Equations of Circles

In grade 10 academic math we look at midpoint of a line segment, distance between two points and then equations of circles. But only equations of circles centered at the origin (sigh). This is a one day thing, that has been quite confusing for some students. I should note that these students have worked with linear equations a lot, but not much else, so these are very different looking equations. In past years, there has confusion despite my best efforts to connect the equation to the Pythagorean theorem/distance formula. I decided to add a little intro activity this time around:

They worked in groups of 4 and got a quick refresher on finding the length of a line segment. They also quickly figured out that the points collectively were leading to a circle. We then defined a circle. They came up with all kinds of properties of circles. When I could, I would provide a counter-example, like a shape that is round, but not a circle. We honed in on "all the points are the same distance from the middle" which we turned into a mathematical definition. Then we "developed" the equation of any circle centered at the origin.

Next, I hopped on to Desmos and asked them them what to do with the equation to make my circle have a radius of 6, or 3, or 8, or 3.5. They got it. Have I mentioned lately how much I love Desmos? I also showed them how to make the circle "move". We had looked at linear equations in the form y = a(x - h) + k, so it wasn't a huge stretch (no stretches involved, actually!) to perform horizontal and vertical translations on our circles.

When we talked about how to tell if a given point is inside, on or outside a circle and they really got it. I love how making a little change can make the rest of the class become seemless.