Tuesday, December 19, 2017

2017 AMC 8 Questions

We finally received our scores. Overall I always have to remind myself that "comparison is the death of joy".  But really I think the kids did very well and when the final stats come out most will be at or above average. I also hope that I get a chance to see some of the same students take it next year and that I'll have evidence how much everyone has grown.

Art of Problem Solving has posted the 2017 AMC 8 problems and solutions at: https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems

To get a feel for this year's set, I tried them out and timed myself. It took me about 50 minutes to complete them carefully.  Since I don't find it as interesting, I didn't use guess and check to speed things up or look at the multiple choice answers unless the question required it. The kids however only had 40 minutes which made it fairly difficult in my mind.  You had to work really quickly and do some intelligent strategic guess work from time to time to finish everything.  I definitely will mention my own timing when we talk about it.  From what I can gather the test was a bit harder this year than 2016 with overall cut scores about 2 points lower for the top 1 and 5 percent overall.

Favorite Parts:

22) In the right triangle $ABC$$AC=12$$BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy]
[My personal preference is always for the geometry ones plus this had a 5-12-13 in it.]   The 3 different ways to solve listed on the site are part of why I like this. Note: its always simpler to look for similar triangles rather than jumping straight to the Pythagorean theorem.

24) Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?

A stealthy number theory problem.  Its actually fun to make out a chart for the first 60 days after which it repeats anyways.   And as often is the case its easier to find the inverse than the positive case i.e. 2/3 * 3/4 * 4/5 = 1 - (1/3 + 1/4 + 1/5 - (1/12 + 1/15 + 1/20) + 1/60)

11) A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?

I just  enjoyed visualizing this one.

Wednesday, December 13, 2017

12/12 End of Quarter

The end of this quarter really snuck up on me.  In my planning process I decided to finally use the video below on  Schumann's enumerative Geometry. What I particularly liked was the triangle puzzles discussed in the video and the fact they linked to a modern discovery.  But on looking through the video for a last time I became a little worried. The end part gets quite complex and I wasn't sure if the kids would be able to follow it. So I started to look back at old Julia Robinson Festival questions and assemble a min-festival we could do in Math Club. Then I remembered that it was the last session and I wanted to do game day to celebrate. So in the end I decided to go with video, take the problems with me in case it looked like a dud as well as the games.

What's especially nice is that there are several natural breaks in the video for pausing and trying the math out yourself. I took advantage of the ones in the beginning and had the kids try out assembling triangles and looking for patterns:

Base shapes:

I brought lots of colored pencils and had the kids draw versions on their own paper. If I had more time I might have precut out base triangles out instead. 


Overall this went pretty well after all. The kids wanted to see the end of the video when I offered them the chance to move on so we did watch the whole thing.  The only other mistake  I made was pausing a hair early the first time and having to explain the rules more than I expected.

In the back half of the day I brought in my usual assortment of board and card games:

  • pente
  • set
  • prime climb
  • terzetto
  • rush hour
These are still popular with the middle schoolers although I really need to pick up a new one before next time. My favorite moment here was one student pulled out last week's skyscraper puzzle to finish working on it today. I really like this display of persistence.

Finally, we also had a club discussion about recruiting. The kids decided to talk to their friends and in front of their math classes as well as one is going to make a PA announcement. We'll see how this effort works. I like that I'm offloading some of this to the kids and hopefully I'll find some 7th graders next quarter.

Thursday, December 7, 2017

Motivating Kids

I recently saw this tweet

The rather interesting gist of the research was how much better US students performed on PISA when given a monetary incentive.  That made me immediately think of my recent success and failures in getting everyone in the Math Club engaged.
If you haven't run an after school club, you might be excused thinking its nothing like a Math class. Of course, everyone is there voluntarily and excited to work on Math problems.  The truth is a bit more complicated.  For one, kids show up for a variety reasons including the dreaded "my parents made me do this."  Secondly, a student's temperament varies day to day.  The club meets after six long hours of school has already taken place. Some days even the best of kids are already worn out. Moreover,  a teacher in a classroom has a whole set of tools to leverage to make kids participate such as grades.  The club is a purely voluntary affair, buy in on everything from talking in front of the rest of the kids, to doing a problem of the week at home is a hard fought battle. Each day I need to find a way to create flow and draw kids into the topic I want us to explore.

