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Thursday, September 15, 2011

Algebra 2: Solving Absolute Value Equations

You know how you can show them this way, all justified and with lots of practice untilblueintheface:

But then a couple days later half of them will do this

and the other half will do this:

So, I stopped teaching it that way. I'm starting with something much like what most of us probably do:

Allison lives at 15 Sycamore Drive, and Sarah lives 8 houses away. Where does Sarah live?

But then, I'm sticking with that model for all kinds of problems.

and

Earlier in the lesson I made them write it out in words, i.e., "the distance from 200 to 3x is 896."

It was more of a pain initially, and not the most effervescent lesson I have ever delivered, but MAN, it did the trick. No more of that autopilot, forget to write two equations, forget that absolute value can't equal a negative nonsense.

This is an idea I stole wholesale from the article "A Conceptual Approach to Absolute Value Equations and Inequalities" by Mark W. Ellis and Janet L. Bryson, Mathematics Teacher April 2011, Volume 104, Issue 8, Page 592.

Inequalities are a natural extension of this concept. Where on the number line are all the values that are more than 896 away? That are less than 896 away?

1. Huh.. that's not the normal way to teach it?

I always think of it -- and when I was forced to teach this in WKU's college algebra class taught it -- as "save the messiest parts for last". In this case, don't handle the absolute value (by splitting it) until you absolutely have to. When calculating, the "messiest part" is getting a decimal value. Do absolutely anything you can to get closer to an answer without getting a decimal, because as soon as you're manipulating decimals in your calculations you're going to get lost.

The same goes in physics and engineering, actually, where "the messiest part" is actually running an experiment to determine some value.

2. Dig it. Have you asked whether they can go the other way? That is, can they go from the picture to the equation? When your students see |x-2|=9, do they "see" a number line, centered at 9 with arrows sprouting both ways?

3. I will be...this isn't the end of this. :)

4. Crust. I'm teaching this tomorrow. Well, if I have time, I'll be totally revising what I do in my free periods and hope it's enough.

Sam

5. Cool!

Seeing absolute value as a distance is great. Have you ever done Dan Meyer's celebrity age guess game?

6. This is awesome. You are already putting the fun back into the absolute value function.

7. Heh, I could have sworn you stole this from CME Project instead :) Way to "look for and make use of structure"!

This method passes what I call the "six month test" -- if you asked them to recall six months from now what they did, would you expect them generally to be okay? The usual "split and solve" method is a horror show down the line.

You also make it REALLY simple to teach absolute value inequalities. No more "less than means 'and' and greater than means 'or'"! Terrific stuff.

I prefer highway mile markers over house numbers, since "eight houses away" is not the same as an eight-count in house number.

- Bowen

8. Oh, highway mile markers is better. Thanks for the tip. As soon as the house numbers made it out into the wild, there were problems, I had to tell them there were only houses on one side of the street that counted by one's, which of course isn't true for any street anywhere. I was also thinking maybe a temperature scale, because then negative values would make sense.

9. Re: Temperatures... They get you negatives, sure, but the questions you ask don't naturally arise with temperatures.

"The temperature today is eight degrees from what it was yesterday." This is not information you could be privy to without also knowing the present temperature. "I am eight miles away from the nearest exit" is information you could plausible have without knowing your present mile marker (e.g. there is a sign saying "Next exit, 8 miles").

10. I really liked this and used it today with my Advanced Algebra 2 class and will use it with my Algebra 2's later in the week. I used Bowen's
"mile markers" instead of houses. We'll see how well it sticks. Thanks for sharing Kate!
--Lisa

11. You can also try years, months, weeks, days, et cetera.

12. Dang. I've already done Absolute Value eqs in Alg2 - but I'll file this idea away for next year!

My neighborhood actually has house numbers in order on one side of the street so I'll be able to use a concrete example! :)

Kristen

13. Good if there is a constant on one side, but it still does not address the original problem |3x-7|=x+1. Students will still make the same mistakes as before. i.e. 3x=7-x+1 instead of 3x=7-(x+1)

14. I was wondering when someone was going to point that out. The kids didn't have any trouble with it. x + 1 is just the distance. Go x+1 in one direction and x+1 in the other. Sure they forgot to distribute, but the answers didn't check out, so they went back and discovered the error.

But honestly, I didn't emphasize these problems with variables on the other side. Because what would that even model? I'll think it's important if anyone can tell me.

15. I really like the visual approach you provide - it gives students an image other than the notation to work with.

I have usually taught absolute value a slightly different way than your first approach - although I can't claim that it has a significantly better outcome...

