Wednesday, February 25, 2015

2/24 Combinatorics

This session I decided to try out some more of the fivetriangle problems. I gave out a double sided sheet with 2 problems on it and let the kids decided whether to start with the easier one or go straight to the more complex problem.

1. Magic Square (easier)
This one was a general success. Almost everyone finished on their own. Although this is fairly amenable to a guess and check attack so its not surprising to see this happen. I ran a small experiment with a bit of competition that was interesting here.  Five minutes in or so, I announced to the group that 2 of the girls had finished this problem.  It immediately engaged the boys to know the girls were going faster but led to more final answer sharing rather than method sharing. So I need a middle path - engagement without corner cutting. 

2. Geometry problem.
This problem's tougher and took more time and individual prompting.  You essentially have to figure out the correct order of steps and then:

  • Recognize an isosceles triangle based on it having two sides that are the radii of the same circle.
  • Determine the rest of the angles in this triangle given one and the fact its isosceles.
  • Find the third side of the larger right triangle that was found in the previous step using the Pythagorean theorem.
  • Find a linear system to solve. The first part based on the radii of the largest circle. The second part based on the unknown radii also adding up to the third side of the triangle found above.

On the bright side I had a few aha moments with several of the kids as they worked their way through it. It also was decent practice for some mechanical skills like using the Pythagorean theorem. On the other hand, I exposed some general weaknesses.

1. Several kids did not know how to find the area of circle. This should have been covered given their curriculum especially for the fifth graders. However, I didn't pre-query if they knew it which is my primary take away.  Since I'm brainstorming activities for Pi day, I'm going to bear this in mind. Maybe we should warm up practicing some basic skills like area and circumference calculations and go forward from there.  
2. Isosceles triangles. Also shaky and theoretically covered. This falls under the same thought about basic skill warmups.
3. Applying the Pythagorean theorem. For example: one student didn't fully realize that c^2 had to be the hypotenuse of the triangle. Lesson: when we worked on proving it, a good warm up would have been some sample applications.  I assumed since everyone said they had seen it but just didn't know how it works that this was already true.

After finishing up the above activities I moved kids individually along to a packet I cribbed from AoPS on beginning combinatorics. I've not really touched this topic this year and its been on my pending list. Structurally this meant there was no great place for a central lecture. It didn't appear necessary here and for the most part the kids had few questions about this material. I'm going to return back to it in more depth and I'll discover if this is really the case. On the bright side, I had a few kids actually lingering past pickup time to work out the last few problems and I also had a few asking for my third backup package of AMC8 problems. Score one for engagement.

For the future I'm going to think a lot about whether the followup activity needs to be mostly self driven or require small assists versus the complexity of the warmup. If I expect to do some group work together then the warmup needs to be more uniformly finishable in 10-15 minutes so I can refocus the kids.  One possibility is to move the challenging problems towards the end. I'm a bit reluctant to do so though because I feel like I get the best efforts at the start of the session.

Wednesday, February 18, 2015

Triangle Number Exercise

Admin note: This is my first experiment getting latex formatting working for formulas so there's going to be more sigma notation than I would use in class. We're also on a break until next week. MathJax TeX Test Page
A triangle number is a sum 1 + 2 + ... + n which is also written $\sum\limits_{i=1}^n i$.

Two cool theorems that can be fairly easily proved are:

  • $\sum\limits_{i=1}^n i$ = $\frac{n(n+1)}{2}$

  • $(\sum\limits_{i=1}^n i)^2 = \sum\limits_{i=1}^n i^3$
The first one is not to hard to see and I expect many of the kids have already at least informally discovered it.  The second however is a bit more complicated and I'm considering whether we can walk our way through it during a  session.

There's a geometric interpretation that makes this fairly obvious:

New: cool triple triangle proof

However, I don't see the kids discovering this on their own but I suppose I could provide cut outs and have them do the transforms manually.

The inductive proof on the other hand might be more easily discovered but I foresee two problems.

1. Most of the kids don't have a good grasp on multiplying expressions like $(a + b)^2 = a^2 + 2ab + b^2$.  We could review this fact again (which we did once during the pythagorean theorem session.

2. Secondly I don't think most kids have done any inductive proofs. So it would probably also require a session just on that to work out the principles using simpler problems. TODO: what would a good set look like?

