# Snug Angles

In Geometry this year, I wanted to write a few more problems to have kids practice with angles of regular polygons… so as I was coming up with a few problems, I realized they had a nice theme to them.

“Which polygons fit together snugly? Which don’t?”

[02 Snug Angles download][Note: There is a typo on #6… It refers to problem 2d, but it should refer to problem 2c]

I made this the day before the class I was going to teach it. But I wanted to have a hands-on “playful” component to this. I asked teachers in my school if they had regular polygon tiles with the same side length… I got a set which included triangles, squares, and hexagons. No pentagons, no heptagons, no nonagons, no decagons, nada.

Of course these tiles were probably produced for lower school kids precisely because they fit together “snugly” at a vertex. But no “play” could really happen if sometimes things didn’t fit nicely together. So — for future reference — I asked on twitter to my math peeps if anyone knew where I could buy regular polygon tiles of all sorts. No links were forthcoming. Sigh.

In class, I expected #1 to be challenging. I wanted students to come up with a reason they had found all the possible regular polygons (of one kind) that fit snugly together. It was nice to see students reason through it, and when we came together as a class, we had a few different cogent explanations. Some involved calculating all possible factors of 360. Some involved recognizing that the more sides you have in a polygon, the fewer of them can fit together “snugly” at a vertex (and the minimum number of polygons that can fit at a vertex is 3).

Although I was expecting #2c to be challenging, I didn’t realize how challenging it would be. I thought I had built a scaffold with the previous problem so it wouldn’t be too hard. What turned out to be the problem? The fact that a regular 7-gon had a non-integer interior angle value. Kids didn’t know that could happen, and that really threw them. Also: setting up the equation was challenging, because kids were confusing “the sum of the interior angles in a regular n-gon” with “the measure of one interior angle in a regular n-gon” (a calculation they had never been formally taught, and were supposed to figure out themselves during this exercise).

I’d say only about half the groups could deal with 2c without any help.

However, all groups ended up being successful. And I just graded their assessments on polygonal angles, and almost every single student got the problem that was similiar to 2c!

The very last question asks students to discover as many possible combinations of regular polygons that could fit together snugly at a vertex. I assigned this as a nightly work problem — and the next day, students came in with lots of great combinations. Unfortunately, I didn’t do anything with this. I should have — but I felt pressed for time.

We could have talked about why 6 polygons were the maximum number that could fit together, or 3 polygons were the minimum number that could fit together. That could reduce our searching! Then I could have asked how people approached the task. Guess and check? Geogebra? Is there a systematic way they could have approached this problem — if they had infinite time and patience — that they could guarantee they had found all possible combinations? Do all combinations need at least one 3, one 4, one 5, or one 6?

Or we could have spent some more time looking at all possible combinations. Some kids noticed — after looking at the comprehensive list I threw on the board after they finished sharing their values with me — that many of the values had common factors: so 3, 7, 42 is one crazy combination that works. And both 3 and 7 are factors of 42. What else could we find?

What I’m trying to say is: the last question was kind of a dumb question to put on the sheet without having a good way to debrief it, and a meaningful conclusion we could have gotten from it. Sigh.

Okay, on to the exciting part. I said I asked on twitter if anyone had a site to buy these tiles. No responses. BUT Christopher Danielson then asked what I was looking for. Kate Nowak jumped on the bandwagon and brainstormed what a teacher might want, ideally. Yesterday, I came home from school and had a box waiting for me. In it:

They are beautiful. And gosh do they smell awesome. Real wood, that smells awesome. I was in heaven when I saw them. So beautiful.

And even more satisfying: you’ll notice that the 3, 7, and 42 fit snugly together!

Now the million dollar question: assuming I had however many of each tile I wanted, what would I do with them? How would I restructure the unit to use them in a way that is compelling? I wanted the tiles initially because I thought some “play” with the tiles would be fun, before delving into the algebra to see the justification of why some work and some don’t work. But I want something more! Something that will have them figure out the 3, 7, 42 connection and gasp! And the 4, 5, 20. And the 3, 8, 24. And the 3, 10, 15. And the 4, 5, 20. And GASP with surprise and horror and delight!

I don’t quite know… But maybe envelopes with index cards in them. And some of the index cards have some configurations they have to “check” to see if they work or not. And some of the index cards have two of the tiles, and students have to see if there is a third tile that works. And for each configuration that works, students get to come to the front of the room, grab those tiles, and check to see if their algebra worked by checking to see if the tiles truly do fit snugly. If they do: they record their discovery on the board for all to see. And by the end of the class, students will have had practice, and in the last 5 minutes, we could all gather at the front, and view some of the weird snug angle configurations together. And see how configurations that are “close but no cigar” don’t work (like 3, 10, 16… which is close to 3, 10, 15). When doing this, we could also talk about why 4, 10, 15 is “worse” than 3, 10, 16 in fitting snugly.

