Paying Teachers To Do Nothing?

I argued constantly during the pandemic that remote elementary education was a waste of time, and all parents wanting to reopen schools should have just pulled their kids and threatened their district’s funding, but that doesn’t mean an elementary school teacher’s job was easy.  But I don’t know enough about the day to day work to comment on it.

I taught mostly high school math during the pandemic, for eighteen months: March to June 2020, summer school in 2020 and 2021, and the entire 20-21 school year.  I did a good job at it, too.  I can attest that for middle and high school, teaching is much harder in remote and required far more time.

Check for Understanding: In person, math teachers give kids a practice problem and walk around the room to determine who gets it, who needs help, what common misconceptions exist. We make sure the kids are all working, check for common misunderstandings to address, give the kids who finish in 30 seconds an additional challenge, and do this all in five-ten minutes.

To say that this task can’t be done in remote understates the difficulty added by remote. In-person, student response and teacher checking  are done simultaneously. In remote, these tasks are sequential. To determine kids understanding on Zoom, teachers had a few options, from unstructured to highly structured, all of which took far more time in an online class of anywhere from 25 to 36 kids.

Easiest method for ad-hoc questions: ask students to put the answer in private chat. For example, put a liner equation graph up and ask for the equation in slope intercept. The question had to be something that doesn’t require math notation, which Zoom chat can’t handle.  In general, getting 90% of students to answer takes on average 10 minutes AFTER the time needed to work the problem. Teachers need a list of names handy to check off each answer. Some kids won’t answer until nagged by name–which assumes, of course, the kids are actually online as opposed to logging in from work or bed or Disneyworld, which you can’t figure out until you’ve called them out by name several times. (I would always mark those kids absent then, despite their Zoom login. Revenge, and it did improve actual attendance.) Despite the problems, I used this method often. My student participation rate was generally over 80%, focus and obsess as I might over the remaining fifth.  So I could ask a question and get close to half of the kids answering quicky, chat them back a followup question while I harassed the rest into responding. It wasn’t perfect but it worked well enough and besides (as I reminded myself frequently) in the in-person version, there were always kids who didn’t work until I nagged them. And inevitably, there were kids who forgot to put it in private, meaning everyone could see their answer, meaning those who just wanted to avoid work could copy the response just to get me off their backs.

Next up for adhoc questions: create a poll, Classroom question (in Google) or a Google quiz. These made it far easier to track who had answered and who hadn’t and teachers didn’t have to go into Zoom logs to figure out who said what. This method also allowed for more than one question, so teachers could get more granularity on misunderstandings.  Still the same math notation limitation and the same nagging issues, delay in response. Moreover, it was really hard to make these genuinely ad hoc. Zoom poll takes a minute, but they’re hard to track outside of Zoom. Google Classroom questions take maybe two minutes, Google forms longer than that. From a practical standpoint, they can’t be really adhoc. So you have to plan ahead, which some teachers do automatically and others (raises hand) find a difficult task. Google forms were great for actual quizzes (see below) but they’re a bit too much work for a simple check for understanding. I never used Zoom polls, used Google forms for quizzes. I used Classroom questions occasionally.

Creating a poll, classroom question or google form quiz can’t be done easily on an ad hoc basis, especially if the question involved formulas that need special font, which most polls don’t allow. So teachers had to either plan and create their questions ahead of time (more hours of work) or create something simple in the moment–again, with response time for each taking ten minutes or so, for the same reason.

Classwork: Both of these methods give no clue as to what errors are being made and in fact, there’s no way online to check for understanding and get a real insight into student thinking. Checking for understanding by its nature has to be quick. Classwork, the bread and butter of math teachers is the other key way to see student thinking, what happens after the “release to work“, whether it be a book assignment, a worksheet, or an activity. For the first year of Zoom–from March to December 2020–I created Google Classroom assignments and students took pictures of their work to turn in.  Teachers using this method have to flip through multiple pages of student work online. This is brutal. I still have nightmares from the time spent reviewing classwork online until, thank the great math gods, a fellow teacher told me about Desmos activity builder and its integration with Google Classroom. As a former programmer, I was able to build my own custom lessons quickly, but for teachers without that skill, Desmos offers a lot of blessed options and googling finds a bunch of others. Desmos and Google Classroom combined were wonderful.  I could build my own activities, assign them to a class, and then see students work as they completed it, catching mistakes in action. If a student never logged in, I could see it. If a student logged in but did nothing, I could see it. I marked a lot of students absent on that basis,  which got them back into paying attention. Huge win.

But there’s that time factor again: either teachers could use their existing curriculum (worksheets or books) and spend hours reviewing work online (in my case, I don’t do homework normally, so this was a big chunk of added time) or they could rework all of their existing curriculum into Desmos assignments, which also took endless hours but at least had something of a payoff.

Assessment: Monitoring test integrity is relatively easy to do in person. (Relatively. And methods got much more sophisticated post-pandemic).  Rampant cheating was a huge issue during remote.  How to reduce cheating? Rewrite tests entirely.

For example, in a paper-based test you could ask a student to graph “y=2x+7” or “y=(x-3)(x+5)”. But Photomath–or, for that matter, Desmos–provides that answer in a heartbeat. Instead, I’d use a Desmos activity and ask students to graph a line with a slope of 2 and a y-intercept of 7, or a quadratic with zeros at 3 and -5. This wasn’t in any way a perfect substitute. Students wouldn’t have to know how to graph a slope of 2 or find a vertex. But at least I could ascertain a level of understanding. There were entire topics that were pointless to teach during the pandemic (exponents, factoring) because there was no way to see if the kids were doing it themselves or photomathing the work. Rewriting the tests still took hours.

