Distance and Midpoints in 3D

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Objective

SWBAT find distances and midpoints for 3D coordinates.

Big Idea

Add another dimension to graphing!

Contextual Note

Today you will mystify your students by taking a regular old coordinate plane and adding the third dimension! Students may be surprised and excited if you build it up and pitch it as a Star Trek-like journey into an unknown world. It is kind of neat to take something that students know so well (the xy-coordinate plane) and add a whole new dimension to it.

Invariably, discussions of the fourth dimension will come up, so you make want to take a quick look at a Wikipedia page on the Fourth Dimension to pique your students' curiosity.

Launch

10 minutes

A 3D coordinate system can be very difficult to visualize, so it is really helpful to have some sort of manipulative that students can hold and rotate around in order to fully make sense of the concepts in today's lesson. I made my 3D coordinate system model out of different colored overhead transparencies. Using card stock also works well, but I think the transparent planes is important.

To begin the lesson, I give students this worksheet and while showing them the model of the coordinate system. I will give some background information about this new system so that we can start working with different concepts. The blank box on the front side is reserved for this background information. Here is what I go over with my students:

  • Points are given by an ordered triple, like (3, 2, –5). The order of variables is (x, y, z).
  • The 3D coordinate system consists of three planes. The plane is named by the two axes that form it. For example, the xz-plane is the flat surface that the x- and z-axes lie on.
  • Instead of quadrants, the 3D system is divided into octants. Octant 1 is where all three variables are positive.
  • Our textbook uses a “right hand rule,” meaning that the octants' numbers increase in the direction of how your right hand can rotate (counterclockwise).

 

Explore

15 minutes

After the background information about 3D coordinates has been disseminated, students are ready to dive in and try some problems in this new system. Questions #1-9 of this worksheet ask students some basic information about octants and planes, but also get them thinking about midpoints and distance in 3D. I will give students about 10-15 minutes to work on these problems with their table groups.

I structured the questions on the worksheet to really build on each other and the utilize MP7 and MP8. Using structure and repeated reasoning will hopefully help students to discover the 3D midpoint formula and distance formula themselves. I discuss more in the video below.

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Share and Summarize

25 minutes

Once most of the class has attempted both sides of the worksheet, I will pull the class back together and we will go over the problems. I will choose specific students who understood the progression of questions in order to explain their thinking.

Teaching Strategy: When going through the distance formula problems on the back of the worksheet, it can be very helpful not to simplify at all. For example, to find the distance from (0,0,0) to (2,3,4), many students start by finding the distance from the origin to the point (2, 3, 0). Leaving it as sqrt(2^2 + 3^2) is much more useful than simplifying it to sqrt(13). The structure of the formula can get lost if you simplify too much. Your students may automatically simplify, but you can leave it in that form when discussing with the class.

Finally, after going through these concepts together, I will assign some problems from the book to give students additional practice with these new ideas and formulas.