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:

visualsequence1 visualsequence2 visualsequence3 visualsequence4

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.




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.


PS. Please spread the word, if you can!


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]

Blermions, Cyclic Quadrilaterals, and Crossed Chords

Last year in Geometry, the other teacher and I weren’t pleased how we introduced the crossed chord theorem. Basically, we ran out of time to come up with a *great* idea and instead had kids measure some things, take the products, and see that the theorem held true. Not our style. We were doing the heavy lifting and kids were making the connection that was served up on a platter.

When we met last week, we decided to rectify this. We brainstormed some general ideas, and I turned those into this activity.

The setup: Kids are so used to looking at “normal” quadrilaterals in geometry. So we thought we’d exploit that. We don’t mention circles. We don’t mention chords.

The TL;DR version: students investigate all quadrilaterals where the diagonals satisfy the property that ac=bd. Students are guided to make a conjecture which we as teachers know will be wrong. Then we show a counter-example to blow their conjecture up. And them bam: they have to try again. Using geogebra and some more encouragement, students discover that all cyclic quadrilaterals satisfy ac=bd. And so the circle emerges out of this investigation of quadrilaterals and diagonals. This is, then, the crossed chord theorem. Which students got at by investigating quadrilaterals. Weird. Now they are in a prime place for wondering why the circle shows up. Proof time!

Here’s the start: we introduce a new type of quadrilateral called a “blermion.” (.docx here)


We had some debate over whether we were giving too much away with this start [1], but we decided we weren’t. (We’re going backwards. The students aren’t deriving the formula. They’re using the formula (which we are calling a “property” of quadrilaterals) to come up with the circle part of the theorem.)

So yeah, we gave kids the ac=bd formula, but in relation to the diagonals of quadrilaterals. And we asked: “which quadrilaterals will this property hold for? We’ll call ’em blermions”

So I ask them to look at the standard quadrilaterals they know — investigating this property using a geogebra sheet — and having them making a conjecture about blermions.


The ggb sheet is here.


So students play on geogebra and come up with some understandings (inductively) about which quadrilaterals are blermions. Then they make a conjecture about all blermions.

This conjecture will fail. Because it is based on students only looking at “nice” quadrilaterals. I want the conjecture to fail. I want to emphasize the point that looking at “nice” examples can often lead to blind spots in your logic.

Students will see it fail when they are asked to drag the four points to specific places (see #5 below). The quadrilateral that results is weird looking. There is nothing that seems special about it. But it does have ac=bd. It is a blermion. Their conjecture about blermions was wrong!

Now students are sent on a chase to find more blermions — and they are encouraged to not just look at “nice” quadrilaterals. They record their results. (If they are stuck, a teacher can have the students fix three points and only drag the fourth point; It turns out you will always be able to drag that point to have ac=bd… and that in fact you can find an infinite number of additional points by doing this dragging of that fourth point.)


At this point, once they have found lots of blermions, students are going to try to make another conjecture about all blermions. I wonder if any student is going to get it. It’s okay if they don’t. At this point, I’m going to have every student plot a different blermion (some “nice” quadrilaterals, but mostly not nice ones). Then I’m going to have them pick any three points and change the color of them. Finally, I’m going to have students go to the “draw a circle with three points” tool, and be surprised by the fact that the circle always goes through that fourth uncolored point.

Why is this good? I hope they don’t get it. Because seeing that every blermion works like this (a circle goes through all four vertices of the blermion) is the key wow factor for kids. It’s strange, because even though I will be giving away this key fact, I think all this play will make this key fact interesting and weird. [2] Once they all see that, they are going to be curious as to how circles even got involved with these quadrilaterals in the first place. And… that is perfect… because then the kids are going to want to know why this happened.

And then we can transition to figuring out how to prove this. Because suddenly the crossed chord theorem is weird and strange and unexpected, and suddenly we kinda want to know why it works!

[1] We had to decide whether students should discover the property ac=bd for crossed chords. Motivating that from a circle and crossed chords was hard. We needed kids to somehow see similar triangles (which felt like we would be giving away too much) or come up with the multiplication idea of the pieces of chords on their own. We had ways to motivate that multiplication, but they weren’t elegant. So we scrapped that.