Grading: Most non-teachers–hell, even teachers themselves–can’t really conceive of how hard it is to grade online. Automation takes care of the multiple choice scenarios, but Google forms allow short answers, and they don’t always exactly match. And never mind the exact matches, how about partial credit? Math teachers routinely give credit for setting up the problem correctly, deducting fewer points for minor math errors, and so on. I bought the least expensive Veikk tablet (love it, and still use it) but I could never find an easy way to mark up student work and save it for return without a lot of extra work. Leave aside that, it is still difficult to keep track of what you’re adding up. You can’t write directly on a google form or desmos, so you have to snip it and make your notes, which you then have to tally up and keep on a separate sheet….and so on. it’s a bitch.

These are essential tasks that went from 2 minutes per instance to an hour or more–each instance, with dozens of instances a week. For teachers (or me, at least), life outside of work was great. But work itself?  This article focuses on life during the early months of the shutdown, but I was able to institute more structure during summer school and by  fall 2021 my school had instituted a formal “bell schedule” with something approximating a normal school day online. It was a lot of work. Teachers coped with this in different ways. The more organized teachers who believe that coverage is the most important thing taught less time online and added far more to the students’ “asynch” hours, believing this would allow the motivated students  to learn more effectively. I did the opposite. But regardless of method, work was much longer and harder.

The only good thing about teaching during the pandemic is that I could do my bit to make life better for students after a government action I vehemently opposed from day one. Meanwhile, moving so much of school online added permanently not only to my pandemic school day, but to my day post-pandemic.

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I remember the day our school closed, asking the head custodian what he’d be doing during the shutdown. “Taking care of mom,” he said. She had cancer. Oh,  so he wouldn’t have to be on campus. He laughed. “Maybe a bit. Not much.”

Bus drivers were definitely furloughed. But in the main,  public education layoffs actually decreased during the pandemic.

Shutting down the schools in March 2020 left hundreds of thousands of people paid a full time salary to do almost nothing. Most non-teaching school lower level staff (attendance, custodial, teacher’s aides) had very little to do. We didn’t even take attendance in most schools from March to June 2020, so those clerks had nothing to do. Custodial staff had to clean if anyone came on campus, but otherwise were onsite doing nothing. Secretaries and clerks had half or less of their usual job. The more highly educated district staff, who are nice, supportive, but ultimately unnecessary staff anyway but ignore that for another time, had pandemic-related assignments, like finding online curriculum to purchase at great expense that we teachers generally ignored. I don’t blame any district or school staff for their long vacation. But they were on light duty at best.

Half of all school employees don’t teach. With the exception of school administrators, most of them had next to nothing to do during the school closures. So closing the schools meant that just under half of all public school employees had their jobs cut in half at least. Schools could have laid off millions of personnel to combine jobs. 

Just one of the many misconceptions deluding all those complaining about paying teachers to “do nothing” during remote education is the fact that teachers were one of only two employment categories whose jobs got much harder and longer during the pandemic. I’ve pointed out endlessly that school closures were primarily a function of parental preference, that teachers’ unions, no matter their pro-closure rhetoric, couldn’t do anything to affect those decisions. There’s mountains of evidence establishing this pattern. But even those who foolishly believe in the evil teachers closed the schools story should remember that if teachers closed the schools, they created more work for themselves, not less.

Meanwhile, does anyone remember the various folks howling about closed schools and lazy teachers demanding that district and support staff personnel get furloughed? Any complaints about the thousands of state government employees getting a long-term vacation at taxpayer expense? Demands that schools collapse jobs to eliminate expensive, unnecessary personnel?

Me neither.

Same Thing All Over Again–But Events Happen

Many, many irritating things happened during the omicron phase, things that sent me into a mild depressive episode. One happy note, however, was that the union obsessive pretense that covid19 is dangerous meant we could have staff and department meetings on Zoom.

Our staff meetings occur before school, so we start the actual school day late. But the meeting start time is half an hour earlier than the normal school day beginning. So I have to get to school half an hour earlier on a day when school starts half an hour late. This induces a cognitive dissonance that eleven years at the same school has never entirely resolved, and every week, I’m at best five minutes late. Zoom meetings allows me to actually leave later in the day and listen to the meeting in my car. On time. Given that two years of school insanity has never once played in my favor, this feels like win.

Anyway.

Our department chair, Benny, was explaining….wait. Before I begin this story, I want to be clear that I’m not really criticizing anyone involved, including Benny. I should also mention, as I have before, that our school is blissfully indifferent to test scores. Admins really don’t care. This exercise I’m about to describe is about as far as we get to caring. Also relevant: since Common Core, juniors take a test that has multiple levels but from a practical standpoint is binary. Students are either “college ready” or they aren’t.

So Benny was asking for volunteers to run brief 30-minute math tutorials designed to help students review topics. We have an intervention time after lunch that can be used for this purpose. Nothing new; we’d done this for the two years pre-pandemic. Except.

“So we’ve identified the kids who failed algebra 1, geometry, and algebra 2 and give them the opportunity to come to tutoring.”

Wait, what? Kids who failed what?

In years past, we had all agreed that kids who failed algebra I and geometry had not a single chance in hell of testing as college ready.  I had argued, unsuccessfully, that we should still tutor those kids and bump their failing grade if they got….better. To use SAT terms: “college ready” is around 600 Math (top 30%). Any junior who was still working on algebra 1 or geometry would be rocking that test at 450.

I know kids in our school who made it to precalc and got a 500 on the SAT math section and did not pass the college readiness standard.

So Benny was suggesting that kids who’d had multiple shots at algebra 1 and geometry would somehow be able to pick up all they’d failed to understand the first time as well as all the topics they needed in algebra 2 in ten 30 minute sessions.

