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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Although I know most of the basics of Geogebra, I have not yet progressed to the stage passed “novice.” However I really want to get there, because *this program is so freaking awesome.*

When I was at TMC14 this summer, there was a sesh run by John Golden, Audrey McLaren, and Jedidiah Butler. They are like Jedi masters of Geogebra (though I know Audrey will play coy and say she isn’t…). When I was there, I learned about conditional objects, and it was awesome. (The google doc they used to help people out is here.) In about 30 minutes, with the help of John Golden and some kind people near me, I was able to make a rinky-dinky geogebra file which has a triangle on it, and has three points on the three different sides. When you drag each point close to where an altitude of the triangle would hit that side, I had something like “WOW!” or “YOU DID IT!” pop up! And if you got all three points close, something like “ALL THREE?! YOU’RE A SUPERSTAR!” show up.

Okay, okay, I wasn’t going to show you it because it’s sooooo dumb. But heck, whatever, here it is. Click on the image to check it out.

Okay, you and me, we both know that file is totally useless a teaching tool. And it is gross looking. By all accounts, I should not be excited by it. But the weird thing was: *I was really proud of it*, and *I wanted to show everyone around me what I created*. Even though I know it was simplistic and useless, I wanted to create a file that did X and I was able to do it! Although it felt dumb to get psyched about it, I was so excited that I could create something that would do what I wanted it to — that I couldn’t do before!

Today I was again inspired by my colleague and friend’s geogebra art, so I wanted to create some of my own.

I was quickly able to make this in 10 minutes [click on the picture to go to the file and mess around with the parameters! cool things happen!]

My goal was to define a curve parametrically and then have — at a ton of points on the curve — a circle to be drawn so it would look like a tube. That ended up looking only moderately neat. So I changed it so that as one traveled on the parametric curve drawing the circles, the radius of the circles would change (based on some formula I fed it). The reason this wasn’t so hard for me? I knew all the commands to do this except for the parametric curve command, which was easy to figure out.

But then I wanted to try my hand at something that would take more than 10 minutes and that would challenge me. I wanted to have something “show” a sphere via the animated drawing of “slices” (ellipses). It was inspired by this beautiful gif, but I knew that was going to be too hard for me to start out with. So I decided I would start out with a simple sphere with slices going horizontally and vertically, with no rotation.

After somewhere between 90 and 120 minutes, I did it! (You can click on the gif to go to the file and play around with some of the parameters.)

Although the image isn’t as cool as the one that took me 10 minutes to create, *I’m way prouder of this*. It is because it took a ton of learning and trial and error in order to figure out how to do this. The set of problems I encountered and somehow figured out:

- I know how to create a single ellipse in the center of the circle, but how do I make another ellipse a certain distance away that still only touches the edge of the circle?
- How do I make the ellipses “width” (minor axis) decrease so that it is fattest near the equator, and almost like a line near the poles of the sphere?
- Without manually typing a zillion ellipses, how do I tell Geogebra to create
*all*the vertical ellipses at once, and*all*the horizontal ellipses at once? - The way I was generating the ellipses resulted in a problem… once an ellipse “hit the pole”(became a point), it would turn into a hyperbola. So I needed to find a way to make sure that once an ellipse “hit the pole” it would disappear.

I figured all this stuff out! So even though the sphere doesn’t look nearly as cool as I’d like, I feel so much more accomplished for it than with the super-cool-looking circles of variable radii drawn on a parametrically-defined curve.

