If I had asked my Principles of Math 11 students “What can you tell me about linear inequalities?” I bet most would have said something about graphing a boundary line and shading one side of it. If I had asked “What does the shaded region represent?” I bet many would not have been able to answer correctly.

Similarly, if I had asked “What can you tell me about absolute value?” most would have said something about changing negatives to positives. Few would have been able to give a satisfactory explanation if I had asked “What does |2 – 7| = 5 mean?”

After reading John Scammell’s recent post on linear inequalities, I realized that I had it backwards. I’d begin by graphing the line. I’d explain that a line cuts the plane into two regions. Together, we’d determine which region to shade. I’d tell students that each point in this region is a solution.

Instead, John’s colleague begins by having students find *x*– and *y*-coordinates that satisfy the inequality. Then, each student plots these ordered pairs on a grid at the front. It becomes clear to students that each solution is a point in a half-plane and that a boundary line exists. I robbed my students of this discovery.

John’s scatterplot reminded me of something from Kate Nowak’s back catalogue. Kate’s activity involves having students guess the number of M&Ms in a container. Plotting the points (guess, **distance** from correct value) results in something like this:

Kate has students write equations and inequalities that model weather forecasting. “Today’s temperature will be more than 10 degrees off from the usual temperature” can be modelled using |T – 68| > 10. In a later post, she has students write an inequality that models this scenario:

She scaffolds this by asking students to explain why |t – 12| ≤ 1993 and |t – 12| ≥ 1993 are not good models and by having students write a sentence that begins with “The distance from…”.

Context is important. Not because it must answer “When am I ever going to use this?” but because it helps build conceptual understanding. I’m guessing Kate’s students can tell her more about absolute value than “it changes negatives to positives”.

By the way, the inequality for music that I find tolerable would be |t – 1985| ≥ 6. I graduated from high school in ’91 not ’85. I’m just not a fan of 80s music.

(The title of this post was “borrowed” from Nat Banting.)

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When I used tolerance for error in manufacturing — “plus or minus 2 mm” — the idea of “it doesn’t matter which direction” clicked.