That’s all my musings for today! I’m going to be chaperoning a trip to Spain in a few days, and that will last two weeks, so goodbye for a while!

# Interested in Presenting at TMC16?

We are starting to gear up for TMC16, which will be at Augsburg College in Minneapolis, MN (map is here) from July 16-19, 2016. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC16-1). It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Saturday, July 16 and 48 one hour sessions that will be either Saturday, July 16, Sunday, July 17, or Monday, July 18). That means we are looking for somewhere around 70 sessions for TMC16.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 18, 2016 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Muir, Jami Packer, Megan Schmidt, Sam Shah, Christopher Smith, and Glenn Waddell

# Clock Puzzle

In our last department meeting, one teacher presented a puzzle/problem for us to figure out.

At 3:00, the hour and minute hands on a clock form a right angle. What is the next time that happens?

The presenting teacher had a pretty darn elegant solution. But I enjoyed working it out using brute force. (That’s pretty much my go-to.) I’m going to type my solution down below the jump.

# Playing with Blocks: Three Dimensional Visual Sequences

During this school year, we now have occasional 90 minute blocks with our classes. I was trying to decide what to do a couple weeks ago with my precalculus class, and stumbled upon the embryo of a good idea. Kids playing with blocks to create 3D sequences. (This idea was inspired by Fawn Nguyen’s site Visual Patterns.)

I got blocks from our lower school math coach. I told kids (either working individually or in pairs) to play around with them until they found a pattern that looked interesting to them. I didn’t want them thinking about the sequence yet… I wanted them to create patterns that looked neat. The only restriction I put on them is that the pattern had to be three dimensional. If it could be represented in two dimensions, I didn’t want to see it.

They made some really nice sequences! Here are a random set of 4 to look at:

I then had students work on filling out this form. It asks them to articulate their “rule” (for building up the sequence) and has them attempt to come up with both explicit and recursive forms to get the nth term. I make it clear to them that if they can’t get the formulae, I’ll give them full marks as long as they show a serious attempt. (Some of the sequences they built involve some mathematical hoops they might not be able to traverse… for example, one group needed to find $1^2+2^2+3^2+...+n^2$ which is lovely, but not something they are going to easily figure out.

[.docx version here]

If I had time, I’d love to do two more things with this.

(1) I think it would be neat to take the photographs of one person’s sequence and give them to another person, to see what they figured out for the explicit and recursive definitions for these sequences. Why? Not only is it sharing more publicly the sequence the kids created, but many of them got a bit stuck on an explicit formula that they do have the capabilities to find, but couldn’t. I think a fresh pair of eyes, and a conversation, could be beneficial for both the original sequence creator and the new person approaching the sequence. (Additionally, there are often many ways to look at these sequences, so even if both got the same formula, there is a good chance they came up with it in different ways.)

(2) Students created a table with the first 5 terms of the sequence in it. I’d love for students to extend the table to 7 or 8 terms in the sequence, and then have students work on finding the first differences, the second differences, the third differences, etc. If students understand that having the same first difference means they have a linear relationship, having the same second difference means they have a quadratic relationship, having the same third difference means they have a cubic relationship, etc., then students who got stuck will have a new tool in their arsenal to find the explicit formula for the sequence. If, for example, they had 5, 9, 15, 23, …, and saw a common second difference, they could do the following:

Since they suspect the relationship is quadratic, they could say: $t(n)=an^2+bn+c$. And then they’d be hunting for the $a,b,c$ to make this the correct quadratic for our sequence. And then use the following three equations, they could come up with the $a,b,c$.

$5=a+b+c$

$9=4a+2b+c$

$15=9a+3b+c$.

In fact, this is an awesome thing to revisit when we get to matrices to solve systems of three variables!!!

UPDATE: One more thought before I lose it! What if I gave students the numerical sequence (e.g. 5, 9, 15, 23) expressed either written out as a list, written out as an explicit formula, or written out as a recursive formula, and had them generate a visual sequence to match it. I’d love to see how many different and interesting sequences might be created that go along with a single sequence!

# Jump in the online math teaching community!

A number of years ago, I had the idea of starting a little program to help those interested in starting a blog do so. And we’ve had some fun variations on a theme in the past few years.

Right now, we’re launching it again … but with an awesome twist!

There are going to be two things going on simultaneously.

Those who are comfortable blogging and tweeting, we’d love for you to sign up to be a mentor for someone just dipping their feet into the online math teacher world! You will be a person that newbies can ask questions to, connect ’em with people and blogs they might find interesting, and be a cheerleader as they get involved.

Those who are new are going to have someone help you out. You will be able to have a trusted person to ask questions to, help you find things that will be interesting to you, and encourage you. And through this, you’ll get to see if the online math teacher community has anything to offer that you want. You’ll get to dip your feet in, with no pressure, and a lot of support!

