Objective

Warm-Up

Investigation and New Learning

Closing

Writing Circle Equations

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Objective

SWBAT write the equation for a circle given different key information about the circle. SWBAT explain where the format for a circle equation comes from.

Big Idea

After thinking deeply about the Pythagorean Theorem and the key properties of circles, students are able to generate circle equations themselves.

Lesson Author

Hilary Yamtich

Oakland, CA

Grade Level

Eleventh grade

Twelfth grade

Subjects

Math

Geometry

Standards

HSA-SSE.A.1

Interpret expressions that represent a quantity in terms of its context.*

HSA-SSE.A.2

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

HSA-SSE.B.3

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*

HSF-IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

HSF-BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

MP4

Model with mathematics.

MP5

Use appropriate tools strategically.

MP6

Attend to precision.

MP7

Look for and make use of structure.

MP8

Look for and express regularity in repeated reasoning.

Warm-Up

30 minutes

Investigation and New Learning

30 minutes

The investigation for the day has two different levels, which is set up to give you some flexibility in how you use this time. There is a good chance that some students will still be working on the portfolio tasks from the first two days (Develop an Algorithm and Lattice Points on Circles). If this is the case, I think it is more effective to give them time to continue with those tasks, while other students get started writing the circle equations. During the warm-up, I get a quick idea about this: any students who need a lot of help with problem (1), (2) and (3) may take more time to work on the tasks from Day 1 and 2, while students who figure out problem (4) can get started on today's tasks.

The goal for today is that every student master Write Equations for Circles Level 1. So once I see that the majority of students have mastered the first 3 problems, I tell students their options: They can continue working on the two tasks from the two previous days, or they can transition to a new task based on problem (4.) I tell them that everyone does need to learn what is going on in problem (4) because that is the key learning necessary to complete Write Equations for Circles Level 1. Students can transition when they are ready.

Letting students decide on their own transition times does open things up to a bit of chaos—and it gets better over time the more I coach students to make these decisions effectively. This means that I circulate constantly during this time and ask students either about the content or about their choice about which tasks to work on.

When I talk to students about their choice of task, it sounds like this:

Have you gotten the big ideas from this task already? It may be better to finish that task on your own time, and get started on the new challenge today when you can work with your team?
Do you see how this task relates to that one? Could you apply these ideas to problem (4) from the warm-up?
How can you make sure you are on track to learn the key new idea today within _______ minutes?
Do you think you need practice with _________ in order to understand _______ better?

As students get started on the Writing Equation task, the big question is:

What number sentence will be true for any point (x, y) that lies on the circle?” In order for students to explore this, ask them to find a lattice point on the circle and show the steps they would use to prove that it is on the circle. These steps basically turn into the equation. This is a good opportunity to use the graphing calculator desmos.com/calculator for automatic feedback: if they find an equation, type it into desmos and see if it generates a circle that fits the requirements. You can make this optional, and present it to students by saying, “How could you use Desmos to check your own answer?

The Writing Circle Equations Level 2 problems are much more challenging and will require that students recall (or understand) two big ideas from geometry:

The line tangent to a circle is perpendicular to the radius of the circle that intersects the circle at the point of tangency
Given 3 points on a circle, the center of the circle is located at the point of intersection of the perpendicular bisectors of the line segments that connect two pairs of these points.

I give student some hints to help them develop these ideas. First, I frame them as questions:

What is the relationship between the tangent line and the circle? How does this relate to a radius of the circle?

How can you find the center of a circle if you know 3 points? What is the key property of the center of the circle? How does this relate to these 3 points?

I give students the chance to come up with these ideas on their own. Even if they don’t figure them out and you end up telling them, it is still a big stretch to translate these geometric ideas to the coordinate plane. So this should be a good challenge for them.