Because I can never precisely replicate angles in these investigations anyway

Good Lord I'm bad at titling posts. Moving on...

Every triangle congruence investigation that I've seen starts with some sort of prompted inductive investigation.

You can do this on a computer, as in the Illuminations lesson. You can use straws, as in Jennifer Silverman's version.

You can also use compass and ruler, as in the Discovering Geometry version of this activity.

There's a reason for this sort of convergence: the activity is solid. It helps kids notice important things.

But I've found this activity unsatisfying for a few reasons:

• I had a hard time motivating the investigation. I always ended up with some version of "Hey kids, what if there are some shortcuts here?" in that tone that clearly implies that yes, kids, there are some shortcuts here.
• Why does ASA work? Why does SSS work? Why doesn't SSA work? Where in the investigative process are we forced to grapple with the "why"? (I'm open to doing deduction to explain the results of induction, but I find it a bit deflating to post-facto explain empirical results that we've accepted as a class.)
• The investigation itself requires, rather than makes necessary, the SSS, SAS, ASA, SSA abstractions. 

In general, the entire investigation feels a bit rigged. I wanted something that felt more natural.

I started with similarity, which we handled in the way that I described here. I asked: "How can we check whether two figures have the same shape?" The kids brought up equal angles. It seemed to work for triangles, but not for quadrilaterals. (I showed them rectangles and squares, both of which have all right angles.) We walked out of there with clear criteria for two triangles being the same shape, and a fuzzy understanding of why that shortcut worked. We did some conventional practice problems, practiced proportions, and the kids were doing OK.

Then: "How can you check whether these figures are the same?"

Of course, we ended up stuck in a brief discussion of what it means to be the same, but we end up with two suggestions: (1) you could cover one figure "perfectly" with the other, or (2) same size and also same shape.

They didn't know it yet, but they were all going to flock to that second definition pretty soon. Because we've set them up perfectly for it, haven't we? They have the AAA criteria for similarity, and they're looking for insurance of congruence. Using the same size/same shape definition, they just need AAA plus some way to guarantee that the two triangles are the same size. From their practice work, they'll realize that any corresponding sides that are equal with similar triangles will end up fixing the sizes of both.

At the start of the next day I provoked them with these images:

"Those two triangles in the circle have to be congruent!"

Well, OK. Why?

"Because the arcs are the same."

"Because the angles are the same."

"Because their sides are the same."

etc.

So I pushed them, and we ended up with our first draft of a congruence criteria for triangles: "An angle and two sides are enough to guarantee congruence."

That required a test. So I drew an enormous angle on the board, and I used a ruler as our side. I pointed out that if that's enough to guarantee congruence, then there should only be one way to finish this triangle. 

And, at first, it seemed like that was the case. There were some skeptics. The bell was about to ring, and for homework I asked them to prepare an argument to settle this issue.

Here's my highlight reel from that next session:

Defeat of the angle and two sides criterion! We showed, using our sketches and compasses, that more than one triangle could fit in per point. (We actually had a whole disagreement about whether there were infinite, two or three triangles that you could make with this starter set of triangle parts, though eventually all was clarified through scribbling and summarizing.)

Here was our summary of the day:

Most important of all, there was a clear articulation of an argument: angles guarantee the shape, and then a side length fixes the size.

We spent the next day tidying up. All we really had was an argument for ASA, but that opened the door for other shortcuts as well. They came quickly, and we made sure that our list of shortcuts was complete, and that we hadn't missed anything. (One kid made a wonderful flowchart that I don't have a picture of.) Then, and only then, did I pass out a photocopy from a textbook and showed them two things:

1. There are clever names for these shortcuts: ASA, SSS, SAS, SSA, AAA, AA, etc.

2. This textbook calls these postulates, and says that postulates are assumptions. Isn't that interesting? Did we just have to assume these shortcuts, or did we prove them?

And we had a brief discussion about that too. And then they got some practice work. And then that was our triangle congruence week.

Addendum: Then we placed with arithmagons for a week. We grappled with finding a general solution for the triangle case. We gave convincing arguments that some square arithmagons were impossible to solve at all, and that others had infinitely many solutions. And then we confronted the difference between triangles and quadrilaterals, and I drew a connection between our work with similarity, congruence and arithmagons: Why are triangles always the exception? That lead to a good conversation about the many interdependencies in a triangle, compared to all other polygons.

Second Addendum: This was all with the Honors class, btw. In non-Honors we've done the similarity stuff but none of the congruence stuff. I'm trying to give that group a lot of informal work that involves noticing side and angle relationships before diving into this investigation with them.

No pretending that the nurturing of America's young elite is the most pressing educational issue of the day, but still...

