Today's lesson is going to combine some geometric concepts with our new 3D outlook to create an equation of a sphere. Like yesterday's lesson, I have carefully structured the questions on the in-class task worksheet so that students can investigate and make conclusions on their own without me just giving them the formula. Today and yesterday's lesson are very geometric in nature, and would make a good extension for Geometry students who like a challenge.
Precalculus students have a lot of background information about spheres so today's task worksheet can be given to students without much introduction. I have students work in their table groups and give them about 15 minutes to work on it. The most important part about the front side of the worksheet is that students are using the 3D distance formula from yesterday to relate the radius to the distance between the center and a point on the circle. While going around the room I focus on that key idea and work with groups who are off track.
While students work in their table groups, I will select students to write their answers for #3-5 on the board. It is up to you whether you choose students to write down correct or incorrect answers on the board. If I notice something that will be a common mistake or see a mistake that would start a good discussion, I will often ask the student to put their answer on the board. It is somewhat anonymous so when we discuss the mistake, we do not have to single out the student since they wrote it while everyone was working in their groups.
These answers will be the jumping off point for our discussion and my hope is that students will see the pattern in these responses (really drawing on MP7 and MP8). Once we discuss the overall structure of the distance formula in 3D and its application to these problems, I will lead a discussion about how every point on the sphere must be equidistant to the radius. Thus, we can replace the specific numerical points with (x, y, z) and make a general equation to represent any sphere.
All students may not get through questions #8 and 9 (about the trace of a sphere) on their own, so I definitely intervene and we talk about these problems. I will choose students who got the questions correct and have them describe their approach to these problems. If you need some tips on explaining how to find the trace of a sphere to your students, you can watch the video below.
When I teach this lesson to my students, there are always a few students who think that the sphere formula should have exponents that are threes. It makes sense- if the circle formula has exponents that are twos and it is a two-dimensional shape, shouldn't the sphere have all exponents that are three? I usually end class by thinking about this question. I will pose it to my students and have them discuss it in their table groups. Then we will share out together. I think this a good summative question for the lesson and gets them focused on the fact that all we used was the distance formula to find this formula.
To end class, I will give students this homework assignment and have them start it in class if there is time.