# Tag Archives: precalculus

Multiple Answer Math tests are my new new thing, and I’m very pleased with how it’s going so far. I thought I’d talk about some of the problems in depth, see if anyone has suggestions.

Most of these questions come from an A2/Trig test I wrote this weekend, focusing on systems of equations, but my tests are always cumulative.

One of the things I really like about this format: I can combine free response and selected responses very easily. So here they had to graph the plane, then answer questions which may or may not have to do with the graph. So I could both test their ability to graph a plane see if they understand how distance works in three dimensions, check out their attention to detail, and see if they remember what a trace is. Query: is “slope of a trace line” acceptable? I’ve never taught 3-dimensional systems before, and the book only said “trace”. But when I was teaching it, I kept forgetting and say “trace line”. I wanted them to demonstrate they could visualize the plane in three dimensions and see the slopes of the lines forming the plane, and I couldn’t find any sample questions. Probably an oddball question.

“a” and “e” contain typos. I originally had a different line, until I realized it’d be too hard to graph on the coordinates I provided. So I changed everything, or tried to. Missed two things. First, I intended “a” to be correct, but forgot to change the constant. That’s okay, it will allow for attention to detail. But “e” is just a kluge question, since I changed the points but forgot to change the distance. Before, it was a test of evaluation; now it’s a more obvious wrong answer.

This question makes me very happy. Transformations, function operations, evaluation, and then a transfer of knowledge test! We’d never done any problems like “e” before. No one squawked, and I even saw some kids solving it graphically.

(I stole this graph from online, but can’t find it any more. If it’s yours, let me know and I’ll provide a link.)

I tiptoed conceptually into linear programming, but we did a lot with feasibility regions and of course, systems of inequalities. I describe my approach for Algebra II, but I step it up a bunch of paces for A2/Trig. I expect them to be able to graph lines and inequalities. They get review during the modeling section, but that’s all.

Another one I just think is elegant because it approaches the absolute value from so many different angles: algebraically, graphically, and then a function conceptual question for good measure.

I use this on both Algebra II and Algebra II/Trig. We math teachers try to beat into the kids’ heads the idea that a function can be defined or expressed in four ways: verbal, algebraic, graphic, and tabular. If this were a multiple choice question, students would just test one value and see what happens. But it’s multiple answer, and plugging in numbers takes a long time. Plotting the points and sketching the lines, on the other hand, works very nicely and very quickly—if you know how to graph those lines.

Every so often you can really mess with the kids’ minds, like this:

None of the “obvious” answers are right. The kids really have to trust their abilities.

This is almost pure concept. I introduced the algebra of rational expressions; we’ll do the graphs later. Well over half the kids correctly selected e, but a lot missed b. Ack.

Here’s a couple that work for either pre-calc or algebra II. The quadratic runs the gamut from conceptual to technical. The circle question is more purely technical, but that’s because there’s a lot to test.

I’m having a much easier time grading these now, once I realized I was actually creating True/False tests.

Still to be resolved, however: I have to distinguish between “left the problem blank because I didn’t know” and “not true”. Right now, I evaluate the test to determine what the student is doing, but in the future I think I’m going to have a field they can mark “T” or “F”. If it’s blank, it’s wrong.

So, for example, take a look at this question again:

Answers A, D, and E are true. The others are false. I give this question 14 points, 2 for each letter.

Almost all my students correctly select A as true, because they’ve built the equation themselves as an exercise and understand the parameters. They likewise know that B is false. Some of them read “maximum” as “initial” and wrongly select C, but many otherwise weak students with good attention to detail get it correct. So even my weak students are likely to get 6 points on these three letters.

Then we get to the tougher ones (they aren’t always in order of difficulty). Students have to understand what elements of the parabola equate to max height, time to max height, and zero height. Obviously, I cover these extensively, but kids have a harder time with this. I don’t just teach them a method. I expect them to know that max height is the parabola’s vertex, so that the x value is time to max height, and the y value is the height.

I had at least 12 students who correctly factored the problem, thus correctly NOT selecting E, but also NOT selecting D. Strong technically, weak on the concept of a “zero”. I gave them partial credit (a point) and yelled at them on the paper: things like “Noooooooooo!” and “Arggggghhhh.” and arrows and question marks and “Yo! What do you think (2t-3) means, exactly!?!”

The vertex questions E and G give students the most trouble, but that seems to be less about concept and more about a reluctance to work with fractions. My algebra II students actually do better than my precalc students because we spend a whole unit on this, as opposed to a few days in precalc.

So an average weak student will get 8-10 points out of 14. Very few students get all 14 points, maybe 8 out of 60. Most get 10-12. If they show their work and I can see they were on the right track with just an algebra error, I give partial credit. Other times, I can clearly see their math was terrible, even if they got the right answer. In those cases, I mark the question correct and then dock them 2 points for bad math.