Some highlights of this wide-ranging interview with Shamus Khan about the development of elites in today's most prestigious private schools:

Books & Ideas: Would you say that this new definition of privilege based on merit rather than birth makes the elite status more acceptable to a society based on equality?

Shamus Khan: In the United States, when we think of equality, we think of diversity, particularly more recently. These last 50 years, a lot of what we think about when we think of inequality is not class but race.

From there he details factors that support the narrative of meritocracy:

We have such residential segregation in the United States that people who come to St. Paul’s or Columbia from wealthy backgrounds come from towns and areas that are totally homogeneous. Also getting into an elite institution is extremely difficult even for privileged kids (Columbia’s acceptance rate is below 8%). So they have to beat out a whole bunch of their peers to get to these places. When they arrive, they are presented with a campus that looks very different from their home environment. It provides an anecdotal support for the idea that this institution is a meritocracy, and therefore it was their hard work, their dedication and their inherent skills and their capacities that help to explain how they were able to get to a place like that.

And I found his take on cultural openness fascinating:

It is the idea that high status people have gone from being snobs to being omnivores, that they have gone from people with very particular cultural tastes (say a taste for classical music) to people with quite varied tatstes (from classical music to hip hop or rap or rock music, to jazz...) [...] The way the new elite distinguishes itself is as being the most inclusive, democratic, open. This promotes the view of the world as a kind of space of opportunities in Thomas Friedman’s sense of the flat world. It is the people who see its wideness who are able to be successful. Who are the closest minded people, the most likely to listen to their very small range of things, heavy metal or country music? It is poor people. 

Food for thought, especially if you find yourself associated with a private school in any way. Read the interview, and even if you don't drop your thoughts into the comments.

"Beg, Borrow, and Steal" isn't great advice, Ctd.

I just cooked dinner. It was very yummy.



I followed a recipe. The truth is that when I cook I almost always follow a recipe. And for good reason. Cooking is hard, and I didn't really grow up doing much of it, but I like making good food. More importantly, when I'm cooking there are almost always other people that I'm trying to feed, and I want to make good food for them.

There's no point in not following a recipe, right? The people who write good cookbooks know what they're doing, and their recipes turn out great.

I've got friends who swear that they never cook from a recipe. They say that they just can't stand it.

I thought about these friends as I was making dinner tonight. I was making a ratatouille and some roasted roots, and I kept on forgetting amounts of spices, or how long things were supposed to stay on the stove for, and I didn't have room for the cookbook in my very small kitchen, so I was constantly running back and forth to another room to consult the book. That was sort of annoying, and (if we're being honest) a little embarrassing. Like, shouldn't I know whether the tomatoes go in with the peppers, and whether that matters?

Last night was annoying too. After picking out the next night's dinner, I made a grocery run. The recipe  I had picked called for eggplant; the grocery store was all out of eggplant. The recipe called for grape tomatoes; the grape tomatoes looked sort of nasty. It would have been really nice if I could've just changed recipes on the fly.

Besides for the convenience of being able to think up dishes on the fly, I'm a little bit jealous of my friends who don't use recipes or cookbooks. They're always making new, interesting creations and combinations of food that I just never think of. It's like, how do they come up with this stuff?

It's not as if I haven't learned anything since I started cooking for myself. Of course I have. To start, I've got this great collection of recipes that I hold in my head (and in a binder). When I read a cookbook, I recognize recipes that I think that I could do and that would taste good. And I'm really much better at managing the kitchen, chopping vegetables, telling when things are done, etc.

But I sort of wish that I could just come up with dishes on my own.

My wife and I have got this idea that we can learn how to do this. Here's our plan: one night a week is going to be Challenge Dinner. For Challenge Dinner, one of us does the shopping, and the other has to use whatever ingredients are offered for that night's dinner. So I can buy my wife broth in a box, parsnip and eggs, and she's got to figure out a way to make that into a decent dinner.

Who knows if it'll work. But wouldn't it be nice to try?

[Original post, here.]

Are these figures the same shape?

Are these lines the same length?

We've got options. We can move one segment until it lines up with the other, and compare them. Or we could get a ruler and compare each length on that. Either way, there's a way to check.

Are these the same shape?

Is there a way that we can check? How about with these two figures? Are they the same shape?

These?

File this under "measure tasks," and see more here, here, here, here, and here. I have nothing much to add except (a) that this is one of them and (b) it went well in class last week. 

There were three major suggestions from the kids:

1. Check the angles. If they're equal, then the figures have the same shape.
2. Check the sides. If they're in proportion, then the figures have the same shape.
3. Take the smaller figure. If a whole number of copies fit into the larger one, then the figures have the same shape.

