What’s in each episode?

Across 6 episodes, students are introduced to the mathematical ideas underlying the following culminating problem (inspired by an exam question provided by Darya Krym):

Anna and a bird are standing at a street corner. Anna walks directly south at a pace of 2.7 miles per hour. Anna’s bird wants to fly alongside Anna as she walks but a wind is blowing from east to west at 4.6 miles per hour.

Draw and label a diagram using vectors to represent Anna’s journey, the wind, and the bird’s journey.

Determine how fast the bird should fly and in which direction it must point itself in order to keep up with Anna.

DOTS Snack Quest in Vector Land, Episode 6, “Apply” question

The idea of the module is to break this question up into bite-size pieces, and present each piece in an episode. Each episode should be slow-and-easy, and take about 15-20 minutes. Descriptions of each episode appear below, along with some notes from our development team intended for you, the faculty.

List of episodes:

0. Pilot. Welcome and introduction for students
1. Crunching the Numbers. Introduces the idea of vectors from one point to another, x- and y- components of vectors.
2. Angling for an Appetizer. Working with vectors in any direction, associating slope with a vector.
   • Notes to the instructor:
     • In episode 2 we are trying to get students to see how the ratio of components is related to direction (this is an intermediate step between components and angles).
     • In the game, students guide DOT in a direction of a given point.
     • In the Practice question, the points all lie on a line. The vectors should all have the same direction (same common ratio of components).
     • The Reflection question is trying to get students to notice a) the points lie on a line, and b) they all have the same ratio of components.
3. Trigo-nom-nom-nom-etry. Finding the direction (angle) of a vector using the inverse tangent.
   • Notes to the instructor:
     • Episode 3 is the longest – in it, we introduce actual angles. It looks wordy, but there is a lot to consider. Your ideas for improvement are welcome!
     • When dealing with compass directions: For simplicity, we always give directions relative to East/West rather than North/South (e.g. 60 degrees North of East, instead of 30 degrees East of North). This is intended to ease the transition to dealing with reference angles, since our reference angles are relative to the x-axis.
     • In this episode, we have avoided dealing with angles of vertical vectors (we have ensured that Vx is nonzero).
     • Step 3 in the Practice section is about converting an angle between -90 and 90 (output of the inverse tangent) to an angle between 0 and 360.
4. Magnitude of the Munchies. Finding the magnitude of a vector. Converting vectors to polar components.
5. Craving Components. Converting vectors from polar to rectangular components
6. Vector Velocity: Nibbles in Motion. Vector addition. Combining velocity vectors to find total velocity.
   • Notes to the instructor:
     • Here we move from position vectors to velocity vectors. We do this without much ceremony, in hopes that students will move naturally between contexts.