He’s also suggesting we give this tutoring to the kids who failed algebra 2. But in a good year, pre-pandemic, 60-70% of the kids taking and succeeding at algebra 2 don’t test as college-ready. Kids who failed algebra 2 were not good candidates for passing the college readiness marker. And tell them that if they succeed at an impossible task, we’ll change their grade but only if. Not just for trying.

I said nothing. All hail Zoom.

“If they go to all the tutoring sessions and make college-ready on the test, we’ll change their F to Pass.”

“What about the students that are marginal but passed algebra 2, trig, or pre-calc? We should give them the same tutoring. And kids who flunked pre-calc or calculus, they’d be eager for that deal.” suggested Pete.

In a good year, pre-pandemic, these were the kids we tutored. We spent time identifying the students who had it together enough to pass three or four years of math with a C+, a B, or even a shaky A, and gave them support. That’s what you always do, if you’re looking to get maximize kids across a finish line. These were the kids who had a shot at passing the test.

“No,” responded Benny. “The test only goes through algebra 2 material. Kids who’ve passed algebra 2 should be able to pass this test without tutoring or an incentive.” 

I said nothing. I didn’t volunteer, either. All hail Zoom.

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In the early days of my blog I would have immediately documented this craziness to provide some insight into how things work. Benny is, like all my colleagues, a progressive Democrat. But in math teacher typology, I’m the woke-conversant social justice warrior. Benny’s the traditionalist (check out our pass rates in this algebra 2 article). I know for a fact Benny doesn’t think his target group is up to the task he’s set. I know this because four years ago, we all agreed that the tutoring pool should be comprised of strong, motivated juniors taking algebra 2 or trig, along with any pre-calc students with Cs. This was the group whose pass rate we might be able to move from 0-10% to 30-50%.

Even more notably, in years past, I would have spoken up to make this very observation. I wouldn’t have been alone, either. Yet no one spoke up.

I’m not sure which is more worthy of comment: I’ve stopped bothering to write about the crazy unreasonable plans that show up in my teacher life, or I’ve stopped pushing back on crazy unreasonable plans that show up in my teacher life.

The first is easier to explain: events, dear boy, events. Both progressives and education reformers upped the nuttiness. The SAT changed to be a much harder test, then became irrelevant. Common Core spent billions on nothing. Two of education reform’s three legs got chopped off.  There was a pandemic and most of the craziness in that era I couldn’t bitch about because it was more central to my location than I usually allow online. Moreover, I try to say it once or maybe twice and then move on, linking to the original article to say “still this.”

The second question is more interesting because it’s not the usual answer. I’m not burnt out. I’ve not given up. I still care. I gritted my teeth and actually picked up my phone to put an observation in the chat yes, while driving, but Benny had set “Chat to Host Only” and so I didn’t end up driving into a ditch while typing a carefully worded but cynical comment.

If I have one Big Idea on high school math instruction*, it’s this: teach less and learn more. Find your comfort limit and develop your skills. Move on if you’re interested.

Kids who struggle with math could productively learn to apply arithmetic, geometry, and a little bit of algebra. The next group, the bulk of high school students, could do a huge amount of math with all that plus second year algebra, basic trig, and some stats. Top tier really should stop at analytic geometry, functions, and more trig. We could teach so much math that calculus could wait until college.

We will never be able to do this, because everything in education is about race. Astonishing, really, why more people can’t grasp that basic reality. Pick any education proposal you like, apply race, and you’ll realize it’s a lawsuit waiting to happen.

But individual schools like mine can still focus on helping kids at every point of the spectrum, even if we have to work through the state mandated structures for class sequencing. In terms of test prep, state tests might only focus on how many students were “college ready”, but we could focus on our average score. Maximize everyone’s score and celebrate that our lowest achievers worked hard to get every question right. This is, in fact, what we did the year before the pandemic. Absent that, we could focus solely on the students who are near the line and increase their pass rates, as I mentioned above.

My first five years at this school, I spent hours advocating for my vision. Chuck, the math coach, and I did our best to find a math path in our existing curriculum to route weaker students through. We differed on end goals–Chuck felt students should fall out by failing, I thought any plan that had students getting Fs by design was absurd–but we both shared the goal of homogenizing our classes so we could teach more content to those who were able and willing. In 2019, we took the top half of algebra 2 juniors, along with the non-honors precalc students and gave them prep sessions. This *dramatically* increased our college ready numbers to nearly half, when in prior years, before and after Common Core, we were happy with low 30%.

In early 2020 we had just started our tutoring groups for the state tests. Juniors still in A1, geo, and bottom half of A2 got skill review on the basics, to raise their confidence and willingness to try on the test. We also prepped the A2/Precalc students as described above.

Then the pandemic shut everything down. No tests in 2020, no prep in 2021 because of a different schedule. We get back in 2022, and the policy is now back to “let’s only give tutoring to kids who have no prayer of passing the test and promise them something they really need but only if they pass the test.”

And only as I wrote this did I realize that Chuck retired at the end of 2019.

Chuck, who endlessly advocated and presented at the district and administrative level. Chuck, whose emails reminding me to be sure to fail more kids annoyed me, but who was at least on the same page. Chuck, who was far more successful than I understood until he left.

Personnel matters.

That’s not the reason I originally began writing this, but once again, writing things down helps me find insight. Need to keep that in mind.

A couple other points.

When I mention events in the title, I wasn’t thinking of Chuck’s retirement but two other major ground shifts. For four years or more, major state university systems and their community colleges have completely abandoned remediation (See “Corrupted College” for details). In all the coverage, left and right, approving and disapproving, of the wholesale abandonment of SAT/ACT requirements in college admissions, no one mentions an obvious fact: grades are worthless. If grades are worthless, then schools up and down the selectivity food chain are going to acquire thousands of students whose transcripts say 4.0 but whose abilities are at the ninth grade or lower level. It seems to me that this will necessarily lower standards dramatically for college diplomas.

