Desmos Activity: Vectors (Day 148)

Chapter 14 of the U of Chicago text contains only seven sections. This means that there's an extra day before the test for an activity.

Meanwhile, on Friday I could have posted a Desmos activity for vectors in general, and Lesson 14-7, Adding Vectors using Trigonometry, in particular. But I didn't -- since I subbed in a Geometry class that day, I posted the activity the class did that day, which wasn't on vectors. In some ways, I cheated you readers out of a vectors activity, and so I'll correct it by posting one today.

Two years ago, the activity I posted on Day 148 was an interesting one -- it directed students to choose a partner and then have the students guess each other's vectors -- but it isn't very pandemic-friendly. While Desmos is known for its partner games (such as Polygraphs), I couldn't find any partner activities for vectors on Desmos. And so today's activity will have to be individual.

One particularly strong set of vector activities was created by Jim Moser. The first activity is an intro to vectors and corresponds to Lesson 14-5, 14-6, and the first part of 14-7 where trig is used to convert between the component and magnitude/direction forms of a vector:

https://teacher.desmos.com/activitybuilder/custom/5e93d6192bd72a0cd9c24871

And his activity where two magnitude/direction vectors are added is his next one, Vector Applications:

https://teacher.desmos.com/activitybuilder/custom/5e98d51d6d540468fc732a74

These two activities are the entirety of today's lesson. There's no need for me to post any worksheets or anything else today.

This is a great time to revisit trig and its role in the Geometry curriculum. Ever since I posted Paige Sheehan's trig activity, the ensuing discussion on Twitter reminded me that some teachers prefer emphasizing sine, cosine, and tangent as circular functions rather than trig functions.

Over the summer, I wrote about giving Shapelorish names to the cosine and sine functions -- names like ringex and ringwye, or wheelex and wheelwye, that show their role as circular functions. But I also pointed out that trig functions are introduced in Geometry while circular functions aren't usually taught until Algebra II, so how can we show students the circular functions first? Fortunately, come to think of it, there is a hint of cosine and sine as circular functions in our Geometry text after all -- Lesson 14-7, where they are used to convert between the two forms of a vector.

When we did Shapelore over the summer, I wanted to cover all of Chapter 14. But we couldn't quite make it to Lesson 14-7 -- we covered 14-6, and then suddenly the schools reopened, and I was about to be placed in a long-term middle school math class. But as we now see, Lesson 14-7 is the key to making the connection between circular functions and Geometry class.

And so let's consider blowing up Chapter 14. We begin by moving up Lessons 14-1 and 14-2 on special right triangles, perhaps right around Lesson 8-7 on the Pythagorean Theorem. This allows both 45-45-90 and 30-60-90 triangles to be taught before volume in Chapter 10, so that last week's questions on equilateral triangular prisms can be taught more naturally. Also, moving 14-2 allows Pythagoras to be proved using similarity, as suggested in the Common Core Standards.

The first two vector lessons, 14-5 and 14-6, can also be moved up. It might make sense to teach vectors closer to translations in Lesson 6-2, since translations and vectors are clearly related. Then finally, what remains of Chapter 14 can begin with 14-7, taking us directly into circular functions -- allowing us to fulfill James Tanton's vision.

Let's look back at Tanton's blog again:

https://gdaymath.com/lessons/gmp/7-7-topic-the-story-of-trigonometry/

We pay close attention to Tanton's videos here. The first three videos fit with Lesson 14-7, since the circular functions take us directly to magnitude and direction of vectors. His fourth video then brings in triangles, and so we're now ready for Lessons 14-3 and 14-4.

With a dozen Tanton videos, it sort of makes sense to split them up half and half -- that is, we show the first six videos in Geometry and the other six in Algebra II. The fifth video does mention the names tangent and secant, the latter of which isn't usually taught in Geometry. Then again, secants (as well as cosecants and cotangents) appear already in Video 4. It's Video 7, on radians, when we've definitely ventured into Algebra II territory.

As for Shapelore, I'm now leaning more towards wheelex and wheelwye. That's because it's the spinning of a wheel that helps visualize the circular functions -- we mark a point on the wheel, and it's the coordinates of this point that define the cosine and sine functions of the angle.

Unfortunately, we can't use only the Shapelorish names of the functions and avoid the mean English names completely. We'd still have to explain why the calculator keys are labeled SIN, COS, TAN. So unless we can come up with Anglish names of the functions starting with SIN, COS, TAN (or use a proper name, such as Tanton's function for TAN), we're stuck with sine, cosine, and tangent.

At this point, we can say that we finally completed the Shapelore project we started last summer -- at least for Chapter 14 of the U of Chicago text. Chapter 15, on circles, perhaps deserves some Shapelore posts of its own. After all, chapter contains many vocabulary words -- central angle, intercepted, measure of the intercepted arc, minor arc, major arc, semicircle, chord of the arc -- and that's just the first page of the chapter!

Over the weekend, I posted twice on Twitter. My first tweet was all about the Katrina Newell worksheet from last week and the discussion we had with stretching and surface area. The second tweet was all about -- not again! -- that viral order of operations problem where we must determine the relative precedence of division (represented the obelus symbol) and multiplication (juxtaposition).