Standard Form of Circle Equations
Lesson 4 of 10
Objective: SWBAT use the method of completing the square to determine whether or not an equation defines a circle and to rewrite circle equations in standard form.
Today’s Warm-Up builds on previous warm-ups. It addresses the same skills and big ideas, but it does so at a level that requires more abstract thinking. For instance, all students are asked to prove the Pythagorean Theorem based on the tilted square method. I really want all students to have the chance to figure this out on their own. It is important to developing a classroom culture that values engaging in mathematical practices as an opportunity.
One option for helping students figure out the proof is to ask one student to write their steps on an individual student whiteboard and let other students walk over to his/her desk to look at one step at a time. This is a good way to encourage dialogue and to not give students too much help at once.
The next three problems involve radical numbers, so that students cannot use a compass to “eye-ball” their answers. They will need to work directly with the Pythagorean Theorem. Because many of my students struggle with number sense, as soon as they see radicals, they grab their calculators and turn everything into decimals. It gets so messy, so quickly. So, I usually write one example of how to square_a_radical on a whiteboard. Then, I refer students to this if they need a reference.
My go to question to ask students when dealing with problems like (2), (3) and (4) is:
How can you use right triangles to help with this? How does the radius relate to a right triangle?
My idea is that in order to find lattice points, I want students to find two square numbers whose sum is the square of the radius. If students easily do this, ask them, “Can you explain precisely how you did that? What numbers were you looking for?” As always, even if students are getting correct answers, it is great to ask them, “How do you know? Why does that make sense? What strategies did you use?”
Hopefully, students know how to do all 4 problems on today’s warm-up, but the 30 minutes of time allocated should give me the chance to assess this. If more time is needed, I may assign students to new partners to teach each other about different problems. This is a chance for students to hold themselves accountable by telling them that they should now be able to figure out all 4 problems. I say, “Use any resource you can think of that will help you really understand these 4 problems. Look at previous warm-ups, talk with partners, work with people at a different table, but you have ________ minutes to make sure you fully understand these problems.”
Obviously, this creates a little bit of chaos, but it helps make students the agents of their learning. I will circulate quickly to make sure that students have made effective choices about their resources. As I check in I will coach them quickly, “What is your game plan? What do you think will help you understand that problem? What learning strategies will you use?” These kinds of questions help me show students that I hold them accountable for their choices.
If things seem too chaotic, one strategy that I can use to quickly get students on track is to make a list of students caught in the act of doing a great job facilitating their own learning.
Today is another great opportunity for differentiating the lesson. The new piece of learning is something that students can teach themselves with coaching and scaffolds, so not everybody needs to work on the new task at the same time. As the class progresses, all of the students will eventually be working on Finding Key Information about Circles.
So far, students have been given three (or four) portfolio tasks in this unit. The extension (Level 2) from yesterday’s lesson is definitely challenging and engaging enough for students to continue working on. Students who need more time with the first three portfolio tasks (Develop a Method to Determine Whether a Point is On a Circle, Lattice Points on Circles, and Write Circle Equations Level 1) can take this time. Students who do not need more time can continue trying to figure out the Level 2 problems from yesterday's class.
Instructional Note: There are more of these challenging extension problems on the back of today’s warm-up for students who want more practice problems. I will use these as a chance to show students how to solve a problem while they still have access to more problems to work on.
The new piece of learning today is for students to use the method of completing the square to rewrite circle equations in standard form. Once students understand the Standard Form of a Circle Equation, they can develop their own strategies for rewriting these equations. This is a good opportunity for students to use technology if they want, because they can type the original equation into desmos, along with their rewritten version, and use the graphs to determine whether or not the two equations are equivalent. I simply make the technology available to them and ask then, “Can you think of a way to use desmos to help check your answers?” (MP5). If students start out using guess-and-check, that is totally fine, because this is a great example of a task that moves from quantitative reasoning to abstract reasoning (MP2).
Today's investigation is a time to push students in their awareness of how they are using Mathematical Practice 8. MP8 shows up when students complete the square, and hopefully start to realize that they always take half of the coefficient of x or y and then square this and add that amount to the opposite side. This is a good example of repeated calculations from which they can generalize. Even if students have already made generalizations about this idea when completing the square, they can freshly realize these ideas when they use the same skill in a new context.
Students who tend to understand these ideas more quickly can work on the Level 2 problems from yesterday. Even if they didn’t start them yesterday, they can take the time to tackle them today, because these will be challenging problems that they can continue to work on throughout the unit whenever they accomplish the day’s task with time to spare.