Making Short Math Tests

A trig student told me he was hanging out with a group of friends, some who’d had me, some who hadn’t. One was bitching about his four page test.

My students snorted. “Ed’s tests are double sided single pages. Once we had a three page test, but only for the space.”

A debate ensued, and those with the widest range of math teacher experience agreed: My tests are shortest, and hardest.

I’m not sure what this means. I don’t try to make my tests difficult. But periodically I’ve perused other teachers’ tests off the copier, and…wow. They are four or five pages. The questions are straightforward. They are typically of what I would call rote difficulty–they could have peeled off a few pages of one of these tests. When math teachers snort about regurgitating algorithms, these are the tests they have in mind.

I used to have more traditional looking tests, but even back then I wasn’t an exact match for typical. Once I started down the multiple answer path, it became even easier to wander miles off the reservation. But without question, multiple answer tests make it easier to assess understanding on multiple topics—thus shorter tests.

This semester, I finally decided to start my class with a functions unit. Regular readers know that I’ve been beefing up my functions curriculum, after initially (as a new teacher) giving it a perfunctory treatment. But I still began the year with linear equations. This last semester start, though, I went back to the textbooks. Why do they always start with functions? I finally started to grasp the logic: beginning with functions allows the teacher to work with transformations, parent functions, mapping, as well as challenging algebra (solving for x in a square root or quadratic function, etc).

So I mapped out a basic plan:

• Function definition: domain, range, criteria
• Function notation
• Transforming functions
• Four parent functions (line, quadratic, square root, absolute value). I told them we’d be introducing lines to ignore them until the next unit.
• Transforming parent functions.
• Solving for input and output

I originally planned to introduce inverses, but the kids were maxed out. This was a much tougher first unit than linear equations, and a good chunk of the lower ability students were struggling with the abstractions. Generally, I was pleased.

Some new questions from my first functions unit test–which was a single page, double-sided.

Notice I slipped in a couple function notation questions? That’s how I save space.

Here’s a mapping question:

Again with the function notation! Am I the only math teacher whose kids simply can’t compute the difference between “f(3) = ” and “If f(a) = 3, a=”???? I do my best to beat it into their heads.

Here’s another way I use space effectively, I think:

So a graph, some free-response algebra, and conceptual understanding. (Most of them DO NOT understand how to read graphs, and missed d.) Time and again, I had to show the students how to write the equation, but they are learning how to isolate. Relatively few order of operations errors.

I didn’t ask them to graph this next one, but again, practice at setting up an equation to find the input given the output. Another plus of doing functions early is an introduction to quadratics, which is a tremendously tough Algebra 2 unit.

Hands down, this next question had the weakest response. The strongest students understood it, but many of the same students who were able to graph the square root were flummoxed by this one. Go figure. But again, notice that I assess several different knowledge areas with the same question.

A New Quiz

I don’t usually discuss my quizzes, which are often relatively straightforward compared to my multiple answer assessments. But I created a quiz on Thursday that I’m really pleased with. It’s my second quiz for linear functions. The students have learned the three different linear forms. The first quiz covers slope intercept and standard form, which are the forms for modeling situations. This one focuses on point slope and creating equations from points, as well as parallel and perpendicular points. We actually did much more modeling of real-life situations than this quiz shows. Usually, my quizzes are a very reliable guide to what the students have done in the previous week, but this was an attempt to see how well they could transfer knowledge and work several concepts in combination.

The quiz itself, I think was cool. I stole the nuggets of two ideas from textbooks, but the presentation and questions are mine own.

I’ve seen this crickets question in both Pearson and Holt Harcourt books. I built the graph on Desmos, and was dismayed that a number of kids counted the barely visible lines, rather than use the points. But most of them didn’t.

Notice that this is a relatively easy question. I didn’t want to focus on the algebra needed to find the y-intercept. I wanted them instead to look at the patterns (the 120 chirps is exactly halfway between 0 and 240), and think about what graphs say vs. what they mean. Most kids confused question c and d, explaining that the temperature was too cold for chirping, or that the crickets died. But after a few pushes, they go…”negative chirps?” which is fun.

Here, I’m just testing their fluency:

Lots of room for self-correction. One student asked me why all her solutions were “Neither”, and I suggested that perhaps she should check her algebra, where she’d handled a negative value incorrectly. Other students plotted the points incorrectly and, because they were only able to find slopes from the graphs, couldn’t catch their mistake–thus giving me an opportunity to reiterate the importance of using different methods to validate and self-correct.

As part of the work leading up to this quiz, they’d derived the Celsius to Fahrenheit conversion algorithm, given two points. I decided to give them the formula to see if they could recognize the errors in verbal description and work a solution using fractions.

And then my favorite:

I got the basic idea from my new favorite textbook series, Big Ideas Math, then played with the goals a bit. Big Ideas has wonderful scenarios.

As always, if you spot any errors or ambiguities, let me know.

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.