roybot

Exeter Math

So I just discovered tonight that Phillips Exeter has all of their homegrown math texts in downloadable PDF form on their website. I downloaded them all and have been really enjoying them. Great reading and lots of stuff I can use. It was hard to head off for my dinner plans. More soon.

Here is the link: Exeter Web Site

Crossing The River With Dogs

So Tedd Herr and Ken Johnson's Crossing The River with Dogs from Key Curriculum Press is definitely one of my favorite math text books ever. I first learned about the book when I was still a student at Keene State where one of my methods classmates was using in a ninth grade class she was in. The book is designed for a problem solving course and I was lucky enough to be able to adopt it for use at John Stark. We don't (yet) have a problem solving course at ASB, but I still use the book whenever possible. The text has an enormous selection of well written problems adaptable for any math class. We have been studying mathematical induction in my Math HL class and so I decided to use the milk lovers problem from the text and extend it into an induction problem for my students.

Tree, Polly, Manas, and Sahil loved milk. They convinced their older brother, Ankit, who did all the shopping, to buy them each their own gallon of milk because they each liked it so much. They all put their names on their full gallons. It happened that on this day they were all really thirsty and each took ten drinks according to a different system. Tree started by drinking half of the milk in her container. Then she drank one-third of what was left. Then she drank one-fourth of what was left, then one-fifth, and so on. Polly started by drinking one-eleventh of her milk, then one-tenth of what was left, and so on. Manas started by drinking one-half of his milk, then two-thirds of what was left, then three fourths of what was left, then four-fifths, and so on. Sahil started by drinking one-half of his milk, then one-half of what was left, then one half of what was left, and so on.
To do: After each had taken ten drinks, how much milk remained in each container? Write a formula for the amount of milk left in each container after n drinks if they had continued drinking in the patterns originally indicated. Prove these formulas using induction or other methods. In what order do the 4 finish their milk? Discuss.

Here is the pages document this problem I created for my students. They are going to use it for a "practice" IA. 

Download Milk Lovers Pages '09 Files

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 cute 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 in the book with no exercises at all (particularly unfortunate since Foerster's claim to fame is his problem sets). Anyhow I used it to create a lesson, along with problems, that I have attached below: Here is the Pages '09 File: Induction 1.pages

Another Proof Without Words

So the first proof without words lesson went so well that I devised a follow-up. The original proof is from Proof Without Words by Roger Nelsen. In some ways I like this one even more because my students often find the initial statement with the inverse trig functions to be completely baffling. It is also nice because it makes the kids dig out some of their geometry skills to justify the necessary angle measurements. Here is the Pages '09 File: The Proof Is In The Diagram 2.pages

The Formula That Killed Wall Street

So Wired has another mathy article for next month's issue "The Formula That Killed Wall Street" Its about a probability formula that banks were using to determine risk. It talks about probability, conditional probability, correlation, regression, and more. I think I will use it the next time I teach probability or just whenever. I am always using articles about gambling when I teach probability but this, and the Netflix article I posted about recently are nice additions.

Proofs Without Words

So this year my IB Higher Level eleventh graders arrived with a full year of Precalculus which has made life a lot easier. I think we have been able to use some of this time to do more cool stuff, although we did get really bogged down doing vectors. Anyhow, I created this worksheet from Roger Nelsen's book Proof Without Words II and used it in my HL class last Friday. Most of the kids were able to figure out the proofs by themselves and the lesson went well.

Here is a screen shot of the first page of a worksheet I created around this idea.

Download The Proof Is In The Diagram - This is a Pages '09 file

Gator Golf

So I am reading, and enjoying Jeff Jarvis's very interesting What Would Google Do? and it has encouraged me to start blogging again. I am gonna try to follow the lead taken in a couple of the other edublogs and post some of my stuff to the net.

So here is a lesson I did this past week: Gator Golf

I love review games. I think I first heard of playing Gator Golf in the book Teaching for Dummies but I can never leave well enough alone so here is my twist on it.

Setup:

Divide the kids up into four teams -do this any way you see fit. Give each team a portable whiteboard to write down their final answers, along with scrap paper and golf pencils (obviously) to make notes if necessary. You will also need to get the Hasbro game Gator Golf. I also went to Home Depot and got some putting green carpet, and some tees and a putting glove at a golf store, you could probably make due without these but I love little details.

The Game

The game is played in multiple rounds

Round 1:

This round relies most heavily on the keynote. In round 1 there are 16 multiple choice questions (the questions I included are around trigonometry). When a question is revealed each team answers it on their whiteboard and then "locks in" the answer by flipping their board upside down. Once three of the four teams are locked in I cajole the last team into finishing as well. At this point the answers are revealed. The team in "control" reveals their answer first followed by the rest of the teams. If the team in control answers correctly they score a tee, if not, the next clockwise team has a chance to steal, etc. After each question is finished control passes to the next clockwise team around the room, the team in control is indicated by the location of the alligator in the slide deck.

