Monday, February 18, 2008

Unequal Methods

Colleagues - I could use some advice. I just graded the Algebra 1 tests on Inequalities and Absolute Value, and they were quite awful. While some of that is due to the distractions of Spirit Week and me being sick a few days, there's more to it, and any reports on successful approaches to teaching inequalities in general and absolute value inequalities in particular would be most appreciated.

A first hurdle is to help students actually understand the number line graphs that they draw of solutions to inequalities. Many use middle school mnemonics about arrows pointing in the same direction as the inequality sign, and draw their graphs from these rules (and go wrong when the variable appears on the right hand sign of the inequality, of course). We've worked on listing a few actual numbers that are part of the solution, plotting these, and then drawing the graph afterward. That helps a good deal after a while.

The next hurdle is to understand the difference between AND and OR inequalities. The most effective approach so far has been a combination of the above mentioned insistence on a list of actual, specific numbers that satisfy the conditions, and lots of problem quartets like the following:
x > 2 and x < 5
x > 2 or x < 5
x > 5 and x < 2
x > 5 or x < 2
This way, we always get one inequality with no solution, one satisfied by all real numbers, and a couple plain vanilla and- and or-inequalities for the same pair of numbers. It's not magic, but it does seem to help to vary one thing at a time.

Then enter the absolute value inequalities, and what a mess they are. There are so many different ways of solving them, and of talking about them, and I've made the mistake of covering several instead of sticking to one geometric approach and one algebraic approach. Now students are, quite predictably, using messy combinations of these.

Geometrically, the absolute value of x - 2 can be understood as the distance of x - 2 from zero, or the distance of x from 2. Last semester, with the Intermediate Algebra class, I relied on the former and had the students set up inequalities such as | x - 2 | > 3 by drawing a number line, placing "x - 2" more than three units away from zero on either side of zero, reading the resulting inequalities ( "x - 2 > 3" and "x - 2 < -3" ) from their sketch, and solving from there. It didn't really stick. I am not sure whether that was due to inadequate repetition or due to this approach being conceptually confusing.

Anyhow, with the Algebra 1 group this semester, I instead belabored the geometric interpretation of | x - 2 | as the distance between x and 2. I taped a large number line under the blackboard and we checked this definition by walking back and forth: -1 is three steps from 2, and sure enough, | -1 - 2 | = 3, and so forth. In order to solve inequalities such as | x - 2 | > 3 I had two students walk three steps from 2 on the number line in either direction, and we talked about what numbers were more than 3 units away from 2. This was difficult for many students (and not only the small group that always tunes out when I use any concrete representations because they think that's too middle school). My hunch is that there's some relation between this confusion and the difficulties Mr. K's students had with the meaning of "more."* Once students did pick up the idea it seemed to stick, but many never really got it. Maybe it's harder to ask when you're confused about what the walking up and down the number line is supposed to mean than when the material is more evidently academic.

I had first hoped to rely on this geometric approach to help students remember the direction of the inequality sign for the two linear inequalities in terms of which they will rewrite their absolute value inequalities, but gave up on that and introduced the approach that follows naturally from the algebraic definition of absolute value. If | x - 2 | > 3 then either x - 2 is greater than 3 or else the opposite of x - 2 is greater than 3.

Now I wish I'd used the geometric approach only for predicting and interpreting answers and stuck religiously to the algebraic approach to setting up the inequalities - because now students are using strange combinations of the two, such as x + 2 > -3 or -x + 2 > -3. In other words, complete confusion, with neither a clear concept nor a clear method to rely on. That's pretty discouraging even before thinking about the many students who did not even acknowledge the fact that there are two solutions to absolute value problems, that a distance can be in either of two directions... So, math teachers, what do I do now?

*We tend to assume too much about students' immediate grasp of the very idea of comparing quantities, let alone the isomorphism from this ranking of magnitudes to a spatial ordering along a line. Bob Moses, with his interest in pre-mathematical concepts that must be in place in order to succeed at Algebra, would presumably have a lot to say about this.


Dan Wekselgreene said...

Absolute value inequalities are a killer for sure. What do you think about trying a graphical approach? The difficulty level would depend on how well your students can reason graphically. But to solve |x - 2| > 3, graph y = |x - 2| and y = 3 and ask "for what x-values is the V above the line?" This can help students see why the shaded part on the x-axis is always two pieces going out, or one piece coming together in the center - and it's based on where the V dips below some horizontal line.

To graph y = |x - 2|, you could have them graph y = x - 2 first and then take the absolute value of each y-coordinate. They can even draw the absolute value of an arbitrary function given its graph. This is great practice - just depends on how much time you have to work on it.

