## Thursday, November 5, 2015

### 11/3 Oops I did it again

This week I did a great job planning to ensure continuity, rolled with how things played out in the room and didn't finish nearly as much as I wanted to. *sigh*  My originally thinking was last weeks practice Olympiad ended with the STAR * 4 = RATS decoding problem which was the most difficult part for most of the kids. So I thought we'd warm up with another example and keep working on the strategies to use. So I found

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I gave this a test before I started and thought it was a bit difficult but I could scaffold the room through the problem and still move onto a divisibility talk with some problems. In reality almost the whole math club finished the problem but it took almost the entire session and I'll have to save the divisibility exercises for another week.   The stickler seemed to be reasoning about where you need to carry and then testing the possible number that could fit. I think taking time to do a harder problem was probably worth it and none of the kids wanted to quit and move onto the second half early so I think were still having fun. We'll see if I bring one these back later this year if a 3rd sample will go quicker.

What was unfortunate about my planning was that I picked a problem of the week that was meant to pair with the second half:

Problem of the Week
Via Dan Finkel:
The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?

We didn't go over testing for divisibility by 11 and  I'm unsure  if the kids all know it or not. I had preprinted the problem so I couldn't change anything on the fly. So as a result I plan to send out a mail with some hint links to: http://www.artofproblemsolving.com/wiki/index.php/Divisibility_rules/Rule_for_11_proof and
https://www.mathsisfun.com/divisibility-rules.html

and hopefully this will be enough for some of the kids to work through this problem which looks much more complicated than it actually really is.

I was pleasantly surprised that 3 or 4 kids solved last weeks problems. When going over it in the room rather than deriving an algebraic equation based on similar triangle and the area formula I opted for a visual proof where you divide the triangle into 16 congruent sub-triangles.  My hope was that this would be more accessible. If I repeat it again I think I'll take more time and pre-print the triangle so the congruence is very obvious and perhaps even have the kids measure the triangles with a ruler to confirm they all are the same.