147 – Concavity Tester in Desmos

How do you introduce concavity? Do this instead:

Bring up this desmos link, and change the title to something less suggestive.

Give students a post-it and instruct them to be silent and figure out what the red parts of the graph have in common, and what the blue parts have in common.

Fill in lots of different functions.

Have fun! One student came up with the marvelous observation that relative maxes are always in the red regions (concave down), and the relative mins are in the blue regions (concave up). She came up with the 2nd derivative test!

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146 – Student Projects in Programming Class

Both are work in progress:




145 – Post IB Day 1 – Three Triangles

IB Math HL students work on the three triangles puzzle. All block!

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144 – The Monty Identity

Had a competitor to the Alexis Identity. This identity broke software at many levels and had to be split up many times. Desmos proof, but be careful, a pretty quick computer is required to even load the desmos link ;-).


So if you every run along sqrt(-14/17), you can replace it with the junk on the right!

Key: (a=2, b=7, c=17, d=3, f=6)


143 – Introduction to Implicit Differentiation

I’ve been having fun with the derivative unit in PreCalc H. Today the students were going to be introduced to implicit differentiation. But as a warmup, I gave them this problem and had them work in groups informally on the whiteboards.

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They have all the tools necessary: product rule and chain rule. No group got there, but many groups got very close and they struggled in different spots. Here’s some examples:

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As you might expect, the chain rule to get d(smiley_face)/dx was the most difficult step. But I think “struggling” with this example before even knowing about the topic was an important step. Students could start bridging the gap before being teacher led through an example. We ended up doing something very similar throughout the entire block. The groups of two would try each example on the standing whiteboards before seeing my work, or putting their work directly in their notebooks


142 – Great Problem – Route to Infinity


Take some time to look at the route followed by the arrows in this diagram.


Then look away and describe the path to a friend.


  • If the pattern of arrows continues forever, which point will the route visit immediately after (18,17)? Explain how you know.

  • How many points will be visited before the route reaches the point (9,4)?

  • Which point will be the 1000th to be visited?

  • Does this pattern “cover” every point (x,y), where x,y? Can you devise a pattern that covers every point with integer coordinates in the x-y plane?

I worked on this problem with some colleagues from the NY Master Teacher program over a google hangout. So much fun and there are a bunch of hidden connections to very interesting math that also happens to be on my various curricula. I’ll be working on this with students soon!

Here’s a example that we were trying out to see if our general solution worked:



141 – Find the general derivative of sin(x)

After asking students to find the first, second, third, fourth, tenth, hundredth, and the five thousand eighty first derivatives of sin(x); I challenged them to write out the general derivative of sin(x). They had some problems with notation, as this is the first time they’ve been asked to generalize such work at this level, but some creative solutions (which were correct!) were put forth. Here’s a programming student in the middle of his thought process. It’s not exactly “correct”, but some fantastic thought going on here:


And a bonus “problem” that some students had when finding the derivative of g(x)=x^3 * ln(pi). Is this really a problem? NO! I’d argue that this student will take to the implicit differentiation that we’ll be doing today much easier than the average student who has memorized the idea that “constants stay out in front when finding the derivative”.

How did I address this problem? It was fun, I told the student an x value, and then asked about the value of pi.

What is pi when x = 3? What is pi when x = 10? What is pi when x = 500? Oh so no matter how much I change x, pi isn’t changing?

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140 – d/dx(tan(x))

What Calculus teacher doesn’t do this? After they know the quotient rule (or the chain rule and product rule), and the derivatives of sine and cosine; have them find the derivative of tangent(x). Heck, have them find derivatives of secant(x), cosecant(x), and cotangent(x) while they’re at it. You get cool derivations like this:

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139 – Khan Academy and Review

The students are in the midst of their IB and AP exams, so I had some missing bodies in class. It was a good day to try some different things. I like to try and check in with Khan Academy every year or two to see if I will be replaced by a website. Not any time soon. The PreCalc H students were working through some of the problems in the Differential Calculus unit and while there were some okay things, there were far to many weird things. It’s quite clear that an actual educator hasn’t had much of a say in what is going on. It’s just so weird. Neat to see that the students naturally brought out sheets of paper.



We spent the rest of the block standing up and working with other humans. How novel. This was great.

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138 – Math HL IB Exam Review

Had a pizza and wings review party after school for the IB Math HL students. The first paper is today, second is tomorrow and the third is a week from tomorrow. Best of luck!