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.

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.

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(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.

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.

A second,

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.

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(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.

Danger Cards

Sometimes I get to school and realize the plan I had for a class is not going to work. Or it will work but be boring for me teach and hence probably even more boring for my students to learn. This was the case a while ago in my IB Math Studies class. I was just going to have the students do some IB problems for practice. Normally I would tie a lesson like this to a review game, but I hadn't done so yet. Luckily I had a prep block and so I got to work.

I knew it would be a group style review game, something akin to Bazinga, a similar activity I hadn't seen until today when I started writing this blog post. Groups would complete a question and draw a card. I also wanted the game to have some treachery, strategy (beyond answering questions), and cross team interaction. What if the cards frequently did bad things? Danger Cards was born.

The first version pictured above definitely looks like it was schemed up during prep before class. Happily, I've tweaked the game through 4 or 5 iterations over the past couple months with various classes and the current version is solid.

I used Illustrator to design envelopes for the cards. With magnet tape they stick to my whiteboard.

And I designed a matching keynote presentation you can download and edit for your classes. I used off the beaten path fonts in this one, so the deck you download will probably look different than mine. You can just change the fonts to work with ones you have.

Set Up

You will want to either make your own envelope decals or download the ones I created here. The labels are in PDF form but completely editable in Adobe Illustrator. Also download the Keynote linked above. Finally, you will want to create a set of index cards to put in the envelopes. You can use my current set pictured below or make your own. The Bazinga post I linked to has some great ideas for stuff to put on the cards as well.

Game Play

Here are the rules from the first slide of the presentation that I share with the kids, annotated with a bit of extra explanation.

• Work with your team to solve the problems on each slide.
  • I try to make teams of 3 to encourage everyone to take an active role.
• Work in your notebooks and then put your final answers on the whiteboard.
  • Everyone writes down the problems and solutions –again to encourage everyone to take an active role.
• Deliver your whiteboard to the answer box.
  • The answer box is just a plastic bin that normally holds paper for recycling.
• Once all but one set of answers are in the answer box, the round is over and the last set of answers must be turned in.
  • Not as hard and fast as it sounds, if a group is working hard, I'll give them leeway.
• Highest Score (by order turned in) chooses a card and views it.
  • They need to be careful not to show their hand! This part of the game makes it really awesome because the kids come up with all sorts of ridiculous strategies to try to fool their classmates.
• Each team (clockwise) can choose to play or pass the card as well.
  • Again lots of strategy because teams need to try to figure out if they actually want the card that was chosen, and if they have already seen the chosen card (or if another team has)
• Each team has one Z-Chip you can cash in to reverse any one decision i.e. to undo your «play or pass» decision or even «forfeit the card you chose»
  • So basically once in the game a team can take back a decision they made. The team that chose the card can also take advantage of this, although the card still effects any team that chose to «play» it. The decision to play or pass a card must happen before the next question round begins.

Before we start the first round each team chooses a Danger Card and gets to «peek» at its contents before it is returned to the whiteboard.

The Cards

I think most of the cards are obvious but here are explanations of a couple of the tricky and unusual ones:

• Peek - Cards that say + Peek allow the team that chose the card to peek at another card of their choice at the end of the round.
• Treasure - Cards that say + Treasure allow the team that chose the card to grab a prize (usually food) from the Treasure Chest. The Treasure Chest is a, usually locked, wooden chest in my classroom. I brought it to school originally for a different (even more sinister) review game I'll talk about in a while, but lately I use it for many games. Kids love it when I open up the treasure chest for the first time.
• Steal Another Team's Points & Swap Points With Another Team - Both of these cards create kind of a mess to execute, but so far I have stuck with them. Basically, the cards resolve in reverse order. So the team that decided to «play» the card last makes the first choice as to whose points to steal, the team that selected the card would resolve last (usually a huge advantage)

Overall

My classes are universally fans of Danger Cards, if you give it a shot in your classroom I'd love to know how it goes.

I found the skull picture from a quick google image search from here

TPIR 04: Freeze Frame

I love to use pricing games from The Price Is Right to teach my students probability. Freeze Frame, although it only involves basic probability, works well for this. Students are generally not exactly sure how many possibilities there are, and have to physically count them or make a list. Usually when we are discussing Freeze Frame as a class I talk about how difficult the counting of options in probability problems can frequently become. 

In prior posts I have mentioned how I usually flip these videos and have students watch them for homework on YouTube and discuss them on a Google Doc. For Freeze Frame I integrated it into a lesson and had students work out their solution in pairs on whiteboards. Everyone also made a «guess» as to the right answer which we finished watching after we talked about the problem.

