pieces and try to create the mosaic. As they do this, I would like them to document their actions:

- What they tried
- What pieces worked together?
- What didn’t work together?
- Qualities they notice about the triangles

- Is there another way to solve it?
- Why do some types of triangles fit together and others don’t?

*Discussion*: (20 min) I would ask all students to pull apart their mosaics and separate the individual triangles. Then I would ask them if they saw any similarities between any of the triangles?

*As this is happening I will write up student ideas on the board. If need be, they may come up to the board and illustrate their thinking.

Assuming they already know terminology for a line, angle, point etc. I will have students pull out the rulers and protractors to assist them in drawing more comparisons. Once we have a pool of properties, we can begin to group the triangles accordingly. Once we are able to do this accordingly by the deduced properties, I will write the names of the triangles on the board (but not yet with their corresponding group). Instead I will ask students what they think goes with each.

*Properties*: (With any luck, we get some or all of the following, although I’m sure I’ll get some other interesting thoughts!)

- 3 equal sides
- 2 equal sides
- No equal sides
- 3 equal angles
- 2 equal angles
- No equal angles
- Right angle
- Obtuse angles
- Acute angles

I will want to pay special attention to that sneaky little purple “Right Isosceles Triangle.” This guy is important because he shows that two properties can hold for one triangle. Maybe we could explore which properties can hold together and which ones don’t (As I’m writing this these ideas are just kind of coming…)

- A scalene can be a right triangle. Why? Because one angle may be 90 degrees, the other two differing, and all sides of different lengths. A scalene cannot be isosceles or equilateral because it goes against the definition of scalene.
- An isosceles triangle can also be right, but can an equilateral triangle also be isosceles? No, the definition of isosceles is EXACTLY two sides of equal length. Although it can be either acute or obtuse depending on the size of the angles
- A right triangle can then be isosceles or scalene. It cannot be equilateral because one angle must be 90 degrees, thus going against the fact that all angles in an equilateral triangle must be 60 degrees.

*Teacher Question*: So then, are triangles actually right triangles? Or does the word “right” just classify a specific type of isosceles or scalene triangle? A right triangle cannot exist outside of one of the two classifications.

Sorry for my tangent. The above discussion over the “right isosceles triangle” may be something for another day! My hope would be to get to the last part of my lesson…

*Discovery*: (15 min) The students would then need to reassemble the mosaic (I will show them the put together puzzle if they need it). With their protractors and rulers I would like them to work on:

Measuring the divided angles in the corners of the square. What is the sum of these angles? What type of triangles have an angle like this?

Measuring the divided angles along a straight line within a puzzle. What is the sum of these angles? What do they notice about all of these sums along a straight line? How does this compare to the sum of the angles within a triangle?

Lastly, I would like them to paste their mosaic together on a piece of paper and write out the angle measures, side lengths, and classification for each triangle. Students should make a note of anything else they notice.

*Reflection*: (last 5 min of class) What is one realization that surprised them today? Can they put anything they’ve seen into another context? How might it relate to something else?

**MY Reflection**: Wow, This was wonderful. I had the initial idea for the lesson because I thought it was so cool how the angle measures across a straight line will add up to 180 degrees. A simple concept, but it helped a lot of things make more sense when I recognized it. I think a lot of times we have this subconscious knowledge that we utilize everyday but don’t fully recognize. Once I started planning out how I would eventually get to a measuring activity, ideas just lead into one another, making this a lot longer lesson that I intended. There is no way I would get through the discovery part in 15 min! For me, this order of events seemed to make the concept clear. Perhaps it should be a day and a half sort of lesson?

Do you have any feedback for Jill or I about the lesson? What would you try?

*Photo credits*: Jill Beauchamp, quinn.anya and bjornmeansbear @ Flickr

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