Solving Inequalities with Learning Disabilites

Are you tired of the reminder that I teach Algebra for kids who arrive in high school not quite ready for Algebra 1? Or that one class is mostly students with learning disabilities and the other class is a contained class where all students have a math disability? I'm reminding you again because context is so, so important. While grading this morning I got frustrated that my students are still struggling with reading inequality symbols and organizing their work. I got a lot of great responses from people about all sorts of issues students face with inequalities, but it wasn't until an elementary teacher chimed in that I got a useful reply to help my students read the symbols. And the only person who even broached my issue with kids who struggle with organizing was an intervention teacher. I love this community because I can ask for help and get it from so many places! Today I really needed to hear from some people who have students like mine. Next time I'll type the hashtag correctly (#swdmathchat, not #swdchat) and get even more help!

Now on to an overview of the unit, followed by an overview of the test and where I plan to go from here.

We started by matching number lines to basic inequalities (one variable, one number, one symbol). We did a card sort and they gallery walked to see the categories other pairs used. They shared the categories they saw and I recorded them. I had them hold up an example of a card that might fit the category (to check if everyone knew the vocab and if the category is well defined). We discussed what open and closed circled might mean and also reviewed what the symbols meant. Everyone recorded the information in the box. Then they set off to match inequality to graph.

For one of my students who especially struggles with symbolic notation I color coded: I wrote the greater than (or equal) symbols in red and the less than (or equal) symbols in blue. I had him color code the symbol first, then check which number line would match. Interestingly, when I wrote the definitions next to the symbol I asked him what word to use - greater, bigger, larger...? He understood greater better than any of the other words. I'm not sure if it's because it's more similar to Spanish or more familiar because it's an academic word but I was glad I asked rather than assuming he'd prefer to use bigger!

In the past I've had plenty of students with misconceptions about inequality symbols, but this year I had students who had trouble identifying the symbols, even with the notes in front of them. Some of them will even turn their paper to copy the symbol, apparently ^ is easier for them to see and write than <. I haven't had any solutions for this until I asked today and Jen said:

@crstn85 from elem perspective: give the bigger number "2 points" (dots), smaller num "1 point" (dot) connect 2 dots to 1 dot to make symbol

— Jen Hudak (@jenhudak4) January 1, 2016

I'm interested to see if drawing the dots on the endpoints and vertex can help my kids see the difference between <, = and > better.

After the card sort we did more practice with just matching. Then given basic inequalities (one variable, one number, one symbol) I had them graph on a number line. This was a nice change of pace since we'd just concluded our study of absolute value equations. They also needed it. This is really a many step process:

1. Identify end point
2. Determine if end point is shaded
3. Determine which direction to shade from the end point

None of these steps are obvious for most of my students. We continued to practiced reading the inequality aloud, picking values to check and then shading as we moved into more complex inequalities.

    

While students noticed that dividing by a negative caused the direction of the inequality to change (and why! Those are student sentences! After some prompting and discussion of course, but students said them!) I really wanted them to be checking their work so I continued to emphasize how inequalities are exactly the same as equations until the final steps.

I have no idea what the IL was, though it appears to be my handwriting?

We practice, practice, practiced. We did a packet based off a CME problem set on the difference between an equation and an inequality. We did compound inequalities (basically double the practice) (check out my colleague's awesome intro via movie ticket prices). We did absolute value inequalities (a little extra work followed by double the practice). Several students continued to ask me what the symbols meant. :( A student had an aha moment about open vs. closed circles on a review day! :) The fifteenth time is the charm? Then we took the test a month after starting this unit and it was not awesome.

Common issues:

1. Equation solving mistakes (integer operation errors, combining unlike terms, not using opposite operations).
2. Terms jumping the inequality symbol (3=x and x=3 may be equivalent but 3<x and x<3 are not).
3. Forgetting to change the direction of the inequality symbol after dividing by a negative.
4. Shading in the wrong direction on the number line.
5. Surprisingly, they made very few new mistakes on compound inequalities and absolute value inequalities. Some kids forgot to change the direction of the symbol for the negative inequality when splitting the absolute value.

Next steps:

1. We need to address this. Mostly it has to do with sloppy work. Some kids will need individual intervention.
2. Easy individual intervention when we address 1.
3. I'm not at all worried about this. It's annoying and we'll talk about it but this is where the errors should be during this unit!
4. I'll show all these kids Jen's dots on the symbol idea and hope that helps. I do think the issue is entirely in translating < and > into words, once they have words they're generally okay.
5. Again, not at all worried. Normal mistakes that kids will understand when they get their test back.

So I made a sheet of equations, inequalities and equations with both x and y (leading into our linear unit coming up next). Each section has four problems using the exact same numbers. I gave them the answers and the goal is to show all work exceedingly neatly. Then to compare and contrast equations with inequalities (it's the exact same process! Except that pesky dividing by a negative thing and the extra work of graphing).