I don’t know if all of what I’m about to throw down here will make sense upon first glance or by skimming. I have a feeling that the flow of the unit, and where each key moment of understanding lies, all comes from actually working through the problems.

But yeah, here’s the general flow of things:

Kids see that all right triangles in the world can be categorized into certain similarity classes… like a right triangle with a 32 degree angle are similar to any other right triangle with a 32 degree angle. So we can exploit that by having a book which provides us with all right triangles with various angle measures and side lengths. (A page from this book is copied on the right.) Using similarity and this book of triangles, we can answer two key questions. (1) Given an angle and a side length of a right triangle, we can find all the other side lengths. (2) Given two side lengths of a right triangle, we can find an angle.

By answering these questions (especially the second question), kids start to see how important ratios of sides are. So we convert our book of right triangles into a table of ratios of sides of right triangles. Students then solve the same problems they previously solved with the book of triangles, but using this table of values.

Finally, students are given names for these ratios — sine, cosine, and tangent. And they learn that their calculator has these table of ratios built into it. And so they can use their calculator to quickly look up what they need in the table, without having the table in front of them. Huzzah! And again, students solve the same problems they previously solved with the book of triangles and the table of values, but with their calculators.

Hopefully throughout the entire process, they are understanding the geometric understanding to trigonometry.

(My documents in .docx form are here: 2015-04-xx Similar Right Triangles 1 … 2015-04-xx Similar Right Triangles 2 … 2015-04-xx Similar Right Triangles 2.5 Do Now … 2015-04-xx Similar Right Triangles 3 … 2015-04-xx Similar Right Triangles 4)

It’s a long post, so there’s much more below the jump…

**Similar Right Triangles#1: The Forward Question **

I am going to begin by having students start seeing every right triangle they see as similar (or, to be technical, close to similar since these only have integer angle measures) to one of the right triangles in this book, titled the Platonic Right Triangles book. (If you want to know where Plato comes in, check out the packet…)

That’s how we will start. So they are going to use the Platonic Right Triangles book along with what they have learned about similarity in order to find missing side lengths of triangles. At the end of this, students are going to measure the height of the organ in the chapel in our school, using what they have learned.

The big idea of this first packet is getting kids to see that all right triangles are similar to this “Platonic” set of right triangles, and given a right triangle with a particular angle, they can use that fact to find missing sides of triangles with very little effort, by setting up a proportion.

**Similar Right Triangles#2: The Backwards Question **

This is the backwards question: Given a triangle with some side lengths, can they find the missing angle? At this point, all they have is their Platonic Right Triangles book. Mainly, the first few pages of this packet has kids try to figure out how they can use the book to find out the missing angle. They’re going to struggle at first, and have various approaches to it, some of which will be more efficient than others. But I see the first four pages of this packet, their struggle, and most importantly *the group and class conversations we have based on their approaches,* to be the key thing here.

Hopefully, at this point, we’ll converge upon the idea that the ratio of sides is a super awesome way to find the missing angle. If they are given the side opposite the angle and the hypotenuse, life is easy. If they are given the side adjacent the angle and the hypotenuse, life is easy. (This is because the Platonic Right Triangles book gives all the triangles with hypotenuse of 1.) But if they are given the side opposite the angle and the side adjacent to the angle, life is hard. And that’s okay. But it would be so much easier if for each of the triangles in our Platonic Right Triangles book, we had a list of all the ratios of the opposite side to the adjacent side.

The important of these ratios gets highlighted with some simple work with this simple Geogebra applet.

From here, we collapse our Platonic Right Triangles book into a simple Table of Right Triangle Ratios. An eighty-nine page book that can be represented in a two page table.

Students hopefully will finish this worksheet seeing the usefulness of ratios in a right triangles, and being able to use a table of values to help them find missing sides and angles.

**Similar Right Triangles #2.5: Why Three Ratios?**

This is a classroom opener. It came about when my co-teacher recognized that we never actually need all three ratios… the only reason we have all three ratios is for *convenience*. And so I whipped this up to get kids to recognize that fact.

**Similar Right Triangles #3: Naming the Three Ratios**

In this packet, students finally get names for the three ratios we’ve been using (sine, cosine, tangent), and hopefully the questions asked will require them to understand they are basically doing what they’ve been doing — but with fancy names. Specifically, I want students to leave understanding that *all a calculator is doing when calculating the sine/cosine/tangent of an angle is that it is going to a super extensive internal table of values and finding the appropriate ratio of sides for that right triangle*. And that *all a calculator is doing when calculating the inverse sine/cosine/tangent of a ratio of sides is that it is going to a super extensive internal table of values and finding the appropriate angle for that right triangle*.

Then there are some challenging conceptual questions that students will hopefully be able to answer, which require them to understand the geometric understanding of the trigonometric functions.

**Similar Right Triangles #4: Special Right Triangles**

I don’t think that special right triangles are all that important, but I know that I ought to expose my kids to them. So I made this…

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Today and yesterday, I had three different meetings with a few different kids who wanted to discuss options. These conversations revolved around:

- Park Effects on baseball batters (sports statistics)
- Understanding why a particular algorithm creates the math art pictures it does
- The Goldbach Conjecture and the Collatz Conjecture
- Time dilation (and time travel)
- How restaurants do their finances and stay in business

Super fun conversations, with kids who just want to learn stuff that they’re fascinated by. For example, the kid who wanted to talk about the Goldbach conjecture said that he wanted to work on proving the Goldbach conjecture (“I will not give up!” he wrote) — and the reason he wanted to do this is because he always had trouble with prime numbers and understanding them. Melting! MELTING!