Something else we always discuss with these Price Is Right problems is what the savvy contestant would do (frequently vs. what the average bear would do.) I mean sure, $1129 is one of the possibilities for the trip to Hawaii, but obviously the trip would cost much much more than that. If I have looked up the stats before hand ​it is also interesting to compare our best savvy contestant's probability to the actual probability of winning for contestants on the show.

Click through to watch a clip of Freeze Frame.​

The Price Is Right has male and female models now, and amusingly for my class the model in this clip had his shirt off. The Price Is Right Web site usually has ​about 5 different playings of each pricing game to watch, so if you like you can find a different playing of the game, but it is down as I write this.

Flip:

It's time for another fabulous pricing game from The Price Is Right. Watch this clip of Freeze Frame (stop the tape before the contestant chooses an answer, about 1:35) and answer these questions:
(a) What is the probability of winning Freeze Frame if you just close your eyes and pull the lever whenever?
(b) Being a savvy contestant you would, of course, not do this, other than deferring to the audience, how would you increase your chances of winning Freeze Frame. What do you estimate your probability of winning to be?
(c) What do you guess the answer is? Finish watching the clip to see if you were right!

The Pyramid Game

The other day, in a post about the lately very funny Family Feud, Dan asked

«Which game show works in the other direction, giving you lots of items and asking you to move one level of abstraction higher to the category that includes them?»

There are probably multiple answers to this question, but the most obvious answer to me is the bonus round from the $100,000 Pyramid, known as The Winner's Circle.

The game is best learned by watching a few You Tube videos (of which there are hundreds). But here are the basics. There are two primary players. One player gives the clues to six subjects of increasing difficulty and the other player is trying to guess the subjects that those clues fit into. The clues must be in the form of a list of items that fit the subject or category. So if the subject was salad dressings the clue giver could say «French, thousand island, or Italian» but they could not describe the category saying something like «It's what you put on a bowl of leafy vegetables.»  ​The clue giver is also not allowed to use charades, synonyms, or prepositional phrases. So, for example, for the category «Things that are quiet» «moviegoers» would be a permissible clue, but since «people at a movie» is a prepositional phrase it would get you buzzed. It's fussy sure, but that is what makes it great.

I have used The Pyramid Game as an occasional segment in my classroom ​since my first year of teaching. At the beginning it was very rudimentary:​

Before I had a projector I played the game by holding up big index cards. I apologize about the Word Art. I was young.​

The first time I played this game in class there was no YouTube so I actually spent a few days video-taping Winner's Circle segments at home, signed out a TV cart at school, and showed some clips during class. Today I would probably flip the introduction part and have kids watch a couple of the YouTube clips beforehand and then maybe show one more in class.

Click through for the clip. Great one because Billy Crystal is the Pyramid master. And because Dick explains the rules as well.​ This video seems to load slow, so cue it up ahead of time.

Students play in pairs and while only two students play at a time it is easy to get the entire class involved because, especially at the beginning, hardly anyone will win the game! After the group loses everyone else can chime in with the clues that were missed in the heat of the moment. The harder subjects can really get you thinking. A great strategy is to try to approach the category from multiple different directions. ​If the category was «Things that you join» you could try the obvious «a club» and then also «pieces of wood» (yes «of» is a preposition, but it is allowed).

There are a whole series of videos showing what not to do in the Winner's Circle. Great for teaching the game.​

Here is one of the first slide deck iterations. A student came up with the category pictured.​

I thought this was pretty good at the time!​

You can also tailor the game to your classroom. I am going to use this game at the end of class a couple days this week, so I will include categories like «Facts about the Unit Circle» and «Trig Identities.» The frequent TV show categories similar to «What Goldilock's Might Say» are also great for classroom use, «What Mr. Roy Might Say» is also usually a hit. Just like on Family Feud and Dan Meyer's clip if you open the door to racy responses the students will run right through it, you have been warned.​

Students initially find this game extremely challenging, but also grow to really enjoy it (You might need to play it a couple times to appreciate this). I have had students get so enamored with the game they devise their own clue sets for classmates to try. They frequently go for the ludicrous «Things a Vegetarian would not Eat», «Bad Jokes», or the impossibly difficult «Words that Rhyme with Orange», «South Dakota towns» etc.

