Going through Christopher Danielson’s new book “Which One Doesn’t Belong?”

Recently Christopher Danielson published an interesting shapes book online – “Which one doesn’t belong?”

As of January 26th, the 3rd version of it can be found via this tweet:

I decided to spend some time today having the kids talk through 5 of the 11 pages. It was an interesting and enjoyable exercise where I purposely played virtually no role at all. If an explanation contained an incorrect statement (about a right angle, say) I did not correct it. The goal was simply to hear their explanations.

I had my older son pick 5 numbers from 1 to 11 at random, and then I went through those slides in the book with each kid individually. Those 10 conversations are presented below. For each slide, the first video is my younger son and the second is my older son.

I think the videos speak for themselves, so I won’t add much comment except to say that both kids found talking about slide 8 to be the most difficult.

Overall my impression is that this was a good exercise to go through with both kids. Both of them stayed engaged for all five of the exercises and found several of the explanations of what didn’t belong to be pretty challenging. For my younger son (who is in 3rd grade), I think one important challenge was translating the mathematical ideas of similarity / difference he saw into words. For my older son (5th grade) the challenge was to provide explanations that didn’t rely on computation.

I would happily recommend spending time talking through the pages of this book to anyone working with elementary school kids. It would probably be interesting for older kids, too, though that’s a little outside of my experience just now.

Here are the conversations for slide 1 (plus the introduction to the exercise with each kid):



slide 3:



slide 7:



slide 8:



and finally slide 11:



A concept I’m struggling to communicate

As I’ve referenced many times, I’ve not taught any of this elementary material to kids before and there are lots of struggles learning (and teaching) this material that are totally new to me. Often concepts that I think will be easy are difficult, and just as often concepts that I think will be difficult seem to be easy.

One thing that I’m really struggling to communicate to my older son right now is the importance of properly labeling diagrams. Maybe this is something that just about every student struggles with, I don’t know, but for sure I am struggling to communicate how important it is.

The struggle with labeling is illustrated well watching my son work through a problem about similar triangles from this morning. He does a good job identifying the geometric ideas in the problem, but has a hard time connecting the dots because nothing is labelled. It was really hard for me to sit on my hands and not help him through this particular difficulty today:


Once he does label one side of the square as “x”, the solution to the problem comes really quickly. I wish could find the right way to emphasize the importance of labeling diagrams properly – hopefully struggling through problems like this one will help get that lesson sink in.


Some fun geometry and a challenging number theory topic

Some days teaching the boys seem to go super well – could be some interesting ideas from them, or just general enthusiasm, but days like this make me really happy that I have the opportunity to teach them math.

Yesterday my son struggled with this AMC 10 problem:

Problem #14 from the 2012 AMC 10 B

It is a pretty challenging problem. We were all pulled in slightly different directions today and didn’t get a chance to do any regular school work until I got home from work tonight. I thought it might be interesting to revisit a different version of this problem and hear him talk through it.

Here’s the problem I came up with and his initial thoughts:


I was interested to see that his initial approach was to try to compute, and the desire to compute was driven by remembering the formula for the area of a rhombus. Drawing the long diagonal of the rhombus gives him a different geometric idea, though. I been trying to emphasize geometric ideas over computation, so I was happy to see the change in approach.

The new approach led us down the path of congruent triangles and then principles of counting. Fun!

So, I stopped the last movie after about 5 minutes. The final part of his solution is here. At the end I ask him to re-explain the geometric idea he’s using ( over counting, I guess), and then showed him an alternate geometric solution.


So that was fun. I love how the geometric ideas pulled him away from the straight computation.

Next up was a new section in our Number Theory book – linear congruences. This is (obviously) a pretty advanced topic, and I actually skipped it with my older son when we went through the same book a few years ago. But, my younger son has found this introduction to number theory to be really interesting, so what the heck.

I just love his enthusiasm when he sees that 2x = 3 (mod 4) has no solutions.


Happy to head into the blizzard on this math high note tonight 🙂