What is this all about?

This is the blog in which I post problems from the UVM Math contest ... decidedly non-trivial problems that both of us can use as class starters for Pre-Calculus and Calculus students.

First, a little background.  These questions are from a yearly contest put out by the University of Vermont for any in-state student. The math department up there has been doing this for years and I've been giving it for years. Each year, I add to our (me and the Mrs.) collection but I just recently acquired a notebook of tests going back to something like '72. These are great questions for students and I figured you all would appreciate if I'd share - judging from the response, you do.

I give them to my seniors as warm-ups, problems to work on while we wait for the morning announcements to interrupt us. Since the SmartBoard doesn't have particularly good formatting options, the best thing seemed to be for me to post them as images.  I'll only post two at a time so you won't be tempted to work on the whole thing at once. I know you people; you have work to do.

Mmmmm. Tasty.

The test itself is 41 questions.  The students have 2 hours and all the scrap paper they need. No calculators, no computers, no Wolfram Alpha, no smartphones, no collaboration; it's just you and this four page booklet ... and a pile of scrap paper.

It's raw, pure math. 

And did I mention, "No Calculators"?

Another thing about this test: "Express answers as a rational number in lowest terms" means that 7.5 is not the answer. They want 15/2 instead.

No exceptions.

"Rationalize denominators. Radicals must be reduced."  This is both a blessing and a curse for the students.  The smart ones realize quickly that everything must reduce, and that helps them stay on track.  They realize that there need to be lots of things cancelling -- how else can the answer be found in a few/ten minutes and expressed as a rational number?

"All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example."  Again, we're not getting a decimal here. The answer must be something you CAN find in 3-7 minutes.

So, time is a consideration, "brain freeze" is rampant, the occasional quiet muttered "You bastard, I gotcha" and you can't discount the missing calculator. It's a challenge and every question is "just on the tip of their brain and then everything cancels and 'viola'."

That "aha!" moment happens often.

The dope-slap happens too, when they forget to reduce 14/10 to 7/5.


For me as a teacher, it's a perfect vehicle for breaking them loose from tech, for giving them a sense of perspective, for giving them a challenge and setting them up against the best of the state.

Most importantly, it's a single test that incorporates EVERYTHING that they know from Pre-Calculus into one single set of questions. To solve each one requires a complete knowledge base and access to both trivia and method, and theorem and idea. Most questions are asked in an odd way that's a fun twist on what you've already done. It's a way for kids to realize that they actually DO know something and that getting a tough answer is really satisfying.

Afterwards, UVM will give out a Best in School and Best in Region, along with money prizes to the top six in the state. Best in School is usually 10-12 correct. (out of 41!) I tell the PreCalc kids that 6-10 is an achievement. The Best-in-State gets 17-20 correct.

The test format: The first one is always some kind of compound fraction and page one (10 - 13 questions) usually contains questions that are "easy" for the students in that they have SOME idea of where to go. They'll make computational mistakes here.

Pages 2-3 are medium difficulty in that the students really struggle to get anywhere that seems "right" and the questions tend to be wordy and confusing.

My brain is melting!

Page four are the ones that students rarely get. As math teachers, we look at an infinite series as easy, but the students don't ... when you have it as a infinite series of triangles, they wilt in the heat.

And don't forget that 2 hour time limit.

And that other problem that took an entire page of algebra and twenty minutes.

And "I forgot the square of 14 was 196, not 216, so that problem needs fixing. Oh wow, all this cancels now. Is that the answer? It seems too simple."

And these are juniors and seniors. In high school. Hating you at this moment.

But loving it when they succeed.