Pythagorean Beans

David Williamson
February 7, 2017

The purpose of this workshop is to introduce basic vector-related concepts:

  • Scalar Multiplication
  • Vector Addition
  • Vector Components
  • Pythagorean Theorem

In order to achieve this, instead of looking at print-outs or computer screens, the students will be divided into four volunteers and the rest will be observers.

The volunteers will be on the second floor, while observers will go to the third floor, to look down on the mezzanine, in front of the UGLC offices.

The floor of the mezzanine will have black tape, to signify the$\ x$ and$\ y$ axes of a cartesian coordinate grid.
Each tile will be considered as one unit, squared.

Next, we will introduce the unit vector (in the$\ x$ direction.)
This will not be referred to as$\ \hat{i}$, as we are more interested in the conceptual understanding than we are with introducing notational conventions.

One student will be the origin, standing on the intersection between the axes.
Another student will be the unit vector in the $x$ direction.

Now, we can talk about scalar multiplication of a vector.
We can move the student two more spaces in the$\ x$ direction, to make a $\begin{bmatrix}3\\0 \end{bmatrix}$ vector.

Next, another student can make a $\begin{bmatrix}0\\4 \end{bmatrix}$ vector, in the same manner.

Once the two vectors have been defined, we can introduce the idea of adding the $\begin{bmatrix}0\\4 \end{bmatrix}$ vector to the $\begin{bmatrix}3\\0 \end{bmatrix}$ vector.

First, the student forming the $\begin{bmatrix}0\\4 \end{bmatrix}$ vector will move three spaces in the$\ x$ direction.

The student at the head of the $\begin{bmatrix}0\\4 \end{bmatrix}$ vector and the origin can then hold a ribbon between them.
This represents a $\begin{bmatrix}3\\4 \end{bmatrix}$ vector.

At this point, we can say that the $\begin{bmatrix}3\\0 \end{bmatrix}$ and $\begin{bmatrix}0\\4 \end{bmatrix}$ vectors form the$\ x$ and$\ y$ components of the $\begin{bmatrix}3\\4 \end{bmatrix}$ vector.

Finally, we take tile-sized square paper cutouts and place these cutouts on the tiles, adjacent to the component vectors, in $\ 3 \times 3$ and $\ 4 \times 4$ squares.
The final volunteer student and the instructor can both take part in the placement of the cutouts.

Then we say that if we want to know the length of the $\begin{bmatrix}3\\4 \end{bmatrix}$ vector, we can use the relationship between the areas of the squares of right triangles in order to find out.

The cutouts are moved from the $\ 3 \times 3$ and $\ 4 \times 4$ squares to a $\ 5 \times 5$ square, adjacent to the $\begin{bmatrix}3\\4 \end{bmatrix}$ vector.

The fourth volunteer is then instructed to create a $\begin{bmatrix}9\\1 \end{bmatrix}$ vector, and we ask the observing students to label the $x$ and $y$ components of the new vector. Then we add the $\begin{bmatrix}3\\4 \end{bmatrix}$ vector to this, to create a $\begin{bmatrix}12\\5 \end{bmatrix}$ vector. The whole group is then challenged to see if they can try to figure out the magnitude of the $\begin{bmatrix}12\\5 \end{bmatrix}$ vector.

Wrap-up: Once the cutouts and ribbon have been collected all the students return to the second floor, we talk a about how these ideas are used in navigation, engineering, and computer graphics.