Classroom Activities
These three activities are by no means exhaustive, but should provide a firm basis for execution of the unit.
The Arbitrary Ruler
As an introductory activity to understanding measurement, I want to give my students blank rulers of various lengths. I will cut these with a band saw from scrap 2x4’s I have lying around, but even old rulers with the print sanded off or pieces of (well sanded!) pallet cut to various lengths should work well. Each of these will be representative of a unit length. Students will then take a measurement of marked distance in the classroom in terms of their unit. From these, they should compute conversion factors to convert from their unit length to that of another student. These conversion factors they calculate can later be identified as dilation factors that transform from their unit length to another.
Once students have taken a measurement with their blank ruler, it will be time for them to subdivide their ruler in some way. They can mark their ruler in smaller intervals based upon a new unit interval, like the width of their thumb or the length of a locker key, in order to construct a more precise measuring instrument. Students should then find the dilation factor that transforms their new unit length into the old one and measure something smaller. Students will pair up and using their new conversion factors, each will predict what the other will measure for a specific object, like a table length. Students can then calculate the percent difference between their predictions and the actual measurements.
What students should get out of an activity like this is that the choice of unit distance does not impact how measurements are made, but does impact how simple they are to perform and how precise they are. Students should see that smaller unit distances produce more precise measurements. Communicating about the units they use is also important, because it allows other people to take their measurements and use them. Finally, I want my students to arrive at the idea that having standardized units of measurements is a smart idea because it simplifies communicating about data.
Transforming the Clothesline Number Line
The next major activity in my unit will begin with students placing numbers onto a clothesline number line. I want them to establish an origin and a unit distance and then use that to demarcate points along the line in both directions with clothespins and note cards. Once students form this large number line, I will have them measure objects along it and perform addition and subtraction by combining and comparing the lengths of those objects. From here, the objects will become vectors that are to be added and subtracted visually. This physical model will appear identical to the way one dimensional vector addition is frequently drawn – as arrows along a number line. Students will perform the computations and then draw the pictures to solidify what they are doing.
The second part is going to be having students model the same addition and subtraction they just did as translations by making a second, identical, number line below the first. This second number line is to be made of bungee cord, not rope, and put under some tension. Be careful with this, however! You can store an extremely large amount of energy in a bungee cord! Students will shift the second number line and use the translation of all of the points on that line to visualize addition. Subtraction will then be viewed as a translation in the negative direction. Students can repeat the exercises from the previous activity or perform multiple sequential translations until they start to see how vector addition and translation by a constant are related to each other.
The third part is to bring dilation into the picture. Since the second number line is made of bungee, increasing or decreasing the tension in the cord can dilate it! Again, be careful that the bungee is not put under too much tension, because that could cause a dangerous condition. Students can use this to model scalar multiplication as a dilation. An interesting way to set this up might be to have a loop of bungee with pulleys at the end. One pulley can remain fixed while the other can slide and be fixed in place with a setscrew. This sort of set up will allow for students to combine dilation with translation, simultaneously modeling scalar multiplication and vector addition.
Students should take away a few different visual representations of addition and subtraction, as well as a concrete visualization of multiplication as a dilation of the line. The purpose is to get them comfortable with the process and visualization of arithmetic as vector operations and transformations.
Two Dimensional Transformations
For this activity, I want my students to see how translations and dilations look in two dimensions. I like to use big sheets of butcher paper because it is inexpensive and a great way to have students quickly generate posters that can be hung up and discussed. Each group of three will be given a sheet of butcher paper, a card and some markers. The card will have a collection of points that the students must first plot and connect to make some sort of shape. Also on the card will be some translations that students need to carry out that are specified by vectors. One will be purely one-dimensional translation. One will be two-dimensional translation. The third prompt will be a composition of two translations. There will also be some scalar dilations and translation-dilation compositions that students will need to plot and draw with their original figure. It will be important for them to label the vector that describes the translation of each point on their posters.
The next task will involve them taking their own drawings and describing the transformations that provide specific instructions on a card. For example, students may draw a card that asks them to describe a transformation that moves the center of their figure to the point (5,5) and triples the size. The group would then describe the dilation and translation that produce that end result. The pictures will be hung up along with the prompt and peer-reviewed during a gallery walk type activity.
The final activity will involve assigning each group a random locker and a random starting point on the floor where my classroom is located. Starting points will be labeled with letters on the ground in painter’s tape. Each team is tasked with describing a vector that points from their starting point to their assigned locker. This vector must be specified in terms of magnitude and direction on the back of the starting point note card. The cards will be shuffled and dealt out to different groups who must then use the vector and starting point on their card to recover the assigned locker. It helps that my school’s floor has tile that approximates a grid laid into it.
The major student takeaways should be a visualization of how vectors are represented in the plane, how they combine, how they dilate and how this all fits together in a physical sense. The activities are designed to be big and engaging, because can be a challenging topic, especially for students who have limited exposure to trigonometry.
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