This Week’s Skill
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality. An equation defines a boundary, which divides the coordinate plane into two half-planes.
This is mathematically correct, sure, but how many novices have you taught who would sit down and attempt to parse that expert language?
The text goes on to offer three steps for graphing linear inequalities:
- Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >.
- Use a test point to determine which half-plane should be shaded.
- Shade the half-plane that contains the solution.
The text offers aspirin for a headache no one has felt.
The shading of the half-plane emerges from nowhere. Up until now, students have represented solutions graphically by plotting points and graphing lines. This shading representation is new, and its motivation is opaque. The fact that the shading is more efficient than a particular alternative, that the shading was invented to save time, isn’t clear.
We can fix that.
What a Theory of Need Recommends
My commenters save me the trouble.
Ask students to find two numbers whose sum is less than or equal to ten (or, alternatively, points that satisfy your 2x + y < 5 above). The headache is caused by asking them to list ’em all. The aspirin is another way to communicate all of these points — the graph determined by the five steps listed above. Rather than present the steps, have students plot their points as a class.
One problem I like is having each kid pick a point, then running it through a “test” like y > x2. They plot their point green or red depending on whether or not it passes the test — and a rough shape of the graph emerges.
Great work, everybody. My only addition here is to connect all of these similar lessons with two larger themes of learning and motivation. One large theme in Algebra is our efforts to find solutions to questions about numbers. Another large theme is our efforts to represent those solutions as concisely and efficiently as possible. My commenters have each knowingly invited students to represent solutions using an existing inefficient representation, all to prepare them to use and appreciate the more efficient representation they can offer.
They’re linking the new skill (graphing linear inequalities) to the old skill (plotting points) and the new representation (shading) to the old representation (points). They’re tying new knowledge to old, strengthening both, motivating the new in the process.
Next Week’s Skill
Proofs. Triangle proofs. Proving trigonometric identities. If proof is aspirin, then how do you create the headache?