A Follow Up (regarding Centers of Rotation)

It’s 10:44pm on a Saturday evening, and I have been thinking about math. Whoooo hooooo! I finally got a chance to ponder how I’m going to attack this question that I posed earlier this week. For those too lazy to click:

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.

Mind Blown

[Cross Posted on the One Good Thing Teach blog]

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:

perpbis

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:

perpbis2

 

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.

perpbis3

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:

perpbis4

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.

Substitution (…and Continued Fractions)

Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:

a_{n+1}=\sqrt{2+a_n} where a_1=\sqrt{2}

So obviously they go like: a_1=\sqrt{2},a_2=\sqrt{2+\sqrt{2}}, a_3=\sqrt{2+\sqrt{2+\sqrt{2}}}, a_4=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}, and a_5=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}

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

\approx 1.414, \approx 1.848, \approx 1.961, \approx 1.990, \approx 1.998, \approx 1.999, \approx 1.9998, \approx 1.99996, \approx 1.999991, \approx 1.999997647

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 \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}} [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.

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.

image1

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

image (3)

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 1+\frac{1}{} where I left the denominator blank. And then I put the small card (fraction side up) in the denominator of the fraction…

image (4)

And I said… what does this whole thing equal?

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

image (6)

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

Ready? READY?

Flip.

image (7)

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 x… and x 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 x=\frac{1+\sqrt{5}}{2}\approx 1.618 (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 a_{n+1}=1+\frac{1}{a_n} where a_1=1… and we saw: 1, 2, 1.5, 1.66666666, 1.6, 1.625, ...

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 \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}?

Let’s say \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}=x

Then we can say \sqrt{2+x}=x

And even simply by inspection, we can see that x=2 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 a_{n+1}=1+\frac{1}{a_n} where a_1=1.

2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, .... WHAAAA?!?!

Lovely. It’s all coming together!!!

You Spin Me Right Round: A Challenge for Geometry Teachers

We are now delving into some interesting coordinate geometry. We’re also beginning to use patty paper. And today the other geometry teacher and I had an awesomely fun conversation that revoles around something you might find it fun to think about.

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.

rot2

 

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!)

image (17)

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?

 

 

Attacks and Counterattacks in Geometry

It’s been a long while since I’ve posted. It isn’t because I have nothing to post about! I’ve just been sooooo busy. This is the first year I’m teaching Geometry, and I’m working with the other teacher to turn it on it’s head. Completely. We haven’t cracked the textbook yet.

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:

define

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.

def1

def2

def3

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:

counterattack

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:

quiz3

quiz2

***

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 languageThings 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:

image

[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)?

Interesting conjectures

Today was my first day of classes — thirty minutes with each class just as a get to know you. Of course you know me. I dive right in and we did math in all of my classes.

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!

20140909_125022

 

[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!]

The Formal Definition of the Derivative, or Why Holes Matter

Lucky you! Two calculus posts in one day. Mainly because I don’t want some of these ideas to disappear in my hiatus from teaching it. This one deals with our favorite topic: the formal definition of the derivative.

\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{(x+h)-(x)}

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
    zzzz

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

However I asked my (non-AP) calculus kids what the h 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 \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{(x+h)-(x)} 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…

f'(x)=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{(x+h)-(x)}

… 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 (x,f(x)) and (x+h,f(x+h)). And since we have removed the limit, we really have a function of two variables.

AvgRateOfChange(x,h)=\frac{f(x+h)-f(x)}{(x+h)-(x)}

We feed an x and h into the function, and we get an output of a slope! It’s the slope between (x,f(x)) and (x+h,f(x+h))!

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

zzzzz1

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: f(1.64,h)=\frac{f(1.64+h)-f(1.64)}{h}. 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 (1.64, f(1.64)) 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.

zzzzz2

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 x-f(x) 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…

zzzzz3 Oh dear missing points! Why? Let’s drag the h value to exactly h=0.

zzzzz4

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: \frac{f(x+0)-f(x)}{(x+0)-x} 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. zzzzz5

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 f(x)=x^2:

AverageRateofChange(x,h)=\frac{(x+h)^2-x^2}{h}

Let’s convert this to a more traditional form:

z=\frac{(x+y)^2-x^2}{y}

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 f(x)=x^2. And I graphed the plane where y=0. We should get the intersection to look like the line f'(x)=2x.

zzsine

Yup. Cool.

I did it for f(x)=\sin(x) also… The intersection should look like f'(x)=\cos(x).

zzsine2