While I don’t normally review tests, I always go through these and give the correct answers and discuss grading decisions.

I strongly recommend giving these a try. They’re lots of fun to make and again, typos are a lot easier to hide.

## Assessing “Upper Level” Math Students on Algebra I

A2/Trig

I am teaching Algebra II/TRIG! Not Algebra II. First time ever. Last December, I gave the kids a packet with the following letter:

Hi! I’m looking forward to our course.

Attached is a packet of Algebra I review work to prepare you for our class. If you are comfortable with linear and quadratic equations, then you’re in good shape. If you’re not, it’s time to study up!
Our course will be challenging and fast-paced, and I will not be teaching linear equations and quadratics in their entirety—that is, I expect you to know and demonstrate mastery of Algebra I concepts. We will be modeling equations and working with applied knowledge (the dreaded word problems) almost constantly. I don’t just expect you to regurgitate solutions. You’ll need to know what they mean.
I’m not trying to scare you off—just put you on your toes! But you should put in some time on this, because we will be having a test when you come to class the first full day. That test will go in the gradebook, but more importantly, it will serve as notice. You’ll know if you’re prepared for the class.

Have a great holiday.

Reminder: My school is on a full-block schedule, which means we teach a year’s content in a semester, then repeat the whole cycle with another group of students. A usual teacher schedule is three daily 90-minute classes, with a fourth period prep. I taught algebra II, pre-calc, and a state-test prep course (kids killed) last semester, and this semester I have A2/Trig and two precalcs.

(Notice that I am getting more advanced math classes? Me, too. It’s not a seniority thing. It’s not at my request. It’s possible, and tempting, to think they noticed the kids are doing well. I know the first decision to put me in pre-calc last year was deliberate, a decision to give me more advanced classes because they wanted a higher pass rate. But I honestly don’t know why it’s happening. Maybe they cycle round at this school, moving teachers from high to low and back again.)

So I said the first full day, and today was a half day, but the kids had a whole packet to work on and I wanted to understand I wasn’t screwing around. If they’d done the work, they’d do fine on the test. If they were planning on cramming, too bad so sad.

I was originally going to do a formal test, but decided to just throw a progression of problems on the board. Then I typed it up for next time, if I teach the class again.

How’d they do? About a third of them did well, given the oddball nature of the test. A couple got everything right. Most of them stumbled with graphing the parabola, which is fine. Some of them knew the forms (standard, point slope), but weren’t sure how to convert them.

Another three passed–that is, answered questions, showed they’d worked some of the packet. The rest failed.

Of the ones who failed, easily half of them had just blown off the packet but have the chops. The other half of that third I’m not sure of.

If you are thinking that kids in Algebra II/Trig should know more, well, they were demonstrably a step ahead of my usual algebra 2 classes. And I think some of them just didn’t know I was serious. Wait until that F gets entered, puppies. Like I told them today: “There’s a lower level option here. Take it if you can’t keep up.” Whoo and hoo.

Pre-calc

I’ve now taught pre-calc twice. The first time, last spring, I was stunned at the low abilities of the bottom third, which I didn’t really understand fully for two or three weeks, leaving some of them hopelessly behind. I slowed it down and caught the bulk of the class, with only four to five students losing out on the slower pace (that is, they could have done more, but not all that much more). So when I taught it again in the fall, I gave them this assessment to see how many students could graph a line, identify a parabola from its graph, factor, and use function notation. If you’re thinking that’s pretty much the same thing I do with the A2/Trig classes, well, yeah. Generally, non-honors version of course is equivalent of honors version of previous year.

I don’t formally grade this; the assessment happens while they’re working. I can see who stumbles on lines, who stumbles on parabolas, who needs noodging, who works confidently, and so on. I was able to keep more kids moving forward in the semester/year just ended using this assessment and a slightly slower pace. One of the two classes is noticeably stronger; half the kids made it through to the function operations before asking for assistance.

This assessment also serves as a confidence booster for the weaker kids. Convinced they don’t understand a single bit of it, they slowly realize that by golly, they do know how to graph a line and multiply binomials. They can even figure out where the vertex should be, and they might have forgotten about the relationship between factors and zeros, but the memory wasn’t that far away.

While I just threw together the A2/Trig course, I put a huge amount of thought into this precalc assessment last fall. I think it’s elegant, and introduces them to a lot of the ideas I’ll be covering in class, while using familiar models.

Part II is just a way of seeing how many of them remember trig and right triangle basics:

If you’re interested in assessing kids entering Algebra (I or II) or Geometry, check out this one–multiple choice, easy to grade, and easy to evaluate progress.