Coin Combinations



What do you notice?

My first observations included, “Hey! Sodas cost $1.50.”

What do you wonder?

The first thing I wondered was, “How many coin combos are possible?”

And immediately after that, “Are pennies allowed?” (Check the picture for the answer.)

And with my teacher hat on… “What strategies would other students and teachers use to answer the “how many combos” question?

Sequel #1

Then I noticed something else on the machine:


How many coin combinations are possible now? Do you think it will be…

  • More?
  • Double?
  • Less?
  • Half?
  • The same?

Sequel #2

The “use exact change” light isn’t lit. How does that affect our original answer(s)?

Post Script

True, true, this is an age-old problem. Maybe even a tired problem. But I still love it, especially in this visual form. Also, I’ve never noticed the coin icons on the payment panel before. That’s pretty nifty if you ask me. 🙂

Tesla Model 3

I’m fascinated by the pre-order hype surrounding Tesla’s latest car, the Model 3.

With a little help from Skitch, let’s turn this scenario into a math problem.


Throw that image on the screen and ask students:

  • How many orders in 24 hours?
  • What info would help you figure that out?

Ideally, after making some predictions (and writing them down!) students make a request for average price per vehicle, and you deliver:



When they’re ready for the reveal…

Sequel #1

Let’s see what else we can do with this…

Pre orders began on Thursday, March 31. Tesla promised a numbers update on Wednesday, April 6. How many pre orders do you think will have been placed by then?

  • Make a prediction.
  • Use math to find a more accurate answer.
  • Explain your thinking.

I’ll drop an update here once we know the answer.


Sequel #2

Tesla aims to sell 500,000 cars per year by 2020. Consider this comment from CEO Elon Musk:


Based on the information in the comment above:

  • Do you think Tesla will meet its 2020 goal?
  • What sort of year-over-year percentage growth will this require?
  • If you think they’ll miss the mark… by how much?
  • If you think they’ll surpass the target… by how much?


Drop an answer to one of the questions above in the comments below. Or, share another idea or two for how this Model 3 craze could play out in a math classroom.

Cue Rockstar Math Edition

Last February I attended and presented at the CUE Rockstar event in Petaluma, CA. It was a blast.

The only way it could have been any better was if it was 100% focused on math. (No offense other disciplines. Math is just my first love when it comes to teaching.)

Well lo and behold, guess what’s coming up on May 14-15?! An all-math CUE Rockstar!

And check out the crew:

Screen Shot 2016-03-17 at 9.26.31 AM

Not too shabby, eh?

If you’re going to be anywhere near Los Gatos, CA in the middle of May (or if you’re willing to make the trek so that you are close to Los Gatos for that weekend), head over to for info (schedule, speaker bios, registration details, etc).

Hope to see you in May!

Points, Lines, and (Maybe Not) Inequalities

One of the things I love about Twitter is the opportunity it provides for feedback on creative work. And thinking about teaching and learning mathematics is some of the most creative work I know. I’m still amazed when folks I’ve never met in real life take time out of their day to help me improve.

Here’s a case in that beautiful point.

Points, Lines, and Inequalities

Earlier this week I created an activity designed to help students make connections between individual points and graphs of linear equations and inequalities.

Here’s the first screen:

screen 1

On its own, pretty simple. But imagine a class full of students doing this. And then imagine directing their attention to a graphical overlay of every student response.


Now we’re talking. Literally. At this point, I’m asking students:

  1. What do you notice?
  2. What do you wonder?
  3. What do these points have in common?
  4. What do they not have in common?
  5. Is there a way we could summarize all of these points algebraically?

And then… Turn them loose on this:

screen 3

My hope is that the previous discussion laid the groundwork for the idea that these are all points where x equals 3, so x = 3 is an (obvious?) way to summarize them algebraically. That’s how it plays out in my head, anyway. More on that in a moment.

Students then move on to the next (rather similar) prompt:

screen 4

(And the activity continues in a similar fashion for a total of six scenarios—four lines, two inequalities. For a closer look, check out the links at the end of the post.)

