Reviewing 7 multiple choice questions from the 2012 BC Calculus exam

I’ve been teaching my son calculus this school year with the goal of having him take the BC Calculus exam in May of 2019. We’ve finished most of the course material (the main gap is techniques of integration which I’ve put off a bit) and I had him take the publicly available 2012 BC Calculus exam this past weekend.

He missed 7 questions on the multiple choice part of the exam, and we went over those questions this morning.

The test itself is here on the AP website:

The 2012 Public BC Calculus Exam

Our discussion of the 7 multiple choice problems he missed is below . . . and, dang, the actual questions are a little out of focus in the videos. I’ll upload pics before each question to clear up that little technical glitch:

Question #4:


Here’s our talk through this problem – I think the error here was simply being a little careless during the exam:

Question #14:


Again, I think the error was a little careless during the exam.

Question #20


This question is asking about integration using the technique of partial fractions, so a topic we have not yet covered. I showed him the basic idea of partial fractions and also how he could have estimated the value of this definite integral.

Question #24


I like this question. It touches on integration by parts and also the fundamental theorem of calculus. Even though he left it blank on the exam, he was able to find the solution here:

Question #85


This question is a “related rates” question. It still gave him a little trouble today.

Question #90


I like this question – it hits on several important properties of convergent / non-convergent series. Although he got this problem wrong on the exam, he does a nice job talking through the ideas in the problem here:

Question #92


This question might be my personal favorite on the exam. The underlying concepts are basically even / odd functions and the definition of the derivative.

My son is able to recognize why two of the properties listed in the question do not have to be true, but recognizing why the 2nd property has to be true was still a little out of reach for him. Definitely a fun problem to talk through, though (and listening to my explanation for the solution as I publish this blog post . . . I definitely wish my explanation was a little less careless):

Sharing an idea from Experiencing Geometry with kids

This week learned about the book Experiencing Geometry by David Henderson and Diana Taimina. Unfortunately I learned about the book through people sharing news about David Henderson’s death. But despite the terrible circumstances, the book was captivating.

This morning I picked an idea from the book to share with the boys. The idea is from chapter 16 and is about drawing a circle through three points in a plane chosen at random.

Here’s the introduction to the problem. My younger son struggled a bit in the beginning to remember the ideas, but they did come to him eventually. That little struggle made me happy that we were looking at these geometric ideas today:

After we’d talked through some of the introductory ideas, I had the boys talk about their thoughts on the geometry in a bit more detail. I was especially happy that my younger son was able to sketch a proof that the perpendicular bisector was equidistant from the two endpoints of a line segment:

I had the boys work through the constructions off camera and then explain what they did. My older son approached the problem through folding:

My younger son worked for about 15 min on his construction – he works in a way that is so much more detailed than me! Here’s his work and his explanation which includes a nice discussion of why the center of the circle is outside of the triangle he drew:

Sharing two of Patrick Honner’s calculus ideas with my son

I’ve been looking forward to sharing two calculus ideas from Patrick Honner with my son for the last week. We were, unfortunately, a little rushed when we sat down and there are a couple of mistakes in the videos below. Even though things didn’t go perfectly, I really enjoyed talking through these ideas.

Here’s the first idea – a twist on integration by parts that Honner learned from the British mathematician Tim Gowers:

Here’s the second idea – a fun surprise when a student made a creative substitution in a integration problem:

So, I stared the project by talking about how to integrate arctangent without using integration by parts:

In the last video we found a possibly surprising connection between arctan(x) and ln(x). Here I introduced the integral from the 2nd Patrick Honner tweet above and showed my son how you solve that integral using partial fractions. The point here wasn’t so much the integral, but rather to show that ln(x) showed up in an integral similar to the one we looked at in the first part of the project:

How I showed the technique that Honner’s student used (though I goofed up the substitution, unfortunately, using u = ix rather than x = iu. By dumb luck, that mistake doesn’t completely derail the problem because it only introduces an incorrect minus sign):

Now that we’ve found two connections between arctan(x) and ln(x), we went to Mathematica to see if the two anti-derivatives were really the same. It turns out the are (!) and we got an even bigger surprise when we found that Mathematica uses the same technique that Patrick Honner’s student used 🙂

Also, in this video I find a new way to introduce a minus sign by reversing the endpoints of an integral . . . . .

A fun discussion about prime numbers with kids inspired by an Evelyn Lamb joke!

Yesterday I saw this tweet from Evelyn Lamb:

It inspired me to do a project on prime numbers with the boys. So, I grabbed my copy of Martin Weissman’s An Illustrated Theory of Numbers and looked for a few ideas:

We began by talking about why there are an infinite number of primes:

Next we moved on to taking about arithmetic sequences of prime numbers. There are a lot of neat results about these sequences, though as far as I can tell, they have proofs way beyond what kids could grasp. So instead of trying to go through proofs, we just played around and tried to find some sequences.

I also asked the boys how we could write a computer program to find more and they had some nice ideas:

Next we played with the computer programs. Sorry that this video ran a bit long. As a challenge for kids – why couldn’t we find any 4 term sequences with a difference of 16?

Finally, we looked at Evelyn Lamb’s joke to see if we could understand it!

It is definitely fun to be able to share some elementary ideas in number theory with kids!

A fun introductory Taylor series exercise with Sin(x) I saw on Twitter last week

Last week I saw a neat “fun math fact” via a Matt Enlow tweet:

By happy coincidence my older son is studying Taylor Series this week, so I thought it would be fun to talk through the problem.

