I wrote a blog post about Triangle proofs, centering around this idea:

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So here’s my scratchwork for a lesson idea: work from the opposite case, can we make a student doubt that different triangles exist? We infuse doubt by assigning students to find non-congruent triangles. When they run up against their sandbox’s boundary– the conditions that cause some triangles to be automatically congruent– they can all of a sudden doubt that triangles can always be made differently.

more here at my shameless plug: http://scottfarrar.com/blog/if-triangle-proofs-are-the-aspirin-what-is-the-headache/

I do massive whiteboarding 4×2 in groups. For me this unit is all about communication and fun. Students do work in groups as well as quizzes in groups, but there is a caveat…. Groups change frequently. 3 choices, they choose, I choose and random.

Our goal is making them better at seeing another viewpoint, right? What better way then encouraging them to listen and share frequently? Especially if they experience competition and discussion about these ideas.

How about they pick groups. They get 4 problems. 5 mins… No writing. Then. Groups change. All must be proven, writing only, except each one must be in a different persons handwriting. Then groups change and these groups participate In a group quiz.

Or. Then. Groups change and all different solutions or methods are discussed and verified or disproven and discussed regarding strengths and weaknesses. I do still like the group quiz idea.

I even test with a hybrid method with part done individually and part done either in a group or individually.

Regarding the interior angles discussion. (Polygons). I am convinced the usual exploration method is flawed. Build the polygons and the formula comes much more organically.

I wouldn’t have put it that way, but the task-based approach is why I’m redesigning my inequalities section this year.

I’ve found that I can use trig identities to provide precalculus students with this experience of NEEDING to know what the heck is going on then and discovering that they having the tools to figure it out for themselves. (Eventually, when they’re all done, I can point out that they’ve done a proof.):

Once they’re very familiar with the unit circle and the graphs of the six trig functions I ask them to type a set of complicated-looking equations into Desmos (The ones I’ve used most recently are at https://www.desmos.com/calculator/vj2budpxkh). They expect crazy graphs and are quite surprised to see horizontal lines (with the possible exception of some weird very narrow spikes which tend add to the motivation for figuring out what’s happening) and the graphs of trig functions they recognize well. It’s helpful in this case that the insights required to make progress are accessible enough (e.g., tan x = sin x/cos x) that someone always thinks of them and no one says “how did you ever think of that”? Also, the fact that we’ve begun with multiple puzzles which turn out to be solvable using similar but not identical insights means that even those who need help from the group the first or second time through eventually experience the satisfaction of proving something for themselves and different people find the keys to different problems.

This year, someone yelled out in the midst of a frenzy of proving, “Whoa, is there some whole other kind of crazy arithmetic of trig functions?” A vast improvement over “When are we ever going to use this?”!

(As a bonus, in this approach where students start by looking at graphs, it becomes completely obvious to them that there are actually values of x for which the identities don’t hold. I find the idea that I harp on during consideration of holes in rational functions, i.e., that two expressions can be equal almost–but not quite–everywhere, gets great reinforcement and is finally driven home.)

I still think that this obsession with linear inequalities is a leftover from the days when “linear programming” was flavour of the month.

Maybe. Certainly a bunch of people on Twitter recommended teaching linear inequalities only in the context of linear programming. But from another angle, it’s part of establishing the coherence of secondary math. Up to this point, we’ve graphed inequalities in one dimension and equalities in two. Apart from the existence of standards, I’m still going to wonder what happens when you mash those two up.

Sandy:

I like to use a “notice and wonder” approach to proofs in geometry. For example, I might give students a diagram of a parallelogram with diagonals and ask them what they notice and wonder. My goal is to get them to “wonder” if whatever they “notice” will always hold true. I often use GeoGebra for this step and give them a chance to explore, but perhaps doing some measuring by hand would create more of a “headache”. Once they’ve determined their conjecture works for many specific examples, we move on to trying to determine if it will always hold true which, if it does, then leads to a formal proof. I also like to throw in problems where a conjecture appears to be true for many specific cases but then students realize it won’t always hold true.

If proof is the aspirin, then doubt is the headache. Doubt is generally absent from proofs that read, “Prove the diagonals of a parallelogram bisect each other.” Students know they bisect each other now. The proof is a contrivance. With your method, students have a moment to wonder if what they’re seeing between their parallelogram and their neighbors will be true of everyone’s.

Mr Ruppel:

The problem was, this took a lot of time (and we had 55 minute classes at the time). This seems like something that could be accomplished through the use of technology (and taking textbooks out of “airplane mode”).

Seems like John Golden’s idea could similarly benefit from some crowdsourced acceleration. Fans of both.

