Exploring Hasse diagrams with kids thanks to Martin Weissman’s An Illustrated Theory of Numbers

I’ve just started the book An Illustrated Theory of Numbers by Martin Weissman with my son:

We are going slowly and are just a few pages in, but I wanted to so a project with Hasse diagrams today because he told me last week that seeing those diagrams in the beginning of the book is what made him want to study the book a bit more.

We started today by looking at the book and exploring a bit about factoring integers:

After that introduction I had the boys read the section on the book on Hasse diagrams (roughly 1 page long) to be sure they understood how they worked. Here’s what they had to say and then a bit of practice:

It turned out that the final exercise in the last video – writing the Hasse diagram for 36 – proved to be a little tricky for my younger son. Because the last video was running long we broke things into two parts. Here we finish the diagram for 36:

We finished up by looking at one of the Hasse diagram exercises in the book. Here the boys wrote the diagrams for 7, 15, 18, and 105.

This project was a nice light touch one. It gave the boys an opportunity to review a bit of arithmetic and introductory number theory. It was also fun to explore this interesting connection between number theory and geometry.

Sharing a Catriona Shearer geometry puzzle with my son – part 18

We went back a few months for our project with Catriona Shearer’s puzzles tonight:

My son was able to solved this problem. He presented his solution here:

After he finished his presentation we went to the titter thread to find a solution he liked. He chose this really nice one:

He explained this solution here:

My plan is to look at one more of Catriona’s puzzles tomorrow and then return to his Precalculus studies next week. I’ve really enjoyed this project and think that the combination of trying to find a solution on his own and then explaining one of the solutions from the original twitter thread has been a great way for my son to study geometry.

Sharing a Catriona Shearer geometry puzzle with my son part 17

We continued our project looking at Catriona Shearer’s puzzles today:

My son had a really nice solution, which had a hidden assumption in it. He recognized that he’d assumed something was true and had a tough time showing that it was true. He got there eventually, though. Math can be like that sometimes!

Here’s his work and the discussion that followed:

Next we looked at the twitter thread to find a solution he liked. He chose this one from Jane Miller:

Here’s his explanation of Miller’s solution and why he liked it:

Sharing a Catriona Shearer geometry puzzle with my younger son – part 16

We dove back into our project with Catriona Shearer’s puzzles today. Here’s the one we tackled:

This one gave my son a bit of trouble, but he was able to find the solution. Here’s how he explained his work:

After he explained his solution I had him look through the twitter thread to find a solution he liked. He chose this one from @chzachau :

Here’s his explanation of this beautiful solution:

I’m always happy when he chooses solutions that are pure geometry. As I’ve said in a couple of prior posts, his instincts (very much like mine) tend to lean towards computation, so I think he really learns a lot from these pure geometric solutions.

Sharing a Catriona Shearer geometry puzzle with my younger son – part 15

My son is really enjoying working on Catriona Shearer’s geometry puzzles, so after a few days off we continued with the project this afternoon. Here’s the puzzle he looked at today:

My son came up with a pretty clever solution – he explains that solution here:

After he solved the problem I had him look at the problem’s twitter thread and find a solution he liked. He chose the solution from Brenda Meshejian. This is the second time we’ve looked at one of Meshejian’s soluitons (and hopefully I’ve pronounced her name correctly this time!).

Here’s Meshejian’s solution on twitter:

Here’s my son’s explanation of this solution:

I started this set of projects because I think the combination of thinking through Shearer’s problems and then explaining a solution from the twitter thread is a great way for kids to learn about geometry. So far this idea has been a really fun way for my son to explore some fun math during the lock down.

Sharing Catriona Shearer’s geometry puzzles with my younger son – part 14

Catriona Shearer posted another terrific geometry puzzle today and I had my son work on it today for a project. Here’s the puzzle:

Here’s how he explained his solution:

Next I had him look through the twitter thread following the problem and find a solution that he liked. He chose this beautiful solution from David Andriana:

Here’s how he explained Andriana’s soluiton:

One of the really fun things about using Catriona Shearer’s puzzles with my son is having him explain neat solutions from twitter threads. His work with Andriana’s solution today is a great example of the educational value that Catriona’s twitter threads have for kids!

Sharing a Catriona Shearer geometry puzzle with my younger son – part 13

We’ve been having a ton of fun for the last few weeks working through some of Catriona Shearer’s amazing puzzles. Last week I saw her share an puzzle from 2018 when she was asked for some of her favorites. Here’s that puzzle:

I thought this would be a great one for my younger son to try, and he was able to solve it.

His solution is a bit computational, so we broke his explanation into two pieces. Here he explains his approach to the problem:

With his approach drawn out now, we moved on to the computations:

Now we took a look at the twitter thread from the original problem to look for a solution that son liked. He chose the solution from Sanjay Singh

This solution was similar to how my son solved the problem, but that gave him a nice opportunity to understand the problem a bit better. Here’s his explanation of Singh’s solution:

I’m really happy that Catriona shared this problem again – it is terrific. It was also great to be reminded how long she’s been sharing these puzzles – it is going to be a long time until we get through all of them 🙂

Sharing Catriona Shearer’s geometry puzzles with my younger son – part 12

Catriona Shearer posted a really nice geometry puzzle today:

I thought this one would be a good challenge for my son and he was able to solve it, though he had a few false starts. He explains his thought process from start to finish here:

After he finished his explanation I had him look at the twitter thread and he liked the solution from M B Patey:

Here he explains that solution and why he liked it:

I feel like a broken record, but I love these problems. Today’s was at exactly the right level for my son and made for a great project!

Sharing Catriona Shearer’s geometry puzzles with my younger son – part 11

Catriona Shearer shared a great puzzle this morning.

This one is a bit more difficult than others I’ve shared with my son lately, but he wanted to give it a try. He wasn’t able to solve it, but we talked about the progress me had and the ideas he had:

Next we looked at a geometric solution given by Phillip Gibbs as well as comment to Gibbs’s tweet from Dr. Rick. These tweets showed a really clever geometric solution to the problem.

Here’s what my son had to say about these solutions:

Finally, I showed my son how to see that the triangles in the picture were, indeed, 30-60-90 triangles. This is a little bit of algebra, but I thought it would be important for him to see why those triangles were there.

I really love this problem and was happy that the thread had some great geometric solutions. Even though my son wasn’t able to get this one solved all the way, I think he learned a lot working through it.

Sharing Catriona Shearer’s geometry puzzles with my younger son – part 10

Today Catriona Shearer posted another great geometry puzzle:

My younger son was able to solve this one and he explains his solution here:

After he finished his explanation we looked at the twitter thread for a solution he liked and he chose this one from @ricardpe:

He explained that solution here:

This is our 10th project in this series. It has been really fun to have my son solve the puzzles and then explain a solution from the twitter thread. I’m really enjoying reviewing geometry with my younger son using Catriona’s ideas!