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.

Assessments with Multiple Answers

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.