While improving math achievement for all students is a worthwhile goal and one I sign on to, my original advocacy for tutoring on state standardized tests was sourced in my desire to help students avoid expensive college remediation and for those of those who needed it, give them a better leg up to pass those remediation classes.

That’s pretty clearly  no longer going to be an issue. Thus, while I still think this approach is crazy and cruel, my concerns about their future in a degraded college system is less acute.

The second shift comes in answer to a question some might be wondering about: Why not Ed for Chuck?  Why don’t I volunteer for Chuck’s work?

My first response is hahahahahaha. Chuck’s job not only needed diplomacy and adminspeak, but also organization and focus. I’m 0 for 4. Moreover, Chuck had spent years teaching at an elite local (public) high school and had transferred here specifically to take on this task. He cared about teaching, but wanted to run a program, and taught less as a result. I care about teaching, and don’t like teaching less.

But the second response came a bit more slowly. Over the past five years, I’ve also undergone a shift in teaching topics, one that utterly gobsmacked me. If you’d asked me a decade ago how I would branch out, I’d have said my druthers were to still teach math, but up my quota of history and English. Ideally, say, in an AP Lang/Lit/US History class designed for bright kids who can read well but hate homework. I also predicted I’d be doing more mentoring.

None of that came true. Instead, I’m teaching with no prep (yay! more money, more variety, less boredom!) and running a program that I don’t talk about because it’s too specific**. But I am making a difference at the individual student level (from remedial to excellent) and the school-wide level with tons of money (which I need***) and  visibility (which I don’t).

Now, we can all agree that I’m an ornery cuss who seeks out the hard way every time. But surely, if I’m not looking for something that falls into my lap, I should take it and run with it rather than beat my head against a wall on an issue that is contrary to stated policy, requires endless handling and maintenance, and gives me no visibility except as a troublemaker?

Don’t worry, though. After I began this piece our district ended staff and department zoom meetings. Back to in-person, where I will inevitably mouth off.

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*Actually, all subject instruction.
**Please don’t speculate, particularly to others. Remember how I wear anonymity.
***Not me personally. For the program.

Five Things I Learned Remote Teaching Summer School

Attendance is a lot better when grades are involved.

Back in March, a younger, more innocent me argued in favor of excusing students who didn’t show up for high school classes after the shutdown. They didn’t sign up for online school. School is an obligation that society shunts on kids, and it has certain boundaries and (yes, really) choices. So if some students didn’t wake up on time and gave up, or decided that a grocery store hourly wage was a better use of their day, I was all in favor of holding them harmless from that decision. Their choice. In my view, they should not be penalized for that choice. No “F”. No delayed graduation. Take the class off their transcript, reduce the credits for graduation.

Silly me, not to anticipate that some districts and unions would band together and decide that if some kids couldn’t perform, then everyone must pass. Some schools froze grades at point of shutdown, others mandated the even more disastrous credit/ no credit for everyone. In both cases, students could skip school entirely and move on to the next class in the sequence so long as they were passing in March. In the second case, students could work hard and learn or never show up and get the same grade. In my particular school’s case, kids who didn’t show were literally missing an entire semester. Didn’t matter. Even in math. They passed with the same grade as the students who attended zoom sessions, asked questions, learned. And don’t try to feed me any shit about how the hardworking kids earned a moral victory, because without college admissions tests there’s no difference between a kid who didn’t learn anything and a kid who did, if the teachers were forced to give them both a pass.

I wrote half an article about this before I realized I was ranting. Peace.

Anyway. In summer school and, I dearly hope, this fall, teachers can give out grades. I had good attendance all summer.

So from now on when you read those stories about absentee students, remember to email the reporter and ask if the students who didn’t show were guaranteed passing grades and knew it.

Brand new students you’ve never met before? Not a problem. 

One thing everyone seemed certain of last spring was that remote learning only worked because teachers had existing relationships with their students. I worried about that, too. Happy news: that turned out to not be a thing.

I don’t do team building exercises, don’t spend lots of time getting to know my kids. None of this made a difference. I had roughly 25 students in two classes, but all but about five of them were taking both sessions (algebra semester 1 and 2). So about 30-32 students total.  None of them were from my school. I didn’t like remote teaching, but I still routinely received the two plaudits that comprise my success metric: probably 25% of my students said “I like the way you teach so much better; I’m really getting this.” and by direct count eight parents sent me a note saying their kids had mentioned how much they like my teaching. I say this not to brag, but because, well,  I’m a pretty darn good teacher and I get a lot of compliments. And that didn’t change in the move to remote. Students who had no idea who I was still thought I was and only saw me on Zoom a couple hours a day liked me a lot better than their last math teacher.

However, explaining is my go-to skill. So if that’s you, then not knowing your students might not be something to fear. If you are a beloved mentor whose influence is based entirely on in-class conversations and bonding, or lunchtime safe space, good luck.

One challenge left to take on: I sit my students in ability-level groups and they work together productively. I didn’t try this in summer school. Again, I’m not a huge fan of getting-to-know-you activities. I just bunch the kids together and tell them to get to it. I’ll have to be more conscious about this if I try it on Zoom.

Zoom Breakout Rooms

According to David Griswold, Google Meet has some nice features, but the list of limitations he rattled off have convinced me Zoom is my bet. I learned about breakout rooms in my other summer job, teaching test prep (they begged me and hey, I could go on vacation and still teach so why not?).