***

Note: it’s amazing how “simple” this sphere image is once you figure it out. Once you create three sliders:

t goes from -5 to 5 [incriments of 0.1]

StepSize1 goes from 0.05 to 2 [increments of 0.05]

StepSize2 goes from 0.05 to 2 [increments of 0.05]

and you enter the following *two *(that’s it!) geogebra commands:

Sequence[If[abs(t – n StepSize) < 5, x² / (25 – (t – n StepSize)²) + (y – t + n StepSize)² / (1 – sgn(t – n StepSize) (t – n StepSize) / 5)² = 1], n, -5 / StepSize 2, 5 / StepSize 2, 1]

Sequence[If[abs(t – k StepSize2) < 5, (x – t + k StepSize2)² / (1 – sgn(t – k StepSize2) (t – k StepSize2) / 5)² + y² / (25 – (t – k StepSize2)²) = 1], k, -5 / StepSize2 (2), 5 / StepSize2 (2), 1]

Then you’re done! Well, you should animate the t-slider to make it cycle through everything without you having to drag the slider!

Seriously, two commands, that’s all it takes. But hopefully from the commands themselves you can understand why it would take me so long to figure out…

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To begin with, an oft retweeted and favorited thing at TMC was this (and heck, I probably retweeted and favorited it):

I heard #tmc14 described this way yesterday. 150 teachers who all believe they can change the world. Here they are. http://t.co/Xa2bz01o9r

—

Glenn Waddell, Jr. (@gwaddellnvhs) July 24, 2014

I *strongly* don’t put myself in that category. That isn’t part of who I am. I don’t teach to save the world. I don’t see myself as changing students’s lives, nor is my goal to have students come back to me and say “your class changed my life.” I don’t blog to change math teaching. I don’t have grand ambitions or even care to think on such a large scale. It’s a nice sentiment, but it’s just not me. I’m like a hobbit, happy and content with my little corner. Working at things with a small scale. Getting an ah hah moment, or altering the way a student sees and understands mathematics, or helping a fellow teacher out with this or that. This is what I enjoy doing.

With that said, a lot of my thoughts around the mathtwitterblogosphere (MTBoS) and TMC in particular have come out of two things:

1. A post by Mo titled “I am a fraud” and as a follow up a post by Lisa Henry titled “Hi My Name is Lisa” which resonated with me and with many others. Some key lines:

Mo: They were so honest, so completely naked, and I, wanting to join in “fit in” offered some of my fears but then as I awoke today I feel dirty. My heart is heavy, because I lied. Well I didn’t completely lie I just shared certain fears and strengths that manipulated people to see me the way I wanted them to see me. We were all skinny dipping but I had a flesh colored bathing suit on “with painted on abs”… And as I enter my 9th year of teaching, I could be entering my last year. There is a high possibility that I could be going into sales and this conference confirms my movement into that field, because I feel so inadequate….. so beyond inadequate.

Lisa: I am not the best math teacher. I am not an amazing math teacher. I have a LOT of work to do to improve. There. I said it. I wrote it in my blog and I am not taking it back. It is there in print. Ever since I have been involved with the Math Twitterblogosphere (MTBoS for short if you are not familiar), I have felt this inadequacy. I see what other teachers are doing in their classrooms. I have tried some things. Even blogged about what I have tried. But for the most part, I haven’t changed a whole lot in my teaching since I started Twitter almost 5 years ago.

2. A flex session held by Lisa Bejarano which was about how we leave TMC and change things — both in ourselves, and in our schools. I have personal notions about barriers involved changing things in my school, nor do I have the presumptuousness to say that I am The Person To Effect Change or that I Know The Right Ways. Because I too feel very much like Mo and Lisa in that I’m not “there” yet. I’m not a master teacher. However when it came to our discussion of how we change things in ourselves, Lisa B. threw up this great chart on the projector:

At first, I was skeptical. I have this bad habit of seeing charts like this and immediately dismissing them. They take something complex and box it in to something simple. But you know, the more I thought and looked at it, the more it made sense to me.

For example, I’m teaching geometry next year, and I have tons of resources, a vision for what I want the course to look like, I think I have the skills needed, I have the motivation/incentives to do well. But I don’t have a way to go through the massive amounts of ideas and resources to actually move forward. I don’t have an action plan. So I’m in the middle of false starts. And it feels that way.

