# Continued Fractions and the quadratic formula day 2

Yesterday my younger son and I did a fun project on continued fractions and the quadratic formula. He really seemed to enjoy it so we stayed on the same topic today.

I started by asking him to recall the relationship that we talked about yesterday and then to make up his own (repeating) continued fraction:

After he’d chosen the continued fraction to study, we looked at the first few approximations to get a feel for what the

Finally he used the quadratic formula to solve for the value of the new continued fraction – it turned out to be $(3 + \sqrt{17}) / 2$!

# A fun quadratic formula project with continued fractions and Fibonacci numbers

My son had a 1/2 day of school today due to the snow storm. Instead of having him work through problems form Art of Problem Solving’s Algebra book, I thought it would be fun to do a quadratic formula project since we had more time than usual.

I was a little brain dead as the spelling in the project will show, but still this was a nice project showing a neat application of the quadratic formula.

We started by looking at a common continued fraction and seeing how the Fibonacci numbers emerged:

Next we tried to see how to find the exact value of this continued fraction – here is where the quadratic formula made a surprise appearance:

Finally, we tried to decide which of the two roots of our quadratic equation were likely to be the value of the continued fraction. We had a slight detour here when my son thought that $\sqrt{5}$ was less than 1, but we got back on track after that:

It was definitely fun to show my son how the quadratic formula can appear outside of the problems in his textbook 🙂

# Exploring Wilson’s theorem with kids inspired by Martin Weissman’s An Illustrated Theory of Numbers

I love Martin Weissman’s An Illustrated Theory of Numbers:

Flipping through it last night I ran into an easy to state theorem that seemed like something the boys would enjoy exploring:

Wilson’s Theorem -> If $p$ is a prime number, then

$(p-1)! \equiv -1 \mod p.$

The proof is a bit advanced for the kids, but I thought it would still be fun to play around with the ideas. There is an absolutely delightful proof involving Fermat’s Little Theorem on the Wikipedia page about Wilson’s Theorem if you are familiar with polynomials and modular arithmetic:

Here’s how I introduced the theorem. We work through a few examples and the boys have some nice initial thoughts. One happy breakthrough that the boys made here is that they were able to see two ideas:

(i) For a prime $p$, $(p-1)!$ is never a multiple of $p$, and

(ii) There were ways to pair the numbers in $(p-1)!$ (at least in simple cases) to see that the product was $-1 \mod p$

Next we tried to extend the pairing idea the boys had found in the first part of the project. Extending that idea was initially pretty difficult for my younger son, but by the end of this video we’d found how to do it for the 7! case:

Now we moved on to study 11! and 13! At this point the boys were beginning to be able to find the pairings fairly quickly.

To wrap up this morning, I asked the boys for ideas on why we always seemed to be able to pair the numbers in a way that made Wilson’s theorem work. They had some really nice ideas:

(i) For odd primes, we always have an even number of integers – so pairing is possible,

(ii) We always have a product 1 * (p-1) which gives us a -1.

Then we chatted a bit more about why you could always find the other pairs to produce products that were 1. The main focus of our conversation here was why it wouldn’t work for non-primes:

Definitely a fun project! There’s some great arithmetic practice for the boys and also a many opportunities to explore and experience fun introductory ideas about number theory.

# What a kid learning algebra can look like

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 $2^{845}$ or $5^{362}$

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 $2^7$ and $5^3$ 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.

# A neat geometry problem I saw from Catriona Shearer

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!

# Introducing the basic ideas behind quadratic reciprocity to kids

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:

I started the project by showing them the chart on Wikipedia’s page showing the factorization of $n^2 - 5$ 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 . . . .

# Trying out Edmund Harriss’s puzzle with kids

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.