## Tuesday, January 26, 2010

My introduction to quadratics has evolved over the years. Recently I settled on having my students do as much graphing as possible and allowing them to make the connections they need to make in regards to the equation of a quadratic and it's resulting graph. I have found that allowing my students to roll their sleeves up and start graphing allows them to become more comfortable with quadratics. This pays dividends when it comes time to solve quadratic equations by factoring, completing the square or by using the quadratic formula.

The order of things goes something like this:

This worksheet usually comes after we have exhausted ourselves on linears and introduces the idea that there are other relationships and, no, they don't all form a line. We do this by measuring various circular objects and calculating the area and circumference. We then graph circumference vs. radius and area vs. radius.

Graph the Magic Number
This worksheet is a little less organic, but still results in a quadratic relationship.

Stretch Factor
What happens when we graph change the "a" value? I give students a series of "Silent Board Games" (borrowed from CPM), where students are given incomplete input/output tables and are required to find the patterns to complete them. Once they are completed, they graph the parabolas on the given coordinate plane. The idea is for them to make the connections between the input/output table, the equation and the graph. Usually students are pretty quick at recognizing they can describe what the graph would look like just by observing the equation.

Graph Given the Vertex
In this worksheet, I give students random points on the plane and have them graph a parabola with a stretch factor of 1 or -1. By this time I expect them to have a grasp of the relationship between the points on a parabola and it's vertex. This allows them to see that if they can graph one, they can graph them all.

Vertical Shift/Horizontal Shift
Both of these worksheets are in the same format as the stretch factor worksheet. I have students complete input output tables and graph. By this time, they are looking for relationships between the equation and the visual interpretation of the graph.

GeoGebra Investigation
This is pretty intensive and I probably need to break it up into smaller more manageable labs. I would like to throw that out there for discussion. The main ideas that I wanted my students to explore are:

• parabolic symmetry
• line of symmetry is the average of the x intercepts
• x value of the vertex and the l.o.s. are the same
• relationship between a, b and the l.o.s.
• similarities and differences between quadratic functions and equations
• solve quadratic equations by graphing
• determining the number of solutions by using the discriminant
Find the vertex
This last worksheet has students take different quadratics and use the equation to find the vertex and line of symmetry. From there they will graph using only the vertex and the stretch factor. I then allow them to use GeoGebra to check their answers.

Worksheets (if you're interested)