85-86 Solving Trig Graphs

Very short post just noting that solving trig graphs with technology like desmos. Without prompting they are using restrictions to cut off the part that is unnecessary (pretty sure they learned that from marbleslides). Good stuff.

 

83-84 Trig Graphs and Cover Midterm

We covered the midterm exam today (40 multiple choice) in groups today. They knew their grade but not which problems that they got right/wrong. In groups of 5 problems, they tried to find the correct answers, and after a couple minutes of discussion for each group questions, I gave the answers and showed solutions for those who had questions.

Worked quite well. Lots of natural discussion and students helping students.

Next the students worked on finding trig graphs that went through given points. After fiddling with it on their whiteboards, they wrote down a set of steps to themselves in their notebooks.

Super interesting seeing the student thinking here. Will continue.

 

81-82 Presentations of the Final Projects in Computer Programming

Some great projects this year (out of only 5 blocks of in-class time).

 

79-80 Transformations of Trig Graphs

As an introduction to transformations of trig graphs, the class worked individually on Marbleslide Periodics. We did wrote down their conclusions for what the a, b, c, and d do for the transformations and gave them specific trigonometric terms (period, amplitude, and frequency). Then the class, in pairs this time, worked on my Match my Trig Graph desmos activity. I had the basics setup on this activity last year, so I tweaked it this year with some goodies. There is now a feature to stop them from just making a “trig” graph with 0 amplitude or 0 frequency, and a feature to show a picture when they’ve succeeded. It’s good practice for them to have to enter in an exact value like π/4 instead of an approximation like 0.7854. Really successful activity this year, students were into trying to find all the pictures.

It’s also fun to see the students working in the activity builder:

 

 

77-78 Make your own Trig Identity

A fun activity to try with your Precalc/Alg2 class when they’re working on proving trig identities is to have them create their own trig identity.

Have them start with a true statement, then modify it by doing much of what you might ask them not to do during a trig proof (work across the equal sign). When they’ve made it properly obfuscated, have them write down the last line on a post it, erase their work and then ask them to prove their identity. If they can prove it then you can switch post-its with other students in the class to try and prove someone else’s “identity”.

Some students always get REALLY into this, and will probably not be able to finish before the block ends. Here’s a student plugging in his identity into desmos to verify that it’s true before he attempted to prove it.

These type of students often come back after a week having worked on some monstrosity of a proof. Honor them!

 

 

75-76 Power Series and Two Surprises!

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.

https://www.desmos.com/calculator/qifn3uchjr

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??):

 

 

73-74 Reassessment Time

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).

 

 

71-72 Match these Trig Expressions

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.

 

69-70 Practice with Exact Values of Trig Functions

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.

 

67-68 Graph of cotangent

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.