Absolute Value Inequalities

Meg is striving for conceptual understanding, but her kids just aren’t getting it. However, they can follow the steps if they memorize “greatOR means ‘OR’ and less thAN means ‘AND.” Can you help her help these students?

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 shuddering 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

Read about her successful PreCalc lesson and leave a comment on the original post: Highs and Lows: Even/Odd Functions and Absolute Value



Writing Proofs

Carl wrote a post asking how to help kids to write proofs:

Students struggled in writing proofs, although most were able to make appropriate guess as to what the proof would show.  Students would largely show an example of any of these statements with different numbers, but wouldn’t know the first steps to actually come up with a proof.

Check out the full post and leave some advice!

Adapting to Failure

Over on Infinite Sums, Jonathan is talking about a lesson that went badly and how he recovered. His recovery is already pretty much set, but his first paragraph is so perfect for this blog I couldn’t not include it.

If you read enough teacher sites, you might get the impression that anyone who has one creates a magical experience during any lesson they teach. It’s like Pinterest guilt or something. But, this is not true. I screw up, all the time. Just a couple days ago something I thought that would be great blew up in my face within 15 minutes of handing it out. How you adapt to failure of this kind is usually a quick way to know how long someone has been in the business. My first year of teaching, we’d be talking Level 5 FREAKOUT. Is this why some people use a worksheet for 10 years? Maybe.

Even though he already figure out a fix for his other classes, head over to Jonathan’s post anyway. Maybe you can help him figure out what to do with those original laminated cards!

PreCalculus Logarithms

 I didn’t bank on the internet not working one day, and I had no back-up lesson plans.  This was a bad flop of an attempted 3-act lesson, along with a few ways to improve it.

That’s how Jonathan describes his post on a logarithm lesson he struggled through.

A few creative students decided to make the “squiggles” and represent a significant change on the scale of the y-axis, but these students did not realize that (a) you really shouldn’t do that between data points and (b) you really, really shouldn’t do that multiple times on the same scale.  So they saw the need for a logarithmic scale, but even after they graphed the points on Desmos, they had no way of making the data scale that way.  Mistake #1.

Head on over to his blog, Hilbert’s Hotel, to read about the rest of the lesson and offer some advice.

Testing and Engagement

A frustrated Justin puts out a call for help:

Screen Shot 2013-10-26 at 3.53.07 PM

I took a testing idea from Frank Noschese and modified it with a suggestion from the lovely Sadie Estrella.  I gave them 8 problems (like and unlike fractions with each of the 4 operations) with the answers.  The directions were to show the work that justified the answer.  That was, I was testing on the concepts rather than the calculations.

In the last two problems, I gave them a fraction and asked them to come up with a multiplication problem and a division problem where that fraction would be the answer.

In my first class, 3 kids flat out refused to even attempt the test.  One girl put hers away and then, while making defiant eye contact with me, told me that she wasn’t done with it yet.  When I explained that she would not be leaving my room with the test in her possession and that if she needed to spend her lunch period with me, that would happen, she gave it back pretty quickly.

Several students left many of the problems blank and, even having the directions explained to them several times in different ways, claimed to have no idea what they were supposed to do.

Last year, I would have taken my frustration at this out on the students, but this year, it makes me question the efficacy of tests at all.  All of the kids who refused to work or left problems blank have been giving full effort for our class work for the past few days and have been doing an excellent job.  I reminded them of that and the general reply was “I know. I just don’t want to do this.”

Head over to his place to read about the entire day and offer some advice.


Taking a Teacher Mulligan

Jeff from Trust Me – I’m a Math Teacher writes about his kids bombing a quiz:

Sometimes, despite trying to do my best job possible as a teacher, I screw up. I’m pretty sure it’s healthy to accept that it happens from time to time.


This happened on a Friday afternoon. I thought about what to do all weekend. I came back to my students on Monday and, in each class, just laid it out for them:

“Guys, nobody did well on this quiz. I’m sorry. I blame myself for that. When nobody does well, that tells me that I probably did something wrong with my teaching. So, I’m not going to include these quizzes in your grade for now. We’ll come back to it next week, I’ll try to teach differently, and we’ll re-take this quiz. Does that sound fair?”

