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Thursday, June 24, 2010

Absolute Value Both Rigorous and in Context

I know I said I was done for the year. SORRY. I am literally sitting around school twiddling my thumbs today. I am ripping this idea off from Dan, but trying to extend it to be appropriate for Algebra 2. Absolute value is one of the first lessons of the year, and in the past my students neither understand it conceptually nor remember an algorithm for solving equations and inequalities with anything like reliability. This feels more like an Algebra 1 lesson to me, but I think it will be necessary.

This is my version... peanut M&Ms were the cheapest/most voluminous things I could find. There are about 230 in a large bag, by the way. Yesterday I polled 50 faculty and staff. In the fall I am going to have to get my butt into overdrive within a day or two to collect at least as many data from students.

I have yet to nail down the details, but the flow will go something like this:

Put up a picture like this.

Ask how far away the houses are from school. Get a few volunteers to describe the mental procedure they used to determine distance from school. Point out that everyone naturally used a difference and absolute value to express distance. And that further, if we can represent distance as absolute value with an equation, we will be able to use it to ask and answer more interesting and difficult problems than our intuition can handle alone. Graph by hand y = |x| by making a table of values. Note the characteristic V shape.

Questions to Answer
Bust out laptops and distribute excel file. As per Dan's original plan, kids will have some choices about what questions to explore and time to flail.

- Who won?
- Rank everybody.
- Top 10 Guessers.
- Any ties?
- Worst guesser?
- Which grade guessed best?
- Which job guessed best?
- Calculate percent error.
(Maybe some/all kids can present aspects of the results on posters we can display?)

Once that's all squared away, I want everyone to explore:
- On average, how good were the guesses?
- Create the scatterplot that displays the characteristic V shape.
- What is the equation of the connected graph of that plot and what do the variables represent?

(This popping up on my screen should not have been, but was, the best part of my day yesterday:)

Follow-on problems once equation is achieved. Solutions using both the graph and the equation.

- What guess corresponds to the average distance from the correct guess?
- What did the worst guesser guess? The best?
- In what range did the better-than-average guessers guess?
- In what range did the worse-than-average guessers guess?

New problems and generalization:
Write an equation/inequality that models the scenario. Make sure to define your variables.
- Today’s temperature is 10 degrees off from the usual temperature.
- Today’s temperature will be within 10 degrees of the usual temperature.
- Today’s temperature will be more than 10 degrees off from the usual temperature.
- If the usual temperature is 68, find values for the three forecasts above using algebra. Show all work at every step.
- Graph the scenario. Indicate the three different forecasts on the graph.

- Write a general expression for the distance between a changing value and a known value. Define your variables.

- Put this equation into words: |x – 10| = 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value equation.

- Put this inequality into words: |x – 10| < 3
- Solve it, showing all work at every step.
- Put this inequality into words: |x – 10| > 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value inequality.

Feel free to poke holes in this or let me know how you would implement it differently. Also I need to get them solving and graphing more complicated equations and inequalities like say 10 = 2 |3x - 4| + 7, so I'd love to hear if you see any natural ways to make that happen. I haven't been able to think of any yet.


  1. Wow! I love this activity. I'll be teaching Alg II next year for the first time in a long time and I'm sure to adapt it for my classes.

    I'm wondering about:
    - What guess corresponds to the average distance from the correct guess?

    specifically, its placement in a section where you seem to want students to begin moving away from the raw data and toward to graph and its equation.

    Am I wrong that one needs the raw data to answer this? As I look at the scatter plot again, I'm starting to think like a student:

    HMMM...there are more low guesses than high guesses...if I only look at the guesses within the margin of error of +/- 300 they kind of cancel each other out, but there are still many more guesses that were more than 300 too low, so I'm guessing the average guess was about 500?

    These are really interesting thoughts, but they come from the way in which the discreet scatterplot represents the raw data, and not the equation that would produce a continuous graph with that shape, so maybe that question would be better posed in a different "section" of the activity.