Previous posts about Explore Math:

http://samjshah.com/2014/02/12/explore-mathematics/

http://samjshah.com/2014/04/11/explore-math-reprise/

http://samjshah.com/2014/04/25/explore-mathematics-part-ii/

The site that launched Explore Math (mini explorations) last quarter for my kids:

http://explore-math.weebly.com/

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*The First Idea*

In Geometry, I want my kids to learn to use multiple tools, and find the tools that are the most useful to them at any given moment. One moment they might need patty paper to trace something. Another moment they might need eraseable (this is key!) colored pencils to emphasize different things. Another moment they might want to pull up Geogebra on their laptops. And another moment, they might need a ruler to draw a straight line. Who knows. So what I did at the start of the year was create geometry buckets, populated with the tools that each group might need at any given time.

I have a different bucket for each group. I color coded most of the items in the bucket (with the exception of the protractors, because I didn’t want to cover any of the angles!). I store the buckets in the room. At any point, kids are allowed to grab them. Sometimes they have to, because they are asked to measure an angle, or draw a circle. When I have them use the giant whiteboards, they have their dry erase markers and an eraser in the buckets. But most of the time, when a kid needs some patty paper, or a ruler to make a diagram, or colored pencils to organize their ideas or annotate a diagram, they’ll just grab the bucket and bring it to their groups. And at the end of each class period, the kids will just put them back.

I thought things would get lost or mixed up. But it’s been a semester, and I just went through the buckets and have found only a few colored pencils were in the wrong boxes, and only a single compass migrated from one box to another. I love these buckets of geometry tools!

*The Second Idea*

I do tons of groupwork in my classes. And I try to switch up groups often enough for some spice, but let them work together long enough so they can learn to work together (I try to do it two times a quarter). However, when kids are in groups, passing things out and collecting things can be annoyingly time consuming. And if my kids know one thing about me as a teacher: I don’t waste time, not a second.

So here they are: something I’ve been doing for the past few years. Folders. Specifically, each group gets one folder.

On the front (not photographed), I have a label with the kids’s names on it. Inside are two pockets. The left hand pocket is for things I normally would hand out. (Mainly: the packets that I make for kids to work collaboratively through.) The right hand side has two purposes: (1) I have kids turn in nightly work sometimes, so they will put it in there, and (2) when I mark up the nightly work, I put it back in there and students collect it the next day.* There are also some “The Dog Ate My Homework” forms for when a kid doesn’t turn in their work. Instead of them calling me over and giving me a story explaining what happened, I just have them fill out that form saying why they didn’t have their work.

One *huge* benefit for having these folders is that it allows me to mix up *where the groups sit each day*.** When I walk into the classroom, kids aren’t sitting down usually. They are waiting for me. I throw down each folder on group of desks, and then kids sit at the group of desks with their folder on it. That way: kids are in different locations each day, mixing things up. The group in back won’t always be in back! Sometimes I give a kid the folders to put down, and sometimes *the power, the sheer power of who sits where, goes to their head. (“Oh, you’re standing by this group of desks? Too bad, I’m putting your folder waaaay over at that far group of desks.”) *Fun times.

Now you might say: each day you have to put in the packets you’re going to hand out the next day? Nope.Well, sometimes. But usually not. In classes I’ve taught before, where I have my ducks in a row, I do a massive photocopying of the papers for the entire unit. I lay them out, and fill up the folders. Then I’m pretty much set for a week or two (or more!). Below is a picture of me doing that today!

That is all. Go back to your regularly scheduled lives now.

**UPDATE:** I forgot to say: I color code the folders for each class. So red folders = my geometry class, blue folders = one of my precalc classes, green folders = the other one of my precalc classes. I also use a lot of file folders to organize things for me. And for those, I use the same color folders for each of my classes. So, for example, when I give a geometry test, I bring a red file folder to class. And then I keep the taken geometry tests in that file — and when I’m going home, I just throw that red file in my backpack so I can mark ‘em up.

*The fact that there is one folder per group also has the added bonus that when one kid forgets to put their name on their nightly work, you know exactly whose it is, because it is in the folder for that specific group (and usually all the other kids put their name on it).

**Someone, somewhere, told me that there was some ed research that suggested that kids sitting in the same spot every day helped them learn better. I have my doubts about that.

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**The TL;DR version:** By learning about angle bisectors, we motivate the need for triangle congruence. We have students figure out when they have enough information to show two triangles are congruent. They use this newfound knowledge of triangle congruence to prove basic things they *know* are true about quadrilaterals, but have yet to prove that they are true. Finally, we return back to angle bisectors, and show that for any triangle, the three angle bisectors always meet at a point. As a cherry on the top of the cake, we do an activity involving salt to illustrate this point.

