Multiple Answer Math Tests

As previously explicated in considerable detail, I’m deeply disgusted with the Common Core math standards—they are too hard, shovel way too much math into middle school. If I see one more reporter obediently, mindlessly repeat that [s]tudents will learn less content, but more in-depth, coherent and demanding content my head will explode.

Reporters, take heed: you can’t remove math standards. The next time some CC drone tells you that the standards are fewer, but deeper, ask for specifics. What specific math standard has been removed? Do students no longer have to know the quadratic formula? Will they not need to know conics? No, not colonics. That’s what you all should be forced to endure, for your sins. In all likelihood, the drone has no more idea than you reporters do about high school math, so go ask Jason Zimba, who reiterates several times in this interview that the standards are fewer, but go deeper. (He also confirms what I said about algebra, that much of it is moved to middle school). Ask him. Please. What’s left off?

Pause, and deep breath. Where was I?

Oh. Tests.

So the new CC tests are not multiple choice, a form that gets a bad rap. I give my kids in algebra one, geometry, and algebra two lots of multiple choice tests—not because I prefer them, and they aren’t easier (building tests is hard, and I make my own), but because my top students aren’t precise enough and they need the practice. They fall for too many traps because they’re used to teachers (like me) giving them partial or most of the credit if all they did is lose a negative sign. Remember, these are the top kids in the mid-level or lower math classes, not the top kids at the school. These are the kids who often can get an A in the easier class, and aren’t terribly motivated. My multiple choice tests attempt to smack them upside the head and take tests more seriously. It works, generally. I have to watch the lower ability kids to be sure they don’t cheat.

We’ve been in a fair amount of PD (pretty good PD, at that) on Common Core; last fall, we spent time as a department looking at the online tests. The instructors made much of the fact that the students couldn’t just “pick C”, although that gave us a chuckle. Kids who don’t care about their results will find the CC equivalent of picking C. Trust them. And of course, the technology is whizbang, and enables test questions that have more than one correct answer.

But I started thinking about preparing my students for Common Core assessments and suddenly realized I didn’t need technology to create tests questions that have more than one answer. And that struck me as both interesting and irritating, because if it worked I’d have to give the CC credit for my innovation.

On the first test, I didn’t do a full cutover, but converted or added new questions. Page 1 had 2 or 3 multiple answer questions and 3 was free-response, but on that first test, the second page was almost all multiple response:


I had been telling the kids about the test format change for a week or two beforehand, and on the day of the test I told them to circle the questions that were multiple answer.

It went so well that the second test was all multiple answer and free response. I was using a “short” 70-minute class for the test, so I experimented with the free-response. I drew in the lines, they had to identify the inequalities.



I like it so much I’m not going back. Note that the questions themselves aren’t always “common core” like, nor is the format anything like Common Core. But this format will familiarize the kids with multiple answer tests, as well as serve my own purposes.


  • Best of all, from my perspective, is that I am protected from my typos. I am notorious, particularly in algebra, for test typos. For example, there are FIVE equations on that inequality word problem, not four. See the five lines? Why did I put four? Because I’m an idiot. But in the multiple answer questions, a typo is just a wrong answer. Bliss, baby.
  • I can test multiple skills and concepts on one question. It saves a huge amount of space and allows the kids to consider multiple issues while all the information is in RAM, without having to go back to the hard drive.
  • I can approach a single issue from multiple conceptual angles, forcing them to think outside one approach.
  • It takes my goal of “making kids pay attention to detail” and doubles down.
  • Easier, even, than multiple choice tests to make multiple versions manually.
  • Cheating is difficult, even with one version.


Really, only one: I struggle with grading them. How much should I weight answers? Should I weight them equally, or give more points for the obvious answer (the basic understanding) and then give fewer points for the rest? What about omitting right answers or selecting wrong ones?

Here’s one of my stronger students with a pretty good performance:


You can see that I’m tracking “right, wrong, and omit”, like the SAT. I’m not planning on grading it that way, I just want to collect some data and see how it’s working.

There were 20 correct selections on nine questions. I haven’t quite finished grading them, but I’ve graded two of the three strongest students and one got 15, the other 14. That is about right for the second time through a test format. Since I began the test format two thirds of the way through the year, I haven’t begun to “norm” them to check scrupulously for every possible answer. Nor have I completely identified all the misunderstandings. For example, on question 5, almost all the students said that the “slope” of the two functions’ product would be 2—even the ones who correctly picked the vertex answer, which shows they knew it was a parabola. They’re probably confusing “slope” with “stretch”, when I was trying to ascertain if they understood the product would be a parabola. Back to the drawing board on that.

Added on March 7: I’ve figured out how to grade them! Each answer is an individual True/False question. That works really well. So if you have a six-option question, you can get 6/6, 5/6, 4/6 etc. Then you assign point totals for each option.

I’ll get better at these tests as I move forward, but here, at least, is one thing Common Core has done: given me the impetus and idea for a more flexible test format that allows me to more thoroughly assess students without extending the length of the test. Yes, it’s irritating. But I’ll endure and soldier on. If anyone’s interested, I’m happy to send on the word doc.

Note: Just noticed that the student said y>= -2/3x + 10, instead of y<=. It didn't cost her anything in points (free response I'm looking for the big picture, not little errors), but I went back and updated her test to show the error.