Instead of isolating the absolute value, then breaking the problem into two problems with differing signs on the opposite side of the equal sign - I recommend that students break the problem into two identical problems which have the absolute value signs replaced by parentheses... but in one of the problems they put a negative in front of the parentheses.

This approach helps students distribute the negative sign a bit more consistently, and also does not let them "forget about" the absolute value signs completely - since they turn into parentheses. However, six months later, I don't know that this approach is any more memorable - and that is why I like your visual.

16. I love this approach! It makes absolute values real instead of some abstract concept that students can't wrap their heads around.

17. Christopher said: "Dig it. Have you asked whether they can go the other way? That is, can they go from the picture to the equation? When your students see |x-2|=9, do they "see" a number line, centered at 9 with arrows sprouting both ways?"

While I admire the visualization of everything that's going on here, I'm inclined to say that what Christopher is describing is nothing short of mathematically crippling.

When you do equations with absolute values, do you visualize the number line?

When you solve an equation like 2x+1 = 3, do you think, "What this really means is that two times the number I'm looking for plus one equals three?".

When you add and subtract positive and negative numbers, do you visualize the number line? (Or dump-trucks filling holes, or something like that?)

When you multiply 253 by 17, do you visualize a huge matrix of apples, 253 long and 17 wide?

Don't get me wrong: Part of knowing a mathematical concept fully is understanding all the ways that concept can manifest. Some of these ways are visual. Some involve the number line. Others allow us to translate between "real-world" concepts like 'putting stuff together' and mathematical concepts like 'addition'.

But algebra was invented ("for crying out loud", I want to add) to allow us to compute answers without having to resort to visualization! What you're proposing puts mathematics back hundreds of years. And while you may feel you are treating your students to the glory of visualization, you're also cheating them of the beauty of a conceptual invention (ie algebra), without which none of modern mathematics would exist!

I know too well that the cute tricks and visualizations that we teach our students do not become stepping stones on the way to mastery: they become permanent crutches. The other day I worked with a 10-year-old who took forever to multiply 8 and 4. When I asked her how she thought about this problem, she said: "Well I imagine four men with '8' written on their heads. Then two pairs of them come together to say '16', and then the last pair comes together to say '32'." .

Now I ask you, is it important to understand that 8x4 = 32 can be visualized in this way? Absolutely. But is this how we want students to compute 8x4, if all they need to know is the value? Absolutely not.

Can't we have it both ways? Keep your lessons rich with visualizations and applications, but don't ignore the pure mathematics of it all. Solving algebra equations is, at this level, about the strategy of isolating the variable. At each step, we remove some symbolic barrier which keeps the variable 'entangled'. If students look at the process of solving an equation in this way, the absolute value function is just one more symbol in the way. "What's the cost of removing it?", they should ask. The answer? Plus or minus.

This sort of understanding will be very helpful when they go on to learn inverse functions and apply them to powers, exponential functions, logarithms, trigonometric functions, etc.

Okay, end of rant! PS, you're a thoughtful math teacher! I love you!

+j

18. Algorithms are powerful and important and shouldn't be ignored and should be taught explicitly. But teaching only algorithms without attending to understanding underlying relationships is what has made math ed such a mess and why most people misunderstand and fear mathematics. I don't want kids to have to draw a number line every time they solve an absolute value equation. I want them to use their understanding of what's going on on the number line give the "split the equation in two and make one side of one negative" procedure some context. If I wanted to just teach procedures without understanding through repetition and reinforcement, I'd go train dolphins.

19. "Algorithms are powerful and important and shouldn't be ignored and should be taught explicitly. But teaching only algorithms without attending to understanding underlying relationships..."

We both want the same thing: understanding. But I'm not sure 'algorithms' are the issue.

I described a way of analyzing the symbolic structure of an equation to help simplify it. You described a way of visualizing an equation on a number line to help simplify it. These aren't creative processes: both ways are equally 'patterned' and 'methodical', or if you like, 'algorithmic'.

And both ways involve 'conceptual relationships': mine are symbolic, and yours are visual. I don't think either of our methods or conceptual understandings is more 'basic' or 'fundamental'. One of the beauties (and challenges!) of mathematics is that there are many ways to think of a single mathematical concept or problem, just like a human being can be seen as a pile of atoms by a chemist, a global threat by an environmentalist, an epic warrior by a poet, a loving protector by a child, etc.

The only way we can hurt children, educationally speaking, is to make them feel that there's only one way to solve these problems, or even that the solutions and methods we show them are the only ones possible. The danger is not the systematicity of the methods themselves, but of the way they are unthinkingly applied.

I think the visualization you teach is very important. I want students to think of the concept of absolute value when distance is mentioned, and to use that conceptual understanding to help them write down an equation to model the distance relationship. Once that equation is written down, I would like students to leave their visualization behind, and use their understanding of symbolic relationships to help guide their simplification and solution of that equation.

