Absolute Value

So I taught absolute value equations in Algebra II. And so far I think things have gone fairly well. I read Kate Nowak’s post on how she did absolute values, and I thought I would change my more traditional introduction to them… but I didn’t. I realized that the way Kate was motivating it (with the distance on the numberline model) was great, but I felt I could still get deep conceptual understanding with the traditional way she eschewed in her post.

So I stuck with that.

I used exit cards to see how they could do… and they were okay.

But after learning how to solve |2x-3|=5 or |2x-3|=-10, I asked kids to solve things like 2-5|5x+6|=5 or something similar. Many students said on their home enjoyment:

2-5(5x+6)=5 or 2-5(5x+6)=-5.

It is unsurprising to me, and yet, it makes me want to throw up. Because what’s coming more and more into focus, and I’m sure you’re going to hear me complain about this more and more in the coming months, is how reliant students are on “coming up with rules” and “applying rules” — without thinking. They desperately want unthinking rules. And this year, because I can’t handle throwing up all the time, I’m vowing to really not give rules to them.

I really got to the heart of this “I LIKE PROCEDURES” thing with them with a true-false activity that I did, using my poor man clickers. I think this exercise highlighted how dependent my kids are on procedures and coming up with simple rules that help them in the short term… but that can hurt them in the long term… It’s a bunch of True-False questions. And when we talked about each one of them, my class saw concretely how reliant they were on misconceptions and false rules. EVERY SINGLE QUESTION led to a great short discussion.

So here they are, for you to use. Sadly, I don’t have the blank slides to share with you, because my school laptop is not with me at home now.

These were great for asking “so who wants to justify their answer?”

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17 thoughts on “Absolute Value

  1. Sam and others

    I am curious to hear a broad opinion on this True-False issue. I have always felt that these kind of questions are very illuminating and I have regularly included them on assessments. This practice was recently questioned by another math teacher at a workshop session I was helping to moderate. I still feel strongly that these are important questions to ask, but I am now wondering whether they should be fair game on graded and timed assessments. Any thoughts?

    • Why not? I have no pedagogical reasons for thinking T/F questions are inherently bad. I just had a whole spate of them on a test on basic set theory, and they really showed me some misconceptions the kids had.

  2. This is what I’ve always disliked about the way students do (or have been taught) math. The want to do it robotically with out thinking. They want to be programmed like a computer so that when they see a problem, they can run through their digital file folder and recall the program used to solve that particular equation. If I had a nickel for ever time I saw students solve 2/x = 4 by multiplying both sides by 2 and crossing the 2′s out. All they see is division and they ‘learned’ to multiply when they see that and whenever you multiply, just cross stuff out. This is why I’m beginning to hate the 40 problem worksheet of old. It almost seems like the point of the worksheet is to make it robotic, which means they aren’t thinking (which is bad).

  3. We’ve been using Always, Sometimes, Never as well as True or False.
    For example, for | x | = x, students learn to test/try a positive number, a negative number, and 0. We also discuss that we are using examples, not doing proofs.

  4. I find that True/False questions can be very useful for probing for understanding. Some students, however, and too quick to pick a side and move on, without thinking deeply about the problem. For this reason, I prefer the Always/Sometimes/Never approach. And yes, having students justify their response is key in developing higher-ordered thinking. For the Always/Never response I would ask, “What makes you think it’s always/never true?” For the Sometimes response I would ask, “When is it true? When is it false?” Analyzing the problem graphically can be very useful here, and you can connect the Sometimes situations with restricted domains.

    I like the way you’ve setup the slides. It is probably far more effective for promoting student discourse than simply looking at a set of problems on a worksheet.

  5. You’ve got to ask: why am I teaching absolute value, and what larger goals can I accomplish in the same breath? I like the use of “covering” or “chunking” here to help with the issues you are describing.

    2 – 5|5x + 6| = 5 becomes 2 – 5F = 5 (where F is your finger covering over the absolute value part). Then solve that, lift the finger, and pray your kids will correctly know |5x + 6| = -3/5 has no solution.

    This covering method works in a ton of places and previews “u-sub” in calculus. The earlier this concept is introduced, and the more frequently it’s used, the better it gets :)

    I also feel that, for practical purposes, complicated absolute value problems are garbage. Show me a situation where anything like 5 + 3|7x – 2| = 10 would come up, and I’ll change my mind! (I’m not saying you’re not stuck teaching it due to testing requirements… but that stinks.)

    • Bowen, I actually do teach it almost equivalently to the way you describe. I don’t think the more complicated problems are worthless , but I agree that the idea and simpler problems are more important.

      My kids hate to think and are so used to procedures that this year is proving hard for me because I’m trying to subvert what they have done for years, in a small way.

      • Your subversion will pay off! I didn’t mean the problems were worthless, just that they’re totally convoluted with no applicable payoff other than heavier lifting :)

        I like the true/false. A local group calls these “Prove or Disprove and Salvage If Possible”. Salvages are fun, asking kids to fix the statement to make it true…

  6. I’m experimenting with translating |x|=3 into +_(x)=3
    instead of |x|=3 into (x)=+-3

    I’m thinking this will save some contortions when I get to inequalities.

    • This could have been used to show that slide #1 had a solution at x=0.

      I was a bit confused how it could be false. Unless you were specifying the value of x.

      Bob
      Algebra 2 teacher
      Navajoland

  7. Although, in general, many teachers or educators do not like T/F questions, I think it is in the way that you write your question. I think it is also necessary to ask students why their answers are such. The T/F above asks for solutions, so I think it is good one. If you ask T/F questions without requiring students to reason or to write their solutions, then I personally think that it is a bad practice.

    • I used these T/F questions to start out our discussion of absolute value in Algebra II. I had them vote their answer anonymously. And then “when we talked about each one of them, my class saw concretely how reliant they were on misconceptions and false rules. EVERY SINGLE QUESTION led to a great short discussion.” It was all about students justifying their answers, and identifying misconceptions.

  8. Thank you so much for sharing your true/false questions! I absolutely plan to use them for discussion this semester.

    I have a very basic comment in regards to the mistake you mentioned your students making, so I apologize in advance! I spend a great deal of time talking about “isolating” the variable. Specifically, we do a lot of practice on just isolating down to the grouping symbol (absolute value bars, parentheses, radical, etc.), which really helps when the kids get to exponential and trigonometric equations. I know they should have this skill prior to Alg II, but, as you pointed out, they simply don’t. I do a mini-lesson where I combine all of the different equations and simply get the kids to get the absolute value/radical/parentheses/trig function alone before they decide what to do with it from there.

    Do you think that might help?

  9. I used this as a warmup/intro to solving absolute value equations the way that Kate Nowak introduced on her blog. My students had seen absolute values before, but hadn’t solved equations that way, so this was an awesome way to get them thinking in absolute values before we got into the solving part. Thank you!

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