Dan's post inspired me to touch up my Keynote* slide deck so that it looked more like the classic TV show from the 80's and so that it would also be as easy as possible for someone other than me to use in class. The deck I am posting here has five rounds ready to go. There are also a few template slides at the end explaining how to easily customize the game for your classroom.​ Finally, I added a rules slide at the beginning to explain the game, but definitely show them some clips from TV as well.

Here is a screenshot of the latest iteration. Click through to download the Keynote slide deck.​

​*In a perfect world I would program this in Flash, but for the time being a slide deck works pretty well, you will just need an additional timer, and you can't really go back if the students pass on a subject.

Lesson Playback: A Trig Foldable

If I am not at the top of my game, my third algebra 2 class can be a challenge. I am usually able to finagle the schedule though so that my third class lands the day after I had a chance to teach a lesson in the first two sections and can hence make adjustments to it with them in mind. This can result in a class that can eclipse the quality of the first two.

Today I wanted to further consolidate our learning about the unit circle. There aren't really many blog entries here so it should be easy to catch up if you are interested. Last class we worked on the trig puzzle I created, most groups managed to solve it and many even worked out the quote. You know an activity is engaging when you get e-mails late at night proclaiming a solution or kids running into class doing the same. The six identities were more of a challenge. They would be today's focus.​

Usually it's 5 Minutes of ❤​.

Today instead of the normal routine we started class with a 3-minute brainstorm.​ ​I always time this stuff on my phone and the alarm is almost always the same song. Currently Eric Hutchinson's lovely Rock and Roll. When the alarm sounds the kids stop working and immediately begin singing along. Always, and no I did not tell them to do this, these traditions just begin. When I turn off the timer it is quiet again. After the three minutes were up I did a quick snowball activity. Basically all this means is the students crumple their paper up like a snowball (kind of foreign in Mumbai) and throw it somewhere else in the room. This is not my favorite activity but kids like throwing their paper at one another, and used sparingly (like once or twice a year) it works. Once everyone retrieved a new paper I set the timer to two minutes and had them add facts to the new sheet they had. ​After this round they passed their paper to a neighbor and added facts for one more minute.

Next I picked one of the front whiteboards (my classroom is a whiteboard paradise) and we did some rounds. I went around the room, student to student, and had each of them contribute a fact they thought were important to the whiteboard map. I scribed. In retrospect I probably could have had a student do this, although they were all busy adding facts they had still missed to their sheets. About halfway through this process I deadpanned that this was even more fun than Scattergories. Although it actually was fun, and everybody was engaged. With no classes up to this point resembling a traditional lecture, my kids had collectively figured out a lot about the unit circle.

Results of our group brainstorm.

​With a board full of information I told my students to indicate on their papers the three most important points we had put on the board and look up when they were done. Next I got them out of their seats and had them indicate their selections using asterisks or checks on the board.

​With a quick review of the unit circle over, it was time for arts and crafts. I had decided yesterday that I would try one of these «foldables» I endlessly see on math blogs, so we gathered round and Katie taught us how to make one of those fortune telling things.

Back to the front I went. I surmised from the previous day's lesson that while few of the students had figured out all six (three pairs) of the trig identities I had included in the puzzle, we could probably come up with them in a group brainstorm. It went really fast actually. Our fortune teller foldable would house this information.

Quick sketch of what the final draft ​should look like.

I split the kids into small groups for the next part of the activity that was a jigsaw. Each group had to sketch a diagram (with words if necessary) that would illustrate why their trig identity pairs were true. I gave them 9 minutes to come up with their diagrams and then each group presented to the others.​ During this time I moved from group to group to give advice and ask questions.

I had thought their might be a few minutes at the end of the block for students to fill in their foldables, but the block was about done. Students snapped photos of the whiteboards and will complete the task for homework.​

Draft explanation of two of the identities.​

So not the most exciting lesson ever, nor the review game I thought I was going to write about today, but a good example of a lesson on a day in ordinary time.​

TPIR 03: Gas Money

Gas Money, which debuted in season 37, is one of the newer Pricing Games on The Price Is Right so you may not have ever seen it before. Although the probability of winning is basic, it is still an interesting game to watch because the contestant can walk away with the money if they choose to. The probability of winning Gas Money, is virtually the same as for Danger Price, the game I talked about last week. The thing is, when I used this game in class my students were all confused. They seemed to think that for some reason the probability of winning increased when cards were revealed. Like it was some version of the Monty Hall problem, a problem we had yet to discuss. The game is also interesting because the theoretical probability of winning is higher than the experimental, a fact I vaguely allude to in the questions but can be teased out more in class.