***

**STEP 1: To motivate the need to figure out when we can say two triangles are congruent when we have limited information, we are showing a problem where triangle congruence is necessary to make an obvious conclusion. **

We’re going to see the need for triangle congruence to show that:

*any point that is equidistant to an angle (more precisely: the two rays that form an angle) lies on the angle bisector of that angle*.

This will be an obvious fact for students once they create a few examples. But when they try to prove it deductively, they’ll hit a snag. They’ll get to the figure on the left, below. But in order to show the congruent angles, they’ll really want to say “the two triangles are congruent.” But they don’t have any rationale to make that conclusion.

Hence: our investigation in what we need in order to conclude two triangles are congruent.

When kids do this, they will also be asked to draw a bunch of circles tangent to the angles. All these centers are on the angle bisector of the angle. This will come up again at the end of our unit.

Kids will be doing all of this introductory material on this packet (.docx)

**STEP 2: Students discover what is necessary to state triangle congruence.**

This is a pretty traditional introduction to triangle congruence. Students have to figure out if they can draw only *one* triangle with given information — or multiple triangles.

We’re going to pull this together as a class, and talk about why ASA and SAS and SSS *must* yield triangle congruence — and we’ll do this when we talk about how we construct these triangles. When you have ASA, SAS, SSS, you are *forced *to have only a single triangle.

I anticipate drawing the triangles in groups, and pulling all this information together as a class, is going to be conversation rich.

The pages we’re going to be using are below (.docx) [slight error: in #4, the triangle has lengths of 5 cms, 6 cms, and 7 cms]

**STEP 3: Once we have triangle congruence, we’re going to use triangle congruence to prove all sorts of properties of quadrilaterals.**

Specifically, they are going to draw in diagonals in various quadrilaterals, which will create lots of different congruent triangles. From this, they will be asked to determine:

(a) Can they say anything about the relationship between one diagonal and the other diagonal (e.g. the diagonals bisect each other; the diagonals always meet at right angles)

(b) Can they say anything about the relationship between the diagonals and the quadrilateral (e.g. one diagonal bisects the two angles)

(c) Can they conclude anything about the quadrilateral itself (e.g. the opposite sides are congruent; opposite angles are congruent)

We have a few ideas percolating about how to have students investigate and present their findings, but nothing ready to share yet. The best idea we have right now is to have students use color to illustrate their conclusions visually, like this:

**STEP 4: This is a throwback to the very start of the unit. Students will prove that in any triangle, the angle bisectors will always meet at a single point.**

Here are the guiding questions (.docx)

And we will finish this off by highlighting the circles we drew at the start of the unit. *Notice we have a single circle that is tangent to all three sides of the triangle. The center of that circle? Where the angle bisectors meet. Why? You just proved it! That location is equidistant to all three sides of the triangle.*

**STEP 5: The reason we’re highlighting these circles is that we’re going to be cutting out various triangles (and other geometric shapes) out of cardboard, elevate them, and then pour salt on them. Ridges will form. These ridges will be angle bisectors. Why? Because each time you pour salt on something flat, it forms a cone. The top of the cone will the the center of the circle. We’re just superimposing a whole bunch of salt cones together to form the ridges.**

The other geometry teacher and I both saw this salt activity at the Exeter conference years ago. Here are some images from a short paper from Troy Stein (who is awesome) on this:

The general idea for this activity is going to be: kids take a guess as to where the ridges are going to be, kids pour the salt and see where the ridges are.

As the figures get more complex, they should start thinking more deeply. For example, in the quadrilateral figure above, why do you get that long ridge in the middle?

My hope is that they start to visualize the figures that they are pouring salt on as *filled* with little cones, like this:

The material I’ve whipped up for this is here (.docx):

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They even blew up balloons and drew triangles with *two* right angles on them (using protractors and rulers).

[Note to self: students cannot tie balloons. Also, they will be found scattered around the student center later in the day if you let kids keep their balloons.]

But we said: *assuming* that you know that *on a plane* the sum of the interior angles of a triangle add up to 180 degrees, can you prove that quadrilaterals have a sum of 360 degrees for their interior angles.

And each group was able to latch onto the idea of *dissection *without me saying anything… breaking the quadrilateral into two triangles.

But then… then… they started to say something that scared me. They said “there are two triangles, and since each triangle is 180 degrees, the quadrilateral is 360 degrees.”

To you non-geometry teachers, this might not seem problematic. But I immediately thought: “oh gosh, these kids think of triangles and quadrilaterals and the like as having some inherent property that can be added to others. They aren’t saying *the sum of the interior angles of the triangle is 180 degrees… *they are saying *the triangle has 180 degrees.*”

So I gave them a follow-up question (which I’m proud of):

“The Blue Triangle is 180 degrees. The Pink Triangle has 180 degrees. So the Giant Triangle (the blue and pink triangles combined) must have 360 degrees. How is this possible? Did we just break math?”

One group had someone who figured it out right away, but the others took a good amount of time trying to figure out where this argument failed. I loved it because it really showed them a misconception they had.

It was the perfect question, because over the summer the other geometry teacher and I came up with the following (which we are in love with) involving triangle dissection:

Finally, to check if each group understood where these came from, we had them write a “triangle dissection expression” for the sum of the interior angles of this pentagon:

Fin.