In December, we’ll pair up mentors and mentees. And during that time, we’ll all work on introducing those new to the online math teacher world to what we have to offer.

In January, we’ll have a 4 week “blogging challenge,” with prompts for both new and experienced bloggers.

If you’re interested in finding out more, or you’re ready to sign up to be a mentor or to get your feet wet checking out the online math teacher world (known as #MTBoS which is the unwieldy acronym for mathtwitterblogosphere), check out the exploremtbos website.

Huzzah!

# Fistbumps

I’m mad I didn’t actually take photos or record any of today’s precalculus lesson. Apologies. But even though this is going to be a textheavy post, I think it’s pretty awesome.

TL;DR: We fistbumped in precalculus. It was awesome. Super complex math got done.

One of the things I’m working on is improving my questioning this year. One of my strengths is scaffolding, but sometimes — in my desire to be super overzealously prepared — I scaffold too much. Today we had our first “long block” (90 minutes) in Advanced Precalculus. This is how class unfolded, after our warm-up.

I asked students to fistbump everyone at their table.

How many fistbumps did you just do?
How many fistbumps just happened in class?

Then I showed them there were many fantastical ways to fistbump besides the standard “clink knuckles” method. Blow it up. Snail. Squid. Turkey. [1] That was a random impromtu aside. But now, next class, I must show my kids the following video:

If you do this in your class, you should definitely have this video queued up. [2]

Then: everyone had 20 seconds of individual think time for this question:

If you wanted to devise an efficient way for everyone in the class to fistbump everyone else in the class, what would that way be?

Kids asked what “efficient” meant. I said “it should be as quick as possible, with the least chance of someone not actually fistbumping someone else.” Now you, friend, take a guess. I have 14 kids in my class. How long do you think it would take my kids to fistbump everyone else with an efficient strategy!

Seriously… reader… take a guess! Good. I’ll reveal the answer in a bit.

After the 20 seconds of individual time, each group shared with each other, and had to converge upon their proposal to the class. We went around. The four groups had three ideas:

1. Line everyone up. The first person fistbumps with everyone else, then leaves. Then the second person fistbumps with everyone else, then leaves. And so forth.
2. We have four groups in our class. The first group goes around and fistbumps with the members of other three groups in order. Then the second group does that with the remaining groups. And so forth.
3. Do the exact same thing as proposal #1, except as you don’t wait until the first person is done fistbumping everyone else. As soon as the first person is done fistbumping with the second and third person (and continues on down the line), the second person starts fistbumping down the line. And so forth.

(I had also anticipated students talking about getting in a “circle” and having one person fistbump with everyone, then another person, then another, etc. It’s organized, but not very efficient. One thing kids asked: can we all fistbump each other at the same time, in one giant mass of fists? I nixed that. I also had kids ask if you could fistbump with both hands simultaneously — to two different people. I said yes! But I didn’t give enough time for students to devise something super efficient with that so that never got turned into a proposal.)

As a class, we decided proposal #3 was going to be the most efficient. So I had them all file into the hallway and try out their fistbump method. I got my stopwatch out. And they went at it, after organizing themselves.

You may wonder what all of this has to do with math. That’s coming. This was just the setup. I honestly think by this point in the class, some kids were wondering what the heck we were doing this for…

So how long did it take them?

Yup. Under 12 seconds! I! Was! In! Awe!

Then each group got out a giant whiteboard and markers and answered the following questions:

How many fistbumps did you just do? What was the average time per fistbump?

Once they answered that question, they called me over to discuss their findings with me. Then I had two extensions:

We have 998 students at our school. How many fistbumps would that be? How long would it take, if we used our efficient method and assumed the same average time per fistbump? [3]

Can you find a method to answer that question?

And clearly, this is where the math comes in. This — in case you hadn’t seen it — is the classic handshake problem.

And from this point on, you have to facilitate class based on what your kids are doing. Some advice?

Advice 1: If kids are struggling, have ’em start noticing patterns about the number of handshakes for smaller numbers of people. Two people? Three people? Four people? Continue working up. Make a table. Look for patterns.

Advice 2: If kids have seen the “rainbow method” or some variation (see below), have them think about the difference between an even number of things being added and an odd number of things being added.

Advice 3: Have kids work on coming up with a single formula that works for even and odd numbers of things being added. Then have them explain why that formula works.

Advice 4: Lead kids to the idea of “double counting”: if we have 4 people, then have each person fistbump with everyone else. Since each person fistbumped with 3 people, there were a total of 4*3=12 fistbumps. However we’re double counting in this, so there are really only 6 fistbumps. (If kids don’t see the doublecounting, have a group of four act it out.)

Advice 5: If a group needs an assist, have individual members circulate to other groups and gather ideas, and then return and share what they found.

