## Sharing a neat problem from Matt Enlow with kids

Yesterday Matt Enlow shared a list of his 100 favorite problems:

I flipped through the problems yesterday and problem #6 struck me as a terrific one to share with kids:

I mentioned the problem to the boys yesterday and today we dove into it. Here are their initial thoughts:

Next I asked them to see if they could say anything at all about what would have to be true if there were powers of 2 and 3 that met the conditions of the problem.

My older son noticed a pattern in the powers of 2 mod 3. That helped us understand some basic ideas about what would have to be true if powers of 2 and 3 differed by 1. We then moved on from that idea to see how the “difference of squares” idea from algebra could help us show that the equation in the problem would probably never be true for an even power of 2 greater than 4. Nice start – now we just had to get to the finish line:

The idea that we were missing at the end of the last video was that powers of 3 only had 3 as a prime factor. Once the boys noticed that, they were able to see that an even power of 2 could never satisfy the equation!

Now we had to look at odd powers of 2. They noticed that roughly the same idea works if the power of 3 was even. There was one little subtle difference in the argument, but luckily both boys were able to explain that bit!

Now we had to look at the case with odd powers of 3 and odd powers of 2. Here I showed them how polynomials like $x^n - 1$ factor. I also shows how the numbers of the form $3^n - 1$ factor when n is odd.

The interesting idea here was that the factorization was always a 2 and an odd number. That showed the product could never be a power of 2. It took a while for us to get to that via the polynomial factoring, but we did get there.

Which then solved the whole problem!

Finally – just to wrap things up, I went to the computer to find powers of 2 and 3 that were “close” together using continued fractions:

I was lucky to see Matt Enlow’s list of problems on twitter yesterday. It is going to be a great resource for me – can’t wait to share more of the problems with the boys.

## Exploring some polynomial basics with kids

I asked the boys what they waned to talk about for a project today and got a bit of a surprise when my 6th grader suggested polynomials. It seems that the topic has just come up in his math class at school and he’s interested in learning a bit more.

To start the project I asked him what he knew about polynomials:

Next I asked my younger son to explain adding and subtracting polynomials, and then to try to see how to multiply them since he said that he didn’t know how to multiply in the last video:

Now I wanted to show an unusual property of polynomials that was relatively easy to understand. My hope was to show my older son something that he’d not seen before but also something that was still accessible to my younger son. I chose to show them a short exploration of a difference table for a quadratic

Finally, I showed how you could use the difference table to reconstruct a quadratic function if you knew the values of the function at three consecutive integers:

So, despite the surprise topic request, this was a fun little project. It was nice to be able to find a topic that you could explore if polynomials were “new to you” and still get something out of if polynomials were a familiar topic.

## Recursive functions and the Fibonacci numbers

My son asked me about recursive functions yesterday morning and I showed him Dan Anderson’s online tutorial:

Even though Dan’s resource covers just about everything ( ha ha ) I thought maybe there was still something we could discuss this morning. So, I talked about the Fibonacci numbers.

First we did a quick introduction:

Next I had both boys pick their own recursively defined functions – and I got pretty lucky with the choices!

Now I showed them one approach you can use to solve these recursive equations. For the purposes of showing this idea to kids I’m not worried about the background details, but rather using the idea for some basic exponent review. (and, sorry, I’m a little careless around 1:30, but luckily catch my error fairly quickly before the whole video is derailed):

Now that we found the neat relationship between Fibonacci numbers and the golden ratio, we finished the calculation and found an explicit formula for the Fibonacci numbers:

We finished up with by checking our new formula on Mathematica. I also showed them a lucky coincidence from twitter yesterday that relates to this project. That coincidence involved this problem posted by Alexander Bogonmlny:

And this portion of the solution posted by Nassim Taleb:

(unfortunately as I tried to zoom in on Taleb’s solution while filming the camera got way out of focus, so close your eyes for the last few minutes of this video 😦 ).

Even if the ideas for finding the explicit solution to these recursive equations is a bit advanced, I still think this is a neat topic for kids to see. It certainly is a fun way to get some nice algebra review.

## Sharing David Wees’s system of equations problem

Saw a neat tweet from David Wees today. We are running around a bit tonight, but I had enough time to have the boys take a quick look before running out the door.

Here’s the problem:

Here’s my younger son talking through the problem:

and here’s what my older son had to say:

I saw in a later tweet that Wees intended to use this problem with younger kids (2nd and 3rd graders), but I’m still happy that the boys were able to talk through the problem and explain their reasoning.

## Working through a challenging AMC 10 problem

My son was working on a few old AMC 10 problems yesterday and problem 17 from the 2016 AMC 10a gave him some trouble:

I thought this would be a nice problem to go through with him. We started by talking through the problem to make sure that he understood it:

In the last video he had the idea to check the cases with 10 and 15 balls in the bucket, so we went through those cases:

Now we tried to figure out what was happening. He was having some difficulty seeing the pattern, so I spent this video trying to help him see the pattern. The trouble for me was that the pattern was 0, 1, 2, . . ., so it was hard to find a good hint.

Finally he worked through the algebraic expression he found in the last video:

This isn’t one of the “wow, this is a great problem” AMC problems, but I still like it. To solve it you have to bring in a few different ideas, and combining those different ideas is what seemed to give my son some trouble. Hopefully going through this problem was valuable for him.

