## A few intro calculus ideas to help explain why we study basic properties of sums

My older son is doing a some review this summer in the Integrated CME Mathematics III book this summer. The topic in the 2nd chapter of the book is sequences and series. I thought it would be fun to show him where (at least some of) this math leads. So tonight we talked about some basic ideas in calculus.

First I introduced the topic and reviewed some of the basic ideas of sequences and series:

Now we used the ideas from the first part to find the area under the curve y = x by approximating with rectangles:

To wrap up we extended the idea to find the are under the curve y = x^2 from x = 0 to x = 2. It was fun to see that the basic ideas seemed to makes sense to him.

I was really happy with how this project went. Putting these ideas together to calculate the area under a curve – even a simple curve – is a big step. It might be fun to try a few more examples like these before moving on to the next chapter.

## 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!

## A quick look at remainders

My older son was learning about the polynomial remainder theorem yesterday and then the Theorem of the Day twitter account tweeted about the theorem:

I took it as a sign that we should review remainders. My younger son doesn’t have a lot of experience with polynomials, so I wanted the main focus of today’s project to be on remainders when dividing integers. Here’s how we got started:

Next we looked at remainders in different bases to see what was the same and what was different:

Now we looked at the relationship between divisibility rules and remainders

Two wrap up, we looked at polynomials. Obviously this part is not meant to be comprehensive as my younger son isn’t that familiar with polynomials. What I was trying to do here was just give a simple overview of the remainder theorem for polynomials, and show that it wasn’t really that different than what we’d just looked at for numbers.

It was definitely a fun surprise to see the polynomial remainder theorem show up in two totally different places yesterday. Hopefully this review of remainders today was a nice exercise for the kids and helped my older son see a connection between division with integers and division with polynomials.

## Playing with Polynomials

We’ll be doing a little bit of review work in the Integrated CME Project III book. Today my son came across an interesting problem about trying to (sort of) match two polynomials. He came up with a nice solution this morning and we talked about the problem when he got home from school today.

The problem goes like this:

Find a polynomial that agrees with $x^3 - x$ at $x = 1, 2$, and 3, and has a value of 0 at x = 4.

Here’s my son talking through his solution:

After he finished his explanation, I showed him my solution to the problem:

To wrap up we went to Mathematical to look at both solutions and also so that I could show him a little surprise:

So, a nice start to this review project. It’ll be fun to work through the book over the summer.

## 3d printing totally changed my approach to talking about trig with my son

For the last two weeks we’ve been playing with this book:

Our most recent project involved one of the pentagon dissections. My son wrote the code to make the shapes on his own. We use the RegionPlot3D[] function in Mathematica. To make the various pieces, he has to write down equations of the lines that define the boundary of the shape. Writing down those equations is a fantastic exercise in algebra, geometry, and trig for kids.

Here’s his description of the shapes and how he made the pentagons:

Next we moved on to talking about one of the complicated shapes where the method he used to define the pentagon doesn’t work so well. I wish I would have filmed his thought process when he was playing with the code for this shape. He was really surprised when things didn’t work the first time, but he did a great job thinking through what he needed to do to make the shape correctly.

Here is his description of the process followed by his attempt to make the original shape (which he’d not seen in two days . . . )

I’m so happy that he’s been interested in making these tiles. I’ve honestly never seen him so engaged in a math project. The original intention of this project was just for trig review, but now I think creating these shapes is a great way to use 3d printing to introduce basic ideas from trig to students.

## Playing with the nonagon tiles

Two of our recent project have involved studying a tiling of a nonagon from the book “Ernest Irving Freese’s Geometric Transformations”

Those two projects are linked here:

Using “Ernest Irving Freese’s Geometric Transformations” with kids

nonagon tiles

After school yesterday I had each of the boys make a pattern with the nonagon tiles and then build the two patterns that were in the book. The videos below show there work. My younger son went first:

Here’s what my older son had to say:

This project was super fun from start to finish. Hearing the thoughts from the boys after seeing the pattern initially was really fun. Building and printing the blocks was a nice geometry / trig lesson. Then having the boys play around with them made for a really satisfying end to the project. I hope to do more like this in the near future.