Room for Improvement

Let’s circle back to Twitter. Yesterday I shared a link to the activity, invited folks to give it a test run, and hoped a few might offer some feedback.

The response proved incredibly helpful. In particular, Bowen Kerins offered not one or two, but nearly a dozen comments about what works, what doesn’t, and how I could make this activity even better.

I have a few takeaways from that conversation about how to make this activity better. I’ll share them here in order to clarify (for myself) and share (with others) some design principles that I think may prove helpful the next time we fire up the old Activity Builder.

I’ll share the first takeaway here, and a couple more in the next few days.

Takeaway #1

The cycle in screens 1-3 (place a point, imagine all the points, summarize algebraically) is missing a crucial step. It begins simply and somewhat informally, which is helpful. And then it invites students to imagine/predict, which I also think is helpful. But then it jumps straight to the summary, without pausing long enough on the “discussion” that I described above.

My current fix. Insert this between screens 1 and 2:

new screen

By asking students to name these points (note: not just their own, but also several others), I think they’ll be more likely to see (or “hear”) the repetition that leads to our algebraic summary later on:

  • “x is 3, y is mumble-mumble…”
  • “x is 3, y is something-something…”
  • “x is 3, y is whatever…”
  • “Hey, x is always 3! So then, x = 3!”

My future fix. This activity uses an existing (and lovely) feature called copy previous, where whatever a student did on one graph screen can be pushed ahead to (or “served up” on) a later one. You’ll see this in action between the screens 1 and 2 in my original activity.

What I would love here is another feature (even more lovely?) where the graphical responses from every student are served up to kiddos on a later screen. In other words, bring the graphical overlay feature that we have in the teacher dashboard into Activity Builder itself. Apply that to my original screen 3, and I think we’re in much better shape.

And the good news? Something like this is already in development. (Side note: I continue to be amazed at the engineering skill—wizardry?—of my colleagues.)

Closing Thoughts (For Now)

I think these proposed changes will prove helpful. But I also think they stretch the activity out a bit. So maybe it’s not realistic (or even helpful) to work through vertical lines, horizontal lines, linear functions, and linear inequalities all in the same activity. Maybe this is two separate activities. Or even three.

I take some comfort knowing I’m not the only one who thinks a narrower focus here could be helpful.

With that in mind, here is my new-and-hopefully-improved activity. Give it a whirl, and let me know what you think in the comments.

Next Time

I’ll share some thoughts on making the middle section (linear functions) stronger.


I almost forgot! You can get the original activity here. Or try it out as a student here.

Sliders with a Purpose

Back in August 2015, Desmos released its Activity Builder, a tool by which teachers (and Desmos folks like myself) can build custom Desmos activities. Over the past few months, I’ve seen a number of Activity Builder screens like this:


In fact, I’ve created a number of Activity Builder screens like that.

However, I’ve also seen quite a few screens like this:


Notice the difference? It’s subtle, but powerful.

The ingredients are largely the same, but the vague objective is replaced by a clear target.

In each case, my followup screen probably looks something like this:


In other words, my ultimate goal here is to invite students to observe and then describe a parameter’s impact on a given graph. But my path to that end has shifted from “poke around and see what you see” to “complete this specific task, now reflect on how you made it happen.”

Here are a couple more scenarios where I’ve found this approach to be helpful:

image4 image5 image6 image7


So, what would you do with this?


P.S. For a closer look at the screens above, go here.

The famously difficult green-eyed logic puzzle

Last Monday I wrote about a riddle involving a bridge, some folks of various bridge-crossing abilities, and a horde of slow-but-blood-thirsty zombies.

Well, Alex Gendler, I’m hooked. I love a good logic problem, and the presentation of these riddles—specifically, the animation and narration—is simply outstanding.

And since I’m hooked, I thought I’d share another in case you’re in (or want to be in) the same boat.

(SPOILER ALERT: The video poses and solves the riddle. Want to think of the “statement” on your own? Stop the video at 1:23. Alternatively, you can watch through to 1:53 if you’d like to know the statement, but leave the how/why to yourself.)

I didn’t immediately think of any extensions to this riddle, but I’d love to hear any you might have dreamed up while working on it.

