A Radical Puzzle

If you search for "Trig Puzzle" in Google one of the first links is to this blog! Pretty neat since I don't update this enough, certainly not enough for my friend Rory. Anyhow, just before Christmas my PLC searched "Radical Puzzle" in Google a few times looking for a Tarsia style puzzle but couldn't find exactly what we wanted. Which was a Tarsia style puzzle that included simplifying radicals as well as basic radical operations (without variables). So we whipped one up that you can download as a PDF. Just click through the picture below.

Click through to download this as a PDF file.

Click through to download this as a PDF file.

The puzzle is a little tricky since there are problems on all 4 sides of the boxes, but the solution will read as expected. If you find the solution phrase too obnoxious you can always edit it in Acrobat. I haven't posted the solution anywhere to keep it off Google, although I know none of our students would ever search for such a thing.

You Are a Jeopardy Writer!

One of my teacher friends approached me a couple of weeks ago with a Jeopardy worksheet of mine I had apparently left around the photocopy machine. She wanted to know if she could use it in her classes, obviously yes. I figured I should post it here as well.


I have presented before about classroom review games and Jeopardy before, but this was a new worksheet I created to encourage kids to come up with their own (hopefully clever) categories and to begin thinking about their upcoming final exam.

Download PDF Download Pages File

It's not enough to just give kids this worksheet and expect them to come up with great categories, trust me. So in addition to handing out the worksheet I showed a few good Jeopardy clips (there are a ton on YouTube) and we worked together to brainstorm a few Jeopardy categories and sample questions together as a class.

One of my students Josh, traps Lobsters for a hobby, and hence became known as the Lobsterman, as you can imagine, I was pretty psyched when another student created a category called "Beware of the Logsterman" for his review. (I still, of course, received many, many mediocre categories, but this pre-work resulted in the overall submission quality going way up!)

Here is the final set of categories I used this year in Algebra2.


If you found this blog because of ISTE I'd love to hear from you! Let me know what else you'd like to see here!

This is Water

My school's graduation is two weeks away and I'm at home trying to write a speech for the ceremony while remaining calm. This is tough. The last time I gave a grad speech I quoted a bit from David Foster Wallace's excellent This is Water Kenyon College commencement address. I will probably draw from Wallace again, the piece is timeless. In the very small cache of documents I saved to bring back from India to New Hampshire there is a worn copy of the entire thing. I've probably read it twenty times, still gets me.

It turns out that a few days ago some folks turned an large chunk of the speech into a video. Click through the picture to check it out. Really great.

Click through to watch the video version of This is Water.

Click through to watch the video version of This is Water.

Here is a link to the full speech.

Crocodile Dentist

I get great review game ideas just browsing toy stores (and way way too many games). Crocodile Dentist was born this way. The game is sometimes hard to find, but seems to currently be available on Amazon.*

Gameplay is in groups, and nearly identical to Danger Cards, a review game I wrote about earlier. But in this one the croc is the star of the show, although the kids still do lots of math.


Set Up

You will need the crocodile (obviously) as well as the Keynote presentation I use for the classes. I also print out worksheets for each round. Below is also the link to the Pages worksheet set I made for the lesson I ran this week. Most of the problems are snapped from Sullivan's Algebra with Trigonometry book.

Download Keynote
Download Pages WS


Game Play

Here are the rules straight from the presentation that I share with the kids, I've added a bit of extra explanation as well.


At this point I jump immediately into the math by having the students work through the first page of questions. Teams hand in their worksheets as soon as they are done. Once most of the teams have handed in their sheets I cajole the final team into turning in theirs. During this part of the game I usually wander around the room making sure everyone is participating.

Once all the sheets are turned in I score them really fast and we discuss any questions that seemed universally unclear. I remind the kids before the game begins that this is primarily about review. Each team gets points for each right answer. The team with the most points goes first in the Croc Round and then play continues clockwise from group to group. (I've tried other turn systems but this seems to be the easiest for me to not mess up during the game.)


