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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.

 
4 Comments

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

 

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Circles in the Coordinate Plane

CCSS-M G-GPE.A.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.

We used part of the Math Nspired activity Exploring the Equation of a Circle and part of the Mathematics Assessment Project formative assessment lesson Equations of Circles 1  for our introductory lesson on circles in the coordinate plane.

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What do you notice?

-There is a right triangle.

-The hypotenuse is always 5.

What will happen if we trace point P as we move the triangle around in the coordinate plane? What will be the locus of points that it travels?

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What can we say about the lengths of the sides of the triangle?

a2+b2=52

Would it be okay to name the lengths of the legs x and y instead of a and b, since they are horizontal and vertical lengths in the coordinate plane?

So x2+y2=52?

Yes. So for each (x,y) location of the point P, we can say x2+y2=52. That is how we describe the equation of this circle in the coordinate plane.

Now I’m going to let you play with a few more pages in your TNS document and then answer a few questions. Screen Shot 2014-03-04 at 8.32.14 AM

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Students played.

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What happens when you translate the center of the circle?

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And then students answered more formal questions about the equation of a circle to ensure that they had looked for and expressed regularity in repeated reasoning. I collected their responses (this was their bell work for the day, but I decided not to immediately show them the results. I took a glance myself to know how they were progressing).

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Next we moved to a few questions from the MAP formative assessment lesson on Equations of Circles.

I am always impressed by the progression of questioning in the lessons. I was particularly interested in how students decided whether the point (5,6) lay inside, on, or outside the circle x2+y2=36.

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One student knew that it lay outside the circle because the point (0,6) was on the circle. He reasoned that if (0,6) is on the circle, (5,6) can’t be. Another student drew the point on the grid and recognized that it could not lie on or inside the circle. Another student used the equation to show that the point (5,6) did not lie on the circle.

What if the point is too close to tell from a sketch of the graph?

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What does it mean for a point to lie on a circle? Another Quick Poll, with the idea for the question from MAP.

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I am learning to ask questions that I think are obvious.

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We looked at the coordinates that they entered, and we changed to the Graph View. I added a Teacher Equation (two, actually) to show the circle. What does it mean for a point to lie on a circle?

An aside: I was in a grade 7 classroom recently. Students were determining the x- and y-intercepts of lines given the equation. I asked the teacher to send a Quick Poll of the graph of a line and have students drop a point on its x-intercept. She was surprised to find out how many students didn’t know what the x-intercept was, and yet they’re supposedly calculating x-intercepts from equations. (Students had dropped points all over the x-axis, but only a few students had dropped a point at the intersection of the given line and the x-axis.)

Next students completed the Mathematics Assessment Project chart about equations of circles to help quell any misconceptions they might have.

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And then another Quick Poll to see how students are doing writing the equation “from scratch”.

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Which brings us a good opportunity to attend to precision. This is where, in the past, I might ask

-do you have parentheses?

-is your radius squared?

-do you have (x+5) squared and (y-1) squared?

And all of my students would have nodded.

But with Navigator, the students determine which are correct and what some need in order to be correct. They see whether their response is leveling up to the standard or not. They find out what to do for their next response to level up to the standard.

We revisited the bell work during the last few minutes of class. Students decided whether they wanted to keep their original responses or revise their response after the lesson. A few changed their responses.

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And I know who to need extra support as we continue to learn.

And so the journey continues with good evidence of what my students know and what my students still need to know …

 

 

 
7 Comments

Posted by on March 24, 2014 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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