Breakout rooms solved a huge problem I had during the spring, when I ran “office hours”. I had no training on Zoom, just used what I saw. I used to have different groups sign in at different times, based on what topic they needed to learn, and it was a huge hassle. Breakout rooms are fantastic. You can set them up ad hoc.

Downside #1–to the best of my knowledge, you can’t add rooms after you’ve started, so I always create a couple extra.

Downside #2–if a kid drops off the line and comes back on, you might not see it for a while. If you, the teacher, are in a breakout room, you don’t hear the sound alert for a new entry. So learn to check the icon (it will say “1 unassigned”).

Downside #3–you can’t peek in on the other rooms. Remember when you were a beginning teacher helping one student out while right behind your back mayhem was breaking out? It feels like that. Except it’s not mayhem, it’s just kids not working.

Still, these are manageable problems. Breakout rooms are your friend.

Collect work right away

I quit assigning homework nearly six years ago. With remote learning, I’m no longer wandering around the room monitoring student work, seeing their progress. Last spring, I just asked students to turn in work via Google Classroom.

I wasn’t obsessive about it; students could skip turning in some assignments. But some students never turned in anything. I’d bring them in for special sessions and establish their level of understanding. Which was a lot of work, but remember, the kids didn’t have to show up at all last spring so I was in “sell” mode.

I couldn’t hold those extra sessions in summer school even if I’d wanted to. Students met with me every day for at least an hour. If they had questions, I held office hours earlier, and they’d come to those. Most kids were also turning in the homework, but at least 10 of the 30-some students were turning in little or nothing, despite coming to class every day and answering questions, demonstrating understanding.

I finally realized that they weren’t turning in work for the same reason they didn’t do homework–because once school ended, they were done. They didn’t think about class until the next day. In short, the reason that I stopped assigning homework all those years ago was still a really good reason.

So what I needed to do was consider this work classwork, not homework. Once I’d explained everything, I didn’t dismiss the class. “Do assignment 2, problems 1-8. DO NOT LEAVE ZOOM WITHOUT TURNING IN YOUR WORK. I will give you a zero otherwise.”

That worked. For some reason, the same kids who were untroubled by zeros for homework would religiously turn in classwork to avoid a zero.

By the way, reviewing classwork adds hours to my week, in case you think it’s all daily walks and a few zoom calls.

Google Form Quizzes

In the spring, I used a Classkick hack as a quiz delivery system. Classkick is a great way to administer several different quizzes to students–upload the quizzes into classkick, which allows you to generate a unique code. You can then give the quiz codes to student groups. Classkick’s value-add is the ability for a teacher to share a quiz view with just one student to help them out with questions.

These were just freeform quizzes, suitable only for regurgitation of the basics. That’s all I was able to do in the spring, and I began summer school using that method as well: build my quiz, convert to PDF, upload versions to Classkick.

But Google Classroom offers a Google “Quiz” option, which I learned was just a google form. With a bit of research, I was able to create my“multiple answer” tests:

Exponents:

Algebraic System

Graphed System

I can weight questions, import images, use images in answer choices. It’s very flexible. Not as flexible as paper and pencil tests, alas. I haven’t yet figured out how to allow students to correct answers or if I want to do that. But it’s a start.

None of this is great.

Pacing is incredibly slow. I’m not optimistic about returning to even my notoriously limited curriculum. If you know a teacher who is bragging about covering everything, that teacher has highly motivated and capable students or a lot of lost kids.  I hate being reduced to one mode of instruction. I know kids are only paying partial attention. their lives have been reduced to nearly nothing. This is a horrible way to teach, a worse way to learn, and shame on the people who think covid19 is a reason to shut down schools.

It’s a terrible thing that fearful people are doing to society, to children, to education. And I’m one of the lucky ones.

 

Evaluating vs Solving

Most math teachers start their year with algebra review. I like the idea of “activating prior knowledge“, as it’s known in ed school, but I never want to revisit material as review. It’s so….boring. Similarly, others “reteach” students if they didn’t understand it the first time and again, no, I don’t do that.

The trick is to wrap the review material in something new, something small. It’s wrapping, after all. For example, suppose the kids don’t really get Power Laws 1, 2, and 3 the first time you teach them, even though you went through them in insane detail and taught them both method and meaning. But you give them a quiz, and half the class is like, what means this exponent stuff? so you grit your teeth, yell at them, flunk most of them on that quiz, and go onto another topic for a week or so. Then one morning write ¾ on the board and ask “How would I write this with exponents?” and through the explanation you take them back through all of the power laws.

But I’m not here to write about power laws, although if you want advice on the best way to teach them, even if it takes longer, there’s no better tutorial than Ben Orlin’s Exponential Bait and Switch.

I’m here to explain how I integrate what we usually call “algebra review” into my course, while additionally teaching them some conceptual stuff that, in my experience, helps them throughout the course. Namely, teach them the difference between evaluating and solving functions for specific values.

Evaluate–what is widely recognized as “plugging in”. Given an input, find the output. Evaluate is Follows P E MD AS rules–well, technically P F MD AS, but who can say that? Note–I am pretty sure that “evaluate” is a formal term, but google isn’t helpful on this point.

Solve–well, technically it’s “plugging in for y”, but no one really thinks of it that way. Given an output, find the input(s).  Follows the rules of Johnny Depp’s younger brother, SA MD E P. (I hope I retire before I have to update that cultural reference). And really, it’s SA MD F P, but again, who can say that?