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Did you watch it? No? Seriously, watch it!

Okay, good. Ever since I saw this video, I wanted to try it in my class. I wanted to have my kids guess the rule. I wanted to be the one saying “yes” and “no” to their three numbers. And today, finally, I did.

In geometry, we’re starting a short excursion into proof and reasoning. Yes, we’ve done a few proofs. And my kids are learning to justify their ideas. But we’re about to embark on a few days where they think about proof, the importance of proofs, assumptions, and other such things. And in our next class, we’re going to start talking about induction and deduction. So today, this was a perfect warm up.

I gave each group a whiteboard. They threw up three numbers. I said yes or no. After they got 3 yes/nos, they were allowed to guess the rule. Then they did it again. Some groups were looking to get the rule in the least amount of guesses. Others were guessing willy nilly. It definitely took longer than I thought. No group got it, but they were making interesting choices with their three numbers in order to figure out the rule.

After about 7-8 minutes of this, I stopped them. I started playing the Veritasium video. It was awesome. Why? Because the random beach people that Veritasium interviewed gave almost identical answers as my kids gave! They saw that the way they were approaching the problem was the same. They heard a few more sets of three numbers that worked/didn’t work, and then I paused the video. Why? Because I heard kids whispering and murmuring that they think they had it. I gave those kids an opportunity to share what they thought the rule was (I did not confirm or deny their guesses). And then I finished the video.

I loved doing this because the kids were totally engaged. And when we start talking about induction and deduction, counterexamples, and keeping an open mind when problem solving, we can use this exercise as an activity we can refer back to.

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I told the groups to mix themselves up — for a change of pace. And then I handed this out.

I told each group that they had *one* opportunity to call me over so I could give them A Big Hint. Then I let them go, giving them giant whiteboards to work on if they wanted.

In my two classes, I had: one group solve the problem explicitly (they had a formula that worked) and one group come up with a recursive solution that really impressed me! The other eight groups were at varying stages of understanding. Most all the groups were gung ho about working, and most all the groups started discovering all these patterns.

Only one of the ten groups asked for The Big Hint, which means my kids have perseverance! I did give varying degrees of mini-hints to kids as I saw them progress, to nudge them this way or that way.

At the end of the second class, a few kids said how they are now braindead, because they did so much thinking. They were exhausted. As a teacher, I call that winning!

I’m still at a bit of a loss as to what I am going to do tomorrow. Since kids hadn’t really finished, I thought I would have them work a bit more. I’m not good at debriefing. Also, many kids have different observations, and I don’t have time to really dwell on this. This wasn’t supposed to even take the whole day!

My plan is to give all my groups collectively A Big Hint, give each group 12 minutes to figure out how to find the *n*th pentagonal number, and then after those 12 minutes are up, I will give them this:

We will go over the triangular and square numbers together as a class. And then WHAM, I leave them with the pentagonal and p*-gon* figures to figure out on their own.

Wish me luck that it goes well tomorrow. I can see it crashing and burning, or being a good wrapup.

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Imagine you’re a geometry teacher, and you want students to **discover** a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to **understand (deeply)** and **articulate why** this method gives you *the* center of rotation.

Although not perfect, I have whipped some stuff up in the past hour that I hope will get to the heart of this question. Of course in my head, I have class discussions, and we gently get at this. These sheets alone don’t get us there. But if you’re interested, feel free check ‘em out.

(Also, since I just whipped them up, there might be some things that need fixing/tweaking…)

There are five of them (all combined).

- Rotations of two points [after this, through discussion, get student to make the connection between
*the perpendicular bisector as all points equidistant from the endpoints of a segment*and*the radii of a circle*] - Rotation of a line segment [after this, through discussion, get students to recognize that they are really considering
*two*perpendicular bisectors… we are looking at one perpendicular bisector to find all possible points which will rotate one end of a line segment to the new point*and*a second perpendicular bisector to find all possible points which will rotate the other end of a line segment to the new point… for both endpoints to simultaneously rotate to their new location, we have to look for the intersection of the perpendicular bisectors!] - Rotation of three points [after this, make it clear that this is
*not*the same as saying that you can trace a triangle on patty paper in two different places and find a center of rotation that will bring the first triangle to the second triangle… in fact, maybe I should have this as an exercise…] - Center of rotation practice
- Rotations of a complex figure

(.docx)

PS. If you’re talking about multiple “center of rotation”s, do you say “centers of rotation” or “center of rotations” or “centers of rotations”? It makes me think of culs de sac, which indeed is the plural of cul de sac. Thank you *Gilmore Girls*.

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Setup: We’ve been talking about *perpendicular bisectors *in various contexts in geometry. But they were just making observations and working on some simple proofs.

Last night in Geometry, students were tasked with the following:

It turns out that #3 is impossible, and #4 is possible with some guess and check. This sets up the background to have kids see something neat.

Then they are asked:

And now they see that for a triangle, the perpendicular bisectors of the sides all meet at a point. And that *is* rare and weird. They then were asked to look at the point that the perpendicular bisectors meet at and the vertices of the triangles and make a conjecture.

Only one student “saw” it. It was fascinating for me that it was so hard for everyone else to see it! Others had conjectures that might have been true for right triangles or isosceles triangles or equilateral triangles… but not that were universally true.

