Friday, January 29, 2016

Extra Reading

There's been a spate of an interesting articles popping up on the web recently. The first one was from David Wees:  There's a lot of excellent advice here.

 I find myself choosing an appropriate task based on some understanding of anticipated student thinking, then imagining how students might approach the task and what they will think about, then considering how to sequence the different strategies student might use toward a big mathematical idea, and then creating the resources to enable me to use the instructional activity in the classroom. This level of planning is sustainable.
This very much meshes how I've been approaching planning recently. I usually start with an area like probability or a particularly activity like "no rectangles" and work backwards to create a sequence around it.  Then I turn to considerations of how the students will react and what should I modify to make the material better and or what I'm trying to emphasize during the exercise.

What I find in addition to this level of planning is I have to evaluate how things played out afterwards. Math Club lessons rarely (if ever) go perfectly especially the first time you try them. Its valuable to be constructively critical about each session and think about what could be improved. Right now, I find most of my areas of focus are on how I react and what I say during a session. When you're in front of a large group of kids, there are a ton of split second decisions that you're continually making based on behavior, kid's questions the pacing that each group is taking etc.

The second article I like was from Ben Blum-Smith:

The biggest takeaway for me was how exceedingly careful they are with people talking to the whole room. First of all, in classes that are 2 hours a day, full group discussions are always 10 minutes or less. Secondly, when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They never ask for hands, and they never cold-call. This means they already know more or less what the students are going to say.
Again this aligns with some of the techniques I've been playing with. Having kids talk in front of the class is really hard for flow. Even if they manage to project loudly and clearly enough for the others to hear, a student explanation is often harder to follow. This doesn't mean its not valuable to do them. Developing group communication is clearly an important skill. But I try to pace these out carefully. I'm also pre-selecting volunteers whenever I have the opportunity to do so based on what I see their work looks like (and also to get everyone up in front) When I do cold call, I'll often specify what kind of answer I'm looking for  "Is there anyone who solved this problem using a regrouping strategy?"  One thing, I'm not doing currently is actively looking to parse mistakes in front of the class. I see teachers doing this in various posts. But for us I feel its so hard for the students to follow along that it can quickly become a one on one session with a room sort of biding its time until you're done. I'm reserving these moments for true one-on-one interactions during the hour.

Finally, I saw this homework sheet for fourth graders (It appears to be from the Continental Math League)

For the most part these problems are solvable by algebraic means. But if you give them out what do you expect pre-algebra students to do with them?  I think the wrong answer here is to have parents setup equation and show how to solve 2(x - 15) = x + 15 for example. I doubt that will stick or provide long term benefit when its so far out of sequence and not embedded within a curriculum. Instead, these are generally looking for bar method or guess and check table like solutions. I tried a few out on my beta tester to confirm that yes these are actually quickly crackable that way.  That said, I question whether it makes sense to do lots of problems like these if you're going to gain the skills to algebraically attack them in a few years anyway. And while developing guess and check skills can be valuable, I find I'm more often fighting over-reliance on it and trying to design questions to encourage other strategies. So for now my instinct is "use with caution."

Tuesday, January 26, 2016

1/26 Recycler

It was another good week for participation. I had 8 responses to the problem of the week and almost everyone found the answer. Interestingly, this is officially classified as 7/8th grade but it was very approachable for my kids.

"Determine the smallest perfect square that is greater than 4000 and a multiple of 392"
This week I had one boy volunteer to demo the solution while we were waiting to go up. (I love that the kids are excited to show off their thinking)  Everyone started by factoring 392. Interestingly, most found the perfect square 7^2 within it but then proceeded to guess and check their way to the solution. This is pretty easy because if you start and multiply 49 by 8^2 you only have to go up through 12^2 before you find the answer double-checking if the square also contains 8 and is thus a multiple of 392.

So if I were revising the problem, I'd push the lower threshold up quite a bit to bring out the emphasis on factoring. The approach I was expecting which no one bothered to do was to multiple one more 2 into 392 so it was a perfect square itself and then just methodically check multiplying it by squares to quickly find 3^2.   I think setting the floor to say 40000 would have encouraged more creative thinking here.

From here, I brought out an activity I found out about from Larry Guth's no rectangle puzzle: activity

The idea is very simple draw a 3x3 grid to start out with and place the most tokens (pente glass beads in my case) on the grid without forming any rectangles.

