Experience First

Students have seen how to use the Law of Sines and Law of Cosines to find parts of a triangle. An alternate method for doing this is by using vectors, which break all the sides of a triangle into horizontal and vertical components. This lesson introduces students to this idea and has them explore scalar operations on vectors in a very concrete context.

The context today is carrier pigeons who deliver mail directly between two locations (how do you train them to do that??) Students informally reason about blocks traveled north, south, east, and west, from a coordinate grid map and how this relates to distance and direction. Students felt comfortable drawing in right triangles to represent the vertical and horizontal components and then the actual path traveled by the pigeon.

In question 4, they begin to reason about how the “direct path” provides the third side of a triangle made up of two other paths. This is then formalized later as the resultant vector.

Question 5 has students think informally about scalar multiplication, having them use their intuition to find the pigeon’s new ordered pair. The point is purposefully not shown on the graph, as we want to challenge students to begin to think more abstractly instead of having to count squares. 

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Formalize Later

While the activity is fairly straightforward for students and introduces all the main ideas of vectors, we wait to use formal notation until the debrief. Students will be introduced to vector notation and component form, as well as the vocabulary words magnitude and direction.

When debriefing question 4, demonstrate how vector w is not only the third side of the triangle (graphical representation of a resultant vector), its component form can also be found algebraically by adding the components of u and v. This helps students to see why w (the second pigeon’s path) is in fact the result of adding two vectors together, both algebraically and graphically.

You will notice in the important ideas that we teach students to use the absolute value of y/x to find the direction as an acute angle (reference angle), and then to adjust for the quadrant. We have found that students do better using this method rather than having to interpret the calculator’s answer in the 1st or 4th quadrant (since the range of arctangent is -pi/2 to pi/2) and turning it into a positive CCW angle. We encourage students to draw a quick sketch of the right triangle made by their vector, and then use reasoning to find the actual direction angle based on what angle is in their triangle and the location of their triangle. The adjustments for each quadrant need not be memorized, though some students benefit from this support.