As an Algebra 1 teacher, I taught students that in order to solve a system of equations with two variables, you needed to have two equations (or two constraints). But I didn't know of any good counter-examples that were accessible to my students where they could experience a situation with three variables and figure out just how far we can get with only two constraints in that context. Enter the Math Forum Problem of the Week,
Math Club Mystery (PDF). In this
scenario (PDF), students, parents, and teachers all go on a field trip to see the movie. We know how many people went on the trip, the total cost of tickets, and the cost of tickets for adults and students. We are asked to figure out how many parents, how many teachers, and how many students went on the trip. This problem turns out to be an accessible example of a Diophantine equation, as well as a great opportunity to talk about unknowns and constraints, and to compare methods such as making a table vs. graphing vs. writing equations to solve problems. Be sure to check out the
Teacher Packet (PDF) to see multiple possible solution paths. A
scoring rubric (PDF) is also available.