The Dentist Round works exactly as described on this slide. This is always intense because the winner of the round will nearly always be suspenseful, although the teams who were successful on the problems will have an advantage. To keep everything moving I usually only let one kid at a time come up to the table where the croc is.


The bonus round immediately follows the dentist round. The winning team plays for candy.


The round is nearly always a blast, because over and over again kids will take "one too many trips to the well" and not get any candy! It's hilarious. Further, if they manage to push four teeth successfully I usually start offering them deals encouraging them to press their luck (and get eaten) good times.

And that's it. Once the candy round is done we jump into round 2 and repeat. If you want you can keep score and have a grand champion at the end, I sometimes do this but it's not really needed. At the end of class I direct students to a Google Doc where they can access all the problems and solutions for additional review. This review game is a blast. If you give it a shot in your classroom let me know how it goes.

Links to Amazon.com are affiliate links when reasonable.

Danger Cards Remix

The other day my seniors, who are reviewing for their IB exam, wanted to play the Danger Cards game for a review session. I told them they if they made it, we could play it. Here is what I did.

danger pic 1

I quickly reformatted my Keynote slideshow to a Google Doc Presentation. The Danger Cards presentation is not fancy or anything anyway so nothing was really lost in the transition. I retitled the question slides with student's names and added blank answer slides after each question slide. Each student was responsible for creating one question slide and completing the subsequent answer slide. At this point I shared the Google Presentation with everyone in class giving everybody edit rights.

I told students that the slide deck needed to be finished by the night before we were going to play the game in class so that I would have time to choose the final set of questions we would use. They were also on the honor system not to cheat and study the questions (or more importantly the answers) their peers had written.

danger pic 2

Although the questions the students came up with were not in any way earth shattering, and the formatting of the slides was not as beautiful as it could have been, for a lesson developed in less time than I have spent writing this post about it, it was definitely a success. Students also had a solid bank of questions and answers (including many we did not get to use in class) that they could use for additional review.

Here is a link to a Google Presentations version of Danger Cards you can make a copy of and use with your classes if you want to try this activity out. Check out the original Danger Cards post for more details about the game.

Scaffolding Trig Graphs - Desmos Kong

Yesterday evening, way too late to be planning a new lesson I was noodling through Sam Shah's filing cabinet looking for something that would allow students to practice trig functions, ideally kind of game like. There is good stuff there, like this graphic organizer from Mimi, but nothing like what I was looking for. For years, here and there, I have heard people mention the graphing game Green Gobs but I have never actually used it myself. Still, downloading software for all of my students to use (that had to be purchased) was going to be out of the question, at least last night. Maybe I could make something similar with Desmos. I have been using Desmos more and more with the kids and loving it. A little while later I came up with Trig Scaffolding.

Anyway, let's get into it. The activity (1) I created is called Trig Scafolding: How High Can You Get? The entire thing is built into one Desmos worksheet. I demoed the activity for my students on the big screen and then shared the link.

I think this is all best explained with screen shots:


When students open up the Desmos Link they see the points of the function they are trying to graph.


Students know they are right if their graph matches up with the points.


The kids can «climb up» to the next graph by turning off the graphs they were working on and turning on the next ones.


I added notes along the way in Desmos to mark students progress and remind them about Desmos's excellent features like the easy integration of sliders. I spent the entire activity circulating and working with pairs of students. I was impressed with the total engagement the kids had through the block and the different strategies they were using to figure out the graphs.


No one picked up on the very thin Donkey Kong theme, but overall the activity worked really well. A couple groups did engage in some mindless guessing and checking, but most of the kids were really trying to reason out the functions with Desmos along with pencil and paper or whiteboards. Although I didn't do so, it would be easy for students to turn in their «solutions» for this assignment by having them save their graph and sharing the link with you.


