Tag arithmetic

An equation with roots of sqrt(5) + sqrt(7)

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 \sqrt{5} + \sqrt{7}. The follow up to that question was to find a polynomial with integer coefficients having a root of 3 + \sqrt{5} + \sqrt{7}.

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 \sqrt{5} - \sqrt{7}. 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!

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Using Mathologer’s “Golden Ratio Spiral” video with kids

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 \sqrt{2}.

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.

Connecting the Euclidean Algorithm with geometry and continued fractions

We are slowly working through this amazing number theory book:

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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!

Project 2 from “An Illustrated Theory of Numbers” -> Playing with the Euclidean Algorithm with kids

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.

Sharing some number theory with kids thanks to Jim Propp’s “Who knows two?” blog post

Jim Propp published a terrific essay last week:

Here’s a direct link in case the Twitter link has problems:

Who knows two? by Jim Propp

Yesterday we did a fun project about card shuffling using the ideas from Propp’s post:

Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Today we did a second project for kids based on some ideas from Propp’s post. The topic today was “primitive roots”. Unfortunately this isn’t a topic that I know well and I messed up one explanation in the first video below. Oh well . . . still a really neat idea to share with kids.

So, I started by introducing the concept of primitive roots by reminding them of the 8 card and 52 card shuffles we looked at yesterday (pay no attention to my explanation about powers and mods at the end. It will become clear in the next video that I goofed up that explanation . . . . ):

Now we looked at some examples of primitive roots with small numbers. These simple examples give a nice way for kids to get a little bit of arithmetic practice and also help them see the main ideas in the problem that we are studying.

After working through these smaller examples, we moved to the computer to continue studying the problem. My older son noticed that the examples that seemed take the longest time to work were primes, but not all primes took a long time. That’s exactly the math idea we are looking at here.

Next we made a small change to the program to study all of the odd numbers up to 1,000 all at once. After correcting a little bug we found that the numbers we were looking for were indeed all primes.

We wrapped up be talking about what was known and what wasn’t known about these primitive roots. I was happy that my older son seemed to be particularly interested in this problem.

Definitely a fun project. It is always fun to find unsolved problems that are accessible to kids (and lots of them seem to come from number theory!). We will definitely have to do some follow up projects to explore the ideas here in a bit more detail.

Lee Dawson’s dart question is great to share with kids!

Saw a great problem for kids on Twitter today:

I had both of the boys talk through it tonight. Their approaches were a little different.

Here’s what my younger son (in 6th grade) had to say:

Here’s how my older son (8th grade) approached the problem:

This is a great problem to get kids talking about arithmetic and also a little bit of number theory. I really loved hearing the boys talk through it.

A terrific volume project I learned from Kathy Henderson

Yesterday afternoon I saw a really neat tweet from Kathy Henderson:

It immediately reminded me of our projects on the volume of a pyramid and a tetrahedron from a few weeks ago:

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Studying Tetrahedrons and Pyrmaids

Comparing a tetrahedron and a pyramid and experiment

We had a hard time finding the volume of the pyramid and tetrahedron by filling them with water because, despite our best efforts with tape, our shapes were not even close to water tight. They were definitely “popcorn tight” though, so we *had* to try out this activity.

Kathy was nice enough to share the handout she used, so designing today’s project was a piece of cake:

So, I had the boys make the shape’s prior to filming – we started the project with a quick discussion of the construction of the shapes. Then we talked about their volume.

My older son thought the volumes would be roughly the same. My younger son thought the one with the rectangular base would have the largest volume.

Next we tried to calculate the area of the base of each prism. Rather than using graph paper as the handout suggested, we found the area of each base by measuring. That gave us a chance for a little arithmetic and geometry practice, too.

Next we went to the kitchen scale to measure the change in weight when we filled the shapes with popcorn kernels. We found *very roughly* the relationship we were expecting, which was nice!

Finally, we revisited the pyramid and the tetrahedron project and looked at the two different volumes using popcorn. We found the ratio of the volumes was roughly 1.96 rather than the 1.7 to 1.8 ratio we found using water.

