Last week I saw another terrific puzzle from Catriona Agg:
I left two copies of the puzzle for my son to work through while I was out this morning. For the first run through I asked him to solve the puzzle as it was stated. Here’s his work and his explanation:
For the second run through I asked him to solve the problem assuming that the radius of the circle was X rather than 5. This was first step in what I was hoping would be an interesting algebra exercise. Here he was successfully able to use the quadratic formula even though the equation he found had 2 variables:
For the last part of the project I wanted to see if he could factor the equation he found in the last video. This turned out to be a significantly more difficult challenge, but he figured out how to do it just as we ran out of space on the memory card!
I suspected that the factoring challenge would be more difficult than simply using the quadratic formula, though I didn’t realize how much different it would be. I might try to find some more challenges that involve multiple variables just to get a bit more practice with these ideas.
Earlier this week I saw a really neat puzzle from Catriona Agg:
We’ve done lots of projects with Catriona’s puzzles in the past, so just search for “Catriona” and you’ll find them.
My younger son spent some time off camera solving the puzzle and then I asked him to walk through his solution. His solution gets the main idea about tangents and circles, and then computes the radius of the semicircles using the Pythagorean theorem:
Typically when we play with one of Catriona’s puzzles I have my son look through the twitter thread afterwards and find a neat solution. I took a different approach today and showed him how to use similar triangles to get to the answer with slightly less computation:
I really like Catriona’s puzzle. I also think that my son’s explanation is a great example of what kids doing math looks like.
Catriona Agg posted this geometry puzzle on Twitter this morning:
I had the boys work on the problem on their own and then talk through their progress.
My older son went first – his solution is along the same lines as most of the solutions in Catriona’s twitter thread, though is reasoning is pretty interesting to hear:
My younger son went next. He wasn’t able to find the solution on his own, but was able to get there while we talked about his work. I’m sorry that I forgot the camera was zoomed in on the paper here. I do zoom out a little over half way through. Hopefully the words are clear even if some of the work is off screen:
At the end of the last video my son had worked through the main idea of the problem. Here he finishes the solution and talks about what he liked about the problem:
As usual, having the boys work through one of Catriona’s puzzles made for a great project. I really liked the algebra / geometry combo that this problem had as I think that was great practice for my younger son. I also think the more intuitive solution my older son had shows how mathematical intuition develops as kids get older.
Yesterday we did a project on this geometry problem shared by Catriona Agg:
I also shared the problem on Facebook and a friend from college shared a solution that I’d not seen before (though looking back at Catriona’s twitter thread, it is there . . . . of course!). So, today I decided to take a 2nd day with the problem and have my son look at this new solution.
But I started by having him explain the “power of point” solution I shared with him yesterday just to get warmed up:
Next I set up my friend Raf’s solution to see if my son could solve the problem using this clever idea:
Finally, since the solution with the full inscribed circle depends on a geometry formula relating the area and perimeter of a triangle to the length of the radius of the inscribed circle, I asked my son if he could prove that formula was true. It took him a minute to find the idea, but he was able to construct the proof:
I was happy to be able to share three different solutions to the problem that Catriona shared. It definitely made for a fun little weekend geometry review!
Yesterday Catriona Agg shared a nice geometry problem on Twitter:
I thought this problem would be a great one for my younger son to work through, so I asked him to give it a try this morning. Here’s how he explained his work:
Usually when I share one of Catriona’s problems I ask him to go through the twitter thread to pick out one of the cool solutions. We took a different approach today as we’d talked about “power of a point” a few weeks ago and I wanted to show him how that idea could be used to solve the problem:
I really like this problem and especially like how different solutions bring in different parts of high school geometry. Thanks (for the 1,000th time!) to Catriona for sharing a great little puzzle 🙂
This past Spring my younger son did a really fun geometry review by studying some of Catriona’s geometry puzzles. I think we did around 20 projects – they can be found here:
20 or so projects is only scratching the surface, though, since she comes out with fantastic geometry puzzles all the time! The one from yesterday is fantastic and I thought it would be great for another project for my son:
His solution to the problem was computational. He explains the main ideas here without going into all of the computational details:
In all of our projects we return to the problem’s twitter thread and my son picks out a solution that he thinks is interesting. Today he picked the solution from @lucythepoet
Here’s his explanation of this solution and a bit about why he liked it:
I think – and have thought for a long time! – that Catriona’s puzzles are great to use with kids. The process of attempting to solve the puzzle (sometimes getting it, sometimes not) and then going to the twitter thread to see all of the neat solutions has been a great way for my younger son to review geometry.
We did a fun series in the Spring using some Catriona Shearer puzzles to help my younger son review geometry. If you search for her name on the blog you’ll find those projects. The idea was to attempt to solve the puzzle and then to go through the twitter thread to find a neat solution to explain.
Today Shearer posted a great puzzle and I thought we’d try the old style of project today for fun. Here’s the puzzle:
My son was not able to solve this one, but here are his thoughts about the problem and some of the ideas that he tried:
This time we went to the twitter thread to find both interesting solutions and help on making progress. He liked two solutions – one from Vincent Pantaloni:
And one from Misty Guthrie:
Here’s his discussion of what he learned from those two solutions:
There are so many great ways to use Catriona Shearer’s puzzles with kids. When they can’t find a solution, the twitter threads are super helpful for seeing how to solve the problem. When they are able to solve them, the twitter threads are terrific for finding different solutions! It is always really fun going through these puzzles!
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.
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:
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.