But what I want to focus on is missing *skills — *and the resulting anxiety. I was thinking about the times in my classroom when I didn’t just take a little step forward but dove right in to make a big change because I had a bigger vision I wanted to accomplish. I can think of two:

1. Implementing standards based grading in calculus (four years ago)

2. Running a class entirely through group-work (two years ago)

Although I advise people to take baby steps, and change slowly, those times I didn’t take that advice were the times I grew the most as a teacher. Those were also the times I felt the most anxiety. Why? Because I hadn’t yet developed the skills I needed. I didn’t know how to organize standards based grading in a way that would work. I didn’t know how to make sure I would catch the conceptual as well as the procedural in this system, nor how I would get synthesis of skills. I didn’t know how to make sure students worked well together. I didn’t know how to create actvities and lessons so that students would have to rely on each other to progress. But you know what? Without jumping in, I never would have gotten the skills needed.

Let me tell you: those were high anxiety times. They required a lot of emotional energy and a ton of time. But they were also times of immense growth for me.

Now to the “me” part.

I don’t feel like I’m a master teacher or close to it. And I feel confident with that assessment of myself, because who knows me better than me?

As part of that, I also often feel inadequate, and sometimes like a fraud. If I look only at the world of my school, I think I wouldn’t feel this way. I’d feel fine. I’d actually have very little incentive to change, because it’s a lot of work for no extra rewards and I am doing well by the kids (see the chart above). But because there is this much bigger world, which I am exposed to (namely the math-twitter-blogosphere), things are different.

I am constantly exposed to many things online. A lot of them are resources and lessons, but sometimes there are ideas about good teaching that I wouldn’t have access too. Like the importance of mathematical discourse (talking, writing), or to question the nature of assessments and what grades mean, or the importance of having students see each other as mathematical authorities. I am not exposed to these ideas in my school constantly, so I would not think that these are things I believe in. But being bombarded by all the stuff out there online and at TMC, and seeing what resonates with me (or what inspires me to change), is helping me (probably subconsciously) evolve my personal, theoretical framework about teaching and learning (thanks Dan Meyer).

And that is where the anxiety comes in. Because now my bar about what is good teaching has been thrust upwards. And now I have to work on reaching it. It isn’t that I feel competitive with others, but that I feel competitive with myself [1]. I have a drive to be my personal best, and to do the best by my kids.

So because of this exposure to great ideas for the classroom, and bigger ideas about what makes an effective classroom, I get caught up in feeling like I’m not doing a good job, and the anxiety hits me. And sometimes this nadir will last for *months*. I don’t feel like I’m doing a good enough job in the classroom. I haven’t given any formative assessments. Kids aren’t engaging in real mathematical discussion. I haven’t improved at all from the previous year. Heck, maybe I’ve even de-evolved. And I get in this cycle of anxiety. It sucks.

But it’s a double edged sword, because it is this anxiety that drives me, that pushes me. I enjoy the intellectual challenge it gives me. And at least for me, it’s this anxiety and this feeling of inadequacy, coupled with my own personal desire to better myself, that provides a productive tension. I recognize in myself that I need those lows and those feelings of anxiety in order to get better. It’s part of my own personal growth process. And as much as I wish I could be confident and grow without the feelings of inadequancy, I’ve come to realize that’s what works for me. At least for me.

I end with a tweet from Jami Danielle who pretty much summed this up for me, and makes this whole post just a bunch of verbal spewage (didn’t I say that at the beginning)?:

[1] I suspect, though I do not know, that all this talk about “inadequacy” and how it resonates with so many people in the MTBoS and at TMC is tied up in some sort of cycle like this. Because of this, I don’t think it’s something to be “fixed” (e.g. how can we make it so when people come to TMC they don’t feel like crappy teachers?). At least I wouldn’t want someone to “fix” it for me.