There is no perfect answer to the problem and I continue to evolve in how I think about this issue. That by itself, is the first and foremost principle.   After each session I try to be critically honest with myself about how well it went and what I could do to improve.  In practice, I  almost always find I do better presenting a topic the second time.   Since I'm continually searching for new material this is something I have to keep in mind. For every really new activity, leverage whatever connection it has to previous ones to inform how it will be done and fall back to more tried and true formats/topics after experimenting.  I don't want to always be on the bleeding edge.

The culture of the club builds on itself.  First that means I always try to emphasize and reinforce when I see notable participation. I'm also ambitious in the sense I want the kids to engage with complex Math that requires a lot of focus.  In my  ideal vision we would just do a challenging problem set that I'd print out each week.  That would in reality be a recipe for disaster.  Instead I'm very mindful of the need to thread in puzzles/games/activities that are particularly playful. This is especially true when starting up with new kids I haven't worked with.

There are several general strategies I'm currently following that are working reasonably well

  • Games and Puzzles are always great as long as they are mathematically relevant.  Often they can be repeated multiple times and kids will develop more insightful strategies.
  • Leverage media. I'm super careful not to show a video most days. But sometimes after working really hard one week, a numberphile video is just the right change in tempo to keep everyone going. 
  • Have the kids use the whiteboard as much as possible. I've written about VNPS before: http://mymathclub.blogspot.com/2017/03/328-vnps.html  This remains an excellent strategy.    
  • I utilize a very minimal  common  routine to get everyone into a Math frame of mind. Mostly this consists of an introduction and talk about what we're planning to do for the day and a group review of the problem of the week.
  • If things don't go as well as I want one week - move on and change things up next time. 
  • Use competition from time to time. I'm also super cagey about this but official contests bring out a lot of energy and focus in most kids. 
  • Shamelessly bribe them with treats. I'm still giving out candy for homework participation. I only give one problem a week and the goal is to have time to think about something interesting over more than a few minutes. When enough work is handed back as a group I bring in treats. The ends seem to justify the means.
  • Talk candidly about where I think things are with kids. If I see a problem or direction I want the kids to go, I'll usually mention it up front. For example, last week I knew we were going to walk through student solutions to the the MOEMS contest. So at the start I told everyone that was coming up and I wanted to focus on listening to each other.
Overall lest this paint a picture of perfection, I still work on motivation from week to week. I'm always looking for other people's ideas on what works and what I might adapt.   Engagement is very near the heart of mathematics teaching, its complex and its not easy.

Looking forward:
Now that I've experienced 4th-8th graders I can definitely see the growth in maturity as kids get older.  Right now I only have 3 eighth graders. If I can recruit more of them, I'm hoping to leverage their leadership potential more. 

Tuesday, December 5, 2017

12/5 Olympiad #1

Today was the very delayed first MOEMS middle school contest day.  As I mentioned before this contest was supposed to be on the same day as AMC 8 so we had to push it out and then I needed some buffer. Fortunately, MOEMS is a low key organization and as long as you get all the contests into the system at the very end in March you can move individual dates around.  I was really curious going in how the kids would do and react to the contest. In looking at the questions before hand I thought this was slightly easier than any of the ones from last year.  My great worry was actually that it would be too easy for everyone.

That turned out to not be the case. When I polled at the end although the kids thought it was easier than AMC 8, they also generally all enjoyed it. That's great since I think its a good format: 5 questions over a half hour gives enough time for most kids to solve what they are capable of solving. And the split over 5 different weeks allows you to parcel the questions out and discuss them in manageable chunks as a group.

Which brings me to the other win for the day. This was probably the best whiteboarding session I've done yet this year. Almost everyone volunteered and there were multiple solutions presented for each of the 5 problems. There was just a ton of enthusiasm. Sadly, I'm not allowed to discuss any of the details of the problems but the kids came up with a lot of good problem solving solutions and really listened to each other.  I'm hoping to extend this streak to next week's whiteboarding and have some more interesting details to record here.

As usual to occupy everyone who finished early I brought a low-key puzzle. In this case I went back to the skyscraper puzzles from https://www.brainbashers.com/skyscrapers.asp  and printed  an easy and hard 6x6 one.

This is my absolute favorite linear systems problem:

Assume that are real numbers such that

Find the value of