Some people (traditionalists, of course) believe that math classes in school should focus only on direct instruction of math, and projects and other group assignments mean less time to study math, so less math would be learned. The teenagers might dislike the monotonous math classes focused only on direct instruction, but, the thought process goes, years later they'll thank you because they actual know enough math to pursue a career while their peers in the less focused math classes aren't being hired.
But I don't follow that philosophy on this blog -- simply because a believer in the direct instruction method doesn't even need to read a blog to find adequate teaching material. If a teacher is searching for the blogs of fellow math instructors online, then that teacher most likely is looking for special projects to implement in the classroom -- not just lecture notes. The most successful math teacher blogs focus on special projects, and so I wish to include such on my blog as well.
I like to gather such activities from various sources, to see which ones I like -- for if I draw only from a single source, the reader would simply go to my source and not read this blog. My main source is, of course, the U of Chicago text. But I also draw from other geometry texts as well as other math teacher blogs.
Now I remember once reading a geometry text and, interspersed throughout the second chapter on logic, there were a number of interesting logic puzzles. Since I'm right now in the logic chapter, why shouldn't I include some logic puzzles as well?
In searching for logic puzzles, I found the website of John Pratt, an astronomy professor from Utah:
What I like about this list of 20 problems is that two of them are identical to the problems I saw in that other geometry text -- Puzzles 4 and 8. And so I decided to take the first ten problems and make them into an activity worksheet.
Now how can teachers use this worksheet? I don't expect any student to discover the answers to all ten problems during a single class period. But perhaps a teacher can divide the class into ten groups and give a different puzzle to each group. Indeed, it's possible to use differentiation here -- since Dr. Pratt writes that he arranged the puzzles from easiest to hardest, a teacher can assign the first few problems to the lower performing groups. Pratt states that elementary students (and hopefully, the lower performing high school students) can solve the first few problems.
I don't include the answers here. Teachers can either solve the puzzles themselves or get the answers directly from Pratt's website.
Now here's the change that I want to mention. As it turns out, the first nine problems listed there today are the same as last year, but last year's #10 is now listed as #11. Here is the new Puzzle #10 listed on Pratt's website:
[I received this puzzle in an email that said it was taken from a standardized test for 14-year-olds.] Albert and Bernard just became friends with Cheryl and they wanted to know her birthday. Cheryl gave them both a list of ten possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, Aug 14, Aug 15, and Aug 17. Cheryl then whispers the month to Albert and the day to Bernard.
Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know either.
Bernard: At first I didn't know when Cheryl's birthday is, but after hearing hearing Albert's statement, now I do know.
Albert: And now I also know when Cheryl's birthday is!
What is Cheryl's birthday?
Now this puzzle was first posted to the web on April 10th, and it went viral as many people posted it on their social networking sites. Pratt writes that he posted this question on the 15th -- so it took just five days for this question to become popular.
The "standardized test for 14-year olds" mentioned by Pratt refers to a test from Singapore. Of course there's been much discussion on the blog about Singapore math -- a curriculum favored by many traditionalists due to its rigor despite being an integrated curriculum.
As it turns out, I've seen a version of this puzzle before last spring. It is a similar brain teaser known as the "Sum and Product Puzzle." The next link contains a statement and solution of the puzzle:
Notice that in describing the solution, the author actually uses Goldbach's conjecture -- the unproved conjecture that I mentioned earlier in the week. Of course, the numbers involved in this problem are much too small to be counterexamples to Goldbach.
I'll repeat the same activity worksheet from last year, although it might be interesting to replace the old Puzzle #10 with Cheryl's birthday problem. The sum and product version of this puzzle might be suitable in an algebra class, especially near the lesson on factoring quadratic polynomials. Once again, I don't give the answers on this blog -- follow the links above if you need any hints.