[2] Here’s the thing. Most things in geometry are presented to students in such a way that their wonderment about the geometric thing is killed. In a proof, the statement to be proved is given up front — and suddenly it isn’t interesting. It might be something really cool, but the exercise around doing the proof doesn’t highlight that. Or — as I’ve blogged before — theorems like the ones involving all the triangle centers… we tell kids to plot the perpendicular bisectors of all three sides of a triangle and they meet at a single point. It isn’t strange and wonderful. They don’t see why that’s weird. They just know we told them to plot the perpendicular bisectors, and they know something will happen because why else would we have them do it? We kill the wonderment of geometry in so many ways.

I want the weirdness and unexpected and unintuitiveness to come back to geometry… that’s where the beauty and curiosity are… and only then have my students work on figuring out why the unexpected happens… and get to the point where the weird and unexpected and unintuitive become obvious and natural. Making the unnatural natural. Yup, that’s the goal. But to do that, you have to first get to the unnatural.

The betterQs blog: A new #MTBoS adventure

For the past few years, I’ve been (sometimes daily, sometimes sporadically) posting on the one good thing blog. Last year I did it every single day. Often times it was a short post, especially in tough days where it was hard to find some little nugget. But what I loved was that it made me reflect consciously on joy and goodness, and pay attention to it. [1]

This year I want to spend some time thinking about how to question well. More specifically, thinking intentionally about what questioning looks like (and how it can be improved) in my classroom — both on my end and on my students’ end. I thought I would blog about it throughout the year, and figured it would be fun to blog with others. @rdkpickle had the same idea! So we figured it was a good idea, and set up a collaborative blog. All this is to say:

Check out the blog! Add it to your feedly or googlereaderreplacement. You’ll get posts by many people delivered right to your door.


But more importantly…You are warmly and heartily welcome to join us, and become an author. The blog just started and we’d love to get as many voices and experiences going on the ground floor.


Read a few posts. Browse a bit. It’s only a few days old, so there isn’t too much to gander at! And consider joining us. (If you want, there’s a tab at the top of the blog that tells you how to join, or just click here. We’ll add you as an official author!)

“But Sam,” you say, “I don’t have time to write every day…”

Silly goose, I respond! You can write however frequently works for you. Once a week? Once a month? Three times a year? The point is to take some time — however much of it — to think about questioning in your classroom.

“But Sam,” you say, “I don’t have a lot to write about…”

Silly turkey, I shoot back! I think it would be cool if you even wrote down a single question that you really loved asking because it provoked discussion. No need to deeply analyze it if you don’t want! Maybe a teacher reading the blog will read that question and think: “YAS! THIS IS EXACTLY THE QUESTION I NEEDED!” And if there were a lot of people just throwing down their good thought-provoking questions, we would soon have an amazing repository.

“But Sam,” you say, “I have a blog of my own! Why don’t I just post it there?”

Silly quail, I reply! You can post anything to do with questioning both on your own blog, and on this blog. No rule against that! In fact, I did that for my first post on the betterQs blog. And that way, someone reading the betterQs blog might get to know you and your own blog!

“But Sam,” you say, “I’m still scared… I don’t want to sign up and then not do it.”

Silly emu, I say. Why not take a baby step and just commit yourself to writing one or two things? Just keep a lookout in your school about how you question, or try to script a good question and see how it goes in your classroom, or rewrite a test question and explain how you rewrote it and why… Baby emu steps. And just see how it goes! You just might think: hey, questioning is something I want to pay just a bit more attention to!

Or, silly emu, don’t worry about signing up! As I wrote a couple years ago: “You should never feel guilty engaging with the community in ways that make sense to you. We’re all coming at teaching from such different places in our careers, such different backgrounds, and such different environments. We all need and want different things.” In other words, you do you.

[1] I also love the fact that because I’ve been using the blog semi-regularly, I can see an archive of so many good things of my own (in addition to seeing everyone else’s good things). On down days, it really helps me remember I’m not as bad as my brain tries to convince me I am.