*Revisiting Richard Evan Schwartz’s Gallery of the Infinite

At the beginning of this year I bought a copy of Richard Evan Schwartz’s Gallery of the Infinite:

We ended up doing a fun project using the book:

Talking through Ricahrd Evan Schwartz’s Gallery of the Infinite with kids

Yesterday I saw the book on our shelf and asked the boys to find something in it that they thought was interesting. The section on bijections caught the eye of my younger son, and we used that section for a project today.

First we talked about the basic idea of bijections and how you could use bijections to tell if two sets were the same size:

Next we talked about a bijection that is pretty challenging for kids to find -> a bijection between the positive integers and the set of all integers:

Finally, we talked through Cantor’s “diagonal argument” which shows that there is no bijection between the integers and the real numbers (and, thus, that the “infinity” of real numbers is somehow larger than the “infinity” of integers!):

Tomorrow we’ll talk through the section of the book that my older son thought was interesting -> the Cantor set.

Talking about derivatives of exponential functions

My older son is starting the section in Stewart’s Calculus book on exponential functions. We’ve already spent a couple of days talking about inverse functions and the topic for today was finding derivatives of exponential functions.

I started by asking how he thought you’d even approach trying to find the derivative of an exponential function. It has been a while since we’ve talked about derivatives, so it took a few minutes before he came to the idea of using the definition of the derivative. Once we began to approach the problem via the definition of the derivative, we found that finding the derivative of an exponential function came down to a single limit:

Next we went to Mathematica to see if we could make any sense of this limit. Without realizing it, I had an error in the code that was causing the code to output numerical approximations. My son noticed the error and had me fix it. Unfortunately fixing that error spoiled the surprise in the answer . . . whoops 😦

Now we went back to the board to finish our computations for the derivative of an exponential function. It is pretty neat to see that the derivatives of all exponential functions are related to each other in a fairly simply way.

This was a fun discussion. The follow up discussion later was a neat problem from Stewart that asked you to show that:

$e^x > 1 + x + x^2 / 2! + \ldots x^n / n!$ for every n. That problem was a nice exercise in derivatives of exponential functions and also techniques of proof.

Using Dan Anderson’s line art program to explore lines with my son

My younger son just started the section on lines in Art of Problem Solving’s Algebra book. By happy coincidence Dan Anderson shared this fun line math art program:

I thought it would be fun to have my son play around with the program and just see what his reactions were.

Here are his initial reactions to the program:

Next I had him manipulate a different part of the program – it turns out that what happens on the screen is pretty complicated to explain!

Finally, even though he hasn’t studied any trig, yet, I had him change the two starting shapes to see how the pictures changed. By luck the changes he made produced a really fun set of shapes to explore:

Dan’s program is a great tool to use to have kids play around with lines. Once we get to the section on equations of lines, I’m definitely going to come back to it to show how the lines are being drawn. I might also use this program to explore parametric curves when we get to that topic in my older son’s calculus class.

Introduction to derivatives of inverse functions

We’ve started a new chapter in our calculus book -> Inverse functions.

After wondering a bit about how to approach this topic, I tried starting out in a different book, Spivak’s Calculus, which has a slightly more theoretical approach.

Now not sure how much the extra theory helped, but we did have a nice discussion about inverse functions this morning.

Tonight I wanted to give a few concrete examples and avoid the theory as much as possible. After a brief discussion, I started with the example of $y = x^2$ and found the derivative of the inverse function:

Next we moved on to $y = e^x$ and $y = \ln(x)$. He already has seen a bit of discussion about $e^x$ and its derivatives, so I let him play with the ideas about inverse functions to see if he could find the derivative of $\ln(x)$ on his own:

Next up were the inverse trig functions. Today I chose to focus on $y = \arcsin(x)$ and $y = \arctan(x)$. I started by showing him $\arcsin(x)$ and how basic trig relations produced a pretty surprising derivative:

Next up was $\arctan(x)$. I let him try this one on his own, but I rushed into it too quickly and forgot to talk about the domain. That led to a bit of confusion at the end, but overall I was happy that he was able to get the general idea.

Exploring Calculus with 3d printing

Last week we were exploring volumes of various solids in the calculus course I’m teaching my son. That subject is a great opportunity to use 3d printing to enhance the course.

In fact, this old video from Brooklyn Tech about 3d printing was one of my first exposures to using 3d printing in math education:

Over the last week we printed 3 different shapes. Tonight I used these shapes as props to go back and review some of the volume concepts we’ve been studying. The ideas were new to my son and he hasn’t quite mastered all of them yet, so the review was productive. The videos are a bit longer than usual as we review some of the concepts, though.

