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The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?

x²+y²=5²

What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

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So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

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And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

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Here are the results of the questions that they worked.

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What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

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And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

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How can we graph the circle, limited to functions?

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How can we tell which points are correct?

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I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

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54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

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I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

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We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

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And so the journey continues, one #AskDontTell learning episode at a time.

 
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Posted by on April 19, 2015 in Circles, Coordinate Geometry, Geometry

 

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Exploring the Equation of a Circle

CCSS-M G-GPE-1

Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

CCSS-M 8.G.8

Understand and apply the Pythagorean Theorem.

8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

We used the Geometry Nspired activity Exploring the Equation of a Circle to begin our exploration of circles.

We also used some ideas from the Mathematics Assessment Project formative assessment lesson Equations of Circles 1.

The TNS document begins by having students observe what they know about the given triangle.

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It is a right triangle.

As students move point P, what happens?

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The triangle is still right.

The hypotenuse stays 5.

The legs change length depending on the location of P.

Some students might say that a2+b2=52, if we let a and b represent the legs of the right triangle.

Then we do a geometry trace of P as we move P.

What path does P follow?

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If we let x represent the length of the horizontal length of the leg and y represent the vertical length of the leg, then we can say that x2+y2=52 for this circle. Alternatively, if we let (x,y) represent the coordinates of point P, then we can say that x2+y2=52. Then we explored what happens as we make the radius of the circle shorter and longer.

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After exploring the equation of a circle centered at the origin, we translate the center in the coordinate plane. Now what can we say about the right triangle that is pictured?

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After the exploration, we used the sorting activity in the Mathematics Assessment Project’s formative assessment lesson.

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And then TI-Nspire Navigator provided a good opportunity for formative assessment – and for students to attend to precision.

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My students left class not only with an understanding of how the Pythagorean Theorem is related to the Distance Formula and the Equation of a Circle, but they also got some good practice attending to precision through the formative Quick Poll that I sent and by categorizing circle equations. This was a much better lesson than I have had in previous years of teaching the equation of a circle.

And hopefully next year will be even better as the journey continues …

 
 

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Completing the Square to find the Center and Radius of a Circle

From CCSS-M: Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

I like the way this standard is worded. I am glad that it is not enough for our students to write an equation of a circle – instead, we must provide an opportunity for our students to derive the equation of the circle (that will be another post), and we have a reason to complete the square (to find the center and radius, which is the topic of this post).

As I was preparing this lesson, I wracked my brain to think of how my students could discover “completing the square” instead of me just giving them an algorithm to follow. I decided to try starting with a circle in center-radius form, expanding that to the general form (or standard form, depending on which textbook you are using) – and then letting them figure out how to go backwards. Of course we had to spend the first few minutes of the lesson going back over how to expand binomials (they were only in algebra last year) – but I didn’t just tell them that, either. I posed a quick poll to start the lesson:

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While they were answering, I was monitoring the student responses.

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I often go on and show students the correct answer (indicated by a green bar) when I show the results to a Quick Poll. In this case, however, I unchecked “show correct answer” before showing the class results. When the vote is split like this in our classroom, my students have learned this means that they have to get up, find another student in the room, and try to convince that student why they chose what they did. I send the poll again to see if anyone has been convinced to change the response.

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And as you can tell, the results were still not that great. So they had to find a different person to convince of their answer. And I sent the poll one more time. And all but a few very stubborn boys got it correct.

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After the Quick Poll, we used the TI-Nspire CAS – for students to enter into the practice of look for regularity in repeated reasoning.

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I felt like they really needed to have a handle on expanding binomials before they could be proficient completing the square.

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Once they had determined how to go backwards (the phrase “divide by 2 and square” was their phrase, not mine),

Completing the Square Notes

 

we looked at a visual representing of completing the square from MathNspired.

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My final Quick Poll was evidence that students were beginning to make sense of why and how we were completing the square to find the center and radius of a circle.

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Even though they will eventually need to do this without technology, we use the technology while we are learning to make more sense of the mathematics.

And so the journey continues ….

 
1 Comment

Posted by on May 11, 2013 in Circles, Coordinate Geometry, Geometry

 

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