I was really unsure how difficult kids were going to find this when I created it, but the scaffolding of the problems seemed to be decent. It is easy enough to adjust anyhow. I think in the next iteration I will add more problems where I place restrictions on the graph, either to make the problems more challenging, or increase the level of scaffolding.

Have you tried anything like this with Desmos? I would love to see it and hear about it!

(1) I called it a game with my first class yesterday and the kids kept comparing it to other games like Danger Cards, I called it an activity in the other classes and there was none of this nonsense. In fact in the other classes the kids told me that it was «a good game!»

Save Kelly!

Rory, who has taken this activity and sprinted with it, has been hounding me to write a post about its creation.



At my last school with the encouragement of my colleagues I picked up the physics classes when the physics teacher retired. My first year I did a lot of lousy book labs although I did manage to do the mousetrap car thing. I knew that moving forward I wanted to add more engaging activities for each topic we examined.

The Save Kelly activity was the result of my sister coming home from college and telling me about some fun stuff she had been doing at college in her engineering program and how maybe we could turn one of the activities into a lesson connecting to Newton's Laws that my students had been studying.

The activity works like this. First students are told the ridiculous back story. That is, they are out hiking with their friend Kelly (2) when suddenly Kelly is abducted by a vicious Pterodactyl and dropped on the other side of an expansive ravine. The students of course must save their friend. All the students have to save their friend is a survival kit, and some pennies. Everyone knows Pterodactyl's can be taken down with pennies right? What follows is from the worksheet I gave the students

Save Kelly!

As you are painfully aware your dear friend Kelly is being held by a vicious pterodactyl across the ravine. You gather your wits and take in the surroundings, first you pull out your survival kit and then you also notice two threads stretching across the pit. You figure that if you can transport pennies to Kelly, they can be used to do away with the pterodactyl.


Your group must use only the materials provided by the survival kit to design a system to transport as many pennies as possible using the twine. You may not use any tools (scissors, knives, etc) only what is in the survival kit, plus one meter of masking tape.

You may design your structure any way you want. With a couple of rules:

  1. Your contraption must start from rest at the designated spot. Once your contraption is set into motion it may not be touched again.
  2. The masking tape must not touch the pennies.

If you find you are running low on supplies you may barter with other teams. You are allowed to run trials, but take care not to squander your survival kit.

The Competition:

15 minutes before the end of the block two official trials will be performed; your rank amongst teams will be based on your best trial. Your score will be determined by the distance your contraption travels in floor squares, multiplied by the number of pennies you transport the entire distance.

The Write Up:

This activity will be written up as a lab report in your lab notebook.

Your write up must include the following components:

  • the necessary pieces of a standard discussion (reread your lab reports guidelines page)
  • sketches of your groups’ final design
  • a clear connection between the activity and each of Newton’s 3 laws as well as any other pertinent physics topics.
  • the results of the activity and how your team worked together to overcome challenges.

You have some options for the format of your lab write up:

For a maximum grade of a good solid B you can turn in a traditional lab report with only the discussion section (and of course the title, page numbers, table of contents, etc.)

For a maximum grade of an A you can turn in an alternative lab report. This written piece could take the form of a news article, a narrative (perhaps from the pterodactyl’s point of view), a journal entry, or something else.

The write up's for this activity were always awesome. A million times more enjoyable to read than 60+ (nearly identical) dreary lab reports. Here is a picture of one I saved.


Overall, it's a pretty good activity, if you are teaching physics definitely give it a shot with your students.

(1) The survival kit basically contained what me and my sister found on a romp through the grocery store, with with some candy and red herrings like penne pasta, manicotti, and marshmallows. It also included a straw and some balloons. Rory put rubber bands in her survival kit as well, but I didn't use to. To make the bags I fed them though my inkjet, which although was probably a dumb idea for the printer, made for some awesome props. Props are huge.
(2) The first year I did this I had a student named Kelly. In future years I always implied Kelly was Kelly Clarkson. Which meant I could play «Since You've Been Gone» over and over again throughout the entire lab. Good times.

