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.
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?
What can we say about the lengths of the sides of the triangle?
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?
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.
What happens when you translate the center of the circle?
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).
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.
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?
What does it mean for a point to lie on a circle? Another Quick Poll, with the idea for the question from MAP.
I am learning to ask questions that I think are obvious.
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.
And then another Quick Poll to see how students are doing writing the equation “from scratch”.
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.
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 …