Section 9.2 of Art of Problem Solving’s Introduction to Algebra is one of my favorite sections in any book that my kids have gone through. The section has the simple title – “Which is Greater?”
One question from that section that was giving my younger son some trouble today was this one:
Which is greater or
I decided our conversation about the problem would make a great Family Math talk, so we dove in – his first few strategies to try to solve the problem resulted in dead ends, unfortunately. By the end of the video, though, we had a strategy.
Now that we’d found that and are close together, we tried to use that idea to find out more information about the original numbers.
I found his idea of approximating at the end to be fascinating even if it wasn’t quite right. It was also interesting to me how difficult it was for him to see that the two numbers on the left hand side of the white board were each bigger than the two corresponding numbers on the right hand side of the board. It is such a natural argument for someone experienced in math, but, as always, it is nice to be reminded that arguments like that are not obvious to kids.
I saw this tweet from Catriona Shearer last week:
It was a fun problem to work through, and I ended up 3d printing the rectangles that made the shape:
Today I managed to get around to discussing the problem with the boys. First I put the pieces on our whiteboard and explained the problem. Before diving into the solution, I asked them what they thought we’d need to do to solve it:
Next we move on to solving the problem. My older son had the idea of reducing the problem to 1 variable by calling the short side of one of the rectangles 1 and the long side x.
Then the boys found a nice way to solve for x. The algebra was a little confusing to my younger son, but he was able to understand it when my older son walked through it. I liked their solution a lot.
Now that we’d solved for the length of the long side, we went back and solved the original problem -> what portion of the original square is shaded. The final step is a nice exercise in algebra / arithmetic with irrational numbers.
Definitely a fun problem – thanks to Catriona Shearer for sharing it!
We are heading out for a little vacation before school starts and I wanted a gentle topic for today’s project. When I woke up this morning the idea of introducing the boys to quadratic reciprocity jumped into my head. The Wikipedia page on the topic gave me a few ideas:
Wikipedia’s page on quadratic reciprocity
I started the project by showing them the chart on Wikipedia’s page showing the factorization of for integers 1, 2, 3, and etc . . .
What patterns, if any, would they see?
Next we moved to a second table from Wikipedia’s page – this table shows the squares mod p for primes going going from 3 to 47.
Again, what patterns, if any, do they notice?
Now I had them look for a special number -> for which primes p could we find a square congruent to -1 mod p?
Finally, we wrote short program in Mathematica to test the conjecture that we had in the last video. The conjecture was that primes congruent to 3 mod 4 would have no squares congruent to -1 mod p, and for primes congruent to 1 mod 4 would, -1 would always be a square.
Sorry for the less than stellar camera work in this video . . . .
Saw a neat puzzle posted by Edmund Harriss last night:
I thought it would be fun to try it out with the boys this afternoon.
I didn’t give them much direction after introducing the puzzle – just enough to make sure that my younger son understood the situation:
After the first 5 minutes they had the main idea needed to solve the puzzle. In this video they got to the solution and were able to explain why their solution worked:
Definitely a fun challenge problem to share with kids. You really just have to be sure that they understand the set up and they can go all the way from there.
Yesterday we did a fun project on a problem I learned from Michael Pershan
That project is here:
Sharing a problem from the Julia Robinson math festival with the boys
Last night I got an interesting comment on twitter in response to my Younger son suggesting that we write the numbers in a circle – a suggestion that we didn’t pursue:
So, today we revisited the problem and wrote the numbers in a circle:
Next I asked them to try to find another set of numbers that would lead us to be able to pair all of the numbers together with the sum of each pair being a square. The discussion here was fascinating and they eventually found
This problem definitely made for a fun weekend. Thanks to Michael Pershan for sharing the problem originally and to Rod Bogart for encouraging us to look at the problem again using my younger son’s idea.
