As a middle school math teacher, I spend a lot of time slogging through review material. Our current textbooks don't challenge the students very much; much of what they see is simply a repeat of what they studied last year with a few new vocabulary thrown in.

So when I get to introduce a lesson on a topic that is a certain distance from what the students have seen before, it's always an exciting day. Today in seventh grade we dove into the graphs of quadratic functions--at a very basic level, granted, but the students always step over the bounds of the textbook and ask great questions:

• Does the parabola ever bend back in on itself?
• How can the slope of a curve increase but never reach straight up-and-down?
• Is it a quadratic function if it has powers of 0? How about -2?
• Will the scales of the axes be the same for all quadratic functions?
• What should I do if the y-intercept of the parabola is inconveniently large?
• Can I put a break in the axes of a coordinate plane?
• Is an expression still a polynomial if it has higher powers of the variable than 2?
• What do the graphs of higher-order polynomials look like?
• What does the function for this (fourth quadrant) parabola look like?