    I can't wait to use this!

    Kevin Feal-Staub

    Math In The News - Integrating current news events into high school mathematics teaching

  2. I was just thinking they would average the column in the table representing each guess's distance from the correct value.

    I see your point though. It would probably make more sense to put it in the previous section. Thanks.

  3. Did anyone attempt a mathematical calculation of the unknown M&Ms?

  4. Sure. By far most people did a visual estimation by trying to line up, eyeball, and count how many little containers would make up the big one.

    But at least three people busted out rulers and did volume calculations. The closest guess used this method. Art teacher. 1.43% error.

  5. I introduce absolute value with the number line and pictures of buildings on it as well. I love this project and was planning on using a verzion of it next year after reading Dan's post. What data did you use to create the scatter plot? What are the labels on your x and y axis? I know this may seem obvious, but I'm missing it. Thanks.

  6. The x is everybody's guess and the y is its distance from the correct value (absolute value of difference between guess and correct value.) If I was being a well-behaved math teacher I would have labeled the axes, sorry.

  7. I like it. After collecting data and before telling them the answer, you should tell them the winner, how far off he/she was, and have them figure out the answer from this information (hopefully they'll see that there are 2 possible answers) i.e. solve a simple absolute value equation.

  8. 10 = 2 |3x - 4| + 7 Hmmm. How do they like fractions?

    3 = 2|3x - 4|

    3 = 2(3)|x - 4/3|
    1/2 = |x - 4/3|

    the distance between the number and 4/3 is 1/2 - so 4/3 + 1/2 and 4/3 - 1/2.

    But not if you lose kids at the first fraction.


  9. Radiolab had an awesome extension on this.

    Francis Galton, he of the Quincunx, regressions, and normal distribution, found that the mean of every person's guesses is consistently better than any one person's guess. Did you find this to be true for this?

    Could be some fun explorations during the statistics unit later- what was the standard deviation of the guesses, did they distribute normally, etc.

  10. To echo all of the other commenters, this is an excellent way to introduce absolute value. Like Kate, I, too, have been mulling over how to contextualize something more "complicated" (from the students' perspective) like 10 = 2 |3x - 4| + 7.

    At the same time, I've been wondering something else: is it really necessary? How obliged are we as math teachers to make everything "real" and, more importantly, are we actually making math less real in the process?

    To the extent that mathematics has evolved to take on multiple personalities--a concrete tool to explore the world that exists, while also the worlds that might--then we would do well to consistently reinforce that sometimes math has obvious applications (e.g. using absolute value to estimate marbles in a jar), while at other times the application may come later, if ever (indeed, that the process itself can be the application: math as a verb).

    In the case of absolute value, we might start by asking, "When is this used in the real world?," and quickly come up with estimations, distributions, binary coding (e.g. on/off switches) in CS. These equations may be simple, but they'll definitely be authentic. With this, we 1) start with something concrete, then 2) "abstractize" it, 3) apply it where possible.

    However, if we make it too specific, and try to cram everything into a real-world context--"When is 2|3x - 4| - 9x + 2 > 9 used in the real world?"--do we risk going too far in the other direction, and eroding our own credibility to discern/promote the dual nature of math? Do we risk focusing only on math as a tool of exploration, but not the object of exploration itself?

    "WCWDWT." It's a great question, no?, and can make math fun. But there's another side, yes?

    Anyway, this is by no means a counterpoint to a real-world approach. I was a math teacher for a few years, too, and recently started Mathalicious(.com) to rewrite all of the middle school math and Algebra I curriculum around real-world topics like these. Personally, I'm totally sold on making math real...where possible:

    Equation of the line between two points: "based on the 16 & 32GB iPads, how much *should* the 64GB model cost?"

    Ratios & proportions: "Usain Bolt is 6'5". Are the Olympics fair?"

    Simplifying polynomials and radicals? Systems of linear equations? What about these?