Side Note

For some reason, I’m reminded of the pirate game, a logic puzzle I first heard during Andrew Stadel’s CMC South talk in 2015.

Can you solve the bridge riddle? (Zombies!)

Last weekend I was at the Interface conference in Osage Beach, MO. While attending a session on Saturday morning, a fellow participant shared this lovely problem with me.

(WARNING: The video poses the problem and solves it. Be sure to stop in the middle so you can give it a try for yourself. It’s way more fun that way. I promise.)

It took me a little while to figure it out, but eventually I did. And it felt pretty great to shave off those last couple of minutes that seemed impossible to shed during my first few attempts.

Then I got to wondering…

  1. Imagine the janitor and professor are even slower (say, 6 and 12 minutes to cross, respectively). How long would it take the group to cross?
  2. Imagine everyone is slower (say, T1 < T2 < T3 < T4 minutes to cross). How long would it take the group to cross? And what’s the winning strategy?
  3. Imagine there are only three folks who needed to cross (to simplify the scenario, let’s say A/B/C who take 1/2/3 minutes to cross, respectively). What’s the fastest they could cross, and what sequence would yield that time?
  4. Imagine there are five who need to cross (A/B/C/D/E who take 1/2/3/4/5 minutes, respectively). What’s the fastest? What’s the sequence?
  5. Imagine there are “n” people who need to cross (A1/A2/…/An who take 1/2/…/n minutes, respectively). Fastest? Sequence?

Some of these questions I’ve answered in my own mind, and some I have not. If you answer one or more of them (or create another extension of your own) I’d love to hear about it in the comments. Cheers!

Side Note

My new favorite YouTube search phrase is Alex Gendler puzzle.

How Many Solutions?

I’ve been thinking about systems of linear equations for the past couple of days. Most of my focus has been on systems of two equations, but this morning I wrote a question about a system of three linear equations that might spark some interesting and revealing classroom conversations:


And now I’m wondering…

  • How many solutions does that system have? Drop your thoughts in the comments, would you?
  • If you put this prompt in front of your students, I’d love to hear a recap of how that plays out.

Resource Countdown • December 24, 2015

For a tiny bit of background, check out last week’s inaugural “resource” post.

Now, on to the countdown!

3… Tweets

2… Posts

22? 30? 50? 100?, by Joe Schwartz . A wonderfully thoughtful post, followed by a flood of wonderfully thoughtful comments.

Good Mathematician vs. Great Mathematician, by Ben Orlin. In shifting from Reeder to Feedly and back to Reeder, I somehow (argh!) temporarily (phew!) lost my RSS feed to Ben Orlin’s delightful Math With Bad Drawings. Resubscribed. It’s good to be back.

1… Book

Several weeks ago, Andrew Stadel suggested I read Weekend Language. I procrastinated for a while, but finally picked it up. It was as good as advertised. I highly recommend it to anyone who wants to become a better presenter.


That’s all for now!

Resource Countdown • December 17, 2015

I’d like to try something new on the blog. Throughout the week, I typically stumble across a handful of tweets, blog posts, and activities that catch my eye.

I’ll gather than up in a countdown style (the Internet loves a good list, doesn’t it?) and share them here. Every Thursday? Some Thursdays? We’ll see, but I’m hoping for the former.

Countdown to what, you ask? The end of the post, I suppose. Or the weekend. Or whatever comes after you read the post. 🙂

Here goes…

3… Tweets

2… Posts

Project Pentagon, by Christopher Danielson. At NCTM Nashville I had the opportunity to tinker with Christopher’s tiling turtles and pentagons. They’re lovely. He writes more about his growing obsession with pentagons here.

Interesting Problem, by Jonathan Claydon. Jonathan raises an interesting question about the role of technology in the classroom, and its implications for the questions we ask students when technology is so readily available.

1… Activity

Steve Leinwand mentioned this MARS task (Calculating Volumes of Compound Objects) during his talk at CMC North. I’m intrigued. Would you use this as-is? Or would you make adjustments? Which ones and why?


That’s all for now!