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.

#### 13 responses to “Assessments with Multiple Answers”

• Audrey

I didn’t have time to read your full post so the answer to my question might be in the part I didn’t read, but one thing I remember about math problems is I could get partial credit on them even if I didn’t get the correct answer. Do you give any partial credit on multiple choice questions? Your questions seem like they would be very frustrating to students unless they understand the problems thoroughly.

• Tort

Just a quick scan of your test:

Problem 10 says to “Select all of the equations that have the solution.” Do you want to write, “Select all of the inequalities that have the solution”?

How are you students doing on these Smarter Balanced types of assessments you are giving?

• Tort

Teacher mistakes happen all of the time and they provide great learning opportunities, especially when we warn our students that we will make them and it is everyone’s job to catch and correct them. The accomplished writers learn to self-edit their work. So, too, with learning mathematics. The kids are accustomed to being passive about this.

• Taryn

Love, love, love! I need to alter some of my algebra 2 assessments to look more PARCC like. Are you willing to share more examples? Thanks!

• Jasper

Testing subscripts and superscripts: g0 = 9.80665 m/s2 * (1 ft) / (0.3048 m) = 32.174 ft/s2

• Jasper

Since HTML superscripts and subscripts did not work…

Testing special cases of subscripts and superscripts: g₀ = 9.80665 m/s² * (1 ft) / (0.3048 m) = 32.174 ft/s²

• Jasper

Regarding question 5:

Please always use units! I can multiply feet per second by seconds to get feet. I can even convert furlongs per fortnight into a fraction of the speed of light, because I can multiply by fractions that equal one. I cannot take a dimensionless number and flat-out assume that the answer is in feet per second.

The first two weeks of my high school chemistry class were spent on a review of basic algebra, labelling of terms in word problems, and dimensional analysis. Mr. Wingo’s students did very well, in part because they had a solid understanding of what their numbers and formulas meant.

Question 5 would be much better stated as:

A projectile is launched into the air. Its height as a function of time can be modeled by the equation h(t) = -16 ft²/sec² t² + 22 ft/sec t + 3 ft

Select ALL of the true statements at right.

h(t) = -16 ft²/sec² t² + v₀ t + s₀, where v₀ is initial velocity and s₀ is initial height

Of course, I expect that you will continue to use proper superscripts instead of ² and proper subscripts for v₀ and s₀.