A big debate broke out between one student who kept on pointing to the picture on the left, and another pointing to the picture on the right:

Meaning, hey, a whole number of triangles fits into the large rectangle on the right. Does that mean that they're not the same shape?

Nice, right?

We settled nothing on Friday, and there's a lot to work with on Monday. We should distinguish between necessary and sufficient conditions. We should create a bunch of examples of pairs of shapes and sort out our thoughts. And, if we think triangles are different than other polygons, then we need to begin to grapple with why that is.

Here's hoping I don't screw it up!

P.S. Why are we doing similarity so early in the year? This is the point in the year when most people would have kids working with congruency, SAS, ASA and SSS and the like. (Or, alternatively, defining things.) So why aren't we doing that? There's a fuller answer to that, I think, but the short version goes like this:

How do you help kids see congruency shortcuts? Do you just list a bunch of combinations of side/angle congruency and then set out to test them? How do you test them? Patterns? Straws? Construction? I don't want to poo-poo all this, but there's a clear path forward for us if we've got similarity in the bag. We can conceive of congruency of two figures as a special case of similarity, with the congruency of one pair of sides fixing the scale factor to 1. And then the question is, what other shortcuts are there out there?

Besides, similarity is really available to us, as I hope this post helped to show.

My (Current) Rules for Teaching Geometry

Presented mostly without comment, and with varying degrees of confidence:

1. Rarely limit our study to just triangles. I'm looking at you, triangle congruence and triangle similarity. You think that AA is a shortcut for similarity? Look at this square; look at this rectangle. Why should triangles be different?
2. Ask kids to prove things that they don't have the tools for proving. For example, ask kids to offer a convincing argument that all these triangles are of equal size.
   
   
   
   That'll create the need for similarity (or the side-splitter theorem, or parallel line theorems) in a jiffy.
3. Vary the problems that we work on between the visually obvious and the visually unintuitive, because one of our major jobs is to understand the relationship between the way we see things and mathematical argument.
4. Treat definitions like theorems. (Thanks, Justin + Lakatos!)
5. Two-column proof, or axiomatic proof, is an abstraction of proof. It is not a natural or normal way to communicate mathematically. The few cases I know of mathematicians working from axioms are remarkable for this peculiarity. (Most often, mathematicians are studying the properties of these formal systems, rather than working within the formal system itself.) So care has to be taken to put formal proof in its proper context, lest kids walk out of the year thinking that mathematicians sit in a room and pump out theorems from axioms all day.

I'd love to hear your thinking about these, as Geometry is a course that I still feel slightly uncomfortable with. See you in the comments!

The First Three Days of Trigonometry

Day One:

We watch this video.

We actually watch it twice. It's cool. We talk about some of the interesting things. I ask them what they were noticing. Some mentioned the shape of the ride, others that there's a bit of shaking in the seats, and others were thinking about what exactly happens when that guy swings off the top of the ride. (No, seriously, go watch the video if you haven't yet. It's fun.)

I ask them my question: How does the height of a person change as they go around this ride?

They sketch a graph in their notebooks. I forget to take pictures of it for the blog. Instead, I remember to take this blurry picture of the board where we recorded the camps:

So that's probably unhelpful. But on the left side we've got the three ideas:

1. The graph of height vs. time will be curvy, but sort of steep in the middle.
2. The graph will not just be curvy; it will literally be composed of half circles, due to the circular motion.
3. The graph will be nice and spiky.

I urged the class to temporary elide the differences between the two curvy graphs, and join brothers in arms to form a single Kurvy Kamp, allied against those who thought the graph was spiky.

Here's something interesting: we actually only had one customer for the spiky graph on that initial go around, when kids were sketching graphs in their notebooks. But as we talked about it some more, a few more voices for spikiness piped up.

We talked about what sorts of predictions spiky vs. curvy graphs make. A few kids voiced the idea that the spiky graph predicted that there was very little time spent at the top of the ride. We watched the video again.

Settled: the graph is curvy, not spiky.

(PS We also had a nice conversation about where people chose to put the zero height. People were trying to avoid having to use negative numbers in their choices of baselines. No one, it should be pointed out, chose the center of the wheel as the zero height. That's a non-obvious choice that needs to be motivated, not just thrust into the situation.)

(Here's a picture:)

Day Two:

A weekend passes, and I'm thinking that we're going to spend a few minutes in class reminding ourselves of what we did last week, and then we're going to try to make things more precise.

I put on that video again. My plan is to ask them to imagine a bigger wheel, and to ask them to graph that, just to get back in the swing of things. But as we're watching that video, somebody just up and says,

"I think we need to be careful about whether we're talking about the peoples' heads or the carts they're in."