Yesterday I returned from a trip and the boys returned from camp, so we were together again for the first time in two weeks. I also happened to see this tweet from Michael Persian:
This problem seemed like a nice one to use to get back in to our math project routine.
Here’s the introduction to the problem and the full approach the boys used to work through it the firs time:
When they solved the problem the first time around, they started by pairing 16 and 9. I asked them to write down their original pairs but to go through the problem a second time without starting with 16 and 9 and see if the choices really were forced. Here’s how that went:
This is a really nice problem for kids. It is easy to understand, so kids can jump right into it. There’s also lots of different ways to approach it. Definitely a fun way to get back into our math projects.
My older son is working thorugh the Integrated CME Project Mathematics III book this summer. Last week he came across a pretty interesting problem in the first chapter of the book.
That chapter is about polynomials and the question was to find a polynomial with integer coefficients having a root of . The follow up to that question was to find a polynomial with integer coefficients having a root of .
His original solution to the problem as actually terrific. His first thought was to guess that the solution would be a quadratic with second root . That didn’t work but it gave him some new ideas and he found his way to the solution.
Following his solution, we talked about several different ways to solve the problem. Earlier this week we revisited the problem – I wanted to make sure the ideas hadn’t slipped out of his mind.
Here’s how he approached the first part:
Here’s the second part:
Finally, we went to Mathematica to check that the polynomials that he found do, indeed, have the correct numbers as roots.
I like this problem a lot. It is a great way for kids learning algebra to see polynomials in a slightly different light. They also learn that solutions with square roots are not automatically associated with quadratics!
Mathologer recently published a terrific video about the Golden Ratio and Infinite descent:
As usual, this video is absolutely terrific and I was excited to share it with the boys. Here are their reactions after seeing the video this morning:
My younger son thought the discussion about the Golden Spiral was interesting, so we spent the first part of the project today talking about golden rectangles, the golden ratio, and the golden spiral:
My older son was interested in ideas about irrational numbers and why the spirals were infinitely long for irrational numbers. We explored that idea for using a rectangle with aspect ration of .
Unfortunately I did a terrible job explaining the ideas here. Luckily we were reviewing ideas from Mathologer’s video rather than seeing these ideas for the first time. I’ll definitely have to revisit these ideas with the boys later.
We are slowly working through this amazing number theory book:
Tonight my older son was out at a viola lesson, so I was looking for a project on the Euclidean Algorithm to do with my younger son. I decided to show him how the Euclidean Algorithm is connected to geometry and to continued fractions.
First, though, we reviewed the Euclidean Algorithm:
Next we looked at a geometric version of the arithmetic problem that we just did:
Finally, we looked at a connection with continued fractions
Exploring the Euclidean Algorithm is such a great topic for kids. There are so many interesting connections and so many interesting math ideas that are accessible to kids. Can’t wait to explore more with this new book!
We are spending a few weeks working through this amazing book:
Currently we are looking at the second on the Euclidean Algorithm, and last night I had a chance to talk through some of the ideas with my older son.
Here are his initial thoughts on the Euclidean Algorithm after reading through a few pages of chapter 1. We worked through the example of finding the greatest common divisor of 85 and 133:
Next we moved on to trying to solve the Diophantine equation 133*x + 85*y = 1. We had already looked at this equation on Mathematica, but had not discussed how to use the ideas from the Euclidean algorithm to solve it.
In this video you’ll see how my son begins to think through some of the ideas about how the Euclidean algorithm helps you solve this equation.
By the end of the last video my son had found some ideas that would help him solve the equation 133*x + 85*y = 1. In this video we finish up the computation and (luckily!) find a solution that was different than then one Mathematica found.
Comparing those two solutions helps to show why there are infinitely many solutions.
I’m on the road today, but hope to be able to talk through some of the ideas from the Euclidean Algorithm with my younger son tonight. The topic is a great one for kids – there are lots of neat math ideas to think about (and to review!). Hopefully we’ll get to explore some of the connections from geometry, too.