Good luck! (Also, remember that the STAR test has very few of these questions on it :)

Anonymous said...

Here are a couple of thoughts. take them all with a grain of salt because (1) i teach middle school, and (2) I have little experience with algebra 1:

- Number lines on the ground, rather than the wall, will allow several groups to work through the problems. You can move your furniture and use masking tape. I'd go outside with some sidewalk chalk. I'd also make up a sheet with leading questions to allow the groups to work more independently.

- That exercise works well for the "How far did you walk" and "How far apart are you"? questions - which will help aha! some of the absolute value stuff.

- I spent time with tables and checking work to demonstrate what happened when you multiplied both sides of an inequality by a negative number. That experience seemed to help a large number, but certainly not all students.

- The one math thing we can thank language arts and history teachers for teaching are Venn diagrams - draw a pair of overlapped circles on the board, and every student will know what they mean. Another math teacher recently told me that she'd had some luck with using venn diagrams to communicate the idea of ands and ors (which is really what they're for, right?). On a side note, i used them to good effect to factor numbers into LCMs & GCFs.

H. said...

Folks, thanks! I'll think and write more later - but it's just so encouraging to find constructive suggestions before the new day has even started!

Anonymous said...

I do lots of different things with inequalities.

One, really simple, is to practice

x < 3 so 3 > x , a lot

Students, even good ones, often have no automaticity with this.

Now, I don't know where which item falls in your curriculum, but with groups that have graphed circles, "the set of all points in the line a fixed distance from a given point" sounds familiar - and I've found some students who readily step down to one dimension.

I assiduously avoid memorizing "and" vs "or" as I find that which one it is depends on the math (iow, when we decide we are doing the mathematical thinking, and then reapplying through some strange rules - better to do the thinking straight away)

The most important thing I do (me), and that I try to push students to do, is to check every solution. I reverse my inequalities, too.

In fact, my "best" low-analysis technique is to break up the number line into pieces, and to check one number from each piece, just to be sure.


Anonymous said...

Many use middle school mnemonics about arrows pointing in the same direction as the inequality sign

oooh ooh - another one to add to the list of useless shortcuts.

Unknown said...

I tried an activity this year for the first time with some success. I gave each student a card with an integers on it and had them form a human number line. I then held a card with a absolute value equation or inequality (I started with equations and then moved to inequalities). I had students step forward if their number made the absolute value equation or inequality true. We then made observations about what numbers on the number line made the statement true. From this activity we were able to generalize that statements such as |x|>3 was an OR statement and that |x|<3 was an AND statement. We observed that |x-2|>3 meant that "some number is more than 3 units from 2".

I also like the idea of relating |x-2|>3 to the graph y=|x-2| and y=3.

Absolute value it is difficult concept for students to grasp. Good luck!

H. said...

First, thank you all so much for all these thoughtful and helpful suggestions! I have a lot of material for an improved unit next year now.

Dan, I had the students solve a set of absolute value equations and inequalities by graphing just before the test - but we didn't do enough of this to solidify the concept. My rationale for adding this assignment at the time was less the belief that this might be a better means toward the end of solving inequalities, and more the general priority of translating between graphs and algebraic information. Students continue to think of "graphing" as a list of things they do to an equation rather than as an alternative representation of the same information - so any reinforcement of the meaning of graphs is good. Next time I'd give this approach much more room.

Mr. K, we have rules against masking tape on floors due to the damage this does to the floor varnish... the tables of unequal values whose signs are changed and then compared is a keeper. I have done a number of examples showing them how 3 < 5 entails that -3 > -5, but it doesn't seem to stick, and having the students practice this on their own would likely be much more effective. As for being grateful to ELA teachers for teaching the kids Venn Diagrams... have you checked what happens if you ask students to diagram an ALL or NO statement? My experience has been that students not only have never seen diagrams other than partially overlapping circles - they even object to the idea.

Jonathan, having students translate between cases with the variable on the left and the right hand side really is so necessary... And checking numbers from each segment of the number line is noted as something to repeat, too.

Alexandra, I'm plagiarizing this - an enormous advantage of the human number line over my paper one is that all the students will constantly be computing whether their own number is a solution instead of just watching the show. Next time!

Meanwhile, checking the long term plan dictated moving on rather than retesting the whole class - we're almost through systems of equations and have started exponential expressions today. It seems a majority of the kids remember that exponents are added when powers of the same base are multiplied... I just hope they will all understand and remember why.