## Monday, October 30, 2006

### Next Lesson: Solving Equations and Inequalities Graphically

This is the final lesson of the unit, aside from review and the test. The objective is for students to apply their understanding of graphical representations to be able to solve inequalities and equations.

After reviewing the homework, students will work on a Do Now worksheet with a graph of a parabola and some dotted horizontal and vertical lines. They are asked to use the graph and the lines to solve equations and inequalities.

Following this, there will be some direct instruction on solving graphically. The idea I want to present is:

To solve an equation or inequality graphically:
a) Write each side of the equation/inequality as a function
b) Graph the functions
c) Find the intersection point(s) (and draw dotted vertical “helper” lines)
d) Determine the part of the domain that solves the initial problem

We will review the absolute value problems they solved in the first unit (i.e. |x + 3| > 4), solving them graphically. I will also show them how to find points of intersection on the TI-83+. We will compare solving these graphically to solving them algebraically. Hopefully, students will see that a "less than" inequality makes a "sandwich" graph on the number line because the tip of the V is dipping below the horizontal line (does that make sense?). A "greater than" inequality makes a "gap" graph on the number line because the tops of the V go above the horizontal line to the left and right of the points of intersection.

After direct instruction, students will work independently to practice these concepts. The homework includes solving quadratic equations/inequalities in the same way. Students will also review for the third quiz (on this, plus translation and transformation of functions).

mrc said...

I feel like the ability to solve graphically, or more specifically to have a visual interpretation in one's mind while solving algebraically, is what separates really good students from the rest. I'm not sure how you teach this, how you really make the connections between the symbolic representation and the graphical representation stick for someone. I've never heard the "sandwich/gap" terminology before, but i like it. In my mind, I kind of imagine that I'm on one side or the other of the horizontal line, and I can only see the part of the graph that "pokes through" to my side of the line. By poking through to my side, it "fills in" the solution set on that part of the number line. Does /that/ make any sense?

Dan Greene said...

I agree with your comment about the value of understanding the link between the graph and the algebra. I remember the first time I taught Algebra 1, and I had students who could graph a pair of lines and find the point of intersection. They could also solve a system of linear equations algebraically. But almost none of them could really make the connection between the graphical and the algebraic, no matter how many times I tried to get them to see it.

I'm not sure how to teach it either, except really thinking through the components of understanding graphical representation and doing your best to create an appropriate scaffolding for your students. I got started on the idea after a couple years of teaching Algebra 1, and realized that students did not really understand what the slope of a line represented. So I designed a whole unit that developed this idea, and tied the visual representation of a steeper line to a larger number to the concept of a greater rate of change.

Last year, in Algebra 2 Honors, I began pushing on the idea of representational fluency even more heavily, after seeing how my AP Calculus students in the past couldn't answer a simple question like "for what values of x is the function positive" without lots of prodding. This year, I'm doing it even more, with the idea that it will just take them time and lots of exposure to really internalize the ideas.

If you are interested, feel free to take a look at the stuff I've posted on ILoveMath.org and send me your feedback. I'd like to get some outside opinions on the quality/effectiveness of my lessons and worksheets.