Here’s the introduction:

My son had some nice ideas about how to approach the problem in the last video, so next we went to the white board to work out the details:

Finally, I asked my son to finish up the details and then asked him for a sort of number theory proof of why 180 multiplied by an integer with all digits equal to 5 was always close to a power of 10:

Definitely a fun little problem – definitely accessible to students learning some introductory calculus.

An introductory talk about power series with my son and a surprise (to me) misconception

I’m wrapping up sequences and series this week with my son and the final topic is Taylor Series. We’d had a few discussions here and there about power series, but it all comes together this week. Looking through some old problems form the BC calculus test, I found a nice one from 2011 that I wanted to use to introduce the idea of error terms.

I intended for the first three parts to be review, but one interesting misconception came up – so the talk was more than just review.

Here’s the introduction to the problem and my son’s work on the first part of the problem. This problem asks you to write down the usual series for \sin(x) and then write down the series for \sin(x^2)

The next question asks you to write the series for \cos(x) and then write the Taylor Series around x = 0 for the function \cos(x) + \sin(x^2).

Here my son wrote the series for the 2nd function in a way that surprised me:

Once we wrote the correct series for the 2nd part of the last question, we moved on to part (c) of the problem -> find the 6th derivative of the function above evaluated at x = 0:

Finally, we looked at the last part of the problem. The question is about the error in a Taylor series approximation. I’d hoped to use this question to introduce ideas about error terms in Taylor Series, but unfortunately I completely butchered the discussion. Oh well – we’ll be covering the ideas here in a much more detailed way later this week:

A fun connection between quadratic equations and continued fractions

My younger son is beginning to study quadratic equations in Art of Problem Solving’s Introduction to Algebra book. So far he’s essentially only seen quadratic equations that factor over the integers. For today’s project I wanted to show him that there are simple equations with fairly complicated (compared to integers!) roots.

We started with a problem similar to ones that he’s already seen:

Next I showed him a type of equations that he’s not see before and we spent 5 min talking about his ideas of how you could solve it:

Finally, for the specific equation we were looking at, I showed him how we could use continued fractions to solve it. As a bonus he remembered the connection between the Fibonacci numbers and the golden ratio and that got us to the exact solution!

Sharing a great Random Walk program with kids

I saw a fun random walk program shared by Steven Strogatz yesterday:

Today I shared the program with the boys. It has 4 different types of random walks to explore. For each one I asked the boys what they thought would happen. At the end we looked at all 4 simultaneously.

Sorry that the starting videos are so blue – I didn’t notice that while we were filming (and didn’t do anything to fix it, so I don’t know why the last two vides are better . . . .)

Also, following publication, I learned the author of the program we were playing with:

Here’s the introduction and the first random walk – in the walk we study here, the steps are restricted to points on a triangular lattice:

In the next random walk, the steps were chosen from a 2d Gaussian distribution. It is interesting to hear what the boys thought would be different:

Now we studied a random walk where the steps all have the same length, but the direction of the steps was chosen at random:

The last one is a walk in which the steps are restricted to left/right/up/down. They think this walk will look very different than the prior ones:

Finally, we looked at the 4 walks on the screen at the same time. They were surprised at how similar they were to each other:

Definitely a fun project, and a really neat way for kids to explore some basic ideas (and surprises!) in random walks.

Sharing Grant Sanderson’s “Surface Area of a Sphere” video with my older son

Last week Grant Sanderson published an incredible video about the surface area of a sphere:

By happy coincidence my older son is spending a little time reviewing polar coordinates now. Although not exactly the same ideas, I think there’s enough overlap to make studying Sanderson’s new video worthwhile.

So, I’m going to do a 6-part project going through the video. Tonight we watched it and my son’s initial thoughts are below. Each of the next 5 parts will be spent discussing and answering the 5 questions that Sanderson asks in the 2nd half of the video.

Here’s question #1:

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We’ve been away from right triangle trig a little bit lately, so I was interested to see how my son would approach this problem. His approach was a bit of a surprise, but it did get him to the right answer:

Screen Shot 2018-12-11 at 4.06.09 PM

The next question in Grant’s video is about how the area of one of the rings on the sphere changes when you project it down to the “base” of the sphere (see the picture above).

I thought that answering this question would be a really good geometry, trig, and Calculus exercise for my son:

Now we get to a really interesting part -> Question #3

Screen Shot 2018-12-12 at 5.29.31 PM.png

Grant asks you to relate the area you found in question 2 – the area of a ring around the sphere projected down to the center of the sphere – to the area of a different ring around the sphere.

Here’s my son’s work on this problem:

Finally – my son answers questions #4 and #5 after a quick review of the previous results. He was a little tired tonight, but we needed to squeeze in these two questions tonight because I have to travel for work tomorrow.

Here’s question #4:

Screen Shot 2018-12-13 at 7.23.15 PM

and #5:

Screen Shot 2018-12-13 at 7.23.34 PM

And here’s his work on those two questions:

Sharing Craig Kaplan’s isohedral tiling program with kids

I saw an amazing tweet from Craig Kaplan this week:

Ever since seeing it I’ve been excited to share the program with the boys and hear what they had to say. Today was that day 🙂

So, this morning I asked the boys to take 15 to 20 min each to play with the program and pick 3 tiling patterns that they found interesting. Here’s what they had to say about what they found.

My older son went first. The main idea that caught his eye was the surprise of distorted versions of the original shapes continuing to tile the plane:

My younger son went second. I’m not sure if it was the main idea, but definitely one idea that caught his attention is that a skeleton of the original tiling pattern seemed to stay in the tiling pattern no matter how the original shapes were distorted:

Definitely a neat program for kids to play around with and a really fun way for kids to experience a bit of computer math!