Proof Update

Things have gone much better the past two days with the trig proofs. Much much better.

Yesterday, I asked for volunteers to put up the solutions. Not every problem was done correctly (most were though). Having the students explain their thinking as they were presenting helped. They did a nice job asking questions of one another. I also stressed multiple methods so we saw different ways of doing things. Some were a bit convoluted, but they worked.

Today was more of the same, except this time I called on the, uhm, hesitant students. I told them they could use a lifeline if they got stuck. One girl clapped and cheered when she did it correctly on the board. One student initially said no. I told him it was his choice. So, of course he then got up and did it.

We have a quiz tomorrow. We talked about how to study for math. I’m still surprised by the number of students whose plan is to “look at” the problems. I suggested to take out a new piece of paper and do the same problems again. Then compare results. I shared that I’ve done the same problems over and over until I understood them.

We’ll see what tomorrows brings in terms of quiz results, but I’m really happy with the way classes have gone the past two days. The students have been doing the math, explaining the math, and asking questions of each other. Yay!

“Teaching” Proofs

We recently began trig proofs. I love trig proofs. To me they are a joyful puzzle.

My students don’t quite share my sentiments.

/understatement

Today I heard: “I don’t know where to begin“, “This takes too long“, or “Show me what to do and I’ll do it“. Friday I worked out some examples. Students worked out examples. They worked together. I walked around and offered guidance. My suggestions included: finding a common denominator, factoring, using the identities (I gave them a sheet with them all listed - I’m not assessing their ability to memorize). Today was more of the same. We’ve seen multiple methods of proving the same thing. I find joy in this - yay, there’s more than one way! The students seem to find this annoying.

Aside from the open-endedness of the steps, what is giving them the most problems is the dreaded f-word.

One student summed it up pretty well, “The darn fractions finally caught up with me.”

I don’t know how to “teach” proofs. They want a step-by-step procedure. Unless I’m missing something, there isn’t one. Just play with it. Try something.

Grasping at Straws

We’ve been exploring the ideas of similarity and congruence in my freshmen classes. Today the students developed congruence by SSS - with straws.

Supplies needed: drinking Straws (1-2 per student), string (dental floss was too slippery), and scissors.

Question 1: Are all quadrilaterals with congruent corresponding sides necessarily congruent?¹
Students began by cutting a drinking straw into four pieces, threading the string through, and tying it off to form a quadrilateral. They quickly realized that they could “move” the angles, so nope, this is not true. Unexpected bonus: discussion of concave/convex.

Question 2: Are all polygons with more than four sides that have congruent corresponding sides necessarily congruent?
Repeat the bit with the straws. Some of them realized they could easily create this by carefully cutting one of the straws from their quadrilaterals. Conclusion, nope, more “moving” angles.

Question 3: Are all triangles with corresponding sides congruent necessarily congruent?
Make the triangle with straws… conclusion, yep, this does guarantee congruent triangles as the angles are “locked”.

We spent the rest of the period trying to explain why this is the case (and formalize the language after a bit). We also discussed whether of not this constituted a proof.

On the way out, one student thanked me for the lesson. He said it was fun to really understand why it worked. Wow. I was thanked. For a math lesson. Wow.

Up next: SAS congruence with straws and pipe cleaners.²

¹We’ve spent the past few days on “if-then” statements and counterexamples as a lead in to this.
²I thought it was time to start sharing.

Image: popping colors via flickr under cc

Sketchpad & Data Tables

I was making a Sketchpad activity and discovered that you can collect measurements in a table.

The full instructions on how to do so are here (search for “Tables”). It’s relatively painless. Select the measurements, go to the Graph>Tabulate menu. There you go! To add more measurements, double click on the table. Done.

I really like this as a way for students to organize and analyze the effects of changes to the figure. I think too often they just click and drag on things without stopping to think about what is really happening. Of course a good worksheet to supplement the activity is necessary. I’m still working on that part.

If you’d like this file, you can download it here. You will need Sketchpad though. Just for fun I tried the “Save As>HTML JavaSketchpad” option. The full features aren’t available (like the table) but here it is (Java compatible browser required). I need to play with this option some more. It is on my list of things to figure out.

Speaking of which, does anyone know how to embed these Java Sketchpad files directly into a WordPress post? My trust aide gave me some code to do so, but WordPress didn’t like it.