Things that get covered in Evaluate/Solve:

• Remind everyone once more that addition/subtraction and multiplication/division run left to right, not one before the other. SAMDEP reinforces that, as I put the S first for the mnemonic.
• “Evaluate”–Evaluating purely arithmetic expressions is middle school math.  At this stage of the game, the task is “evaluate the equation with a given value of x”.
• “Solve” –Solving is, functionally, working backwards, to undo everything that has been done to the input. Right now, they know how to “undo” arithmetic and a few functions. They’ll be expanding that understanding as the course moves forward.x
• Hinted at but not made explicit yet: not all equations are written in function format. I believe that, given an equation like  3x + 2y = 12 or x² + y²=25, the terminology is “given x=4, solve for y” or “given y=3, solve for x”, but I’m not enough of a mathie to be sure. Feel free to clarify in the comments.
• As I move into functions, this framework is helpful for understanding that evaluating a function must have one and only one answer, whereas solving a function given an output can have more than one input. It’s also useful to start capturing the differences between absolute value and quadratics, which aren’t one to one, and lines and radicals, which are.
• The “PE” in PEMDAS and SAMDEP stands for exponent, but in fact the laws must be followed for every type of function: square root, absolute value, trigonometry, logs,  and so on. Informally, the “E” means “do the function” or “undo the function”, depending on whether evaluating or solving. So evaluating y=4|x-5| -6  with x=1 means subtract 5 from 1 (the “parenthesis), then take the absolute value (the “exponent”), then multiply by 4 and subtract 6. Solving the same equation would be adding 6, dividing by four, then undoing the absolute value to create two equations, then adding five in each one. (This is more complicated in text than explaining it with calculations on a promethean.)
• YOU CAN’T DISTRIBUTE OVER ANYTHING EXCEPT MULTIPLICATION. This one is important. Kids will change 2(x-1)²  to  (2x-2)² to 4x-4  with depressing speed and while many of them will make the last mistake in perpetuity, I’ve found that I can break them of the first, which also helps with 3|x+5| not turning into|3x+15|. For some reason, they never distribute over a square root, but plenty will try to turn 3cos(3x) into cos(9x).

Here’s a bit of the worksheet I  built.

I have found this prepares the groundwork for an indepth introduction to functions, which is my first unit. So when they’ve finished Evaluate and Solve, followed by Simplify ( more on that later), the functions unit:

• Definition of a function
• Function notation
• Function Transformation
• Parent functions–I start with quadratic, square root, absolute value, and reciprocal, and some of my approach is at the bottom of this article.

So by the end of the unit the students can graph f(x) = 2(x-1)² – 8 , as well as find f(3) and a if f(a)=10, and understand that the x and y intercepts, if they exist, are at f(0) and f(x)=0. They can also do the same for a square root or reciprocal function. Then I do a linear unit and a quadratic unit in depth.

Function notation, particularly f(a)= [value], is much easier for the students to understand once they’ve worked “evaluate” and “solve” with x and y.

This also helps the students read graphs for f(4) or f(z)=7.

Back in January, a Swedish guy living in Germany, as he describes himself, read the vast majority of my blog and then summarized his key takeaways and some critiques. His 6 takeaways are a pretty good reading of my blog, but he’s completely dismissive of my teaching and pedagogy, saying I’m mathematically naive and often, due to my ignorance, end up creating more confusion teaching needless information to my students. He explicitly refers to The Evolution of Equals and The Product of Two Lines, but I suspect he’d feel similarly about The Sum of a Parabola and a Line and Teaching With Indirection.

I’m really sure my students aren’t confused. I get pretty decent feedback from real mathematicians. There are legit differences between teachers on this point that approach religious wars, so there’s that.

Besides, these sort of lessons do two things simultaneously. They give weaker kids the opportunity to practice, and the top kids get a dose of the big picture.

Yes, it’s been a while since I’ve written. Trying to fix that.

 

Great Moments in Teaching: The Charge

Friday, two weeks from the end of school, and it’s rally schedule: chop off fifteen minutes from each block for a screaming session in the gym. It’s fourth block, my trig class, and although I try not to have favorites, this semester has been a bit low on students with energy and ability. But even the goof-offs in this class can remember the basics of trig, have put some effort into memorizing the unit circle, reciprocal values, the occasional Pythagorean identity,  know the difference between sine and cosine graphs.  And only two cheaters. The top kids are amazing, enthusiastic, and driven–and there are lots of them, many of whom I just taught Algebra 2.  So a fun class, and really the only one with a genuine personality this semester.

I had given them some extra time to finish up a test from the day before, and it’s now just 35 minutes to rally.

“OK, I want to cover a couple things to set up Monday. Let’s….”

“NOOOOOOO!!!!” the blast of complaints hit me. I turned around and glared.

“Come on! It’s Friday! You can’t make us learn something new!” Tre, who last had a math teacher that wasn’t me in freshman algebra, put on his most ingratiating grin.

“It’s so hot, and my brain hurts. Please, no more math!” Patti slumped dramatically.

“QUIET!” I turned back from drawing a cosine graph to bellow them into submission.

“I just want to introduce a couple of interesting properties and get you thinking, once again, about…oh, for christ’s sake.”

“WHAT??? What happened?” the students crane their heads forward to see the object of my irritation. I was growling at a student whiteboard sitting on a desk.

“Oh, some student used a fricking sharpie to draw a self-portrait.” and I held up the board so the class could see the penis.

“HAHAHAHAHA!” TJ was cracking up and I whirled at him furiously.

“You know, we use these white boards every day, and if I can’t get the sharpie off, it’s ruined. You think it’s FUNNY that students destroy my stuff?”

TJ was genuinely puzzled. “No. You just called him a dick. Like, without saying so. That was cool.”

“Fine. Ruin the fun of yelling at you. Take one more ounce of joy from my day.” I grinned at him and sprayed cleaner on the board.

“Ain’t no cleaner taking off sharpie,” Ahmed sympathized.