With those techniques in hand the following would be workable:

Let $s_1 = (\sum\limits_{i=1}^n i)$


$(\sum\limits_{i=1}^{n+1} i)^2 =  (s_1 + (n+1))^2$

Which expands to:

$s_1^2 + 2*s_1*(n+1) + (n+1)^2$

Then since we also know $s_1 = \frac{n(n+1)}{2}$ we can substitute that into the second term.

$s_1^2 + 2*\frac{n(n+1)}{2}*(n+1) + (n+1)^2$

which simplifies to

$s_1^2 + n(n+1)^2 + (n+1)^2$

And then using the distribute law to factor out n+1 you get:

$s_1^2 + (n + 1) (n+1)^2  = s_1^2 +  (n+1)^3 $

That's enough to use the inductive principle i.e. each successive term means adding the next cube and I think I could move the kids through steps via a worksheet type approach.

Tuesday, February 10, 2015

2/10 Olympiad #4

Olympiad days usually go well and this one was no different. (Perhaps this means I should introduce more competitive tasks on regular days despite my instincts)  To start I gave out my first worksheet. After a math session with my son, I decided  that the curriculum is neglecting how to convert repeating decimals back into fractions and that would make a fun warm-up exercise. I started by asking how many already knew how to do this. The most amusing answer: "I've memorized all the conversions".   So I made up a sheet that walked the kids through some examples and then made the transition to transforming some geometric series into a simple fraction i.e.

1/K + 1/K^2 + 1/K^3 ...   = 1 / K - 1  for K > 1

This worked out really well. After  short bit of lecturing at the start where I demoed converting .11111...,  Many of the kids had a bunch of aha moments.

We then did the Olympiad. I haven't graded the results yet but from glancing at the sheets as they were working it looks like they did fairly well overall.

Once again, on review as a group I was struck by the spectrum of algebraic thinking.  

Given a problem:  A group of dimes and nickels add up to X and there are Y more dimes than nickels. The most common technique was to create a chart and effectively brute force test the first few possible combinations.  This works fine in a MOEMS test because they create problems that  are very amenable to this and other guess and check strategies.

Addendum: I tried this out on my son and it was very interesting. He spent about 30 seconds and guessed the correct answer. I then asked him if he could do it algebraically and he proceeded to setup the equation and solve it. So maybe I'm underestimating the kids. The problems are structured so that the guess and check alternative is so easy that it might not pay to setup an equation. I'd have to give them some problems that require fractional answers to know for sure. Moral: structure your problems carefully. Provide an easy guessable answer only if you want to encourage that.

One other note, I had an extra parent volunteer at the club today and that was tremendously helpful. I'm hoping to have more volunteers in future weeks. In general I'm finding my bandwidth starts to max out at 10:1 and extra adults are extremely useful.


Pi day is coming up soon and I definitely want to do something themed for it near that day. Maybe we'll explore alternating series to derive pi/4 a bit.

Also I've been fiddling with the idea of doing an algebra readiness assessment at the end of the year. The district doesn't do anything officially that I find interesting and I wonder if the parents would appreciate it.

Friday, February 6, 2015

Field Testing

Whiteboard in the hallway at My Office

I like to try out potential problems on my coworkers. Actually sometimes I put ones up that I've found that I think are fun and probably too advanced for the club.

Tuesday, February 3, 2015

2/3 Finish what we started

My main goal for this week was to get to all the problems we didn't do last week after our KenKen digression.  So I started reviewing the AMC8 6-10 problems. These again didn't feel very deep and the kids had a pretty easy time outlining solutions to each other.

One note: Given a problem like

There are four more girls than boys in Ms. Raub's class of 28 students. What is the ratio of number of girls to the number of boys in her class?

Most of the kids tend to decompose the problem by saying if equally divided there would 14 of each and since there are four more then we should add and subtract 2 from 14 to get the two subtotals: 16 and 12.   I'm surprised that no one tends think in terms of equations i.e. x + x -4 = 28 solve for x.  This is a general trend through most of the problem solving. I'm not sure if its developmental and post algebra this all changes and if its worth emphasizing looking at the more algebraic approach.

After the AMC problems I had everyone choose between some practice Maths Olympiads or some problems from the five triangle site. This worked out pretty well. Almost everyone stayed on track and finished at least one sheet. The only drawback is that it doesn't allow for group solution sharing. I'll have to think if there is any way to do both without an extra person.