NYS Integrated Algebra 2/Trig: neat regression problem
The list of performance indicators (ie, the informal curriculum) for New York State’s new Algebra 2/Trig course is packed. Overpacked.
What should come out? Sequences and series, normal curve, probability, regression, etc. Save those for another course (I’d be happy to put in 6 – 8 weeks of the stats/probability related stuff into precalc. A real unit, you know? And that’s the only place sequences and series belong — leading straight towards limits).
But for now, I don’t get my say. If I were taking the test, I might stubbornly get everything else right, and refuse to answer the questions I didn’t think belonged. Really? Yup. That’s the sort of thing I did as a student. But I’m teaching this stuff.
So the neat regression problem? It’s detective work. Take an old Regents Exam (I’m using the August 09 Math B and August 09 Integrated Algebra). Print out the score conversion chart(s). And reveal the Great Truth: NYS Education Department sets 2 scores, “pass” and “mastery,” and lets the calculator supply the rest. And that’s a starting point many of us can use.
Task 0(a): Look at the conversion chart. Identify the four fixed points – (0,0), (p,65), (m,85), and (t,100).
Task 0(b): Plot the data. They should see it’s not linear. I’ll want them to notice for Math B that it takes 11 points to get from 82 to 90. If you used the August 09 Integrated Algebra, you might want them to notice that it takes 14 points to move from an 80 to an 85.
Task 1: Hypothesize which sort of regression they used. This is hard. I’m skipping this with my kids. If they did it they’d see it is not linear. If they see Integrated Algebra, prepare for an elementary discussion of concavity and that ugly inflection point. But, and here’s the neat thing, but it is guessably a cubic regression.
If they use Math B, very tough. They may see that it is concave down (not in those terms) and wonder if it is a square root or some fractional power.
Task 2: Someone suggests (the kids if you do Task 1 and are fortunate, otherwise the someone will be you) that the State performed a cubic regression. Attempt to recreate what they did, and recreate the conversion chart. Put away the State’s chart. Find a, b, c, and d (well, d=0) for . Continue by filling in values for y for integer x values from 0 to the maximum possible. And then take out the State chart, and compare.
Task 3: Get ready for a real discussion of fairness vs unfairness, and of what other methods they could use to create the conversion charts. Maybe the kids want to experiment? (Three line segments is easy).
By the way, it’s nice to know that four points gives four equations with four unknowns, and that we can actually fit the cubic exactly. In my case, my kids have fit quadratics to three points, so I’ll want to discuss the extension.
Also, to create the graph (that I hope appears correctly) I used (0,0), (30,65), (68,85), and (87,100) for August 2009 Integrated Algebra, I used Google to calculate and the other big numbers, I used an on-line 3 eqn/3 unkn calculator to fit the curve (no regression unless necessary for me!) and I chose Function Grapher to graph my curve.
And I’ll let you know, maybe, how my kiddies react to the problem.