So I’ll start off with the low so I can end on a high note. What are the three most dreaded words for a math teacher?
Absolute. Value. Inequalities.
I’ve tried 147 ways to teach these. This year I went with let’s shade our number line first, then write the inequalities to solve. Talked about kids on leashes at the mall can be five feet in front of mom or five feet behind. Talked about restraining orders mean you need to stay five feet away from me in both directions. Talked about how when we shade in between two numbers, we’re going to write our expression in between the two numbers. Talked about how I’ve shaded all the numbers below or to the left or, you know, less than -5, so I better write it as < -5 even though the original says >. Sent them home with homework which maybe 10% did. Next day: practice problems on individual whiteboards. (powerpoint file here)
Chaos ensued. Like I never even mentioned number lines the day before. Or that we’ve talked about writing inequalities when we’ve shaded between two numbers for two days before that. Maybe I really did forget to mention the need for two equations yesterday? That’s probably it. So let’s see why what we did wrong is wrong. Let’s see the correct way. Let’s go around to every group and have them say it needs to be < -7 instead of >. Have every group tell me to put the expression between the two numbers for < since I’ve shaded between the two numbers. Try another round.
Maybe 15% less chaos. Finally towards the end of eighth period (third section of Alg II w/ Trig. Which, yes, did I mention that? Second time around to see these absolute value guys.), I gave up. “Some teachers say a good way to remember is that greatOR means ‘OR’ and less thAN means ‘AND.'” I could feel Tina quietly weeping from a distance as I said it. I even made sure to say “some teachers” so that they wouldn’t think that I would ever say something like that. What happened next?
“Oh, that makes sense now!!” Maybe 80% got the next two examples correct, which was a vast improvement over the previous classes. “But they won’t remember it six months from now!” True. But obviously they didn’t remember my method 2 minutes later, so….
Scorecard: Conceptual learning: 0 Tricks: 1
High Point: Intro to Even and Odd Functions
We’re describing graphs this week in PreAP Precal. It’s the first time they’ve seen even/odd graphs, so I decided to let them try to figure it out by showing this totally awesome powerpoint.
You have to download it to appreciate the awesomeness of it. I used some emphasis tools to make the reflect across the y-axis and rotate around the origin. Then the next day we brought out our individual whiteboards and drew graphs based on descriptions. (File here) We were rockin’ and rollin’ and then I hit them with this one:
I know, it totally hurts my brain that f(6) is greater than k(-2) when it’s never increasing. I even told the kids I didn’t get it the first time I saw it and said it was a “did you eat your Wheaties?” question. (Of course this is when someone from central office popped in to observe and half the kids only have 2 points plotted.) I even had one or two groups in each class come up with the correct graph! One period was begging for more to try, but sadly I had to tell them no because we had to take a quiz so we could have enough grades in that category. Ugh. And next week we start on our shortened schedule so we have 5 less minutes in each class MWF. Double ugh.
Final scorecard: Conceptual learning: 1 Tricks:1
Delta
September 1, 2014 at 12:56 pm
I sympathize; in my stats classes students routinely get tangled up the same way with equivalent “within” statements (e.g., “almost all data points are within 3 standard deviations from the mean”, “there is a 95% chance that the sample mean is within the margin of error from the population mean”). I’d definitely caution against using the “lesser than means and” trick, because the full term for greater is likewise “greater than”, and it’s setting up students for failure whenever they run into that proper phrasing.
One of our foundational problem is that students never get taught or drilled on the basic logic of IF/THEN, AND or OR statements in the first place, so much of what we do in math is built on quicksand without that preparation. Because of that and time constraints, we’re usually cheating them out of proper definitions and logical proofs, and trying to find a shortcut to some intuition without proper justification. (Link)
My first stab at a solution (within these constraints) would be to check some numerical inequalities to build intuition: Which of the following are solutions to |x| <= 3? (a) -2, (b) 2, (c) 4. And another for greater-than; justify in writing if needed. Then document the pattern and run an introductory "warmup exercise", e.g.: Rewrite the absolute value inequality as a compound inequality with "OR" or "AND", and sketch the solutions. (a) |x| 7, etc. Get them to focus on that as an important skill before they get distracted by other “algabra-y” manipulations. That might take some extra time but unfortunately we’re doing triage over not really knowing the logic here.
Delta
September 1, 2014 at 12:59 pm
Hmmm, HTML ate some of my suggestion there at the end. I’ll try again:
Rewrite the absolute value inequality as a compound inequality with “OR” or “AND”, and sketch the solutions. (a) |x| < 5, (b) |x| > 7, etc.
Meg Craig
September 1, 2014 at 2:13 pm
Totally agree with you on the need to build in logic earlier. Something I did not do well when I taught geometry and wish I could go back and do better!
I like the idea of concrete examples first. Maybe have each person in the class try a different number, then plot the ones that work on a number line to see the graph take shape?
Delta
September 1, 2014 at 6:39 pm
^ That seems like it could be a good idea. I do find that in lieu of proper proofs, falling back on raw numerical examples does seem to help.
Brian
December 29, 2014 at 8:48 am
Love your approach with the even/odd functions, it seems so intuitive. Odd (no pun intended) that I never took the approach that you took on your Powerpoint. Thanks Meg.
HERSHEY’S TORTE! (and Alg II Unit 1) | Insert Clever Math Pun Here
May 28, 2015 at 11:20 am
[…] Next up, absolute value. I almost feel bad sharing this because I cannot teach this well. […]
Merry
November 30, 2018 at 7:39 pm
Thank you for sharing the good way to introduct even and odd functions. But I can’t download the resources you shared. When I clicked, it said the link was not valid. Could you share the resources with me? Thank you so much.
Meg Craig
January 1, 2019 at 9:19 am
All the links seem to be working on my end. What specific file did you need?