Don't forget our August-September MathAnagram. No responses yet but this mathematician deserves our recognition. 'Relatively' speaking, this mathematician was truly unique and, perhaps, the last of a dying breed.
Sometimes I found that a coordinate problem was a good way of reviewing the geometry and algebra needed to refresh memories after a 2-month layoff. Here's a fairly straightforward one that admits many different approaches and might be used to set the tone. Encourage students, working in groups, to find at least THREE different methods. This will extend the thinking of those who "solve" it rapidly and sit there complacently. This problem is a bit more appropriate for the students who completed Algebra 2 although Geometry and Algebra 1 methods are possible.
To reiterate: The problem itself is not particularly challenging. The purpose here is to provide review of several ideas, methods, theorems and strategies.
Given points A(0,0) and B(12,0). Determine the coordinates of all points C(x,y) such that ∠ACB is a right angle and ΔACB has area 18.