Golf Round:

After the 16 questions are up I tally the tees that each team earned to determine each team's golf position. Sort of like in Hole In One on The Price Is Right (another post for another day) The team with the highest score gets to putt closest to the gator (but not too close). I usually wait until after the scores have been tallied to reveal the gator which always goes over very well. Prior to this just the green is on the floor and so the kids' curiosity is piqued. Everyone gets one putt, they score one point for hitting the gator at all, and three if the gator devours and spits out their ball. It is really important to allow the kids to have some fun golfing while also not getting so bogged down in the golfing part that no additional math gets done.

Round 2:

Each team gets a few minutes to peruse the longer form questions from the round two document (included in the zip file) then I have the teams in reverse order of score select a question (or two based on time) that they want control of. After this selection each team gets a few minutes to work together on their questions and write up solutions on the classrooms whiteboards (not the mini ones) for everyone else to see. If teams have extra time they can try to solve the questions their team did not select in hopes of stealing another teams questions. Once time is up I score each question, one putt per correct answer and we golf once more. This golfing round goes much more quickly since there are a maximum of 10 puts, probably much less.

Final Round:

If there is time, I will have teams wager part of their score on one final putt. To determine the grand champion.

I actually didn't give any of the teams any prizes for winning the golf game this year, just the glory of being champions. Candy would work fine but is not really necessary, I would certainly advise against giving any sort of bonus points as a prize.

Other Thoughts:

I hadn't played this game in a few years, since first gator had died, but I saw a new release of Gator Golf recently and picked up a copy. The game takes a bit of work to set up but always goes over very very well. I love tweaking rules and things so am always changing the games around to try to make them better. This time for example I had red tees and white tees mixed in my golf hat and when teams scored a point during round 1 they drew their tee out of the hat, red tees counted double. 

Whatever you do with the rules be sure to tell the kids what they are ahead of time and stick to them, kids go crazy when you change the rules in the middle of a game. Also during round one don't hesitate to stop the game and "go over" the tricky problems. This is a review game after all.

Files: This file includes the keynote slideshow I used for round one. Lately when I am making a slideshow for a game I create a Pages document with all the questions to begin with so I have included that document as well. I also included a copy of the Round 2 questions. The questions for this game are all about Trigonometry covering through trig equations but the game can easily be adapted for most other topics. More soon.

Gator Golf Zip (7.7mb)

Imaginary Friends

So the cartoon blog xkcd.com is often hilarious and today's post is no exception.

Cube Cards v1.2

So, since I am presenting on the Rubik's Cube at the NESA conference in Bangkok later this week, I found some time to revise my cube cards. You can download version 1.2 here

Changes

• fixed a few spelling errors and rewrote text for clarity in a few spots
• reworked the layout of the cube diagrams on cards 5, 6, and 9
• corrected an error my mom spotted on card 5
• rearranged the pdf file so that they could conceivably be printed front to back

Incidentally, these cards along with everything else on this blog are published under the creative commons license, click on the CC Badge for more information. Feel free to edit the master files, remix, and rework anything as much as you want. If you come up with something clever I would love to see it. It would also be great if you could join the mix too, consider posting a great lesson or other activity you have developed for everyone to share.

A New Math Study

So a new study is forthcoming from the government on the state of math education in the US. You can read an interesting article about it in the NY Times here. It suggests that common ground must be found between constructivist and more traditional direct instruction approaches. This is no surprise, I like to think I am pretty constructivist, but I still find myself at the board sometimes helping the students get their heads around what we have been learning or solidifying connections between the different areas of a topic we have been studying. My current batch of students are surprisingly eager to learn how to do everything by hand (1) so sometimes I will also get up front and show them how to do certain problems on their calculators and encourage them to try these approaches.
These math wars have been going on for a long time. I fondly remember my first year teaching at John Stark in NH. Right out of college and gung-ho I had all my kids write weekly math journals (2). The best outcome of this was a long dialog I had back and forth with one of my students, Matt Corsetti. Matt had found some articles from the California group Mathematically Correct on the internet and was wondering what was up since we were doing some of this "bad" stuff they talked about in our class. It was cool to discover how fruitful discussions could be with students. I find these conversations, whether about learning, the classroom environment, or whatever, keep me connected, inform my teaching, and improve my practice. This is something I try to do as much as possible now, even though I am not currently doing it through journals.

(1) This is probably because the IB has changed and now includes includes a non-calculator part so none of us are completely sure exactly what kinds of questions we need to be ready to do by hand.

(2) This, by the way, was not the most popular set of assignments I ever came up with and not one I would necessarily recommend for first year teachers -although now the lugging of journals could be done away with and kids could write blogs which teachers could evaluate painlessly with an RSS reader