Today I pose a challenging Bell work problem. I want my students to have the opportunity to engage in similar reasoning to the flow of yesterday's investigation. I will encourage my students to work together on this problem. The definition of a sphere is deliberately parallel to our definition of a circle from yesterday's class. I added the third dimension variables for the center and a random point on the sphere. I also included the term space since a sphere is a 3-D figure.
I use this problem to start class today because I want students to focus on the structure of an equation and how it relates to the geometry of an object. I also want to promote students' understanding of how reasoning in 2-D can be applied to thinking in 3-Dimensions (or more).
I expect that my students should need 3-4 minutes to develop the equation. I plan to have a student or two share their work and his/her reasoning. One question that I will be sure to ask is, "How did you decided to include the (z-j)^2 term to your equation?"
Yesterday, we learned the standard form equation of a circle. Today, I want students to learn the general equation and to gain confidence in working from one form to another. I plan to start by giving the students several equations in standard form. Instead of students doing all the problems I give each group one problem to expand. Then, I will ask them to expand the equation fully. I will give my students several minutes to complete the task. Then, the students put the results on the board to compare and discuss.
During this discussion I will name the equation as the general equation of a conic. I share the formula that includes an "xy" term so that the students see the entire general equation. I explain that we are going to only work with equations that do not have an "xy" term. I will conclude this presentation by encouraging students to write the general equation down in the circle portion of their reference sheet. We will then discuss some ways to interpret the equation to determine if an equation is a circle. We use the expanded equations to see that the general equation represents a circle if the parameters A and B are equal.
I want to take some time today to discuss the two forms of the equation of a circle with students. Which equation would be most useful: the general or standard form? Most students will choose the standard form since it helps you to identify the center and the radius most easily.
To challenge them, I plan to give the class an equation in general form and ask how to use algebra to rewrite the equation in standard form. I have several problems for converting circle to standard form. We work the first problem as a class.
If the students have difficulty converting from general form to standard form, I will help them to review what we did at the start of this section. Working backwards through this process can be difficult. So, as a class we will look at how to use completing the square to complete the task. I typically take my time doing this. The idea of working backwards is a key idea in understanding how integration methods work in calculus.
After going through the process, I take some time to review what was done and answer questions. I give the students another general form equation to work in groups. I move around and answer questions or correct errors as the students work. I make sure to discuss example 3 of the resource since the coefficients on the squared terms are not 1. I will show the students the first couple of steps and then let them finish the problem.
After answering questions about the last example, I give students the Circle Problems Worksheet. The problems on this worksheet ask students to write equations for circles when information is given about the circle and to convert equations from general form to standard form.
My students usually struggle with Questions 4 and 5 on this worksheet. For some reason, they find it difficult to visualize the location of the circle. I remind students that drawing a sketch will help with this problem. I also explain that if the circle is tangent to the y-axis it is like a ball sitting up against a wall. If it is tangent to the x-axis it like a ball on the floor. This helps students understand the idea of tangent to the circle. Even these hints are sometimes not enough. I need to remind some students to draw a radius on their sketch. If they do not draw the radius to the point of tangency, I will ask the students if there is another radius that would help us find the length of the radius. This is usually enough to get the student going.
The last three problems are challenging. When students ask me for help, I respond by asking what need to be the case for two circles to intersect? We may discuss ideas such as:
I conclude this lesson by asking the students to turn in an Exit Slip. For today, there are two questions. The first is for students to ask a question that I have not been able to answer for them. The other question is for students that are understanding. I want to challenge those students and bring back the bell work idea from today. Students may answer both questions it they want.