Induction by Contradiction

So for years I used Proof By Induction, but never really understood why it worked. This frustrated me, and so I set out to discover the «proof» for proof by induction. I searched far and wide in all my textbooks and just kept finding the domino analogy to justify the three steps. Sure the analogy is memorable, but to me it never seemed like a proof. So after looking up induction in nearly every book I have, I found a decent explanation in Paul Foerster's Precalculus. He uses Proof By Contradiction to develop induction and the method is both clear and logical. Unfortunately this great induction lesson has been relegated to an appendix, and there are no exercises at all (particularly unfortunate since Foerster's claim to fame is his problem sets). I used Paul's explanation to create a lesson, along with problems, and I have attached both below. I'm not teaching induction this year, but induction came up in a conversation I was having with my friend Bryan who is. The worksheet below is updated from the first time I taught this four or fives years ago. The last two times I taught induction, however, I turned the worksheet into HW problems to fit my Exeter style problem sets, these, and a whole strand of induction problems I used in subsequent problem sets are included here.


Passion, Trigonometry, & Locked Doors

The other day Rory posted a new problem solving method with a 75 letter acronym. I have no idea what it was, but the first letter was this.

P: What problem in my local community ignites my passion to the point of action?

On Friday at school I had a long conversation with John, a programmer who is working with ASB to design some gamification stuff. Looking around my classroom he saw the unit circles everywhere and exclaimed his love for trigonometry. «I need you to come talk to my students!» I exclaimed after he went on and on about how he uses trig all the time in the programs he writes. I wonder if he learned most of that trig when he really needed it for the coding, or it was all from tenth grade (I suspect the former).


Just like today when I finally learned how to break open a locked door with a credit card. I was standing in the doorway above, admiring my newly installed chin up bar, thinking «maybe tomorrow I will be able to do one chin up» when I closed the door to discover two things: first, the door could still close with the chin up bar in place (this should have been obvious) and, second, the door was now locked!

With no key, and a lock that was real (not one of those US bathroom locks that comes with a pin) it was time to hit the net. I had, of course, heard of using a credit card to open a locked door, but always figured it wasn't really possible with a real door or with a real lock. Luckily my computer was not in the locked room, so off to YouTube I went. Gold immediately, with a young man (maybe 12) clearly demonstrating how to open his front door with a credit card. I watched it a few times until I was convinced that I too could get into my locked bedroom, and I did. Here is the clip. John's desire to learn trigonometry was ignited by his passion for video games. I needed to learn lock picking to get into my bedroom. When passion is ignited learning follows.

TPIR 05: Safe Crackers

I have mentioned before that I am a huge fan of The Price Is Right. I wasn't always a huge fan of the game Safe Crackers though. I can remember watching TPIR as a little kid and being scared when Bob would walk towards the big doors at the back of the set. Frequently when this happened the doors would open to reveal Safe Crackers (and the terrifying?!) Pink Panther Theme.

I am, thankfully, no longer afraid of Safe Crackers, or Henry Mancini, but Safe Crackers remains a great pricing game to analyze. Since I like to introduce problems before math formulas I would definitely use this before I taught students about permutations. The kids usually have no trouble figuring out that the lock has 6 possible permutations and that only 4 of them are really viable. Occasionally a couple of students do even better. Here is what I share with the class.


Here is a link to the You Tube clip

The Price Is Right is the longest running game show on American TV, part of its lasting appeal is due to its wide variety of pricing games. Many of these games can be analyzed with probability theory. Take a look at Safe Crackers. What is the probability of a contestant winning this game? What about a Safe Crackers aficionado?