Click through to watch some clips.​

Gas Money is one of the newer games from The Price is Right. Watch a clip of Gas Money and then use what you know about probability to answer these questions.
​(a) What is the probability of winning Gas Money if the contestant plays blindly (i.e. just guessing)?
(b) From season 37 through season 40 on The Price Is Right, Gas Money was won only 6 times and was lost 42 times. Explain why is this curious? What might be the explanation for this?

Trig Puzzle

Generally reinventing the wheel is not the way to go with a math activity but I am a glutton for punishment sometimes. Years ago teaching pre-algebra I had made students a handwritten puzzle worksheet in which to solve they had to cut out the pieces and reassemble them into a box making sure to align the sides using exponent rules. This, I think, might have been inspired by a similar Pizzazz worksheet.

These ladies were on fire during this activity. Just outside my classroom are these awesome booths where students can work, love em.​

Anyhow, in one of my recent field trips through Sam's filing cabinet I remembered downloading something similar called a Tarsia puzzle. I would create one for my trigonometry students I figured. They had mastered the Unit Circle and I wanted to have them use the unit circle to have them think about some of the basic identities (odd, even, co-function etc.) I figured one of these puzzle worksheets could serve this purpose well. It might also be a nice bridge between the trig we had done so far and the next lesson I had originally planned about identities that I feared would be too hard. It turns out there is a free program to create these Tarsia puzzles (great resource here), but alas it is Windows only and at home I only have a Mac. Nevertheless the idea was gnawing at me so I decided to go for it and make my own.

I found the PDF I had downloaded from Sam's site (I can't seem to find the exact link) and opened it in Illustrator. From here I was able to delete all the original equations and add new ones I created in Math Type. Before I entered any equations into Illustrator I made a list of the six identities I wanted to focus on and made eighteen pairs that students would have to match. Next I drew the final shape for my puzzles' picture on paper and entered the equations to create the key. From here it was relatively easy to create the student version of the worksheet because I just cut up my key and entered my triangles into Illustrator. It was initially challenging to get the text rotated and oriented properly but by the time I had entered a few triangles of data I was a pro.  ​Additionally, I decided to add a couple layers to what was already a puzzle. First, I did not tell my students what final shape the triangles would be assembled into, and second I added a quote (that connects to the shape) that would reveal itself when the puzzle was completed. Further the quote has blanks that need to be filled in, making it even more challenging. I also figured that the blanks in the quote would make it more difficult for students to work the puzzle backwards. The quote also made it really easy to check to see if the puzzle was properly solved.

I've used this activity with two of my three algebra two classes so far and it has gone great. It is a little bit more difficult than I intended but in one class one group was able to crack the whole thing during the time allotted but just barely. Another group stayed behind after class to finish it. ​During the lesson I moved from group to group and helped students make connections between the puzzle and the unit circle, great lesson for a Friday afternoon math class.

Click through for a PDF version of the puzzle, editable in Illustrator if you have it.​

Trig Speed Madness

So I am now about a week in, to what has become a mild case of unit circle madness here at ASB. I got the grand idea of posting students completed unit circle worksheets on the wall with their times and blocks when they completed the task correctly in under 3:40. The length of «I Knew You Were Trouble.» I had told students to say «done» when they finished the task so that I could tell them their time to write on their paper. Of course, on the first day of this I wasn't even looking at the time when at 2:10 or so Camille shouts that she is done, and she was, correctly. Up to the wall she goes. And it's on. Everyone wants to be on the wall, and everyone on the wall wants to post a better time than they did previously. It's not unusual to spot kids practicing their unit circles at break and lunch.

Completed Unit circles from C & E blocks. C block managed to have every student in the class successfully complete their challenge today!

​I encourage the students to compete against their personal bests and not against each other. But of course they do both. The fastest time I have ever ever seen goes to Malavika who has been killing it all week. 1:12 then 1:02 then 1:01 and yesterday an unbelievable 54 s. The entire class to a moment to gasp.

54s Wow.​

​But that was yesterday. Everyday is great new fun with the kids. This morning I was in my classroom losing a fight with the school printer when I realized I was about 6 copies short of the Unit Circle quiz sheets I would need for class (that was, of course, about to begin). «Oh, I might be able to help you with that Mr. Roy» says Ruby from a few feet away. «Huh? You have blank copies of the Trig Speed form?!» «I might» she says. «Well do you have 6?» Indeed she does! «I made some to practice.» I am of course laughing, because this is both awesome and ridiculous, a hallmark of ASB. And Ruby achieved a personal best when she finished her unit circle in 1:57.