Mike was really excited about the activity and I believe brought it to Math Night at his son's school. I'm happy to report it worked really well for us as well. We spent about 15 minutes experimenting to find a maximum and trying to prove why that was the case. I then went over the pigeon hole principle idea and we did the extension to count how many combinations of layouts there were. Generally my students do not have much exposure to counting problems so I was only able to get them to partially develop this on their own. I was pleased that they knew the multiplication principle of combinations. This is definitely an area I'm planning to work on more in the future. One thought is just to go through some of the material from the AoPS pre-algebra chapter. I'll look around the web more to see if I can get some other source material.

For the back half, this week I decided I wanted to reuse my triangle number worksheet from last year:
I made two small revisions this time. First  I added the multiple by 8  investigation. Secondly, I switched the graphic out to one that more clearly suggests the geometric interpretation of the general formula.  I was hoping that by stressing that there was a geometric approach out there I could channel the kids who found the pairing strategy to find it. Most of the room found the pairing approach again but the hinting didn't quite work. I did have several kids play with the idea of using the triangle formula direction i.e. 1/2 base * height.  Due to limited time I ended demoing the double triangle approach right before we had to go. If I repeat (and I totally will since I love triangle numbers) I will try to probe why this doesn't work more actively. In fact I may circle back next week to this idea since its an interesting avenue.

Generally what I think is necessary here is just more playful geometry exercises. This is weak point in the curriculum and kids need more exposure to playing with shapes and figures. I plan to do something with Pascal's triangle sometime soon. I'll be very curious to see if the kids can pick out the triangle numbers hiding in its sides.

Also during this exercise I had what the more I think about it was an exciting exchange with one boy who was struggling with the idea of abstraction. About 3 questions down I ask the kids to move from a particular solution for a specific triangle number to a general formula. This idea that the answer was not a number proved more challenging for him.  We talked a bit about what does a function mean, and how can it be an answer in of itself. I'm hoping this was a conceptual breakthrough moment.

Finally, I decided to have the kids investigate the 4x4 and 5x5 grids for the no rectangle problem as the problem of the week. Now I'll have to work out the solution myself before I see them again.

Tuesday, January 19, 2016

1/19 Third Olympiad

There was lots of good stuff this week. First off, I had 9 kids work on the problem of the week including several of those who just joined. That meant I could pre-select one of the newbies to show his solution on the whiteboard.

Before we reached that point, I took advantage of the recent largest prime number discovery and had a quick math chat about Mersenne primes and  274,207,281-1  More Info.  Everyone seemed interested so I'm tempted to go with a prime number based activity next week.  Also before I had the volunteer work on the white board I reminded everyone about working on listening to each other and made a few comments on how the problem had a lot of unconstrained parameters that seemed important but we wouldn't need to find in order to reach the solution.

Here' the original video: which does a very entertaining job of explaining the solution.

At this point we were ready to go with the latest (and slightly delayed) Olympiad. I was tempted to hold it off another week but that risked bunching too many of them in February.  Again we had seven new students. So this required discussing how the Olympiads work, the basic rules, and why we do them. Overall this year's procedure works much better than how I approached them my first year

1. I always remind everyone to read carefully / double check their work.
2. Check up front for kids missing pencils.
3. Bring a light activity for those who finish early so they are occupied and don't distract the test takers who are left. I also tend to move the early finishers to one corner of the room to further insulate them from those still working. 
4. Always go over the problems right after finishing. (Don't wait a week) Kids really want know if they found the right answer. So they are super motivated to listen to each other show their solutions. 
5. Always try the test out first yourself on the same day so you don't forget the problems. I tend to look at the problems more from the perspective now of what do I think will be harder for the kids to solve.  If I have time, I'll even practice some explanations for areas that I think might be more difficult.

This set was about average in difficulty and based on a quick glance as kids handed them to me, I think everyone did really well this time.  Interestingly, there was no regrouping or distributive law problem, the first time I've every seen that omission. Instead, the easiest problem was an odd take on Pythagorean triples that I found a bit flat. When we went over it, I actually asked if anyone recognized the numbers. I had one boy spot the 3-4-5 tuple which let me go off on a small riff on the subject and draw everyone's favorite picture:

I'll have to take a look at the scope and sequence for Fourth Grade again because I'm hoping we can do some geometric inquiries into the proof the of the theorem later on.  I have least 2 ideas since last year on alternate approaches the kids could look into. This I will definitely want to coordinate so that everyone is at least familiar with the theorem.