This scenario reminded me of one I wrote about a couple years ago. I felt like I’d screwed up, I must not have taught the lesson well, I took the blame and we set about correcting the problems they’d struggled with on the test.

All this was fine, until I started reading their test corrections.  The first question on the page asks “How did you study for the test?”  Page after page had answers such as “I didn’t” or “I read my notes” or “I flipped through notes right before the test.”

Turned out I needed to take some of the blame, but the students needed to take some responsibility too, with 2-4 days between classes they had to do more than flip through their notes to be ready for an assessment.  I wondered where Jeff’s students fell on the spectrum of blame. Three days later he added a comment to the original post:

Good news, though! The students re-took the quiz this week and did SO MUCH BETTER!

I’m thrilled for him and his students, but I am left wondering – when is it the students’ job to ask more questions, practice more on their own and take more responsibility for their learning, and when is it the teacher’s job to go back and spend more time on past lessons? Do you expect students to study on their own time or should they have mastered the material during class? What’s the difference between needing to review and having misconceptions? Can you tell whether the students would have done better with more independent practice or if they’re totally lost just by looking at an assessment?

An Equity Puzzle

Should we be teaching kids to “do school” or changing our definition of “doing school” to include the boisterous students or something else altogether? I always have a few students who struggle to be in the classroom but this year I have one in particularly who failed everything last year and I have no idea what to do with this year. Advice here or over on Dan’s post appreciated!

Work in Pencil

Twenty-six of our strongest sophomores from last year have been given to me as juniors, to take algebra 2 and pre-calculus in one year. About twenty are serious and ready to work. About six are boisterous and unable to hold their attention on listening to one person for more than about 90 seconds—but they do quality work when they work. All twenty-six would clearly (to me) be bored in our regular algebra 2 class. The knee-jerk reaction: the ones who know how to stay focused can stay, the other six have to go. But that’s the reaction that has put thousands of students, primarily low-income students of color, in courses below their ability level for decades. So the right thing to do is keep them all, and work explicitly on classroom behavior skills. And the only time that doesn’t feel good is when the first twenty students keep telling me…

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Still there?

Are people interested in continuing to interact with this blog? Did it make your transition out of google reader?

Found a worthy post this week, My Struggles by pispeak

1. Building Numeracy.  I know we all think about it/talk about/play with the idea.  We have bootcamps and special days in our classes to “review” or “refresh” these old topics.  But how can I build strong basic numeracy (I’m talking arithmetic, decimals, fractions, estimation) in every lesson?

2. The Checked-Out Student.  No matter how interesting or how important the lesson is, I often have a student or two who is just not invested.  He or she is usually not even going to try. […] How do I positively and effectively get that student to buy in?  I don’t have any great answers to this question.

Head over to the original post to leave some feedback.

Leave a comment here if I missed a post that needs a signal boost. Definitely ignored this project for several months…

Iterating – A 3D Graph Failure

Jonathan of Infinite Sums shares his vision for a 3D graphing project, how it didn’t pan out and why it was worth it.

A lot of teachers are scared of unknown, expensive (timewise) endeavors because potential failure means the concept flops and you now have several days less than you did before. But if you want to improve, failure is something you should embrace. There are a ton of misconceptions and issues hidden in this project that I would’ve known nothing about if I didn’t have 60 kids pilot the idea for me. I could have ironed out the construction issues if I spent the time to build an example, but I lack the visual miconceptions and organizational problems that slammed into them.

Check out the full post to find out his recovery plan and/or offer some advice and/or congratulate his risk taking.

Complex Instruction and Group Worthy Tasks

Complex Instruction and Group Worthy Tasks on Curiouser and Curiouser

I’ve been struggling with reaching diverse learners in my very heterogeneous math classes.

Simply put, students “don’t fail to participate because they are too shy or don’t want to participate. They don’t participate because other children in the group see them as having nothing to offer…In short, they have low academic status within the group.”

But it’s not just about the behavior. It’s about best practice. I have students with a WIDE range of abilities in the same classroom. Everyone deserves access.

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