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***

I’m going to make a short post inspired by Twitter Math Camp 2013 (TMC13), rather than TMC14. Both @calcdave and I led morning sessions for precalculus teachers. Through that morning session, some nice end-products were created — an organization for the curricula, actual classroom activities — and you should feel free to check them out here. [1]

@calcdave and I brainstormed how we could get people in the morning session to know each other, but make sure we have math content in that activity. We came up with *Rational Function Headbandz*, which was inspired by this post on *the agony and dx/dt*.

**The setup: **There are a bunch of cards (they could be index cards). On the front of them is an graph of a rational function. On the back is the equation of the rational function. The cards are attached to ribbons or headbands, so that when attached to the forehead only *other* people can see the graph on the front of the card — not the person wearing it. Sort of like this image below. You can re-imagine how to create these cards/headbands so they work for you.

**The Goal: **Since this was an introductory activity, participants picked one of two goals for themselves… (a) to figure out as many features as they could of their rational function and to sketch a graph from those features, or (b) to figure out the equation of their rational function.

**To Play: **I put all the cards/headbands on the table, and *covered up the graph with post-its* so the participants couldn’t see the graphs. I wrote on the post it if the graphs were graphs I considered sort of challenging, pretty darn challenging, or wow-you’re-going-for-it challenging! Then they attached their headbands to their head, and had someone else remove the post-it note.

Before starting they were told the following things about their rational functions:

- All the graphs are of rational functions.
- Some might be plain old polynomials. (Rational functions with the a 1 in the denominator!)
- If written in the most factored form, none of the terms has degree of more than two
- If written in the most factored form, most of the coefficients are really nice

Each person carried around with them a notebook, and they were allowed to ask *up to three questions* about the graph to each person (and a get to know you question to each person!). The rub? All questions had to be answered with a single word or a single number.

A valid question: “How many holes does my graph have?”

A valid question: “Is my rational function a line?”

A valid question: “Does my rational function cross or kiss the x-axis at x=3?”

An invalid question: “What is the coordinate of the hole?” (Because the answer will have two numbers as an answer — an x-coordinate and a y-coordinate.) You could instead ask “What is the x-coordinate of one of the holes of my graph?” and then follow up with “For the hole with x-coordinate BLAH, what is the y-coordinate?”

After three questions, they move on to a different person. Then another. Et cetera. From these questions they were supposed to gather information about their graph, and possibly about their equation.

You stop the game whenever you want. Everyone looks at their graphs and equations, and ooohs!, dohs!, and aaahs! result.

And then if you have time, you can debrief it with students by talking about what they thought was important information to gather in order to sketch or come up with the equation for the graph (holes? x-intercepts? y-intercepts? vertical asymptotes? horizontal asymptotes? slant asymptotes? end behavior?). And then if you had time you could have individual students present their graph, their thought process, and their solution.

**Our Graphs: **We really varied the nature of the graphs because we were working with precalculus teachers and we didn’t know their ability level with the material. And also I know I emphasize in my class working backwards from the graph to the equation, but that isn’t a standard thing taught. So I would highly recommend creating graphs of your own based on the level of work that you’re doing in your class.

**Trouble Spots: **One thing that was challenging for us when we played this was what someone does when they have figured out their own equation/graph. They came to us and we confirmed. But then what? We should have anticipated this because we had such varying levels of difficulty for graphs. I wonder if a good solution would be to then try to figure out the equation for the rational functions of others when they are being asked questions.

Another thing to keep in mind is that this will take a longer time than you think. We used this as a get-to-know-you activity, and so that extended everything even more. (In your class, your students probably won’t be using this as a get to know you activity.)

**Alternatives: **Just as I adapted this from a teacher using them for trig functions/graphs, these can easily be adapted for other topics. Some initial ideas:

Geometry vocabulary review: Students have a vocabulary word on their heads. They only can ask questions with one-word answers. (e.g. “Does it have to do with parallel lines?”)

Polynomial graphs (instead of rational function graphs), or even just parabolas [update: Mary did this!], or even just lines.

Students have derivative graphs on their heads, and they need to come up with a sketch of the original function (for this they should be allowed more than one-word answers).