When we do our math homework on Google Docs (to prepare for discussion the next day) students write each notes back and forth on the document (and I chime in as well). Here are some notes they wrote about this problem:


Egg Hunt

Easter was my favorite holiday as a kid because we would always have these sick Easter Egg hunts at home and then again at my grandparents' house. My parents even managed to hide Cadbury Eggs for me and the sister when we were in the desert in the middle of Morocco one spring break.

egg pic 1

Cue one of my favorite non-math classroom events (1). Today before school I made it to the classroom early and hid about 20 plastic eggs (2) for the students to find. I stopped putting candy in them years ago, because students never find all the eggs and candy forgotten for months causes problems. We have the hunt at the beginning of class and then at the end of class the kids hide the eggs for the next group.

The students get really clever with their hiding spots and we always try to come up with seemingly «impossible» places to hide the eggs. Every class wants the class that follows them to find less eggs than they did of course! There is an egg, that never gets found, inside the Rubik's Cube on this alphabet poster for example. Good times, and totally worth the 10 minutes or so of class time this sacrifices. Happy Easter.

egg pic 2

(1) I have considered putting equations in the eggs and other such schemes to work math into this lesson but so far have kept it a simple hunt.

(2) If you don't have a set of plastic eggs (and who doesn't?) now is a perfect time to buy them because they are probably virtually giving them away at Walgreens and other such stores.

A Trig Assessment

Ever since Daniel started his series on assessment a couple weeks back I have been meaning to post some questions from my own assessments. Here is a one from a quiz we took in Algebra 2 with Trigonometry on Friday.

quiz pic

Proofs are difficult for students in general. What I particularly liked about the second part of this question was that although we had been talking about various trig identities created by the unit circle and the unit circle itself for a few weeks (see the previous blog posts), this important identity hadn't come up. I was saving it for the quiz.

I like giving students questions like this on quizzes because they challenge my students do more than just regurgitate some facts that they have previously memorized. Khan Academy can assess knowledge of basic facts (and simple applications) pretty well. I want my students to be faced with questions where they don't necessarily know what to do, and they might even take a wrong path.

This reminds me of Andrew Wiles' comparison (1) of mathematics to the exploration of a dark mansion

«one goes into the first room, and its dark, completely dark, one stumbles around, bumping into the furniture, and gradually you learn where each piece of furniture is, and finally after 6 months or so you find the light switch, you turn it on and suddenly it’s all illuminated you can see exactly where you are.»

Now, this proof was not Fermat's Last Theorem, but I think it was still challenging. My kids, of course, really hate that I put questions like this on their quizzes and tests at the beginning of the year, but they grow to appreciate them. And boy are they excited when they figure them out. They know, of course, which questions are the ones making them think. Here is a bit of their work.

quiz pic2

A second,

quiz pic3

These really exceeded my expectations, I was thinking maybe a few of my students would get the question right but it was a much much higher percentage.

(1): If you haven't seen this documentary about the Proof of Fermat's Last Theorem from BBC's Horizon program go watch it –really awesome. I always show it to my kids when I get a chance.

Automating Our Jobs Away

Interesting article with lots of good links from Quartz via NextDraft about how middle class jobs are more and more rapidly being replaced by computers. And in the case of the example below, the jobs that are left are being run by computers…

«In a gleaming new warehouse in the old market town of Rugley, England, Amazon directs the actions of hundreds of “associates” wielding hand-held computers. These computers tell workers not only which shelf to walk to when they’re pulling goods to be shipped, but also the optimal route by which to get there. Each person’s performance is monitored, and they are given constant feedback about whether or not they are performing their job quickly enough. Their bosses can even send them text messages via their handheld computers, urging them to speed up. “You’re sort of like a robot, but in human form,” one manager at Amazon’s warehouse told the Financial Times. “It’s human automation, if you like.”»

I frequently hear some variation of the quote «Any teacher who can be replaced by a computer should be.» it is easy for me to fathom adequate online math courses with hundreds of students in them. I think I could even teach an ok online math course to a large group of students. And what about if (or better when) Khan Academy begins offering certificates for course completion for courses such as Algebra 1 or Calculus and those students are shown to be as adept as the ones in the traditional classes? Maybe us math teachers will be replaced too. I think I have a job at least until June 8th.