“Dude, this is Kaboom,” Tre said. “Kaboom’s the bomb.”

“Best cleaner in the known universe.” I spray the board and let it sit. All my kids know I love Kaboom. I tell new teachers about Kaboom, an essential teaching tool. When the kids write F*** in Sharpie, it’s so incredibly satisfying to wipe the obnoxiousness out of existence with one spray. Lesser challenges–gang graffiti, pencil sketches, soda spills, even small patches of gum–all disappeared.

“I hate students, dammit.” I turned back to the board. “I mean, don’t get me wrong. I love you all. But I just hate students. Ruin my stuff, treat it like crap….” I stop, because students breaking my stuff can put me in a foul mood in a hurry.

“It wasn’t us!” Matteo protested.

“Dude, it was you.”

“Screw you, Furio, how do you know?”

“Cuz you’re a dick! That’s your picture!”

I laughed, feeling much better. “Look, back to work. So you know how there’s a line, and then we can square a line, or multiply it by another line, to get a…”

“Parabola,” a reasonable amount of the class chorused, but I could hear talking.

“Shush, whoever’s talking. What happens when we square the cosine function? Take a look at the function and let’s just square what we….BE QUIET BRIAN..see. Cosine starts at…QUIET.” I turn around, wait for quiet. “Cosine starts at what, Furio?”

“1.”

“So 1 squared is..?”

“1”.

I mark (0,1) in a different color, and move to the next hashmark. “Cosine is zero at pi over 2, zero squared is…QUIET.”

Most kids were paying attention, but there was this low level nattering that rose up every time I turned to the board.  But we got through the first one quickly.

“So here’s the square of the cosine function. What do you notice?”

“It’s a cosine graph!” Vicky.

“Sure looks like it. Period? Amplitude?” and we identified all the parameters for a cosine function graph.

So the square of the cosine function can also be expressed as a regular cosine graph. Amplitude and vertical shift, one half, period one half the usual.”

Ahmed said with faux judiciousness, stroking his chin, “Ah, but how do we know this? It might just look like a cosine graph!”

“Good question. We can see the key points work, but maybe that’s just a coincidence. So pick a value and let’s plug it in. QUIET!”

“How about pi over six?”

Carla was impressed. “Wow, when you double the value, it becomes something entirely different.”

“Yes….QUIET!!! I’m always surprised at how the alignments happen. So now let’s go on to the sine function. What do you all think will happ….QUIET!”

“Jesus Christ, Eduardo and Brian, will the two of you shut.up.? NO! Stop the innocent ‘who me?’ crap. Three times in the past three minutes. I tell you to be quiet, turn to the promethean, turn around and there you are yapping again. Do I need to move you?”

Eduardo (Manuel‘s younger brother) and Benny look abashed, hearing the edge in my voice. I was mad at myself more than anything these two had done. Note to new teachers: don’t push through without attention. Constantly shushing is a sign you don’t own the room..  Don’t push through, stop when you need to. And it wasn’t an accident I’d picked two of the top kids in the class to shut down; it showed everyone else I was serious, if the unusual edge in my voice wasn’t enough.

By now I was furious with myself, and boy, do I get global in a hurry. My rotten students ruin my whiteboards and never shut up. I’m an idiot who decided to teach something complex 30 minutes before the weekend. And there are times when I’ve decided it’s not worth it and call it quits–call a pop quiz, put a problem on the board as an exit ticket, something. But deep breath, act like nothing happened, and push on, vowing to give it one more shot before I bail on an exit ticket activity.

“Wait.” Joanie, probably my top math student this year, sat up and scowled at the graph dots. “How can that be a cosine, too? That’s weird.”

“What kind of cosine function? What’s different?”

“It’s reflected. So cosine squared is cosine, and sine squared is negative cosine?”

“Looks like it.”

“But what’s the point of this?” Vicky asked. “Since squaring a sine or cosine function just takes you back to cosine, why do it?”

“Well, math applications will quite often require you to square functions, so it’s good to know how they behave. However, I really just want you to think about exploring functions. Up to now, you’ve been working primarily with transformations or known formats with parameters you can just plug in. But now we’re investigating functions that aren’t familiar with. Notice, too, that we did this all graphically with a minimum of evaluation.”

“So just for fun, what if we add the two functions we just created?”

“Here they are together. So let’s add the five primary points.”

TJ puzzled. “They’re all one? Really? That’s weird.”

“Yeah, but you can see it in the graphs,” Juan observed. “They’re equal at one-half, at opposite ends at one.”

I join all the points.

“So the graph y= cosine squared plus sine squared is always….”

“One!” the class chorused.

And then I threw out casually, oh so casually, “And cosine squared plus sine squared is…”

“One!…”

The pause was the best part. I looked down, and waited as the recognition grew, until by god, the entire room was shouting in approval, clapping and stomping.

It’s one of those things that maybe you had to be there. But in half an hour, at the end of a day, in hot weather, right before a rally and a weekend, I’d not only gotten those kids to apply their knowledge of trig graphs in a new approach, but draw a connection from graphic to algebraic. They hadn’t recognized the familiar equation because their minds were in “graph” mode, and only when I asked about a Pythagorean identity, using almost exactly the same words, did they realize that they already knew what the graph would show. But not until then.

And they thought it was really cool that I’d pulled them around to this recognition.

Literally, a minute of stomping until I waved it down. “All right! Thank you. Remember during the first week, when I told you I’m a stickler for understanding the connection between algebraic and visual representations? Here you go.”

And then, “But what about tangent? What happens when you square that?”

Ten minutes left and I’ve got them asking questions. I realized I haven’t had to shush them once.

And just as the bell rings, we established that tan²(x) + 1 = sec²(x).