[1] One thing I worked on in a group with four other people is how to get students to understand inverse trigonometric functions (a topic we collectively decided was challenging for students to wrap their heads around). I blogged about the result of our work here. I used it in class this past year, and although I didn’t use it completely as intended, it did really push home the meaning of what sine and cosine were graphically (the y- and x-coordinates on a unit circle corresponding to a given angle) and then what inverse sine and inverse cosine were graphically (the angles that are corresponding to a given y- or x-coordinate). Check it out!

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This was prompted due from a 30 minute mini-sesh that Justin Aion had around his 180 blogging adventure this year. For those not in the know, a 180 blog is something teachers started doing a couple years ago — posting once a day. (It is called a 180 blog because there are supposed to be — though I definitely don’t have — at least 180 school days in an academic year.)

The difference between regular blogs and 180 blogs are that 180 blogs tend to be a single snippet, every day. Sometimes it is just a photograph. Sometimes it’s just a paragraph. Sometimes it’s a brief reflection. And you know what? *You know what?*

I kept a 180 blog last year too. And I just realized I never mentioned it on this blog, nor did I ever give it a post-mortem or reflection. So tonight, the first evening of TMC14 inspired by a mini-session, that is what I am going to do.

My 180 blog all started because I have an incredible colleague and friend at my school who I know would get along with this community of math teachers online like gangbusters. I wanted to bring him into this world, but it stressed him out too much, and moreso, he didn’t have that much time. I took a stab at ensnaring him by showing him the idea of the 180 blog. It has a low barrier of entry. It involves only 5 minutes a day. And it has a basic structure to it that he could routinize: make a post each day. He agreed! We would both keep a joint 180 blog!

And thus: the very cleverly named ShahKinnell180 blog was born. (Click on the image to be taken there!) [1]

Back to Justin’s TMC talk. He spoke about how he wanted his 180 blog to be centered around *reflectiveness*. And I think that many people do use them for that. However I was 100% sure that reflectiveness *wasn’t* something I was looking for.

Besides getting my colleague/friend involved in this online math teacher world, I think my reasons for wanting to do this are as follows:

- I wanted a little archive of my teaching life. So the only rule I had in making it was that I would post a picture every day, and a few words. Nothing expansive, nothing overwhelming. I had in mind those people who take a photograph of themselves everyday for a year, and then splice them all together, resulting in this whole pastiche of the passage of time? I revel in the fact that I now have this little slice of my teaching life all beautifully laid out. Visual. Chronological. And what I kinda love the most: just like the blog is filled with snapshots of things that happened to me-as-teacher (usually from my classes, though not always), the blog itself is now a snapshot of who I am as a teacher.Although I haven’t done this yet (why not??? well I didn’t even think about writing about it here until after it was done for a whole year! so who knows where my head is at), I would love to send it to my parents. Heck, it’s a great way for non-teachers (wow, this could be awesome for teachers-to-be too!) to see a depiction of what people in our profession do, what we get our kids to do, what we think about, what experiences we have. It’s like a regular blog, but less reading — perfect for skimming and being non-threatening!
- I wanted something to keep me on the lookout for the good. My brain constantly tells me I am not good at what I do. And I am someone who can obsess over what’s not going right and just skip over the juicy deliciousness in front of me. (I was that kid in high school who would take a test, get stuck on one or two questions, and leave saying I knew I did horribly on it… not because I was being modest, but because I would focus totally on what I didn’t know, instead of seeing things in perspective.) All this brings me back to a few years ago when I was a contributor on the “One Good Thing” blog (my posts on that blog are here). If you don’t know about that blog, it is a collaborative blog where teachers just write something good — anything good — that happened. Big or small. The tagline to the blog is: “every day may not be good, but there is one good thing in every day.” That’s some powerful stuff. And you know what? Because I was posting on that blog, I had a shift in my mindset. Even in my worst days, especially in my worst days, I would force myself think back through the day for something good. And heck if I couldn’t find something. And then I started paying more attention to the good that was happening
*when*it was happening (I would think: “Heck yes! I need to blog this!”).I wanted my 180 blog to remind me that I do good things in the classroom. Even when I feel like I’m stagnant, when I’m not innovating, when my kids are lost and I’m at fault… I wanted my 180 blog to keep me on the lookout for things that I should feel proud of. Not every post is a “feel good” post on my 180 blog, but the point is: I was constantly on the lookout for something I would want to post about or an image I wanted to save from the day. - Finally, and probably least important to me, I wanted something to keep me accountable to being a good teacher. This probably sounds a bit weird… but as a regular blogger, I noticed I would get extra enthusiastic about something when I knew I was doing something or creating something and realized I could blog it. When my classroom wasn’t the only audience, and when what we did just disappeared in the temporal aether. Perhaps a 180 blog would help me do the same?

