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.
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.
It is a right triangle.
As students move point P, what happens?
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?
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.
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?
After the exploration, we used the sorting activity in the Mathematics Assessment Project’s formative assessment lesson.
And then TI-Nspire Navigator provided a good opportunity for formative assessment – and for students to attend to precision.
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 …