By happy coincidence my older son is studying Taylor Series this week, so I thought it would be fun to talk through the problem.

Here’s the introduction:

My son had some nice ideas about how to approach the problem in the last video, so next we went to the white board to work out the details:

Finally, I asked my son to finish up the details and then asked him for a sort of number theory proof of why 180 multiplied by an integer with all digits equal to 5 was always close to a power of 10:

Definitely a fun little problem – definitely accessible to students learning some introductory calculus.

]]>I intended for the first three parts to be review, but one interesting misconception came up – so the talk was more than just review.

Here’s the introduction to the problem and my son’s work on the first part of the problem. This problem asks you to write down the usual series for and then write down the series for

The next question asks you to write the series for and then write the Taylor Series around x = 0 for the function .

Here my son wrote the series for the 2nd function in a way that surprised me:

Once we wrote the correct series for the 2nd part of the last question, we moved on to part (c) of the problem -> find the 6th derivative of the function above evaluated at x = 0:

Finally, we looked at the last part of the problem. The question is about the error in a Taylor series approximation. I’d hoped to use this question to introduce ideas about error terms in Taylor Series, but unfortunately I completely butchered the discussion. Oh well – we’ll be covering the ideas here in a much more detailed way later this week:

]]>We started with a problem similar to ones that he’s already seen:

Next I showed him a type of equations that he’s not see before and we spent 5 min talking about his ideas of how you could solve it:

Finally, for the specific equation we were looking at, I showed him how we could use continued fractions to solve it. As a bonus he remembered the connection between the Fibonacci numbers and the golden ratio and that got us to the exact solution!

]]>Today I shared the program with the boys. It has 4 different types of random walks to explore. For each one I asked the boys what they thought would happen. At the end we looked at all 4 simultaneously.

Sorry that the starting videos are so blue – I didn’t notice that while we were filming (and didn’t do anything to fix it, so I don’t know why the last two vides are better . . . .)

Also, following publication, I learned the author of the program we were playing with:

Here’s the introduction and the first random walk – in the walk we study here, the steps are restricted to points on a triangular lattice:

In the next random walk, the steps were chosen from a 2d Gaussian distribution. It is interesting to hear what the boys thought would be different:

Now we studied a random walk where the steps all have the same length, but the direction of the steps was chosen at random:

The last one is a walk in which the steps are restricted to left/right/up/down. They think this walk will look very different than the prior ones:

Finally, we looked at the 4 walks on the screen at the same time. They were surprised at how similar they were to each other:

Definitely a fun project, and a really neat way for kids to explore some basic ideas (and surprises!) in random walks.

]]>By happy coincidence my older son is spending a little time reviewing polar coordinates now. Although not exactly the same ideas, I think there’s enough overlap to make studying Sanderson’s new video worthwhile.

So, I’m going to do a 6-part project going through the video. Tonight we watched it and my son’s initial thoughts are below. Each of the next 5 parts will be spent discussing and answering the 5 questions that Sanderson asks in the 2nd half of the video.

Here’s question #1:

We’ve been away from right triangle trig a little bit lately, so I was interested to see how my son would approach this problem. His approach was a bit of a surprise, but it did get him to the right answer:

The next question in Grant’s video is about how the area of one of the rings on the sphere changes when you project it down to the “base” of the sphere (see the picture above).

I thought that answering this question would be a really good geometry, trig, and Calculus exercise for my son:

Now we get to a really interesting part -> Question #3

Grant asks you to relate the area you found in question 2 – the area of a ring around the sphere projected down to the center of the sphere – to the area of a different ring around the sphere.

Here’s my son’s work on this problem:

Finally – my son answers questions #4 and #5 after a quick review of the previous results. He was a little tired tonight, but we needed to squeeze in these two questions tonight because I have to travel for work tomorrow.

Here’s question #4:

and #5:

And here’s his work on those two questions:

]]>Ever since seeing it I’ve been excited to share the program with the boys and hear what they had to say. Today was that day

So, this morning I asked the boys to take 15 to 20 min each to play with the program and pick 3 tiling patterns that they found interesting. Here’s what they had to say about what they found.

My older son went first. The main idea that caught his eye was the surprise of distorted versions of the original shapes continuing to tile the plane:

My younger son went second. I’m not sure if it was the main idea, but definitely one idea that caught his attention is that a skeleton of the original tiling pattern seemed to stay in the tiling pattern no matter how the original shapes were distorted:

Definitely a neat program for kids to play around with and a really fun way for kids to experience a bit of computer math!

]]>Here is my son’s work on the first series:

Here’s his work on the 2nd series:

Here’s his work on the 3rd series:

This has been a fun topic to cover. I’m excited to start on power series tomorrow!

]]>Here’s that project:

Introducing the boys to Futility Closet’s Paradox of the Second Ace

Today we continued the project and calculated the two probabilities in the “paradox.” These calculations are pretty challenging ones for kids, but even with the counting challenges, this was a really fun project.

I started by reminding them of the problem and getting their thoughts from yesterday:

Now we calculated the probability of having a second ace given that you have at least one ace. It took a while to find the right counting ideas, but once they did the calculation went pretty quickly. The counting technique that we used here was case by case counting:

Next we moved to the 2nd problem -> If you have the Ace of Spades, what is the probability that you have more than one ace? The counting technique that we used here was complimentary counting:

Finally, I asked the boys to reflect on the problem – was it still a “paradox” in their minds or did it make a bit more sense now that we worked through it?

I really loved talking through this problem with the boys – thanks to John Cook for sharing it and to Futility Closet for writing about it originally!

]]>My reaction was that it would be fun to turn this into a project for kids, but this one would need a little introduction since conditional probability can be incredibly non-intuitive. During the week I came up with a plan, and we began to look at the problem this morning.

Here’s the introduction – I asked the boys to give their initial reaction to the seeming paradox:

Next we looked at an example that is slightly easier to digest -> rolling two dice and asking “do you have at least one 6?”

My younger son had a little trouble with the conditional probability, so I’m happy that we took this introductory path:

Next we moved to a slightly more difficult problem -> rolling 3 distinct dice. I used a 6x6x6 Rubik’s cube to represent the 216 states. To start, I asked the boys to count the number of states that had at least one six. Their approach to counting those 91 states was really fascinating:

Finally, we looked at the analogy to the 2nd ace paradox in our setting. So, if you have “at least one 6” what is the chance that you have more than one six, and if you have “a six on a specific die” what is the chance that you have more than one six?

Again, my younger son had a little trouble understanding how the cube represented the various rolls, but being able to hold the cube and see the states helped him get past that trouble:

Tomorrow we’ll move on to studying the paradox with the playing cards. Hopefully today’s introduction helped the boys understand

]]>I started off tonight’s project by asking my son to talk about the Harmonic series and give two different “proofs” for why it diverges:

Next I pulled one of the exercises out of his book. This series is a slightly difficult integral – and we’ve skipped the techniques of integration section for now – so this part was a nice integration review, too:

Following the approach via the integral test, I wanted to show how the comparison test works. The comparison test is actually the topic for tomorrow, so this was new material. He did a nice job thinking through how the test would work:

Finally, I picked a random problem from the book and asked him to work through it. It turned out to be a pretty neat problem asking why the integral test wouldn’t work on a particular series:

It is fun going through sequences and series with my son. It is going to take a bit longer than I initially thought, so we’ll probably be working through the ideas here through mid December. We’ll come back to techniques of integration after that.

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