For the rest of the class, to get them there, I did the following:

This was a huge setup for my “one good thing.” There were gasps, and one student said, and I quite, “MIND BLOWN.”

This weekend they are going to try to figure out what the what is up?!

PS. Yes, I am fairly certain that the setup of having students see the rarity of perpendicular bisectors meeting at a point, as well as having them look and fail to see something inside a set of seemingly random points was crucial for the big reveal. In fact, the fact that they *didn’t *discover it on it’s own was so powerful when they ended up seeing it.

**UPDATE:** The file I used is here.

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where

So obviously they go like: ,, , , and

So great. Awesome. NOT. *Booooring*. So I showed them the decimal expansions:

WHOA! This is getting closer and closer to 2… Weiiiird…

And then I say I can show them this will continue, and we can find a way to show that [where the pattern continues *forever*] will practically become 2.

*DIGRESSION WHICH IS ACTUALLY WHY I WANTED TO BLOG ABOUT THIS*

To do this, I start with something else. I don’t know why, but I really wanted to show them a continued fraction first, to get the point across easier than with the square root. This was the continued fraction.

I went through a frenetic mini-lecture, and I think I had about 40% of the kids along with me for the whole ride. I’m not sure… maybe? But later a kid came by my office, and **I thought of a better way to show it**. Hence, this blogpost, to show you. (I have seen teachers use this method when teaching *substitution* when solving systems of equations… but I have never used it myself. I’m dumb! This is awesome!) This is what I did when showing the kid how to think about this in my office.

First I took a small piece of paper and I wrote the infinite fraction on it.

Then I flipped it over and on the back wrote what it equaled… Our unknown that we were trying to solve for.

I emphasized that *that card itself* represented the value of that fraction. The front and back are both different ways to express the same (unknown) quantity we were looking for.

Then I took a big sheet of paper and wrote where I left the denominator blank. And then I put the small card (fraction side up) in the denominator of the fraction…

And I said… what does this whole thing equal?

And without too much thinking, the student gave me the answer…

Yup. We’ve seen that infinite fraction before. That is !

Ready? READY?

Flip.

**THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. **That flip was the single thing that made me want to blog about this.

Now you have an equation that you can solve for … and is what you’re trying to find the value of. This equation can easily be turned into a quadratic, and when you solve it you get (yes, the Golden Ratio). And it turns out that is close to what we might have predicted…

Because in class, we (by hand) calculated the first few terms of where … and we saw:

And when I drew a numberline on the board, plotted 1, then 2, then 1.5, then 1.66666666, then 1.6, then 1.625, we saw that the numbers bounced back and forth… and they seemed to be getting closer and closer to a single number… And yes, that single number is about 1.618.

COOL! [1]

BACK TO OUR REGULARLY SCHEDULED PROGRAM

So after I showed them how to calculate the crazy infinite fraction, I went back to the problem at hand… What is ?

Let’s say

Then we can say

And even simply by inspection, we can see that is a solution to this!

Fin.

[1] What’s neat is that yesterday I introduced the notion of a recursive sequence that relies on the previous *two* terms. So soon I can show them the Fibonacci sequence (1,1,2,3,5,8,13,…). What does that have to do with any of this? Well let’s look at the exact values of where .

. WHAAAA?!?!

Lovely. It’s all coming together!!!

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*Here’s the impetus/setup.*

We have just finished talking about translations, and we’re moving on to reflections and rotations. When we introduce rotations to kids, we give them a backwards problem pretty early on: here is a figure and here is the rotation of the figure. Try to find the center of rotation. Use patty paper and guess and check.

This is an *awesome *exercise (inmyhumbleopinion) because it has kids use patty paper, it has them kinesthetically *see* the rotation, and it gives them immediate feedback on whether the point they thought was the center of rotation truly is the center of rotation. Simple, sweet, forces some thought.

This is our intro. We have a few more exercises in our problem packet similar to this. But then we move on. Le sigh. This felt wholly unsatisfying to us, because at its heart we left our kids hanging. We never get at the obvious question. How do I find the center of rotation *without* guessing and checking.

How do we take this introduction and make it mathematics?

**The Challenge**

[Baby challenge: Figure out how to find the center of rotation, given any figure and a rotated figure. This is not totally simple, okay? I’m calling it a baby challenge only because the next challenge is *sooooo* hard!]

Imagine you’re a geometry teacher, and you want students to **discover** a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to **understand (deeply)** and **articulate why** this method gives you *the* center of rotation.

What do you do to achieve these twin goals?

(This is what the other teacher and I were discussing today, and having loads of fun doing it!)

You *don’t* need to answer this question in the comments. (Though you’re welcome to throw down any ideas.) I’m actually not looking for advice. (We’re well on our way to coming up with an answer to this.) I just thought it would be a fun thought exercise for you, if you like thinking of lesson planning/curriculum design. This is the type of stuff I love thinking about — when I have time!

This is backwards planning at it’s most fun, in my opinion. I have a deep result that is abstract and hard to grasp. I have very concrete 9th graders who I want to get from knowing almost nothing to discovering, understanding, and marveling at this great mathematical insight. How do I get from Point A to Point B?

