We’re working on power series in Calc. Every year I love setting them up with the following series to find the convergence/divergence of:
They work on whiteboards, wander their way through the tough algebra and to the answer (converges everywhere), and are (generally) very happy to have made it through a tricky problem unscathed. Inevitibly, a student asks what the power series looks like. YES.
It all leads to an amazing conclusion. That mess of a series with alternating terms, odd polynomials, and factorials leads to …… (check the graph to see!)!
And then it also leads to my favorite desmos graph (ever??):
It’s the last two weeks of the quarter. Time for nearly every desk to be filled afterschool (I only allow reassessments on one day a week).
Neat activity: as an introduction to proving trig identities I asked them to match these trig identities:
Then when they finished the matching (through algebra and substitutions, no technology allowed), I assigned each group one proof of the match.
We’ve done some special right triangle practice in the first quarter, but when you remove the context of the application (it was converting rectangular complex numbers to polar form) the students struggle. I don’t want them just memorizing or using some finger trick to get these trig ratios, I want them having some idea on how they can be derived. Many of them “knew” this stuff when they did it last year in Algebra II. but I tested their carried over knowledge by giving an 8 question quiz on sine, cosine, and tangent of different radian values and 8 minutes. The median score for this pretest was 0/8. But after discussing some strategies for finding the values, the median score for the post test at the end of the block was 7/8, and the median score for a similar quiz at the beginning of the next block (no homework on this stuff was assigned) was 8/8. They’re much more confident at solving these type of questions and most of them don’t need the entire 8 minutes, I’d bet most are done in half that time. I don’t love the idea of having to practice speed, but I really only care that they’re fast enough at finding these values. If they can “jump” that high, then I don’t care if they can “jump” higher. This all sort of fits in with Michael Pershan’s most recent fantastic post, found here. These students need to be somewhat proficient at calculating these exact values without a calculator because all their future standardized tests (SAT, ACT, AP, or IB) will have some sort of non-calculator portion, let alone the goal for deeper understanding. Where their knowledge fits in with the deriving the answer vs memorizing the answer vs at what step did they memorize the answer,… I don’t know. Here’s my key for the second quiz. Although I’ve been teaching math for more than a decade, I do NOT have these values memorized. I use my memorized facts (the two triangles, SOHCAHTOA, and the quadrants) to calculate the values if I need them.
We are working on reciprocal trig graphs in precalc, so I had them draw the original graphs and find the multiplicative inverse of the function values to find the reciprocal trig graph.
See the open circle for cotangent at pi/2? Can you guess why they put it there? These kinds of thoughts are REALLY good signs for students. Love it.
Here is the Precalc H group using the last 30 min or so at the end of class to start their take home.
Also some motivation from a student on why take homes might be a good idea to use in your class:
I think a great way to start off induction proofs is to lead them into what they’re going to have to prove. If I raise 6 to any (integer) power and then subtract 1, what is this always divisible by?
(Student does some scratch work).
5? Oh yea? Prove it!
Also if there’s a better use for tons of whiteboard work, then I don’t know what it is….
I also end up putting the clearest student work directly in the notes. Provides some of them some motivation to be clear because they all want to be in the “official” notes.
We’re co-seating AP BC students with IB HL year 2 students. 90% of the curriculum is the same. We’re now working on some of the 10%. The IB kids are working on their papers (Internal Assessments, about an 8-10 page exploration on a math topic of their choice). Some really awesome topics, hopefully they haven’t bit too much off with their problem choices.
The BC kids are learning the calculus of parametric and polar equations. So cool to see the students learning something new with very little teacher interaction. Awesome environment.