## Friday, February 20, 2009

### Updates and A Challenge Problem From Our First Contest

• There are some 'new' math blogs (in some cases, new to me!) that I wish to bring to your attention. One of these was just started by one of my former students who is still in high school. She made excellent contributions to MathNotations last year. I'll post a link shortly. Look for creative thinking, challenging problems and an engaging writing style!
• Look for the Math Problem of the Day in the right sidebar. This is a 'gadget' made available by Blogger. Nice problems which are accessible to advanced middle schoolers and secondary students. They change every day so try them and check the solution link which is provided.
• There's a wonderful fraction calculator out there on the Web to which I will post a link and publish an article in a few days. If you haven't seen it, it will blow your mind! Imagine seeing the decimal expansion of any rational number to any desired number of places (within limits) instantly and that's not all it does!
• Yes, Pi Day is coming so I will post links to previous articles I've published and other excellent resources out there.
• Some of you know there's an excellent free resource of Singapore Math assessments for primary grades and more. You can download these and use them for your students. they make wonderful Problems of the Day and discussion points for your next department meeting! I'll post a link in a few days. Wait till you see the level of thinking and the content in the Grade 3 assessment, for example.

The following was question 5 from our contest. I'll leave it up for you to try. Feel free to comment or solve. This question proved to be of moderate difficulty for the teams. One has to be very careful about adhering to all the conditions regarding points P, Q and V. Have fun with it!

Problem 5 (2 pt question)

The graphs of y = 2x+3 and y = -x2 + bx + c intersect in 2 distinct points P and Q, where P is on the y-axis. Let V denote the vertex of the graph of the parabola.

(a) Determine all values of b for which the points Q and V coincide.
(b) Determine all values of b for which Q and V are distinct and the slope of line QV equals 3.

Totally_clueless said...

Alternative solution to the Math problem of the day (2/20/09):

To prove: P= prod_(i=1)^(k) (m+i) is divisible by k! for all m > 0.

Proof:
As mathematicians like to do, I will just pull something out of thin air to prove the assertion.
(Consider.....)

P/k! = (P.m!)/(k!m!) = (m+k)!/(k!m!) = C(m+k,k) = n, where n is an integer. QED.

TC

Dave Marain said...

TC--
Nice, elegant, and much more satisfying than the indicated inductive proof which appeared to have some flaws, not to mention notational issues!

Another approach based on fundamental divisibility concepts (remainders, etc), which should be part of a middle schooler's repertoire, is the following idea:
(i) Given any set of 2 consecutive integers one is divisible by 2;
(ii) In any set of 3 consecutive integers exactly one is div by 4 (again consider remainders!);
(iii) In any set of 4 consecutive integers, exactly one is div by 4, etc., from which the factorial statement follows easily.

Now the general proof of the assertion I'm making does require induction or an argument similar to yours, but the idea behind this argument may be more accessible to 7th-10th graders.

Gee, if these Problems of the Day generate this much discussion, I'll be able to take a break from writing my own investigations for a while!

So how do you feel about the contest problem I posed? Straightforward, mechanical, prosaic?? There is a catch in there if one uses a particular algebraic set-up so I'm interested in reactions. I will post the answer to the 2 parts shortly...

Dave Marain said...

Silly typo in my divisibility argument which I'm sure everyone noticed!
In any set of 3 consec. integers, exactly one is div by 3...

Totally Clueless said...

Hi Dave,

Well, you could say " In a set of 3 consecutive integers, at most one is divisible by 4"

Also, I like the way the contest problem was structured. The students have to use the information provided in the preamble to calculate c, and then move on to the rest of the problem. The two parts have their own nuances, and the solutions are gratifying (non-messy!!).

Had you posted a very similar problem on the blog earlier?

TC

Dave Marain said...

tc--
I've had several similar quadratic function problems on the blog although none quite like this one. I wrote this from scratch with the intents that you observed. I re-worked the problem many times to include those little twists.

My underlying purpose in writing these problems is to provide learning experiences for students. I believe good problems can 'teach' as well as or even better than all the explanations we provide. I should mention that 2-3 teams obtained the correct answer to both parts, although I didn't ask for an explanation so I'm really not sure if they recognized all of the subtleties. A couple of teams didn't pick up all of the clues, particularly that P and Q have to be distinct points so that Q cannot be (0,3).