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.
Fun experiment: At the very beginning of the vectors unit, just after describing what vectors are, ask the students what they think should be the answer if I add these two vectors. Stop talking, and take pictures of what they think “adding” could mean for vectors. Some students have clearly heard of it before (some in physics class), but I love the variety of answers presented. Not necessarily an intuitive idea…. Just wait until we get to vector multiplication!
I was stumped on an IB probability problem using Poisson. Twitter to the rescue!
Here’s the problem:
And here’s the helpers:
And here’s my work:
Gave the precalc classes a fun problem. Find the distance from the origin to a point in 3D. As you’d imagine, the toughest part is the visualization. Here’s some of the student work:
Showed this video to the computer programming class:
Some fantastic math(s) and computer programming topics related:
- Traveling Salesman Problem
- Quintic easing
- Parametric Equations
- Random Physics Awesomeness
So cool, not alone the fantastic power of the kids seeing some code in the video and thinking… hey I could do that:
Tis the time of year when the IB students learn some different stuff than the AP BC kids.
AP Calc BC
Cross-sectional volumes. This quick demo worked just fine. Also had a bunch of 3D printed models to show, but the paper on board technique worked nicely.
IB Math HL
We also did this exact same example in PreCalc H that day (who is also doing prob/stats). A great motivator for why just about everyone in the room should know how this stuff works. I’d bet everyone in the room has a family member who has already been, or will be affected by cancer.
Here’s the setup:
A blood test has been developed to detect cancer. The probability that the test correctly detects someone with cancer is 0.97. The probability that the test correctly identifies someone without cancer is 0.93. Approximately 0.1% of the population has this cancer.
Question: You walk in to the doctors and take this test. It comes up positive. What is the probability that you have cancer?
What a great lead-in to Bayes Theorem. Give it a second, what would you guess the answer is?
Here’s the Bayes method of solving the problem. A bit strange and abstract. Hard to handle.
Still, an amazing result. Only 1.4% of the people who test positive actually cancer???????
Lets make this more concrete: Take a population of 1,000,000 and walk through the actual amount of people who have cancer etc…
I like that so much better. Same math, but so much easier (for at least me) to understand.
Gave this task to a student who had already seen the BEAM problems. Can you come up with a strategy for reflecting a point over a circle (inversion through a circle)? Each point outside the circle has to map to one and only one point inside, and vice versa.
Here’s his thinking on a whiteboard:
He is using the angle between the two tangents to uniquely map the image inside (and outside if necessary) the circle.
Here’s his desmos sketch. Really cool.
Had a little of extra time with the precalc students so the worked on the BEAM entrance problems. Really great, love it. Do it with your students (they’re designed for 7th graders). Love how students used problem solving techniques on problem number 3: They were making the problem smaller so that it was easier to understand.