Well, see, now that's interesting. Pretty soon we've found ourselves confused about how the motion of the people and the carts are related, and I ask them to whip out their notebooks and draw graphs that keep track of three heights: the height of the heads, the height of the seats, and the height of a notch on that little inner wheel near the center of the ride.

The class was pretty much split between these two possibilities:

This is a round that the folks who thought that everything peaks synchronously definitely won. Their arguments were clear and convincing, and mostly hinged on the fact that you could connect the heads of the people, the seats of the carts and the edge of the wheel with a straight line. 

And then somebody said that they thought the edge of the circle was going faster than the inner wheel. That's interesting too, and the class was split on whether they were buying it or not.

At the heart of all this? A deep confusion about how to square the fact that there is constant rotational speed with the idea that there could be non-uniform speeds at different points around the ride. I think that the difficulty of pairing easy, clean constant circular motion with messy, more subtle notions of speed was at the heart of the spiky/curvy debate the day before.

Things are basically settled by the time I ask students whether they'd give a Gold Medal to the guy who runs the outer ring of a racetrack in 10 seconds, or the guy who ran the inner ring in as much time.

My initial plan was to just urge them to figure out ways of being more precise with our measurements of heights, but I was glad for the way that class went. In the last few minutes we start talking about getting exact values for the height, but we run short on time.

They say that they know about special right triangles, so I give them a homework assignment on special right triangles.

Day 3:

The thing is, yeah, they don't really remember special right triangles. I figured they wouldn't, so I give them this Warm Up problem set at the beginning of class:

So we talk about that. 

And then I ask how we can get more precise with our graphs. Enough is enough with this messing around. Let's settle this.

We decide to set the initial height at 0. And then it repeats every 6 seconds or so. So at least we've got those two points. And what about half-way, at 3 seconds? Well, yeah, that would be the maximum height. What about half-way to that, at 1.5 seconds?

Another unexpected surprise from a student. He gets up and says, look, I think it would be best if we split up the Ferris Wheel into eighths. And it takes an equal amount of time for a seat to go from one of those sections to the next. And then we should just number the heights around those eighths.

I ask him, for clarity: "Is a Height of 2 half of the Height of 4?" He says that it is. And I ask him if a Height of 1 is half way to the Height of 2.

So I ask him to graph it.

Me: You understand, this is spiky, right?
Kid: Yeah. But maybe spiky is right!

I love it. Brilliant.

And now we've got a real argument going on. What's so fascinating to me is how fervent the spiky folks are. They're not just excited by that whole "split the wheel into eighths" argument. They're also tossing out things like "Well the wheel is moving constantly and so the height is changing at a single rate!"

But I've been patiently waiting for this moment for three days. I tell the kids, correct me if I'm wrong, but I think that our disagreement basically comes down to this:

[Except, imagine this without all triangles and angles and stuff.]

When you do half a turn here, does that mean that you've gone up half the height?

And we start busting out triangles, and the bell rings.

A number theory problem that I modified a bit for primary schoolers

This is a problem that was a huge hit with some 5th and 6th graders, and an OK day with some 4th grade kids.

It's a straightforward rip-off this problem from PCMI:

The sum of 4's divisors are 7. The sum of 5's divisors are 6. Etc.

Except that a lot of cool stuff happens along the way. I don't want to dish out any spoilers, but there is a lot of cool stuff to dig into.

We're talking about divisors, summing and multiplication here, so it's a good workout for the elementary school brain. There are also ample opportunities for spotting patterns, and a few good chances to offer justifications. It's a ton of fun when it works.

The only thing you need to work around is function notation, but that's not a big deal. 

Anyway, I do love this problem, and until you dig into the math you're just going to have to trust me that it has connections with exponents, adding fractions, .9999... = 1 and all sorts of other rich elementary school (and high school and number theory) topics. It's absolutely and truly one of my favorite functions and problems.

Only issue: it was definitely not a hit on the first day of class with my 4th graders. They struggled with finding divisors and their multiplication isn't strong enough to make the sorts of connections that make this a blast. It was OK, and there were a few cool moments when I pointed some things out, but overall it was a bit disappointing.

But that's more an issue with my taking an unnecessary risk on the first day than with the actual problem. I imagine that I'll bring this back into class once the kids are up to snuff, operations-wise. Good practice for all sorts of operations, and a lot of patterns to sniff out too. 

Let's hang out a bit in the comments. I'll give you two prompts: (1) Would this work with your elementary school students? Why, why not, etc. (2) Can you think of more "higher" math topics that would do well in a small-person classroom?