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:

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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.

Sharing Ben Orlin’s Math with Bad Drawings with kids -> talking dice and p-values

A few weeks ago I ran across a copy of  Ben Orlin’s Math with Bad Drawings at a book store:

Last night I asked each kid to pick a chapter in the book to read so that we could talk about those chapters for a project today.

My younger son picked the chapter about dice – hardly a surprise as he’s been fascinated with dice forever! Most of the dice you see in this video can be found at the Dice Lab’s website if you are interested in more information about them. Here’s what my younger son had to say about the chapter and about dice this morning:

My older son picked the chapter on p-values – gulp! This topic is pretty advanced and once that isn’t super easy to explore with kids. But I gave it a shot.

First, here’s what he found interesting:

Next I designed a little experiment on Mathematica. For this experiment I wasn’t using p-values but rather confidence intervals – this was just for simplicity, but was still also not super easy for the boys to understand.

In my experiment, I picked 30 numbers from a normal distribution with mean of 5 and standard deviation of 10, and we looked to see if we could tell (statistically) if the mean of the numbers was greater than 0.

What we found was that roughly 25% of the time, 0 was in the 95% confidence interval of the mean. Also, roughly 2.5% of the time, the lower end of that confidence interval was greater than 5 (so we excluded 5 from the confidence interval roughly 5% of the time!).

Hopefully this little experiment helped the kids understand how you could find “wrong” results every now and then:

I love Ben’s book – definitely a fun read and although it isn’t specifically meant for kids, there are plenty of ideas in the book that can be shared with kids.

Introductory U-substitution

We’ve been doing a little bit of work in the integration chapter of our calculus book. Today the topic was u-substitution. I was expecting the topic to produce a great deal of confusion, but things went pretty smoothly. There will be plenty of time for confusing integration problems later, I guess, ha!

When my son got home from school tonight I had him try out three problems from the u-substitution section. The first two are straightforward and the third is a bit of a surprise.

Here’s the first one: \int (1 - 2y)^{13} dy

For the next problem I chose a trig integral: \int \cos(x)^4 \sin(x) dx

Finally, I chose a problem that isn’t obviously a u-substitution problem when you look at it: \int x \sqrt{x - 1} dx

Again, I was worried that we’d get off to a bit of a rocky start, but thinks went well today. Going to have him work through a bunch of problems tomorrow to get a little more u-substitution practice.

Playing with some zometool fractals

My older son had an event yesterday and so the Family Math project only had my younger son. I asked him what he wanted to do and he chose another zometool project from Zome Geometry.

This time the project was about fractal-like shapes.

Here’s the first one he built -> a Sierpinski tetrahedron:

Next up was a fractal tree:

Definitely a fun way to spend the morning!

An exploration of Euler’s method

My older son is studying calculus and my younger son is studying algebra this year.  I’d run across some problems on Euler’s method looking at old BC calculus exams, but we are still a long way from talking about Euler’s method in the calculus course.  However, as my younger son begins to study lines, I thought it might be fun to do some visuals on slope fields and just touch on Euler’s method as a way of talking about slope.

Here’s how I introduced the topic:

Next I gave them a neat visual example -> wind speed and direction on the earth

Now I moved to Mathematica to talk about slope fields in general. The two specific slope fields we looked at here were (in calculus language) dy/dx = y and dy/dx = x.

Finally I had the boys choose their own fields and try to describe them ahead of time:

Definitely a fun project. The mathematical idea behind Euler’s method isn’t that hard and kids can understand the concept pretty easily. I really had a lot of fun exploring the ideas of slope fields with the boys this morning.

Our first 5 weeks of Calculus

Here are the 11 projects calculus-related projects that we’ve done to start the school year:

(1) Working through Calculus with my older son

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The concepts in this project were based on ideas discussed in Grant Sanderson’s amazing video series “Essence of Calculus”.   The specific video we were working with here was the “Geometric Derivatives” one.

