18 of the 24 precalc students had a field trip. The rest of them designed stuff in Desmos and then we went to the innovation lab and lasered and printed their designs. I showed how to create this coaster design for about 5 minutes, and then let them play.

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

Used this workflow to get their designs ready to print. The poster design was rendered in high definition with fragmentarium (but originally designed in desmos).

This student is using their “superhero” powers to expand (x+h)^3 (pascals triangle and binomial expansion). Fancy right?

One of the algebra 2 teachers uses this picture to explain what happens when you divide a little number by a big number and vice versa. Cool to see that it stuck. The students were using it to determine limits to +/- infinity.

Here’s the “Michael limit test” for limits to infinity. Plugs in 100 for x. Useful idea!

Introduced Limits with a couple of Desmos Activities:

Worked quite well.

Have you ever tried to swim in a lap pool with your eyes closed? How long were you able to go without hitting a lane divider? I can get about 5 or 6 strokes in before I hit and need to correct my direction. I’ve done some triathlons, and one of the hardest parts of racing is swimming in a straight line. You can train all you’d like on lanes, looking down at a lane marker to go straight, but swimming in open water is a different challenge. The thing that worked best for me was to take some number of strokes, say 10, and then take a look to make sure you’re pointing in the right direction. As your muscles get more tired you tend to wander in different directions.

I’ve been asking my students for quarterly feedback for 4 or 5 years, and I’d put it in the top three changes that I’ve made that have most affected my teaching. I use the feedback to keep me honest. It’s hard to open up to anonymous feedback from teenagers, you think the worst is going to happen. But I’ve found that not only do they give marvelous feedback (“course” correction, do you see what I did there), but they tend to appreciate the addition of another data point that you give a damn, and that their input matters to you. There is so much good stuff that they have to say, and if you provide them time, space, and importantly, optional anonymity; they will hand you pure gold. It doesn’t have to be a long feedback form, my quarterly feedback form is only 6 questions:

Here are some quotes from this past feedback session, for some context, these are Juniors and Seniors in advanced math classes.

I love how in depth they think about how they learn best, and they definitely don’t all agree on their favorite methods. I love how they give me constructive feedback and compliments in the same response. I also deeply appreciate their pushback on thing that we need to work on as a class. And this isn’t some royal “we” going on here, they often see changes that they themselves can make to improve their learning (not that they always take themselves up on their own advice!)

An important part of this feedback cycle is to acknowledge their responses publicly. I like to try and get the gist of each question and write down my takeaways. I also think it’s useful to take a comment that I disagree with and explain my thinking. For example, there is a group of students who would rather I was more flexible with my reassessment policy. I explain to them that I wish I had a time turner because then I could provide each and every student as many opportunities as they needed to prove their knowledge on a topic.

I hope you can find a time to try something similar in your classes. It’s hard to not focus on a negative bit of feedback, but I’ve found that I’ve gotten ever so slightly better at seeing the big picture. You gotta bang into some lane dividers to keep your path.

Now that I’ve gotten more adept at the laser cutter and 3D printer, it’s been fun to produce some visuals for the students and their projects. Some help them visualize a complicated scenario, for example breaking a truncated icosahedron into pyramids to measure it surface area and volume. Some help them verify calculations like looking at the math behind logarithmic spirals and nautilus shells. Some are just motivating to play with and figure out the underlying math (Escher’s Circle Limit Puzzle). It’s been a ball having them all working independently and just trying to help through sticking points or provide some guidance for direction of study. I’ve also provided some programming and excel work to help visualize tough problems. We live in a great time, technology truly is helping these students solve problems that they want to solve.

Here are a bunch of the topics that they’re researching!!

A homework example asked the student to find the square of the size of a vector and then find the dot product of the vector. Same answer. Weird. Student asked, is this always true? Why, LOVELY QUESTION!

I just want to shout out to the wonderful tweeps who responded to my question:

I’ve collated and linked their responses in a google doc. Thank you all. The responses came from HS teachers, College professors, industry professionals, and a former student. There was so much gold in these responses that I promised to talk more about the AMAZING uses of how dot products are used next class. I haven’t even touched on how Machine Learning uses dot products!

Also my investigations led me (back to) MarioKart because a website claimed that the boost pads used dot product to calculate the boost amount.

I’ll leave you a gif from GameCube’s MarioKart Double Dash and you can decide for yourself:

To introduce the dot product to the precalc class, I had them first find the angle between two vectors using our old trig methods. Interestingly they used right triangle trig to get the answer; I didn’t anticipate that. Lovely!

To link it in better with the future (how did they not anticipate the future !!!????!!!), I asked them to solve it with law of cosines as well:

Then I had them work in pairs on the following Desmos Activity: Vector Investigation ?????????????????? where they were answering questions based on the dot product but it was called B in the activity. There are a couple of reasons that I decided to not tell them what was being measured, and after doing the activity I think it was the right move. Still if you want to try out the activity with the labels dot product, here is a link to the same activity. Some great observations and guesses from the students, many went down the path of thinking about it in terms of quadrants instead of angles, but I think they were able to build some intuition for what the dot product was measuring.

Here are some sample responses from the first slide.

After the first three slides, the groups had a very good idea that the mysterious B thing was somehow related to the angle between the two vectors (and not related, directly, to the quadrants). We still have a bit of work left to nail down dot product, but it was a nice start. To finish off the day we solved the original problem, but with dot products. (Quite a bit easier). I’m still pondering if I’d like to show the proof of how the dot product is related to the law of cosines, or to bag it.