    In the end, perhaps there's no right answer. Maybe the best we can do is to stand up and say, "When it's real, I'll help make it real. And when it's not--or when we just don't know--I'll help redefine reality."

  11. Cogent points, @Karim. I was trying to think of a reason for them to know how to solve 2|x+3|-7=10+5x-8 etc, besides 'they might see it on the state exam.' And I think it's mainly that I just want them to have the broad knowledge about all kinds of functions to have all the tools to solve anything. Think about for any equation, it's the same process: isolate a function and apply an inverse to both sides. (Quadratics are the exception, or any other time factoring is required to make progress.) The rest is details, like checking that none of your apparent solutions cause trouble in the original equation, or finding all the solutions to a trig equation.

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  13. I hear you, and wish I had struggled more with these kinds of questions when I was a teacher. (When I left the classroom for a different position, I was only starting to get these feet wet). I think you hit it on the head with your comment, "I just want them to have the broad knowledge...", which is similar to a coach saying, "I want them to feel the joys of being in shape" (which reminds me)."

    Perhaps there are three main responses to the student question, "When will I need this?," which will then inform how we allocate our time/attention/interest/passion:

    A. Now (concrete, real-world problems: math as tool)
    B. Who knows?? (abstract, brain-exercise: math as object/puzzle)
    C. In June (test-prep: if not in A or B, maybe not math at all)

    Often, our dilemma seems to be how to teach more abstract equations in real-world ways. But is the premise that our students won't dig it if we don't? If we only taught in an algorithmic/drill-and-kill way, then it--indeed, anything--would probably be a tough sell.

    Still, anyone reading this is probably not that kind of teacher, right? Which is to say, if we're honest with ourselves and our students--if we discern between application/mystery/lame in an upfront and authentic way--then hopefully we'll develop enough cred that they'll cut us some slack when we have to, for instance, teach 2|x+3|...just because we have to. If soccer practice were all wind-sprints, players would revolt. If it were all scrimmages, they wouldn't last the half.

  14. I created this. Which I think could help with student understanding.

  15. I got tired of taking breaks this past year (winter break, spring break, some school districts have a second winter break to correspond with ski trips the families take) so I decided to utilize the internet to avoid losing valuable instructional time. Below is the link to the introductory lesson my Algebra I students received on absolute value this past spring break. The presentation within the video lesson can definitely be improved (far too wordy, for one) but in its most basic form the kids just "got it".

    This is the first of a series that eventually covers absolute value inequalities.

  16. I hope by "Put into words" you don't mean "The absolute value of ..." but rather "The distance between x and 10 is 3".

    This makes things a bit tricky with things of the form |x+3| = 9, but then you can read it as |x - -3| = 9 and say "The distance between x and -3 is 9".

  17. I did this lesson two weeks ago in my bilingual Algebra 2 class and it was excellent. Thanks for organizing it this way. The only thing that didn't go easily was the technology. I think I was the first teacher to grab the laptop cart and I don't think it had really been updated since last year so the computers were feeling neglected. Other than wrangling fussy laptops, it was an awesome lesson.

  18. Kate - I love this lesson ... and we start absolute value in 2 weeks.

    I'm thinking about using a picture of candy corn in a bag from Estimation 180 ... emailing it out to as many staff and students as I can to collect their guesses. I'll use a google form to collect their responses. From there we should be good to go.

    Any thoughts since you last did this lesson that would help the lesson flow well?

  19. Hi Caren - There's something satisfying about holding the physical container. But I don't know how much you lose by doing it with a photo. Not that much, is my guess. Good idea - very efficient!

    The hardest part will be getting them to come up with "subtract the guess from the correct number of candies" as a way to measure how wrong everyone was, and "take the absolute value of that" as a way to describe how far away from correct they were. That's a very important part, and I remember struggling with helping kids to come up with it without giving it away. You might want to put some forethought into how that conversation is going to go.

    Also take a look at this, and this.


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