Quadratic Relationships in a High School Class

Every module begins with a tripodal base.

• What the students are expected to learn.

• What the students are expected to do.

• Engaging and keeping the students' interest. Modules should be open-ended.

What students are expected to learn

• Solving problems that require several steps and combine several different tools.

• The geometry of intersecting circles.

• The algebra of intersecting circles.

• The vagaries of real measurements.

What students are expected to do

• This module would begin by reviewing or, better yet, referencing the work with linear functions. Since this is a mathematics class, students should be expected to determine the distance from each detector to the earthquake from the usual bare graphs rather than using a program.
  

• Students should also use graphs drawn with compasses to check their algebraic work below.

• The new mathematical technique is finding the intersection(s) of two circles using computations like the following. Note it is important for students to do these "by hand" rather than use a program like the one in the current Virtual Earthquake pages.
  

  From here on, the computation is straightforward using the quadratic formula. The result using real data is usually two points of intersection for the two circles.

• In theory, with three detectors and three equations there should be a single point of intersection BUT with real data this almost never happens. Because of measurement error or some other problem, the three circles are very unlikely to intersect in a single point.

  This is a great occasion to discuss measurement error. This is the point at which the current Virtual Earthquake pages stop. Apparently they use a model rather than real data. With real data there are other potential sources of problems.

  • The speed at which the waves travel depends the medium through which they are traveling.

  • The real problem is three dimensional.

• This is a very open-ended situation. Some students might read more about earthquakes; others might work on the three-dimensional problem, which can be solved using the same techniques as the two-dimensional problem; still others might develop computer- or calculator- based programs to do all the arithmetical work involved.

What do we need from the Virtual Earthquake

Both less and more. As in the preceding example (middle-school linear relationships) we need a bare module that does not do all the computational work for students. But, in this case, we also need more information, or more focused links, leading to more information for students who want to go beyond the material in this module. In particular, we need a discussion of the simplifications involved in the model behind the Virtual Earthquake pages.