I loved doing this activity. Kids got into it. They felt ownership and camaraderie. Kids were up and moving. Because we had a long block, kids had time to play and productively struggle with the ideas. And most importantly: I didn’t overscaffold. I built up motivation and then sprung a good open-ended question for kids to work on.

[1] If you don’t know what I’m talking about, clearly you’ve never hung out with middle school students.

[2] queue is such a strange word, right? 80% of the letters are unnecessary. “q” is the same pronunciation as “queue.”

[3] The answer is around 18 hours. What I loved is that when a group got that — after we got 12 seconds for our class — I was like “come ON guys? does that make sense? it would take almost a whole day with no breaks? REALLY?” I wanted them to see the answer was kinda absurd. But it is right, because although it might seem absurd on the surface, each time we add more people, we’re making the number of handshakes grow pretty darn fast! (Follow up? How fast? Let’s make a graph! Ohhhh, quadratic? PRETTY! And grows super quickly for higher numbers, unlike linear graphs.) Turns out the answer is much shorter than 18 hours. I had a misconception that someone helped me see on the betterQs blogpost! I liked admitting to my class i was wrong!

# Getting to know you…

For the past few years, I’ve had students fill out an online survey for their very first nightly work assignment. It’s to help me get some of the logistics out of the way (their nicknames, making sure they read and understand some key things in the course expectations, making sure they know to have their name on the back of their calculators). But it’s mainly for me to learn about my kids.

I’ve found the questions on the survey are simple and nonthreatening enough that I get interesting responses. However I find that I do get way more extensive and thoughtful answers from the upper level grades than from freshman.

Here is a link to the survey if you want to check it out.

The thing is… I get tons of interesting information about my kids. They let me know about some horrifying thing that happened in fifth grade math that they still remember, or an amazing feeling they got once some abstract concept snapped into place, or about some lifelong passion of theirs that I wouldn’t know about. Perhaps the most important question — in terms of the information I get from it — is this one:

It’s kinda amazing. The phrasing of the question implies that there is something they are nervous about and are invited to write. (It’s so different than “Are you nervous about math this year? If so, why? And if not, why not?”… It’s like asking “What questions do you have?” instead of “Do you have any questions?”)

I’m not going to copy and paste responses, but I will share some types of responses:

• keeping up with the material / keeping up with classmates / falling behind
• test anxiety
• fractions
• coming across as annoying to classmates
• memorizing formulas
• explaining my reasoning in words

They really open up given the opportunity, especially considering I had only met them for 30 minutes before I asked them to fill this out. And if a kid came to you and had told you they were nervous about any of these things, you would know as a teacher precisely what to say!

So what I do, once I get these surveys, is I write back individually to each kid. The emails aren’t long, but they do talk about things that students specifically referenced in their survey. Here’s one from a couple years ago:

Howdy [Stu],

I’m reading through the surveys that you guys filled out for precalculus, and I wanted to respond to you, just to say hello! I’m thrilled that you’re going to be in our large band of precalculetes for the year. I’m excited about everything! We’re going to be doing a lot of exploring and making a ton of connections. I love love LOVE math and have since I was in high school, and I want to extend myself to you. If you ever feel overwhelmed or unclear about things, and they just are staying foggy, never hesitate to email me to set up a meeting. (Of course, I think you should first try to ask a colleague, because they often are better resources than I am.)

You noted that you’re nervous about keeping up with the workload. It is going to be a solid amount each night, but I very much try to keep it reasonable and I also try to make sure it is all relevant/important. I don’t assign 10 of the same types of problems, but rather I assign a couple of them and expect students to try extra problems if they need extra practice. But please let me know if the workload is getting to be too much for you. Last year I asked for feedback periodically on the workload and for the most part kids said it was fine, except in the third quarter when I think I accidentally asked for too much — and when kids told me, I was able to be more conscientious!

You also said that you don’t talk a lot at first, but you will. I saw you talking in your group! I think maybe because this is going to be a group-based class, you’ll find you’ll come out of your shell pretty quickly! But if you’re painfully shy, definitely talk with me. I’ve worked with kids who are shy before and we’ve come up with ways to help get over that so they can delve into the math!

Glad to have met you, and I’m looking forward to an enjoyable year.

Always my best,
Mr. Shah

It takes up a long time to write to every student. I have smaller class sizes that most of my friends, because I’m in an independent school. But still… I only get 5-6 emails written in an hour.

Why do I do this survey? Mainly because I love reading their responses. Especially to these two questions:

I also take the time to reply individually because I hope — though I never really know — that it helps make me more approachable. I pray that it implicitly tells my kids hey, I care. And early in the year, when I stumble through not remembering their names and want to crawl in a hole, this is such an important sentiment to get across.

So in this survey are some of my better questions, and how I deal with them.

[Cross posted on the betterQs blog]