About educationrealist

19 responses to “Multiple Answer Math Tests

  • Portlander

    I had a sadistic biology teacher in high school. He’d give multiple choice tests where a, b, c, or d, could all be correct. The only saving grace was he only did it that way occasionally and it would be only a small section of the test. To the students’ howling complaints he justified it by saying he had a college prof that did the same thing, except with the options for e) all of the above, and f) none of the above.

    Those things are h*ll. It’s all knowledge based. They take the g right out of the test! 🙂 You can’t finesse it with “I know b & c are wrong, ergo a must be right.” Or, “I know b is correct so I won’t even read, much less think about, a, c, and d.”

    • educationrealist

      You know what’s funny? My kids, because they *aren’t* the top achievers used to gaming tests, don’t complain about it. I’m teaching an A2/Trig class next. I wonder if they’ll complain?

      There’s no question that some of the things I develop work because my kids have such low expectations of math and school in general. Thanks for the feedback; it will prepare me in case I ever run into it.

  • Vijay

    Which grade is this, I forget?

    I think this test selected to cull the top 10 or 20% of the kids for college math and engineering study.

  • Mark Hampton

    !his is a fantastic math test! I deeply appreciate all the work and thought you put into it. Would you be willing to send the .doc file? I wondered if you had completed any other tests this way? As a long-time teacher, I am always looking to spread around people’s good work. Thanks in advance.

  • Φ

    On question 9 (the sandwich – burrito question, but it isn’t numbered), the student does not appear to shade the region correctly. She shaded the region consistent with y > x – 3, not y < x – 3 (what the student wrote, and the question expected).

    • educationrealist

      Actually, it’s right. The wording was tricky. She wants 3 OR FEWER THAN 3 less. I should have mentioned; we discussed that in class quite a bit. Doing it again, I’d change the problem, because the wording was hard to do no matter what. I spent a lot of time thinking about it while building the test, and finally got tired of it.

      So if she bought 5 sandwiches, she could have 2 or 3 or 4 or 5 burritos.

      • Φ

        Okay, got it. But in that case, the equation is wrong, is it not?

      • educationrealist

        Don’t think so. The equations should be:

        8x+12y <= 120

        All that said, as I mentioned in the piece, I was trying to come up with a way to assess them on inequalities without making them do everything themselves. a) time constraints b) the “fill in the missing pieces” is something you see in common core assessments. This was my first go-round at it. So I’m not convinced of its excellence, and I will never give it in this form again. However, I just looked at it again and still see the same flaws, nothing new. It’s possible that I’m missing some huge error, but my errors are generally on the typo or double-entendre side.

      • Φ

        I think this is a really good problem too. But the student graphed y > x-3, i.e., the number of burritos is no fewer than three fewer the number of sandwiches. so if she buys 4 sandwiches, she could buy 1 or more burritos. but she wrote the equation y < x-3, meaning 1 or fewer burritos in my example.

      • educationrealist

        Well, thanks! I appreciate it.

        and oh, good heavens. You’re right. She got two signs wrong that I missed, not one. I see what you’re saying now. I was looking at the shading.

        If you could see what I was grading that day, you’d know why. At least half the kids only had one or two equations (which I was pleased at) while she had them all. In context, the actual position of the signs was small potatoes!

  • JM

    How did you decide on the balance between single-correct-answer and multiple-correct-answer questions? It appears to me that a multiple-answer question requires more thought & time than just one single-answer question. Given a limited time for the test, what’s an appropriate mix?

    I think your sample questions here are quite well conceived. I wonder how well chosen the PARCC/SBAC questions will be, and whether delivering these online will work as well as on paper.

    • educationrealist

      The first test, in which they were mixed response, was just a pilot. I hadn’t really thought through how I would work it. All subsequent tests have been multiple answer only.

      Yes, they are more work. In fact, I’ve realized that grading them means giving as much credit for rejecting an answer as accepting one. So what I’ve done is total up the number of selections. If I had 10 questions with 5 possible answers each, there are 50 answers. Then I give two points to each. THEN I curve it upwards, because it’s pretty hard.

      Thanks! I’m glad you like the questions. They are probably a little too non-obvious. I don’t dislike the CC tests I’ve seen, but they’re hard.

  • Assessments with Multiple Answers | educationrealist

    […] 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. […]

  • JR

    I stumbled upon this while looking for some hard systems of inequalities questions for my Common Core algebra students. This is a great test, and I honestly think it does more than the Engage NY assessments. If I could get any other work that you have done, lesson plans, assessments, I would greatly appreciate it. Im a new teacher (3 Years) in the Bronx and always looking for different ways to reach my students.

  • Designing Multiple Answer Math Tests | educationrealist

    […] got the idea for Multiple Answer Tests originally because I wanted to prepare my kids for Common Core Tests. (I’d rather people not use that post as the primary link, as I have done a lot more work […]

  • What I Learned: Years 4-7 | educationrealist

    […] Here’s my first post on the topic, but I’ve revisited it often. They allow flexibility way beyond the usual multiple choice–I can mix and match between freeform and formatted response, or include both in the same question. I can create one question with varied procedural tasks, or one question that dives deep into one situation. They also allow me to greater access to student thinking. […]

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