In Modeling Linear Equations, I described the first weeks of my effort to give my Algebra II students a more (lord save me) organic understanding of linear equations. These students have been through algebra I twice (8th and 9th grade), and then I taught them linear equations for the better part of a month last semester. Yet before this month, none of them could quickly generate a table of values for a linear equation in any form (slope intercept, standard form, or a verbal model). They did know how to read a slope from a graph, for the most part, but weren’t able to find an equation from a table. They didn’t understand how a graph of a line was related to a verbal model—what would the slope be, a starting price or a monthly rate? What sort of situations would have a meaningful x-intercept?

The assessment confirmed my hunch that I haven’t been wasting my time. I tried to focus on problems they could solve in multiple ways—up to and including plugging in the answers. I wanted them to be able to approach a problem “as if it were their money”, as I kept telling them when they were figuring out how many power bars and gatorade they could buy for $20.

Here’s the assessment, with the percentage of 83 students who answered correctly (click to enlarge).

Comments:

- Questions 1, 3, 8, and 10 were at or just above the “random guess” percentage. Everyone screwed up the first question, flipping rise and run (answering -2). 3, 8, and 10, however, were answered correctly by students who received a C or higher.
- Question 11 makes me want to beat my head with a student whiteboard. Only 62% of the class knew what the slope of y=-10x + 7? Really? Some of the students who got it wrong then went on to accurately identify the slope from a table and pick the right equation in slope intercept form. It is to weep. But anyway, my guess is that 20% of the kids who got it wrong actually know it, leaving 20% who still aren’t sure. And that’s bad enough.
- Very proud am I of the results for question 13, since we hadn’t done anything like that in class. Apart from the “read carefully” hint, I gave no assistance. Of course, I didn’t include the midpoint for AM, which would have cut the success rate in half. But it still shows the students are thinking and not giving up.
- While the system of inequalities questions are barely at 50%, that’s a huge improvement over the semester final.
- The students had trouble with question 19. This suggests that many of them are still just plugging the values into the equation rather than reading the slope from the table, since the slope is a fraction—and at least a third of the class still can’t multiply fractions.

I curve my multiple choice questions on a 15 point scale, so 85 and up is an A, 70-84 a B, down to 40-54 a D. A student could pass with a D- by answering 8 questions; I think getting 40% of a test that would not (for those students) include any gimmees is worth a low passing grade. On that scale, the average score was 70%—and that’s a first. Every other assessment, including the two on linear equations last semester, had average scores in the D range.

Only two As and four Bs. Many students who usually nail Bs got Cs, while the students who usually failed also got Cs or solid Ds. This too makes me think I’m on the right track. It tells me that some of the B students are used to memorizing methods—and since I didn’t give them one clear-cut method, they had a bit of trouble. The D and F students, who couldn’t memorize or even really understand the methods, were really able to benefit from the instruction and had solid results. (For many students, I consider a passing grade of D a great achievement, and tell them so.)

Several of the B students who did poorly came up to talk to me about it, and that, too, was revealing. Most of them understood realized that this new approach was exposing a weakness on their part and, instead of complaining, talked about their difficulties and asked how they could improve.

We’re doing quadratics in the same way—just spent three days learning how to create table values from descriptions. For some reason, the only quadratic word equations I can think of involve geometry—but then, most of my kids need four or five seconds to remember the formula for a triangle, so the review is win-win.

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