For today’s math project we are doing a 2nd project from George Hart and Henri Picciotto’s Zome Geometry:
I asked the boys to pick three shapes from the section on Archimedean solids. Here’s what they picked:
Shape 1: A Truncated Icosahedron
You start with a triple length icosahedron:
They you truncate it:
Shape 2: A truncated dodecahedron
Start with a dodecahedron with sides made from two short blue struts and 1 medium blue strut:
Now truncate it:
Start with an octahedron with side lengths equal to 3 long green struts.
For today’s math project I asked the boys to pick a project from George Hart and Henri Picciotto’s Zome Geometry:
My younger son picked a project about “Rhombic Zonohedra” which led to a terrific discussion about quadrilaterals and 3d geometry:
Ny older son picked a project on stellations of a dodecahedron. He was a little confused by the directions, but sorting out the confusion led to a great discussion.
I wish every kid everywhere could have the chance to play around with a zometool set.
We are spending this week studying applied max / min problems in Chapter 3 of Stewart’s Calculus book (we are using Edition 3). The chapter is a mix of theory (Rolle’s theorem, the mean value theorem), ideas about finding max and mins of functions, and finally applied max / min problems. We discussed the theory a bit, and I’m planning on circling back on the theory later in the year, we’ve already informally covered ideas about max and mins, so I wanted my son to focus mainly on the applied max / min problems.
The problem he’s presenting here goes like this:
Find the largest rectangle that can be inscribed in an equilateral triangle if one of the sides is on the base of the triangle.
So, a pretty standard problem, I guess, requiring a bit of geometry.
Here’s his solution:
After he presented his solution I gave a few comments to help him understand
(i) a slightly different way to approach the geometry (this is closer to a “fun fact” rather than an important piece of learning calculus).
(ii) one idea about finding max / mins of functions that might help him simplify some of his calculations, and
(iii) using the second derivative to tell if you have a max or a min
The next section in my son’s Calculus book is Newton’s method. I think it is a neat topic, but I chose to do a high level overview today because I wanted my younger son to join in. He’s learning algebra this year and I think (obviously) that the calculus details would be both over his head and not interesting to him.
We have looked at Newton’s method before in this project:
Exploring Newton’s method with kids
and I used the Mathematica code from A. Peter Young at UC Santa Cruz in this project, too. That code can be found here:
The page from A. Peter Young at U.C. Santa Cruz that gave me the Newton’s method code for Mathematica
So, here’s the high level overview I gave for Newton’s method and, more generally, the problem of finding roots of equations.
One fun thing that came up at the end of this video is that my older son noticed that Newton’s method might not always find the root you were hoping to find.
In the next part of the project we explored the idea my older son brought up at the end of the last video -> Are there cases where Newton’s method might not work as expected?
Next we looked at the function . We used Newton’s method with an initial guess of x = 3 to try to find approximations to the root x = 2.
Finally, we explored Newton’s method for complex numbers. This part was just for fun and to explore a few pretty pictures.
Today the second my son was studying in his calculus book was about using first and second derivatives to approximate values of functions.
I had him work through 6 problems in the book – 3 of which had answers in the book so he could check his work. Then we moved on to talking about the problems he did for which he was unable to check the answer.
The first problem involved finding the approximate value of using a linear approximation:
The next problem was finding the value of again using a linear approximation:
Instead of working through the 3rd problem, I asked him to use a quadratic approximation to find the approximate value of .
We’ve been talking about ideas in the related rates chapter in our calculus book for the last few days. My son has struggled with the ideas in this chapter much more than I expected he would. Fortunately many folks on twitter who have much more experience teaching calculus than I do have told me that this section often gives students a lot of trouble.
