Two Final Problems

Trig Problem 2

For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.



The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.




Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.

Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

Triangle Mosaic

Holy cow, have I been busy.  Sorry for the lack of new posts.  What makes it worse is that I have had several guest posts to get up that students were kind enough to send me weeks ago.  In addition to scads of things that I want to write up for myself!


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:

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

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

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)

Fabulous! You have one example of everything.

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

Introduction: (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:

1. What they tried
2. What pieces worked together?
3. What didn’t work together?
4. Qualities they notice about the triangles

**If students solve the mosaic, they should focus on:

1. Is there another way to solve it?
2. 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

Hopefully, they will see comparisons between the word “Equilateral” and the same angle and side measures, “Right” and the triangles with 90 degree, or right angles, “Scalene” and the triangles that depend on their individual scale/measure, although “Isosceles” doesn’t work too well, but it can be the odd guy out.

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

Solid Unit

As our semester in geometry drew to a close, we investigated solids.  Rather than dispense formulas (Go, Kate!), we tried to follow the Van Hiele levels.  Play and touch to get some understanding and visual recognition, sorting to start thinking about characteristics, and summarizing findings in definition-like descriptions.  Then we started thinking about measures.  Surface area is so natural, especially combined with the idea of a net.  But what about volume.  It's very interesting to have students sort the power solids by volume.  The sphere and hemisphere are very subtle.  I find it's also very common for students (college math majors) to be unable to remember formulae.  "Isn't there one with a 4/3?"

We filled them with water, and with no instructions from me, they immediately set out to try and verify their conjectured order.  (It's not as messy as you'd think.  The top of the solids container makes a pretty good tray.)  Two methods come up:  adopting a unit, and measuring each of the shapes in terms of the smallest, and filling one and trying to pour it into the next.  The different methods lead to noticing different things.  It seems like the groups that adopt a unit notice more numerical relationships from the data, and the groups that directly compare notice more of the geometric properties of the solids themselves.  ("Where does the water go?")

Usually from this data, you can suggest the idea of comparing solids with similar relationships.  Cone, Sphere, Hemisphere, Cylinder; Triangular Prism-Pyramid, Cube-Square Pyramid, Cylinder-Cone; Square Prism-Rectangular Prism-Cube, Small Triangular Prism-Large Triangular Prism or Hexagonal Prism.  Brilliantly designed little set.  (Although it does get into the experimental error that Dan Meyer cautioned about (was excited by?) in his TEDx talk.)  We compare the exterior of the solids and the water compares the interior.  Significantly different for the smallest objects.  Students have suggested measuring then by immersion, but we have yet to try it.

So this brings us to the boundary of informal and formal argument/reasoning.  How can we relate the volume of the prism and pyramid.  I do like models that fit together, but then that's just one example.  Of course, then, I tried to model it in geogebra.

Webpage or geogebra file.




It didn't help most of the students.





So I tried again:

Webpage or geogebra file.





This was helpful.  Or far more helpful, anyway.





Both sketches make use of Cavalieri's Principle to show equivalent volume.  We got at that in class by doing some block building, where each student had the same number of blocks per level.  This we extended into understanding the volume of a generalized cylinder.

Resources:  One of my favorite resources for this kind of classical problem is David Joyce's Java implementation of Euclid's Elements.  Book XII is the one you need for these problems, especially Proposition 7 and 10.  Our department's java wiz David Austin is the one who connected us to those.  David A's visualization work is literally inspiring, and worth checking out.

We finished this all by building with polydron tiles a plethora of polyhedra.  I love how, left to their own devices, students invent regular polyhedra, antiprisms, various truncations, and completely original solids.  (Unfortunately these are pretty expensive, but they are durable and usable by kids as young as 2nd grade.  Cf.  ETA Cuisinaire.) I set them the challenge of building a polyhedron with volume between 1 and 2 liters as a fancy new container for a boutique.  The need for actual measurement and estimation as well as decomposition and formula use makes this quite a challenging problem.

Extension:  as I was thinking about this and looking for resources, I came across Archimedes' proof that a sphere is 2/3 of the circumscribed cylinder.  Famously, this is the relationship that Archimedes wanted put on his tomb.  I took the translation from the Archimedes' Palimpsest that was posted at Cut the Knot (an invaluable geometry site), added some clarifying comments and made it into a handout with an accompanying geogebra sketch.  The sketch isn't really for visualization, but allows the reader to experimentally test some of Archimedes' unjustified claims.  (All correct, though.  Were it today, the justification of the steps would be left as an exercise for the reader.  Pretty good exercise.)

Riddles and Reasoning and Math Teachers at Play 14

When is a carnival full of problems? Besides like every circus movie ever?

The new carnival is up, hosted this week by Susan Van Hattum at Math Mama Writes.

My favorites include the Math Recreation post on origami and a clever lesson using statistics to catch cheaters which also uses bad jokes.

The bad jokes thing reminded me of a lesson I use with riddles about reasoning.

Reasoning and Riddles
The framework David Coffey and I use for reasoning, based on the NCTM process standards of course, is:
Mathematicians are engaged in reasoning when they:
-Make sense of something (sorting, understanding a problem, interpreting a representation)
-Make a conjecture about something (initial answer, plan of attack, possible relationship)
-Make an argument for something (justification, verification, proof)

I then give the students a list of riddles and ask them to figure out the answers. As we look at their answers, and more importantly, how they got their answers, they generate lots of examples of making sense, making conjectures, and arguing for why their answer fits.
(General riddles and Halloween riddles are posted at my faculty page. Click the links for the pdfs.)

We then explore a more math-centric riddle (it's usually a geometry class):

Four Sided Riddle

1) Taking the clues for a mystery shape in order, put a checkmark next to the last clue you need to know exactly the type of shape that the mystery shape is. Then explain your answer.

1. It is a closed figure with four straight sides.
2. It has two long sides and two short sides.
3. The two long sides are the same length.
4. The two short sides are the same length.
5. One of the angles is larger than one of the other angles.
6. Two of the angles are the same size.
7. The other two angles are the same size.
8. The two long sides are parallel.
9. The two short sides are parallel.

2) Using one less clue than your answer to number (1), draw a shape that satisfies all those clues BUT is different than the mystery shape, or explain why this cannot be done.

There is also a nice Van Hiele connection here as students at different levels approach this task very differently.

Dinosaur Comics are perfectly qwantzian. Click the cartoon to see it full size, click the link to get to the web comic's home. (T-Rex does not always subscribe to human norms of taste and good form, obviously, so, at your own risk.)

Eventually I'll work all my favorite webcomics in here.