Maya Quinn, Shannon McClintock, Josh Britton, ed, & Jennifer:

As much as I like the lesson ideas you all are suggesting, I’m more interested here in the underlying theory of task design motivating each of them. Tasks don’t generate other tasks; task design frameworks do.

My 1st year teaching, I essentially did this. Choose three points that make y>2x+5 true. Graph them. Write them down. Share them out with the whole group.

The problem was, this took a lot of time (and we had 55 minute classes at the time). This seems like something that could be accomplished through the use of technology (and taking textbooks out of “airplane mode”). Have all the kids graph 2 points that work and two points that don’t. Automatically display the graph with all the students’ points. See if students notice anything. This is a case where technology increases sharability and speed of getting to the WTF…

Proofs… @Joshua: Never thought to see a Spenser quote on a math blog. This gets at the problem for me: Hawk doesn’t need proof, and a lot of our students don’t either. Maybe they can all appreciate proof, though?

I’ve approached it a lot like Sandy, extension of noticing and wondering. The headache almost has to be their own conjecture. The problem (if it is) with dynamic geometry is that the examples are so convincing. It’s basically a million examples (usually literally uncountable) that all verify. How much proof do you need?

So what do mathematicians get out of it? For me, it’s a top level of knowing and understanding. If you can prove it, usually you really understand it.

Background. I have found that, once students are convinced that the sum of interior angles for a triangle is 180 degrees, it is not hard to bring them on board for two methods to compute a convex n-gon’s sum of interior angles (methods of Proof which can be unpacked, compared/contrasted, have their assumptions questioned, and so forth):

1. Connect one vertex to each non-adjacent vertex, thereby creating n-2 triangles, which each have interior angles summing to 180 degrees. Summing across all constructed triangles gives the sum of the convex n-gon’s interior angles; i.e., (n-2)180 degrees.

2. Construct a point in the interior of the convex n-gon, and connect all n vertices to it, thereby creating n triangles, which each have interior angles summing to 180. Summing across all triangles gives n180 degrees; but we have overcounted by the 360 degrees around the constructed point, so we end up with a total of: n180 – 360 degrees.

Luckily, the two formulae agree: (n-2)180 = n180 – 360.

Foreground. A number of students (at many ages and stages!) quite like to doodle. I have noticed that few of them are able to freehand draw a 7-point star (at the least – few have seen it done) and many become suddenly curious about such a figure’s construction.

Here is a rough picture of how to draw one: http://i.imgur.com/70npKaT.jpg

I have used colors to indicate how the pen[cil] or marker moves; in order: red, orange, yellow, green, blue, purple, black.

In the past, my students have gravitated towards sketching this picture [give them a chance to try and freehand it on their own, first!] and this may pique their interest in stars.

This is a good opportunity to explore stars. As far as the next topic mentioned (“triangle proofs”) a nice question is, what is the sum of the interior angles of the 7-star?

In fact, maybe it is best to step back and examine a 5-star. Then a 6-star, a 7-star… and can they figure out the formula for an n-star? (There are, of course, opportunities to use technology here: perhaps graphing software that uses angle measures to draw a shape, which means computing by hand certain interior angles for a particular star.)

Note that stars are not convex figures (why?) so we ought not assume the previous techniques will necessarily* work in finding their sum of internal angles. Still, method 2 does work quite nicely: Draw a central point, and connect it to each vertex; how many triangles does this make? How does this help? Etc.

All of this business gives a potential opportunity to talk about proofs more generally (what are our assumptions? – e.g., that the sum of any triangle’s interior angles is 180 degrees, or that all stars are “star convex”; how are we reasoning from one step to the next in the proof for a fixed n?; how are we reasoning in providing the formula for general n?; why did we start with a 5-star and not, say, a 4-star? etc).

I cannot say this is the right task for All Students, but it can be a good direction in which to press further their knowledge of interior angles (which might already involve at least the two methods above, though this is usually pressed further by asking for the interior angle of a regular n-gon or some such thing) while emphasizing important components in proving (besides assumptions: strategies/heuristics, pitfalls such as overcounting, generalizing- to name a few) and one can certainly take it even further (e.g., examining Koch snowflakes).

Though I recognize the technology component could be well-implemented, I might reiterate that the doodling aspect appears to stay with [some] students. (Notebooks and scrap-paper tell no lies…) Once their interest in a geometric figure is real and personalized, exploring the shape and playing with its properties become more meaningful.

MQ

*An n-sided polygon always has interior angles summing to (n-2)180 degrees. (See the classic, and funnily titled, “Polygons Have Ears” by G.H. Meisters.) A nice task is to draw a concave figure C for which neither method 1 nor 2 above can be used directly to find the sum of interior angles; for curious students, a reasonable follow-up question could be: What is the fewest number of sides C can have – and why?