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We started off the year with a very conceptual beginning, focusing on the importance of *words, definitions, *and *classification. *As you might have remembered from our first day activity, we have also been sprinkling in a good amount of conjecturing. [1]

I want to share one activity that I thought was not only was engaging, but led to really interesting discussions in my classroom.

***

**Part I: Defining & Counter-attacking**

On the second day of class, I had each of my geometry groups try to come up with a definition for the following words:

This is actually *really* challenging. I mean: you yourself, try to define a triangle without looking it up, or even more challenging, a polygon. Before starting, groups were told that other groups would try to find fault with their definitions, so they should be as specific and clear as possible.

Some things different groups wrote (all are problematic for various reasons):

Circle: “A circle is a closed figure where all the points are an equal distance from the epicenter, and starts and ends at the same point.”

Triangle: “A triangle is a 3-sided, 3-angled shape with straight lines that connect to the endpoints. Also, all the angles add to 180 degrees.”

Polygon: “Any shape that has exclusively straight edges that are all connected, and has to have at least three angles.”

Each group then passed their three definitions to a different group. And that new group was tasked with finding a **counter-attack** to these definitions. What this means is they needed to draw something that satisfies the definition they were given, but is *not* a circle (or triangle, or polygon). Those trying to counter-attack were allowed to read the definitions they were given *in any way that seemed reasonable to them*.

We then had a class discussion. Students publicly posted their group’s definitions (they were written on giant whiteboards), and then those with counter-attacks were allowed to present them to the whole class. When the counter-attacks were presented, it was interesting how the discussions unfolded. The original group often wanted to defend their definition, and state why the counter-attack was incorrect. Then a short spontaneous discussion would occur.

At the end of the discussion, I found myself often being arbiter and passing judgment on each counter-attack: “yes, this counter-attack works, because …” or “no, this counter-attack doesn’t work, because…” I felt the kids needed to know (a) whether the counter-attack really did satisfy all parts of the group’s definition, and (b) whether the counter-attack was using a fair reading of the group’s definition. When I said the counter-attack was valid, the group who found the counter-attack was elated! And when I said the counter-attack was invalid, the group who wrote the definition was elated! It became a bit of a spontaneous contest.

**What was awesome was the subtleties they ended up talking about when trying to find the counter-attacks.** When talking about the circles or polygons, for example, they realized that we have to say this is a 2D figure, otherwise there are many other curves that would work. When talking about triangles, saying the figure had three angles was problematic because there are 3 *interior* angles and 3 *exterior* angles. For triangles and polygons, students realized how crucial it was to say that the figures were closed. I was so impressed with how they were really trying to attend to precision in this task.

***

**Part II: Understanding The Textbook Definitions**

Eventually, we looked at a textbook’s definitions for these three words.

It took us a while to understand these definitions, and why the particular language was chosen. The polygon definition was especially challenging — especially the second half!

***

**Part III: Taking Things Further**

I started the next class with the following DO NOW:

Although I thought this would be easy for them, it was interesting to see that they found this challenging and abstract.

We also came up with the following two questions for a mini-quiz we gave:

***

The ending of our first unit involved students coming up with their own definitions for the bunch of quadrilaterals (kite, dart, square, trapezoid, rectangle, rhombus, convex quadrilateral, concave quadrilateral, isosceles trapezoid, parallelogram). This opening activity was designed to make that exercise easier when we got to there. Specifically, it was designed to show them that *clear and precise language is important to communicate your ideas*, and *it isn’t easy to come up with clear and precise language*. Things that we “think we know” are really quite hard to pin down… Like what a circle, triangle, and polygon are.

***

UPDATE: I found some examples of “counterattacks” that my kids drew for various definitions, so I figured I’d scan them in for posterity… We came up with more when we were having our discussion.

And here is a random picture I took in class with two of the whiteboards:

[1] I’m finding this to be a really rewarding thing to have sprinkled in. I’m learning it’s challenging for students to be able to try to make a potential conclusion from a number of examples. But in fact, isn’t that a crucial skill in mathematics? We see a number of examples of something, we decide on a very plausible conjecture, and then we try to reason out why that conjecture is true (or come to realize it isn’t true)?

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The class I’m most nervous about this year is Geometry. It’s my first time teaching the subject, and it’s my first time teaching 9th graders. I had planned a paper folding activity which would get the students *noticing* and *wondering*. And importantly, then conjecturing.

Here are a quick snips from the paper folding activity.

The big idea was to get kids to realize that no matter which two points you chose initial to make that first fold, you’ll always end up with some things that are true in your final figure. Each student at a group folded two different pieces of paper — differently. Each group then looked at all their paper folds and made some great observations.

The couple that stood out to me:

When you make the folds, you will always end up with a *pentagon*. Some of the kids had a really small side, so they didn’t “see” the pentagon, but with a little prompting they did.

The five sided figure has *three *right angles. The two at the top (the corners of the original sheet of paper) and the point at the bottom. The point at the bottom is *always going to be a right angle. *

As a class, people shared their group’s observations. And then we focused on that third right angle. THAT IS WEIRD.

And so I sent the kids off to try to come up with some reasoning for why that third angle is always a right angle.

