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* *Reflection: Student Led Inquiry
Distance and Midpoints in 3D - Section 2: Launch

When I taught parametric equations this year, both of my classes thought of a 3-dimensional coordinate system as a way to tackle our original parametric equations problem that had three quantities. So when it came time to actually use 3D coordinates, my students had some background information about how this would work.

Visualizing this system is very difficult, so it was really important to have these 3D models that students could use. Students were constantly using them to decide what octant a point was in or where which plane they were referring to for a specific problem.

*Students Already Thought of This!*

*Student Led Inquiry: Students Already Thought of This!*

# Distance and Midpoints in 3D

Lesson 1 of 9

## Objective: SWBAT find distances and midpoints for 3D coordinates.

*50 minutes*

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.

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#### Launch

*10 min*

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).

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#### Explore

*15 min*

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 min*

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.

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Distance and Midpoints in 3D
- LESSON 2: Equation of a Sphere
- LESSON 3: 3D Vectors and the Dot Product
- LESSON 4: The Cross Product
- LESSON 5: Planes in Space
- LESSON 6: Lines in Space
- LESSON 7: Unit Review: Math in 3D
- LESSON 8: Unit Review Game: Lingo
- LESSON 9: Unit Assessment: Math in 3D