## What a kid learning math can look like -> working through a pretty challenging geometry problem

My older son had a really neat geometry problem on his enrichment math homework. The problem is this:

Let $A_1, A_2, \ldots, A_n$ be the vertices of a regular n-gon, and let B be a point outside of the n-gon such that $A_1, A_2, B$ form an equilateral triangle. What is the largest value of n for which $A_n, A_1,$ and B are consecutive sides of a regular polygon?

His solution to this problem this morning surprised me because his starting point was “what happens if n is infinite?”

I asked him to present his solution tonight. It certainly isn’t a completely polished solution, but it is a great example of how a kid thinks about math

## Packing circles into a square

In the last two weeks – completely by coincidence – I’ve done two fun project with each of the boys related to circle packing.

The first was with my younger son and inspired by a Robert Kaplinsky tweet:

Sharing Robert Kaplinsky’s pipe stacking problem with my younger son

The second was with my older son and inspired by one of the problems on the Iowa State “problem of the week” collection.

The problem was about the limit of the area of circles packed into an equilateral triangle:

A terrific problem to share with calculus and geometry students from the Iowa State problem collection

Today I wanted to combine the two ideas and look at two ways of packing circles into a square. First I introduced the problem and we looked at the problem of packing circles in a square stacked directly on top of each other. As in the Iowa State problem, we found a surprise in the area covered by the circles as the number of circles approaches infinity:

Now we moved on to the problem of “staggered stacking.” In this video I introduce the problem and let the boys try to figure out why this problem is a little bit harder than the stacking problem from the last video:

Now we began to try to solve the “staggered stacking” problem. Turns out this problem is really tough! There are a lot of things about – the number of circles is much harder to calculate – but we were able to make some progress on some of the easy cases:

Now we tried to calculate how tall the stack of circles is. I think the algebra here is close to the edge of my younger son’s math knowledge. But he does a great job of explaining how to calculate the height. The nice thing is that he remembered the main idea from the project inspired by Robert Kaplinsky:

Finally – we put all of the ideas together. There are a lot of them, but with a little math magic, they all fit together really well!

So, a fun project following the neat coincidence of seeing to other problems related to circle packing recently. I think all of these problems are great ones to share with kids learning geometry and also learning calculus.

## A terrific problem to share with calculus and geometry students from the Iowa State problem collection

This is another problem from the Iowa State problem collection:

Yesterday I looked at one of the number theory problems with boys:

Some of the Iowa State “problems of the week” are great to share with kids

Tonight I tried another terrific (though very challenging) problem with my older son:

Here’s are his initial thoughts about the problem:

Now we rolled up our sleeves a bit and started to solve the problem. His first thought about what to do was to try to solve the problem with one inscribed circle and then with three inscribed circles:

The problem with three inscribed circles was giving him trouble so we moved on to a new movie and sort of started over on the three circle problem. While he was re-drawing the picture he was able to see how to make some progress:

Finally, having solved the problem with three circles, he moved on to solving the problem in general and found the surprising answer:

I really like these problems. Obviously not all of them are going to be accessible to kids, but the ones that are accessible are really amazing treasures!

## Seeing ideas about substitution for the first time

My son had an interesting problem on his enrichment math homework this week, and it gave him a lot of trouble this morning:

Tonight I thought it would be good to talk through the problem since I think the main idea he needed to solve it was new to him.

Here’s the introduction and some of the ideas he tried this morning:

Next we took a look at the equations on the computer and talked about some of the ideas we saw:

After looking at the graphs of the equations on the computer we came back to the whiteboard to talk about substitution.

Finally, having worked through the introductory part of u-substitution in the last video, I let him finish off the project on his own.

I can’t remember talking through this topic previously, but it was fun. It is always neat to be there when a kid is seeing a math topic for the first time.