Here’s the project:

Working through Calculus with my older son

(2) What a kid learning calculus can look like

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The problem here is finding a line that is perpendicular to a parabola. Here’s the project:

What a kid learning calculus can look like

(3) A beginning Calculus example

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In this project my son works through a common introductory calculus problem about finding the largest area for a rectangle with a given perimeter.

A beginning calculus example

(4) An Introductory implicit differentiation example thanks to Patrick Honner

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I mentioned on twitter that we were working on implicit differentiation and Patrick Honner gave me a good challenge problem to share with my son:

An introductory implicit differentiation example thanks to Patrick Honner

(5) Working through a challenging calculus problem

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This was a pretty challenging problem I found in the calculus book we are using – Stewart edition 3.   The problem is about a light casting a shadow as it shines on ellipse.

Working through a challenging calculus problem

(6) Struggling through a related rates problem

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This was a neat trig problem about finding the 3rd side of a triangle changes as the angle between the two known sides changes:

Struggling through a related rates problem

(7) Learning about approximations

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For this project we explored how to use calculus to find approximations to values of some common functions:

Learning about approximations

(7.5) Playing around with Newton’s Method

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This was a project that I did with both my younger and older son.  The topic was Newton’s method.  I didn’t want to skip it because it was a neat application of calculus, but I also didn’t want to dive too far into the details and thought my younger son would enjoy learning a bit about it, too.

Playing Around with Newton’s Method

(8) Applied Max / Min problems

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This project showed how to use calculus to find the largest square that you could inscribe in an equilateral triangle (with one side on the base of the triangle).

Applied Max / Min Problems

(9) A fun calculus project -> folding a circle wedge into a cone

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This project was a classic -> finding the largest volume you could make by folding up a wedge of a circle into a cone.

Folding a circle wedge into a cone

(10) Finding the Area of a Circle using Riemann Sums

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The motivation for this project was showing how a seemingly impossibly to sum Riemann sum could actually be understood as giving the area of a circle.

Finding the Area of a Circle using Riemann Sums

Finding the area of a circle using Riemann Sums

We’ve moved on to chapter 4 of my son’s calculus book -> integrals. The first sections are talking about Riemann sums.

I was a little surprised at the initial difficulty my son had with the examples. I think one of the problems – maybe the main problem – was how many different things you had to keep track of to evaluate these sums. Once we studied the ideas a bit more and saw that the sums could be really be broken down into keeping track of lengths and widths of rectangles, the ideas seemed to make a lot more sense to him.

Last night I decided to show him a Riemann sum that was different than the polynomial examples he’d already worked through. For this one – finding the area of a circle – he knew the area, but the sum was pretty complicated.

Here’s his work finding the sum:

Next we went to the computer

Finally, I thought it would be fun to show him a little surprise -> the Riemann sum that you would use to find the volume of a sphere is actually pretty easy to evaluate by hand.

I think that’s going to be it for specific Riemann sum background work. Looks like the next sections introduce integrals. Excited to dive into this topic!

A fun calculus problem -> folding a circle wedge into a cone

I’m a few days late publishing this exercise – my son finished up the section on applied max / min problems last week. But I thought his work on this problem was fascinating and wanted to publish it even if it was a little late.

So, last week my son came across this max / min problem in his calculus book:

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It gave him a little trouble and since I was on the road for work it wasn’t so easy to help him. We went through the problem when I got back from a trip -> I thought it would be fun to start from the beginning and actually make some cones before diving into the problem.

Next we started down the path of trying to work through the problem. Here’s how he got started:

In the last video he was able to write down an expression for the volume of the cone in terms of the angle of the wedge. In this video he writes down a variant of that expression (the square of the volume) and gets ready to find the maximum volume:

Now that he has a relatively simple expression for the volume squared, he finds the derivative to find the angle giving the maximum volume:

Finally – he calculated the maximum volume. The expression for the angle is a little messy, but the maximum volume has a (slightly) easier form.

Overall, I think this is a great problem for kids learning calculus. It also pulls in a little 3d geometry and 2d geometry review, which was nice.

With this section about applied max / min problems done, we are moving on to integration 🙂