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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

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