Here’s the first shape: y = Sin(x) revolved around the x-axis:

The next set of shapes were not volumes of curves rotated around the x-axis, but rather shapes where the slices were squares:

Finally, we looked at the curve $y = \sqrt{x - 1}$ from x = 1 to x = 5 rotated around the y-axis. Here my son remembered how to calculate this volume by looking at the slices parallel to the x-axis, but struggled a bit with when the slices were cylindrical shells – so we spent a long time on that 2nd part.

I’m happy that we have the opportunity to explore these shapes with our 3d printer – it definitely is incredible to be able to hold shapes like these in your hand!

A fun creation with our facets

My older son is working on a different math project this morning, so once again my younger son was working along. While cleaning up a little bit yesterday we found our old collection of “facets” – so I asked my son to build something for the Family Math project today.

He built a really neat shape:

We have done two previous projects with the facets (including making a big circle 🙂

Our Facets have arrived!

Our second facets project!

They are definitely fun to play around with!

A neat geometry problem I saw from Catriona Shearer

I saw this tweet from Catriona Shearer last week:

It was a fun problem to work through, and I ended up 3d printing the rectangles that made the shape:

Today I managed to get around to discussing the problem with the boys. First I put the pieces on our whiteboard and explained the problem. Before diving into the solution, I asked them what they thought we’d need to do to solve it:

Next we move on to solving the problem. My older son had the idea of reducing the problem to 1 variable by calling the short side of one of the rectangles 1 and the long side x.

Then the boys found a nice way to solve for x. The algebra was a little confusing to my younger son, but he was able to understand it when my older son walked through it. I liked their solution a lot.

Now that we’d solved for the length of the long side, we went back and solved the original problem -> what portion of the original square is shaded. The final step is a nice exercise in algebra / arithmetic with irrational numbers.

Definitely a fun problem – thanks to Catriona Shearer for sharing it!

Volumes of Revolution

We just started volumes in Calculus today – it is one of my favorite topics. Learning this material is one of my favorite memories from high school.

My son seemed to enjoy it, too. We worked through the 6 examples at the start of the section together this morning. When he got home from school I had him try a few examples on his own.

He picked two problems from the book. The first problem was finding the volume when the area between $y = x^2$ and $y^2 = x$ is rotated around the x-axis:

The second problem he chose asked him to find the volume of a “cap” of a sphere – this is both a really neat problem and a pretty challenging one. I was surprised that he chose this one, but it was fun to talk through:

For the final problem we worked in Mathematica and made a 3d picture of the volume created when you rotated the curve $y = \sin(x)$ between $x = 0$ and $x = \pi$ around the x-axis:

I’m excited to pick a few more of these shapes to print as we work through this topic.

Working through a differential equation problem from the 2015 BC calculus exam

I’m using old problems from the BC Calculus exams to make sure I’m pitching the course at the right level. A few of the old questions have surprised me, but many are really nice problems.

This differential equation problem from the 2015 exam was a nice way to explore some basic ideas about derivatives with my son:

We started by reading the question and then drawing in the slope fields:

The next question asked you to find the 2nd derivative in terms of x and y only, and then asked you to talk about the concavity of solutions to this differential equation in the 2nd quadrant:

The third part of the question asked about a solution to the differential equation at a specific point – in particular if that specific point was a maximum or minimum:

Finally, a pretty neat question about a linear solution to the differential equation. Unfortunately I forgot to zoom out after reading the question – hopefully my son’s words explain what he’s doing:

I thought this was a really nice introductory differential equation problem – it was nice to see that we could talk through it even though we haven’t really talked about differential equations formally, yet.

Using a John Allen Paulos problem to discuss probability and expected value with my younger son

I saw a neat problem from John Allen Paulos earlier in the week:

Today my older son was working on a different math project, so I thought I’d use Paulos’s problem for a nice project with my younger son.

I started by introducing the problem (and forgetting to zoom out after introducing it – sorry about the middle 3 min of this video . . . .). Despite the filming goof, you’ll see my son head down a path that illustrates a common counting mistake.

Now we found that our probabilities didn’t add up to 1, so we tried to found out where we went wrong. Fortunately, my son was able to track down the error.

The next part of the project was discussing the expected amount we’d win playing this game. I didn’t define “expected value” but my son was able to come up with a good way of thinking about the concept.

Finally, we went to the computer to write a little program in Mathematica. This part of the project turned out to be a nice lesson in both simulations and in statistics.