And this is where I had my good moments. *This is not an easy question* for the very first day of class. And with the 8-10 minutes they had, no group completely made a perfectly sound set of reasoning. But so many were making statements that were getting them closer to the answer.

A few groups noticed that the two folded triangles looked like they were always similar. (Indeed, I had totally missed that when I first did this problem…) Some kids were able to come up with the logic that if the triangles were similar, then they could actually prove the bottom point was 90 degrees. Loved it! (As of yet, no kid or group has proved that the triangles are similar.

And a few groups were also noting that when you unfold the paper, there is something really remarkable about the bottom of the page, with the two creases emanating from that point (which is the vertex of that 90 degree angle when folded). They noted that somehow — by folding up the two triangles — they are splitting the bottom of the page (a 180 degree angle) into three angles, where the middle angle is always 90 degrees.So they have a 90 degree angle, and then two smaller angles that add up to 90 degrees. I said: that’s awesome. Now how do you know that three middle angle is 90 degrees, no matter where the creases are? (As of yet, no kid or group has explained this *halving*.)

They are getting there.

Even though I was nervous about this being too hard and even though I wanted to provide more hints and more structure… I let things be. I wanted it to be tough. It wasn’t about the answer, but about the process of talking with each other and thinking and persevering and reasoning.

In fact, I wrote on the board that kids can ask me for one hint to help them if they got stuck. No group asked for a hint!

I am excited to see what happens when I next see them (Monday). I am having them try to figure this out at home, and write up whatever they could discover. If they could figure out the reasoning, great. If not, what did they think about and attempt.

My one good thing from today was watching my kids think aloud and struggle aloud and come up with really interesting ideas. Ideas I thought they might come up with, and ideas that were fresh to my eyes!

Here are all our paper foldings!

[Note: I am posting this both on the one-good-thing blog and my blog, because it both is a good thing and because it deals with teaching my classes!]

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I see that expression and my mind goes to the following places:

- Doing a bunch of tedious algebraic calculations for a particular function in order to find the derivative.
- I “see” in the expression the slope of two points close together.
- I envision the following image, showing a secant line turning into a tangent line

And I think for many teachers and most calculus students, they think something similar.

However I asked my (non-AP) calculus kids what the stood for. Out of two sections of kids, I think only one or two kids got it with minimal prompting. (Eventually I worked on getting the rest to understand, and I think I did a decent job.) I dare you to ask your kids and see what you get as a response.

What I suspect is that kids get told the meaning of and it gets drilled into their heads that they might not fully understand what algebraically is going on with it.

It was only a few years ago that I came to the conclusion that even I myself didn’t understand it. And when I finally thought it all through, I came to the conclusion that **all of differential calculus is based on the question: how do you find the height of a hole? **I started seeing holes as the lynchpin to a conceptual understanding of derivatives. I never got to fully exploit this idea in my classes, but I did start doing it. It felt good to dig deep.

The big thing I realized is that I rarely looked at the formal definition of the derivative as an *equation*. I almost always looked at it as an expression. But if it’s an equation…

… what is it an equation of? An equation with a limit as part of it?! Let’s ignore the limit for now.

Without the limit, we have an average rate of change function, between and . And since we have removed the limit, we really have a function of two variables.

We feed an and into the function, and we get an output of a slope! It’s the slope between and !

Let’s get concrete. Check out this applet (click the image to have it open up):

On the left is the original function. We are going to calculate the “average rate of change function” with an x-input of 1.64 (the x-value the applet opens up with).We are now going to vary *h* and see what our average rate of change function looks like: . That’s what the yellow point is.

Before varying h, notice in the image when *h* is a little above 2, the yellow “Average Rate of Change” dot is negative. That’s because the slope of the secant line between the original point and a second point on the function that is a little over 2 units to the right is negative. (Look at the secant line on the graph on the left!)

Now let’s change *h.* Drag the point on the right graph that says “h value.” As you drag it, you’ll see the second point on the function move, and also the yellow point will change with the corresponding new slope. As you drag *h*, you’re populating points on the right hand graph. What’s being drawn on the right hand graph is the average rate of change graph for all these various distances h!

Here’s an image of what it looks like after you drag *h* for a bit.

Notice now when our h-value is almost -3 (so the second point is 3 horizontal units left of the original point of interest), we have a positive slope for the secant line… a positive average rate of change.

The left graph is an graph (those are the axes). The right graph is a $h-AvgRateOfChange$ graph (those are the axes).

Okay okay, this is all well and dandy. But who cares?

**I CARE!**

**We may have generated an average rate of change function, but we wanted a derivative function.** That is when *h* approaches 0. We want to examine our average rate of change graph near where *h* is 0. Recall the horizontal axis is the *h*-axis on the right graph. So when *h* is close to 0, we’re looking at the the vertical axis… Let’s look…

Oh dear missing points! Why? Let’s drag the *h* value to exactly *h*=0.

The yellow average rate of change point disappeared. And it says the average rate of change is undefined! 0/0. We have a hole! Why?

(When *h*=0 exactly, our average rate of change function is: which is 0/0. YIKES!

**But the height of the hole is precisely the value of the derivative. Because remember the derivative is what happens as h gets super duper infinitely close to 0.**

We can drag *h* to be close to 0. Here *h* is 0.02.

