I like today's lesson because it brings back a topic that we learned in a past unit, but in an unexpected context. Students will use parametric equations to allow us to write an equation of a line in three dimensions.
Students have so much experience with equations of lines, but looking at them in 3D may throw them off. I try to build a little conceptual understanding of lines in 3D and the nature of their slopes in order to build a bridge to finding their equations.
I get started by showing students slide 2 of this PowerPoint and have them name three other points that are on a line given by two points. The purpose of this is to get them thinking about the slope and how the x, y, and z coordinates must increase or decrease in the same ratio. Students are also asked to describe the slope. Obviously they cannot write it as a single fraction, but they can at least say how the x, y, and z are changing. I usually give students 10 minutes to think about these questions and how they could generalize an equation to represent any point on the line.
I don't expect all of my students to be able to figure out the need to use parametric equations to write the equation of a line, but I can get them there by framing the discussion in a specific way. For slide #2 of the PowerPoint, students were given two points on the line and were asked to find three more. I usually start by writing down the first two points and then extending the pattern to add more. The list will look like this:
Students will see that the x, y, and z coordinates all have their own pattern and it is related to the slope of this 3D line. If I noticed a student using parametric equations, I will have them share their thinking. If the class needs a little nudge, I will ask them to just make an equation to represent the pattern for the x-coordinates. Usually many students can describe the x pattern with an equation and I will ask them to show their work. Then, I will have them finish up the equations for y and z.
It is interesting that students will not all get a unique equation for the line - we can write many different equations to represent the same line. In this image you can see three possibilities - the first two are very obvious but the third one is not. I wrote the third set of parametric equations to see if students could make sense of why it would produce the same line as the other two sets. I talk more about this in the video below.
Next I move on to slide #3 and have students find the parametric equations for the line passing through the two new points. I will choose a few students to share their different sets of equations and we will talk about what is the same about all of the parametric equations and the reason for it. Next, I will demonstrate how we can solve each parametric equation for t and set them all equal to each other to produce the symmetric equation for the line.
The last slide of this PowerPoint gives one more problem but it is slightly different - students will have to find the equation of a line given a point and a parallel vector. Students are also challenged the write the equation as parametric equations and as a symmetric equation. I will end the lesson by having a student explain their work.
Finally, I will assign a homework assignment from the textbook to summarize what we learned.