Over the last few days my son had worked through maybe 15 of the problems in our book – so I just picked one that he hadn’t done yet. Here’s the problem:
Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of per minute. How fast is the length of the third side increasing when the angle between the sides of fixed length is
Here’s the introduction to the problem and his initial thoughts:
(oh, and I should say at the outset, we’d not looked at this problem before and I didn’t realize that we needed a calculator, so I just banged out a few numbers on my computer just to speed things along. Most of my calculations were right, but one at the end of the 2nd video was wrong. Whatever I typed in produced the answer of 0.2 m / min, but if I’d actually typed in what we had on the board I would have found 0.4 m / min. That typo by me led to a little confusion in the 3rd video when we went to check our answer. Sorry about that error.)
Here’s the next part of his work.
There’s one part of this problem that we’d not really talked about carefully yet -> the angle numbers in the problem are given in degrees, but in calculus you need to be using radians. Since the main focus of this problem / session is related rates, I just explained that fact quickly in the video.
Again, sorry for the typo by me in evaluating the value of the answer. Fortunately we decided to check the answer – which was also an important conceptual calculus exercise.
Here we checked the answer from the last video and found the change was double what we were expecting. This was unexpected!
We checked the prior math off camera and found the typo. Luckily it was easy to find since we’d filmed the project. Hopefully this was a nice (and accidental) way to show that checking your answers is important 🙂
I stumbled on a neat, but challenging, calculus problem in my son’s book yesterday afternoon. We talked through the problem and then I wanted to revisit it today by having my son solve it from scratch.
Here’s the problem and his solution – It was too bad that the whiteboard didn’t leave enough room for the picture:
Following his presentation, we talked through a few of the math / calculus ideas in the problem:
Finally, I wanted to show him a different solution involving a u-subsitition. I was just doing this on the fly and didn’t realize how confusing a u-substitution would be. But we got through that part (and the upside is that I’ll remember that this topic is a lot more tricky than I think when we eventually get to it!):
Definitely a fun problem, and one that really forces you to think pretty hard about tangent lines and derivatives.
Today I decided to revisit an absolutely terrific number theory book – Martin Weissman’s An Illustrated Theory of Numbers:
I started first by talking about regular old integers and then introduced the idea of Gaussian integers. The boys have seen complex numbers before, so although the concept of of Gaussian integers might be new, they’ve done computations with imaginary numbers before.
They had lots of interesting thoughts and ideas – the ones I chose to explore in the future sections were addition, multiplication, and what “primes” might look like:
First up – we talked a little bit about the geometry of Gaussian integers, and what adding two Gaussian integers would look like:
Next we looked at the geometry of multiplication. It is a little harder to see what is going on here, but luckily in the previous video my older son had thought to compare the length of the points (or their distance from the origin). So, that approach at least gave us a nice place to start:
Finally, we took a look at the concept of prime numbers, and found a few regular integer primes
Today I talked a little bit about implicit differentiation with my son. We are following the progression in an old copy of Stewart’s Calculus book that I have (the 3rd edition) and section 2.6 discusses implicit differentiation.
After we had an introductory discussion this morning, Patrick Honner sent me this message:
Since my son had a half day of school today and I’m going to be working late tonight, I talked through the problem with him when he got home from school.
As I mention in this video introducing the problem, he’s not done any implicit differentiation problems on his own, yet, but you have to start somewhere!
By the end of the first video he’d found a value for the derivative, but now we had to interpret what he found – wow is it interesting listening to a kid trying to wrestle with calculus ideas for the first time!
To wrap up, I showed him some potential puzzles that are hiding behind the scenes in calculation – though I didn’t resolve these puzzles for him.
My older son is studying introductory calculus this year and so far (roughly the first two weeks) has been mostly learning about limits and derivative rules.
I wanted to try out a simple max / min problem just so he could get a peek at an application of some of the ideas he’s studying. I picked the standard problem about building a fence.
Here’s the first part -> what is the largest area enclosed by a rectangular fence if the perimeter is 40 feet?
Here’s the second part of the problem – now the rectangular fence has one side up against an existing wall. What’s the the maximum area now?