But that is not infinitely close. So this is a good approximation. But it isn’t perfect.

And **this is why I have concluded that all of differential calculus actually reduces to the problem of finding the height of a hole. **

Here are three different average rate of change applets that you might find fun to play with:

one (this is the one above) two three

In short (now that you’ve made it this far):

- Look at the formal definition of the derivative as an equation, not an expression. It yields a function.
- What kind of function does it indicate? An average rate of change function. And in fact, thinking deeply, it actually forces you to create a function with two inputs: an x-value and an h-value.
- Now to make it a derivative, and not an average rate of change, you need to bring h close to 0.
- As you do this, you will see you create a new function, but with a hole at h=0.
- It is the height of this hole that is the derivative.

PS. A random thought… This could be useful in a multivariable calculus course. Let’s look at the average rate of change function for :

Let’s convert this to a more traditional form:

Now we have a function of two variables. We want to find what happens as *h* (I mean *y*) gets closer and closer to 0 for a given x-value. So to do this, we can just visually look at what happens to the function near y=0. Even though the function will be undefined at all points where y=0, visually the intersection of the plane y=0 and the average rate of function should carve out the derivative function.

If this doesn’t make sense, I did some quick graphs on WinPlot…

This is for . And I graphed the plane where y=0. We should get the intersection to look like the line .

Yup. Cool.

I did it for also… The intersection should look like .

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**They deal with u-substitution.**

But here’s the thing. **For my kids, it’s just a random method to evaluate an integral. They don’t conceptually understand what is going on… **what this changing of variables is doing.

When I thought deeply about this, I realized what truly is happening is that we are transforming space… From the plane to a much convoluted plane. But it is through our particular choice of that makes the change in space beautiful, because it turns something that looks particularly nasty and converts it into something that looks rather nice. Ish.

Here is a screenshot from one of my geogebra applets illustrating this (you can click on the screenshot to be taken to the applet):

We start with a pretty ugly function that we’re integrating. But by using this substitution to morph space, we end up with a much nicer function. I mean, throw both of these up and ask your kids — which one of these would they rather find the integral of. They’ll say the one on the right! The u-substitution one. Although not perfect [2], it’s pretty kewl.

The applets are here:

And the applets are dynamic! You can change the lower and upper bounds on the graphs and the lower and upper bounds automatically change on the graph! But because math is awesome, the areas are preserved!

Some things I maybe would have done with the applets in my class:

- Let kids play with the applets and get familiar with them.
- For the first applet (starting simple), have kids count the boxes and estimate the area on one graph, and then do it on the other (careful though! the gridlines are different on the two graphs!). Whoa, they are always the same!
- For the first applet (again, starting simple), ask them to drag the upper limit to the left of the lower limit. Explain what happens and why.
- The second applet is my favorite! Put the lower limit at x=0. Drag the upper limit to the right. Explain what is happening graphically — and that tie that graphically understanding to the particular u-substitution chosen.
- In the second applet, can students find three different sets of bounds which give a signed area of 0?
- In the fourth applet, have students put the lower and upper bounds on x=6 and x=7. Have them calculate the average height of that function in that interval (the area is given!). Do they have visual confirmation of this average height for this interval?Now Looking at the u-graph, the bounds are now u=8 and u=10. Have them estimate the average height of that function in that interval (again, the area is given)! (The average height “halves” in order to compensate for the wider interval. It has to since the areas must be the same) Have students do this again for any lower and upper bounds for this graph. It will always work!
- In the fifth applet, have students put the lower bound at x=0, and have them drag the upper bound to the right. What can they conclude about the areas of each of the pink regions on the graph? (Alternatively, you can ask: you can see from the graph that the signed area on the original graph will never get bigger than 1, no matter what bounds you choose. Try it! It is impossible! Armed with that information, can you conclude about the pink regions in original graph?)

I’m confident I had more ideas about how to use these when I made them [3]. But it was over a year ago and I haven’t really thought of them since. **But anyway, I hope they are of some use to you. Even if you just show them to your kids cursorily to illustrate what graphically is going on when you are doing u-substitution. **

***

[1] Though I bet if you asked a class *why* they can use “substitution” when solving a system of equations, *what the reasoning is behind this method*, they might draw a bit of a blank… But that’s neither here nor there…

[2] What would actually be perfect would be a copy of individual Riemann Sum rectangles from the graph “leaving” the first graph, then in front of the viewer stretching/shrinking their height and width for the appropriate graph, and then floating over to the graph and placing itself at the appropriate place on the axis. And then a second rectangle does that. And a third. And a fourth. You get the picture. But even though the height and width morph, the area of the original rectangle and the area of the new rectangle will be the same (or to be technical, very very close to the same, since we’re just doing approximations). In this sort of applet, you’d see the actual *morphing*. That’s what is hidden in my applets above. But that’s actually where the magic happens!

[3] I recall now I was going to make kids do some stuff by hand. For example: before they use the applets, kids would be given lower and upper x-bounds, and asked to calculate lower and upper u bounds. And *then* use the applets to confirm. Similarly, given lower and upper u-bounds, calculate lower and upper x-bounds. Use the applets to confirm.

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