The really neat part about the answer discussion was I was able to call on almost every single kid in the room, including all of the new students. So I think everyone has been up to the whiteboard and discussed their work at least once now.

For my filler activity for the early finishers  I chose a followup congruent shape worksheet from Matt Enlow:

Just like last week, these worked really well drawing everyone in.  And since we had about five minutes to spare at the back end of the hour, everyone had a chance to start working on it.

Finally: I went back to the UWaterloo site for the problem of the week:

I think this one will prove easier for everyone despite its classification.

Friday, January 15, 2016

Sometimes its Harder the other way Part 2

I spent some time thinking about what initially looked like a very simple triangle congruence problem last night which I've outlined below. Given a perpendicular angle bisector of a triangle, show that it intersects the bottom at its median. Turn the problem on its head, (Given a perpendicular bisector of a triangle that intersects the median prove its perpendicular) and it suddenly become quite a bit trickier since you lose the obvious way to show the triangles are congruent.

What I like here is that its very similar structurally to this problem:
Both problems have the same asymmetric complexity. Both are solvable by essentially reflecting above or below the original figure.  Even better, this version is not as hard the previous one which would make  for a very nice progression if presented in sequence.

(Apology: I tried doing the whole explanation in Geogebra which doesn't scale as well as I intended. I recommend zooming in to read the figures)

Tuesday, January 12, 2016

1/12 Human Calculators

This week Math Club started up again although sadly without my son who had a fever and stayed home.  There are seven new kids to get to know and I'm on a good start memorizing their names. Hopefully after another week or so I'll have them down. We started this session by going around and having every kid introduce themselves, their class and if they were new why they joined and if they were returning what their favorite activity from last session was. There was a lot of "My parents signed me up" from the new kids which hopefully isn't a sign of true motivation. Based on the work afterwards I'm not too worried.  It was also interesting to see what problems struck a chord:

  • The toothpick / matchstick problems from Martin Gardiner were mentioned several times.
  • The logic grid puzzle came up. I knew everyone likes those from last year.
  • Interestingly one kid mentioned the practice math relay.
I then did the club charter talk. As I mentioned last week the 3 key points I like to emphasize are:

  • Basic behavior expectations. As I told the kids "You guys were great last session so hopefully I won't have to mention these again."
  • The importance of listening to others when they are demonstrating their work and not just listening but actively thinking/evaluating their strategies.
  • What to do when you're stuck. I had the kids especially those from the first session volunteer ideas for this part of the talk.
From there I jumped into the icebreaker which I chose because it gets everyone moving. I decided to call it A.L.U. after the arithmetic logic unit chip but I didn't tell the kids straight off the meaning of the acronym.
The rules are pretty simple. Everyone lines up initially sitting . I had 3 groups of 5.  The goal is to get the last person in the line to stand up. Each turn the team uses the following rules.

1. Anyone may sit down.
2. The person in the front of the line may sit up or sit down.
3. Everyone else may only stand up if everyone in front of them is standing up but if they do so the people in front must then sit down.

This ends up being pretty exciting for the room. After the first iteration I had the kids break into two groups instead and told them to think about what we were doing and how it related to arithmetic.
After letting them brainstorm for a few minutes I hit gold. One boy volunteered: "Does this have something to do with the binary system?" I wrote down some of the beginning iterations using 1 for standing up and  0 for sitting down,


And from their we had discussion about a binary counting. I wish I had more of these type activities in my back pocket (with movement) They are always appealing. Perhaps, for another time, I may have the kids design adders using logic gates.

For the back half of the session, I did end up using  Matt Enlow's congruent shape worksheet:

This was highly popular and kept the room humming.

My only regret was not having enough time for a linked tangram/tetris activity.  Finally I gave out a problem of the week based on an AoPS video I happened to watch recently:

Bobo the wonder clown is walking across a train bridge when he hears a train whistle behind him. He’s 45% of the way across the bridge. Whichever way he walks he’ll get off the bridge just as the train arrives. If the train is going 60 mph, how fast is Bobo walking?

I like this problem because of the conceptualization necessary to solve it. We'll see next week how high the participation rate is. Overall this week had great flow. I came away feeling exhilarated.

Wednesday, January 6, 2016

Preparing for the new Quarter

Next Tuesday will be the first session of Math club for this quarter. As a result, I'm in the last stages of planning this week. One monkey wrench for me is that the next MOEMS Olympiad was released yesterday. Fortunately, they are very lenient about when you administer the contest so I will not have to do it on the first day back. Which is lucky because I like the first day to be relaxed and focused on setting expectations.


