Thursday, May 5, 2016

5/3 Square Roots

This week's theme was inspired by some of  questions the kids had while working on the Pythagorean Theorem about finding square roots. I have a small collection of subtopics around square roots and I thought we could go over some of them in Math Club as a group.

To start things off one of the girls showed her (really well done) solution to the Problem of the week:
See (https://drive.google.com/open?id=1R2gM2gyvIyrSdRK-y_wlYXrAZKTGY3KYcD63UZeZfVM)  which used an external equilateral triangle rather than the interior one I used.  Participation has been winding down a bit more than I like so I'm going to think if I can come up with an incentive to encourage a few more kids to work on the problems.

From there  I used a warm up riddle from Joshua Greene's BlogNumberPlay
"You’re creating a new coin system for your country. You must use only four coin values and you must be able to create the values 1 through 10 using one coin at a minimum and two coins maximum"
Everyone split it up into several groups of 3-4 kids and after 5 minutes of brain storming the kids found 3 different solutions. Having all the groups writing and verifying their answer simultaneously on the whiteboard made for some extra excitement.

Next I bridged into the square root discussions.

Firs we talked about bounding a square  root by using landmark numbers and then binary searching.
So for example the square root of 60 should be between 20^2 and 30^2 (which are easier to calculate) and then you an try 25^2 which is slightly larger, then perhaps 23^2 to finally arrive at the answer of 24^2 < square root (600) < 25^2 .

We then talked a little about irrational numbers.  As I found last year most kids at this age when asked to define a irrational number mention repeating decimals and often can't define the converse. So after having everyone come up with their ideas. I gave a formal definition i.e. the ratio of two integers.

With this preamble out of the way,  I mentioned most square roots are irrational. I had prepared two different proofs to show for why the square root of 2 is irrational.

1) Even/Odd proof by contradiction: http://www.math.utah.edu/~pa/math/q1.html

2) Geometric proof by contradication: http://www.cut-the-knot.org/proofs/sq_root.shtml#proof8ppp

I knew that these were both going to be a stretch, So I started by talking about how a proof by contradiction works and what we were assuming: that a rational number existed equal to the root and therefore at least one of the numerator and denominator were odd and how we were going to prove both were even.  We also worked through the basic rules for multiplying first:

1. even times even = even
2. odd times odd = odd
At last, I walked through the proof twice. Even so I think this was the most complex topic for the year. I decided to skip the second proof based on the room's reaction. The first was more than enough to absorb. I'd still like to present this next year but I'm thinking a worksheet format might work better where each step was guided.

Finally for the last piece of the lecture I asked if anyone knew how square roots were found before we had calculators?  As expected no one knew the answer. I explained that I was going to show one of the old algorithms for fun but I didn't expect anyone to memorize it.

I then reworked out the square root of 600 to a few decimal places by hand on the board.  Again if I repeat next year I'll have the kids try it once with some simple values like the square root of 2 or 3.

My original plan was to have the kids try out some  Sample AMC8 questons for the back half of the session:

https://www.artofproblemsolving.com/wiki/index.php?title=2014_AMC_8_Problems

But I thought everyone could use a break so I ended up handing out another sudoku puzzle. This occupied everyone to the end of the hour.

Overall:
1. I need to tinker with the format a bit. I think a work sheet for some of the square root material would be great for next year. I want to keep the proof but perhaps lead up to it with a simple proof by contradiction if I can find one.

2. I'm  going to come back to the sample AMC8 material in the next next few weeks. I'm toying with turning this into some kind of team contest.

3. I really want a lighter game/puzzle for next week to switch things up.

Problem of the Week: April's problem from http://www.moems.org/zinger.htm