I don’t have any grand pronouncements from the experiment. I definitely didn’t learn anything about teaching from keeping the 180 blog. I am almost certain I will not return to the 180 blog for teaching ideas, or to see how a particular lesson went. I definitely did not become a better teacher because I kept the blog. (At least not in any tangible way.)

But here’s the thing: looking at this experiment on the whole, I am **beyond thrilled I started my 180 blog and kept up with it. **Why? Because when I have moments (be it days, weeks, or even month-long-stretches) when I feel like I’m not doing a good job, I simply can pull up the blog and browse through it and recognize:

*I don’t do the same things every day. I am thoughtful about stuff a bunch of the time. I have pretty great kids who do some pretty great and possibly hilarious things that are worth recording/remembering. *

Which them reminds me: I’m lucky that I get to do what I do. I enjoy thinking about what I get to think about. I really do enjoy working with kids (which definitely needs reminding because… well kids are rarely easy). And that: if this is my job, if this is what I get to do and get paid for it, then things are pretty great.

So as I noted, I was blogging mainly to archive. And archive I did. I have no desire to archive again next year. However I had been thinking at the TMC14 session I was at: is there anything that could get me to do another 180 blog?

And I dawned on the answer. I could create a 180 blog around *one* specific thing I was working on as a teacher. And this 180 blog would force me to stay accountable.

Examples:

- I’m not an expert at deep questioning in the math classroom. So I would be forced to blog about one question I asked, if I had time write about some of the context in which the question was asked, and what happened when I asked it in the math classroom. I would then briefly evaluate whether the questioning was good and/or if there was a better way to have asked the question.
- I am trying to make groupwork the central way kids in my classes learn. So I could write one blogpost each day about how I facilitated some part of groupwork — either in the planning of the class, during the class, or after the class.
- I am trying to be more conscientious about formative assessments. So I vow to have one formative assessment each day in one of my classes (not even all of them! just one!). It doesn’t have to be even a big one… even a 10 second “thumbs up if you get this, thumbs to the side if you’re slightly confused, and thumbs down if you’re totally lost” counts.
- I struggle with wait time. So each day, I vow to record with a timer how many seconds I wait after one question (only one question!), and I post the question and the wait time on the blog.
- I know I’m terrible at “closing” class. I have kids work until the end, we rarely take the time to summarize what we did, the big questions we tackled, the big questions we have lingering. Very often it is: “Eeep, sorry, we’re out of time. Check the course conference for your nightly work. Missyouloveyou!” Okay, maybe not the missyouloveyou part, but you know what I’m talking about. So blogging about the close of one class each day.

I’m not saying I’m going to do anything of these. If I do, it will definitely only be *one* of them. But the idea is that it is targeted about something I want to improve upon, and doing it will hold me accountable.

[1] As a follow up, my colleague who did the 180 blog with me blogged many — but not all — days. But heck if he’s not been so inspired that he’s starting Geogebrart, his own blog about making art with geogebra which has been knocking my socks off this summer. Once you peruse his entries on our 180 blog and you peruse his new Geogebrart blog, you probably understand why I feel lucky beyond belief to get to work with this guy!

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