This is such a great project and I’m super happy that Kathy Henderson shared it yesterday. Working through the project you get to play with ideas from arithmetic and geometry. With a larger group you probably also get to discuss why everyone (presumably) found slightly different volumes.

So, a fun project that was relatively easy to implement. What a great start to the weekend 🙂

Happy Pi Day

For Pi day today we explored the amazing near integer e^{ \pi \sqrt{163}}

I started by showing the boys the numbers as well as just how close it was to being an integer. I measured the closeness both in terms of the decimal expansion and in terms of the continued fraction expansion of the number:

Next I asked the boys to each take a turn finding another number relating to \pi that was either nearly an integer or nearly a rational number. It turned out – especially with my younger son – to be a really nice way to discuss properties of powers of numbers.

The number my younger son found was 100 * 314^{\pi}

The number my older son found was 3.4 * \pi^{\sqrt{2}}

So – obviously just for fun – but still a neat way to talk about numbers and continued fractions. And a pretty fun number at the start, too 🙂

Sharring problem A1 from the 2017 Putnam with kids

We had a snow day today and I finally got around to sharing a neat problem from the 2017 Putnam Exam with the boys.

When I first saw the problem I thought it would be absolutely terrific to share with kids:

I started off the project today by having them read the problem and spending a little bit of time playing around:

After the initial conversation the boys, I triehd to start getting a bit more precise. The first sequence of numbers they knew was in the set was 2, 7, 12, 17, . . . .

They were not sure if 4 was in the set or not. My first challenge problem to them was to show that if 4 was in the set, then 3 would be in the set.

My next challenge question was whether or not 1 would be in the set.

Now we moved on to one of the number theory aspects of the problem – is 5 in the set?

During this conversation my younger son noticed that we had found a number that was 1 mod 5.

Finally, we talked through how you could find 6 from the number my son noticed in the last video.

I’m really happy with how this project went. This problem is not one (obviously) that I would expect the kids to be able to solve on their own, but most of the steps necessary to solve the problem are accessible to kids. It was really neat to hear their ideas along the way.

Comparing double stuffed Oreos to thin Oreos

In 2015 we did a project comparing double stuffed oreos to regular ones:

Do double stuffed oreos have double the stuffing?

As I said in that blog post, I’d seen a few teachers discussing the idea, but I don’t remember who originally shared the project. So, to be clear again, the idea for this line of study isn’t mine, but I’m happy to have some fun with it.

Instead of revisiting the prior project today, I tried something slightly differnt -> comparing double stuffed oreos with thin ones. The prep work for this project proved to be a little harder than I was expecting because the double stuffed oreo shells were really fragile. So, if you want to repeat this project, be prepared for lots of broken oreo shells!

To start I introduced my son to the problem we were going to try to solve today and asked for his thoughts. The problem was to try to find the ratio of the volume of stuffing in the double stuffed oreos to the volume of stuffing in the thin ones.

Our original intention was to weigh 10 of the crackers from each of the 2 types of cookies. We were able to get only 8, though. The 8 thin crackers weighted 22 grams and the 8 double stuffed crackers weighed 33 grams.

Sorry the writing was off screen.

Next we moved to weighing the full cookies. I didn’t communicate really well at the start of this video, and confused my son a bit. Eventually we got back on the same page weighed 4 cookies of each type.

The 4 double stuffed cookies together weighed 60 grams. That led us to conclude that the stuffing weight was approximately 27 grams for 4 cookies.

Finally, we repeated the process in step 3 with 4 thin oreos. We found that the 4 cookies together weighed roughly 30 grams, meaning the total weight of the filling was 8 grams.

So, my son’s guess of 4 to 1 for the ratio of the filling weight was pretty close. Turned out be 27 grams to 8 grams for 4 cookies, or about 3.5 to 1.

Definitely a fun project. I haven’t done much in the way of introductory statistics for kids – this project definitely gets kids engaged!