The moral? We need it all. And most of all, we need thoughtful application of the tools we teach.

20. Such a great way to teach absolute value. You are so right students are used to just being given a procedure and what they need is to think!!

21. Another idea to use is rooms in a hotel. Many times they're in order :)

22. Hi, Kate!

I know this several months late. I just starting creeping up on your blog a few weeks ago, and this post is what really sucked me in. I think this is brilliant, and I actually used it in my College Algebra class this week. We played a version of Around the World with it. I talked about it here if you're interested:

http://peterson-epsilon-delta.blogspot.com/2012/02/around-world-with-absolute-values.html

Thanks for all your ideas. You're such an inspiration to me as a math teacher.

Rebecka

23. For many of us Algebra II teachers, absolute value is one of the first topics we teach. For me, this is the one topic in Algebra II that I am determined to do a better job with this year!! I feel like I teach it well every year, but they just do not remember it over the course of the semester. I can't figure it out!! Therefore, I am looking for ideas now to use this September.

I see your original post on this was last September. As I'm sure you would agree, one way to decide if a teaching method worked is to see how well them kids do with the topic, not just on the chapter test, but also on the year-end test. So here is my question: Could you, or any of the others who commented, share how well your students retained the information using this method...or any other method? Rebecka, did it work well in College Algebra? Maybe we could repost this somewhere and get a discussion going!

24. Hi, Kelley--

In answer to how it went in College Algebra: for the student whom I could convince to draw and solve, it went splendidly. Honestly, those students solved nearly 100% of the absolute value questions on that unit test perfectly. The problem was...too many of my students had seen the "3 cases" before and tried to remember what they had learned from high school. Which is fine, if they would have remembered it correctly. But they didn't. Not a single one. I just got a bunch of nonsensical answers, especially when it came to the inequalities.

So, in the future, I'm going to try to find ways to encourage/force students to solve via drawing. I adore this method. It gives a much deeper meaning to absolute value than just "making everything positive."

Now, do students remember it at the end of the year? I think that depends mostly on how much they use it throughout the year.

That's my two cents!

25. Thanks, Rebecka! I know that I definitely need to do something differently this year, and it seems like drawing it out is worth a try.

I do have the kids continue to practice it using the three cases all year in preparation for the end-of-course test, so it always surprises me when they still don't remember it after months of practice. This tells me that I'm not teaching it well and something needs to change. I'm so excited to try something different!!

I'm still curious, Kate, how your kids did with it by the end of the course. Will you use this same method again this year?

Thanks again! Sorry about the typo in my last post. I promise that I don't ever say "them kids"!

26. I had a similar experience to Rebecka. It worked well for the kids who engaged with it, right throughout the year. But I also had a bunch who had well-meaning tutors, friends, etc try to teach them shortcuts... disastrously.

I don't have Algebra 2 next year, and I'm not sure to what extent this shows up in the Precalc course I will be teaching. I would want to do it this pictorial way again, but I think perhaps the magic will happen if we can connect it to the algebra-only algorithm.

27. Thank you! I'll definitely try some visuals.

I am most wanting to "fix" the kids who simply drop the absolute value bars and pretend they aren't there, like your one example above. I was brainstorming yesterday about a little demonstration involving a beanbag marked with an X that is in a Ziploc baggie with another object. I would stack a few more objects on top of the baggie and discuss how to get the variable/beanbag alone by only touching one object at a time. The baggie would represent the grouping symbol (absolute value bars, radical, parentheses). We would then discuss the following: (1)how each item needs removed in some way in order to isolate the beanbag, (2)how you would start with the item farthest away from the beanbag, and, most importantly, (3) how you can't just reach into the baggie and take out the beanbag because that would require you to mess with two objects at the same time. I would really emphasize that there needs to be some way to rewrite the equation without the "baggie" before you go on!

I thought this may also make a point to my sweet kids who would still solve 3(x+1)-2=12 by dividing both sides by 3 and getting (x+1)-2=4. It would encourage them to look at the -2 first because it is further away from the x.

Any thoughts?

28. Love this. In figuring out how to improve the context I came up with this, going to try it out tomorrow.

"An anxious motorist on the Long Island Expressway calls 911 on a cell phone, but her battery dies before before the dispatcher gets all the information. Before the call gets cut off, the dispatcher hears: "Please help, there was an accident on I-495! We are 8 miles from the mile 54 marker..*garbled*" Luckily the dispatcher has two emergency vehicles at the ready.

- Represent the highway with a line, and indicate where the accident might have taken place.

- Write an equation that will help the dispatcher know where to direct her 2 vehicles.

Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.