This time around I ended up with 18 students and a wait list of 2.  7 of these kids are new this session. Once again it mostly skews towards fourth grade with a few fifth graders thrown in. I assume that core group will probably now continue into next year. So I'm half expecting to be imbalanced again towards fifth grade in my third year.  I also have mostly boys unlike last year. So I sent some emails to the teachers I know  asking if they'd encourage more girls to apply next quarter. We'll see if that helps the gender imbalance. We'll also find out how manageable the larger group size is.


There have a lot of interesting puzzles I've seen go by in the few weeks. @Five_triangles has been tweeting a few.  Then @CmonMattTHINK  had this cool  worksheet:   

@daveinstpaul uploaded this amazing sketch recently. This shows the relationship between a 30-60-90 triangle and a hexagon. I hadn't seen/thought of this connection before and I find it really beautiful. I almost have enough extensions on this theme for a day now see (

Finally in my intro mail I gave out an encoded message for the kids to crack:

"nqougeq kg eykb uocm'f nwjkql acylkql, w'e oggvwjx sglnylr kg fqqwjx qhqlzgjq slge oyfk kweq yf nqoo yf kbq jqn vwrf nbg nwoo mq igwjwjx cf. ws zgc ylq ymoq kg lqyr kbwf eqffyxq, fqjr eq y jgkq myuv nwkb zgcl syhglwkq kzpq gs uyjrz."

I'm waiting to see if anyone solves it or not. [Update: One student so far has already written back after cracking it]


One of the ways I'm trying to scale up is by being more organized about volunteers. So in my intro letter which resembles the one I sent last time: I'm using a shared google doc spreadsheet for volunteers. This seems like the easiest way to encourage people to commit to a particular date and to cover all the dates. Hopefully, everyone who pre-committed will actually signup and I will find enough people to round everything out.

First Session

With so many new students coming in I'm thinking about the importance of the initial charter talk.  I have my skeleton notes here:  As with last session the 3 key points I aim to cover are basic behavior, listening to others when they talk and handling when you get stuck.

 Beyond that I want to balance out the rest of the time with fun activities. I have a binary addition algorithm exercise  I tried that involves using people as digit/places which I'm considering doing again. I'' see if I find another binary system problem to pair with it.  Alternatively, I'm also thinking about more tangram based puzzles that would pair well with Math Enlow's worksheet.

Saturday, January 2, 2016

An Update on DreamBox

I've now watched about a month more of my son's play with DreamBox. So this constitutes an update to my prev. post:

New Interface

I had wondered previously what happened after all the stories in an area were finished. Did they become replaced with a new set? Clearly there weren't enough to last through the 8th grade, nor did they look appropriate for older children. I just found out the answer. Somewhere past the beginning of 3rd grade, you switch environments to the intermediate one.

In this environment, the lessons are now named and the pathway is made more clear to the student. There are no more stories and cartoon characters but the lessons remain the same. Unfortunately, the naming convention is a bit limited. On first glance, my son had 4 different cards with the same lesson name and no way to distinguish between them. 


  • The interface remains compelling. Its surprising how far badges etc. go as motivation.
  • The manipulatives used in the various lessons are very self explanatory. There's very little second guessing what the game interface wanted you to do.
  • I like the emphasis on various techniques like number bonds, landmark numbers etc.


  • The lessons were not very adaptive. There was way to much work on comparison tests i.e. is 123 > 132. Supposedly the intermediate level will start using pre-tests which may help here.
  • Some of the multiplication activities are too open ended. Asking for a multiplication fact for a given product doesn't mean a student will be guided to think about all the different combinations. A devious one (like my son) might just pick 1 times X = X every time for example. This required me to intervene and say lets find a different product that equals the same value.
  • I also found the practice multiplication facts to be a bit too randomized. There was no systematic introduction to families of multiples. In our case, coverage for multiples of 8 seemed light and I supplemented along the way where I found his knowledge thin. I'd like to see this track particular facts more carefully. 

Generally speaking I learned lots of things watching my son work his way through this material that I'm not sure the system picked up on.  This makes a reasonable approach for this part of basic arithmetic but don't treat it as completely hands off. Spend time watching how your child  and realizing where you need to supply feedback and extra practice / explanation.