## Finding Cos(72)

My older son is learning trig out of Art of Problem Solving’s Precalculus book this year. Yesterday he was working on the “sum to product” section, which derives rules for expressions like Cos(x) + Cos(y). It reminded me of one of my all time favorite math contest problems:

Today I thought I would show him my solution to that problem. What we go through probably isn’t the best or easiest solution, but I think it is an instructive solution for someone learning trig.

We started by talking about the problem and how some of the ideas he was currently learning could help solve it:

At the end of the last video we’d found a nice equation that we derived from the original problem:

$\cos(36^o) - \cos(72^o) = 2 \cos(36^o) * \cos(72^o)$

Now we used the double angle formula to simplify even more and find a cubic equation satisfied by Cos(36):

Now we tried to find the solutions to the cubic equation we found in the last video. This part gave my son a bit of trouble, but he eventually got there.

Now we were almost home! We just had to compute the value of Cos(72) and we’d be able to solve the problem. That involved one last application of the double angle formula:

I think solving this problem from scratch would be far too difficult for just about any kid just learning trig. But, the fun thing about this problem is that the ideas needed to solve the problem are all within reach using elementary trig identities. So, I think that working through the solution to this problem is a nice exercise for kids.

## The cube root of 1

After a week of doing a little bit of practice for the AMC 8, my older son has returned to Art of Problem Solving’s Precalculus book. The chapter he’s on know is about trig identities.

Unrelated to his work in that book, the cube root of 1 came up tonight and he said “that’s just 1, right?” So, we chatted . . .

First we talked about the equation $x^3 - 1 = 0$:

For the second part of the talk, we discussed the numbers $\frac{1}{2} \pm \frac{\sqrt{3}}{2}$ and their relation to the equation $e^{i\theta} = \cos(\theta) + i\sin(\theta)$

Finally, I connected the discussion with the double angle (and then the triple angle) formulas that he was learning today. You can use the same idea in this video with $\cos(5\theta)$ to find that $\cos(75^{o}) = (\sqrt{5} - 1)/4$:

So, a lucky comment from my son led to a fun discussion about some ideas from trig that he happened to be studying today 🙂

## Using 3d printing to help explore a few ideas from introductory algebra

Last spring I was playing around with some different 3d printing ideas and found a fun way to explore a common algebra mistake:

Does (x + y)^2 = x^2 + y^2

comparing x^2 + y^2 and (x + y)^2 with 3d printing

Today I decided to revisit that project. We started by looking at the same idea from algebra:

Does $x^2 + y^2 = (x + y)^2$ ?

At first we talked about the two equations using ideas from algebra and arithmetic.

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Now I asked the boys for their geometric intuition and then showed them the 3d printed graphs of the two functions.

This part ran a little long while my younger son was stuck on a small but important point about the graph $z = (x + y)^2$ – I didn’t want to tell him the answer and it took a couple of minutes for him to work through the idea in his mind.

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Next I showed them 3d prints of $x^3 + y^3$ and $(x + y)^3$ and asked them to tell me which one was which. It is really neat to hear the reasoning that kids use to go from shapes to equations.

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For the last part of the project I asked the boys to come up with their own algebra “mistakes” for us to explore. My older son chose to compare the graphs of $\sqrt{x^2 + y^2}$ and $x + y$.

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My younger son chose the two equations $x^2 - y^2$ and $(x - y)^2$. Changing the + to a – in our first set of equations turns out to have some pretty interesting geometric consequences – “it looks sort of like a saddle” was a fun comment.

One especially interesting idea here was exploring where $x^2 - y^2 = 0$. We used Mathematica’s ContourPlot[] function to explore those two lines because those lines weren’t immediately obvious on the saddle.

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I’m happy to have had the opportunity to revisit this old project. I think exploring simple algebraic expressions is a fun and sort of unexpected application of 3d printing.

## Talking inverse functions with my older son

My older son was stuck on a question about inverse functions from his Precalculus book:

Let $f(x) = \frac{cx}{2x + 3}$

Assume that $f(f(x)) = x$ for all $x \neq -3/2$. Solve for $c$.

We began by talking about what was giving him difficulty and then moved on to solving the problem.

Next we moved on to looking at the function on Mathematica. It was a little unlucky that the scale was different for the x- and y-axes, but I think the pictures still got the point across.

After we finished talking I posted about the problem on twitter and John Golden made a neat Desmos version of the problem:

## Starting AoPS’s Precalculus book this year

Since we stopped home schooling about 2 years ago I’ve mostly been doing projects with my kids rather than covering new content. This year I wanted to get back into content and have decided to do a slow walk through Art of Problem Solving’s Precalculus book with my older son.

Today this problem gave him a little trouble:

Find the domain and range of $f(x) = 1 / (1 + \frac{1}{x})$

We talked about the problem when he got home from school tonight. Here’s what he thought about the domain:

Next we talked about the range which is a more complicated problem (at least I think so anyway):

So, I’m happy that he was able walk through this explanation after school today. WE spent a while talking about it this morning and I was hoping that the ideas wouldn’t slip out of his mind during the day.