OK, so there’s a problem that goes like this:

Sketch the circle whose equation is x^2+y^2=100. Using the same system of coordinate axes, graph the line x+3y=10, which should intersect the circle twice – at A =(10,0) and at another point B in the second quadrant. Estimate the coordinates of B. Now use algebra to find them exactly. Segment AB is called a chord of a circle.

OK, what have the kids seen before this? They have been introduced the equations of circles that are centered at the origin. Simply, we did this by asking them to plot points that were say 5 units away from the origin and we discussed from the standpoint of the pythagorean theorem. This happened about 3 pages prior to this question. We’ve also done a few problems about circumcenters and the smallest circle that fit around a right triangle. However, this is the first circle of a radius 10, I believe. This is also the first time, they are asked to graph a line and a circle on the same coordinate axes.

This is also the first time they are being asked to approximate the point of intersection. The point does end up being a lattice point (what we refer to as a lattice point is a point that has integer valued coordinates), so the students probably will be able to come to class with the answer, which happens to be(-8,6) if they did a decent job of graphing it well.

What they have not done before is algebraically substituting in an expression for y. So this is the interesting part – to see if anyone thinks to do this. Most student probably won’t. In the past, many student remember solving systems of equations in algebra class, but they get confused when they see the quadratic terms. Once prodded in the right direction to ’substitute’ however, they realize that the linear equation is easy to solve for x, and see what they can do. This quadratic is actually not that hard to solve, pretty simple factoring involved and no quadratic formula, so it’s’ a nice first attempt at this type of problem. I will report back with what happens in my class tomorrow.

This problem also introduces the terminology of a chord of a circle. I generally like to have the students come up with a definition at this point so that we can formally have one to move forward in the problems as well. Overall, lots of good stuff in this problem, mostly generated by the students, even the substitution idea sometimes.

If you’d like to try this problem it is # 4 on p.49 at my website. You can get pdf files for the entire curriculum there. Have a look.