The kids rushed out to the rally. Rallies are my one Bad Teacher thing: I don’t go. I checked the whiteboard, Kaboom had wiped out most of the damage. Then I walked to Starbucks just completely charged, reliving the math and the applause. All the yelling, all the grouchiness, wiped away. I’d killed.

I keep telling you: Teaching is a performance art.

 

The Evolution of Equals

High school math teachers spend a lot of time explaining to kids that all the things we told you before ain’t necessarily so. Turns out, for example, you can subtract a big number from a smaller one.  Fractions might be “improper” if the numerator is larger than the denominator, but they’re completely rational so long as both are integers. You can take a square root of a negative number.  And so on.

Other times, though, we have to deal with ambiguities that mathematicians yell at us about later. Which really isn’t fair. For example, consider the definition of variable and then tell me how to explain y=mx+b. Or function notation–if f(x) = 3x + 7,  and f(3) = 16, then what is f(a)? Answer: f(a) = 3a+7. What’s g(x)? Answer: A whole different function. So then you introduce “indeterminate”–just barely–and it takes a whole blog post to explain function notation.

Some math teachers don’t bother to explain this in class, much less in blogs. Books rarely deal with these confusing distinctions. But me, I soldier on. Solder? Which?

Did you ever think to wonder who invented the equal sign? I’m here to wonder for you:

Robert Recorde, a Welsh mathematician, created the equal sign while writing the wonderfully named Whetstone of Witte. He needed a shortcut.

“However, for easy manipulation of equations, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words “is equal to”, I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.”

First of his examples was:  or 14x+15=71.

Over time, we shortened his shortcut.

Every so often, you read of a mathematician hyperventilating that our elementary school children are being fed a false concept of “equals”. Worksheets like this one, the complaint goes, are warping the children’s minds:

I’m not terribly fussed. Yes, this worksheet from EngageNY is better. Yes, ideally, worksheets shouldn’t inadvertently give kids the idea that an equals signs means “do this operation and provide a number”. But it’s not a huge deal. Overteaching the issue in elementary school would be a bad idea.

Hung Hsi Wu, a Berkeley math professor who has spent a decade or more worrying about elementary school teachers and their math abilities, first got me thinking about the equals sign: 

I don’t think this is a fit topic for elementary school teachers, much less students. Simply advising them to use multiple formats is sufficient. But reading and thinking about the equals sign has given me a way to….evolve, if you will…my students’ conception of the equals sign.  And my own.

Reminder: I’m not a mathematician. I barely faked my way through one college math course over thirty years ago. But I’ve found that a few explanations have gone a long way to giving my students a clearer idea of the varied tasks the equals sign has. Later on, real mathematicians can polish it up.

Define Current Understanding

First, I help them mentally define the concept of “equals” as they currently understand it. At some point early on in my algebra 2 class, I ask them what “equals” means, and let them have at it for a while. I’ll get offerings like “are the same” and “have the same value”, sometimes “congruent”.

After they chew on the offerings and realize their somewhat circular nature, I write:

8=5+2+1

8=7

and ask them if these equations are both valid uses of the equal signs.

This always catches their interest. Is it against the law to write a false equation using an equals sign? Is it like dividing by 0?

Ah, I usually respond, so one of these is false? Indeed. Doesn’t that mean that equations with an equals sign aren’t always true? So what makes the second one false?

I push and prod until I get someone to say mention counting or distance or something physical.

At this point, I give them the definition that they can already mentally see:

Two values are equal if they occupy the same point on a number line.

So if I write 8=4*2, it’s a true equation if  8 and 4*2 are found at the same point on the number line. If they aren’t, then it’s a false equation, at least in the context of standard arithmetic.

So if the students think “equals” means “do something”, this helps them out of that mold.

Equals Sign in Algebraic Equations

Then I’ll write something like this:

4x-7=2(2x+5)

Then we solve it down to:

0=17

By algebra 2, most students are familiar with this format. “No solution!”

I ask how they know there’s no solution, and wait for them all to get past “because someone told me”. Eventually, someone will point out that zero doesn’t in fact, equal 17.

So, I point out, we start with an equation that looks reasonable, but after applying the properties of equality, otherwise known as “doing the same thing to both sides”, we learn that the algebra leads to a false equation. In fact, I point out, we can even see it earlier in the process when we get to this point:

4x = 4x+17

This can’t possibly be true, even if it’s  not as instantly obvious as 0=17.

So I give them the new, expanded definition. Algebraic equations aren’t statements of fact. They are questions.

4x-7=2(2x+5) is not a statement of fact, but rather a question.

What value(s) of x will make this equation true?

And the answer could be:

• x= specific value(s)
• no value of x makes this true
• all values of x makes this true.

We can also define our question in such a way that we constrain the set of numbers from which we find an answer. That’s why, I tell them, they’ll be learning to say “no real solutions” when solving parabolas, rather than “no solution”. All parabolas have solutions, but not all have real solutions.

This sets me up very nicely for a dive back into linear systems, quadratics with complex solutions, and so on. The students are now primed to understand that an equation is not a statement of fact, that solutions aren’t a given, and that they can translate different outcomes into a verbal description of the solution.

Equals Sign in Identity Proofs

An identity equation is one that is true for all values of x. In trigonometry, students are asked to prove many trigonometric identities,, and often find the constraints confusing. You can’t prove identities using the properties of equality. So in these classes,  I go through the previous material and then focus in on this next evolution.

Prove: tan²(x) + 1 = sec²(x)

(Or, if you’re not familiar with trig, an easier proof is:

Prove: (x-y)² = x²-2xy+y²

Here, again, the “equals” sign and the statement represent a question, not a statement of fact. But the question is different. In algebraic equations, we hypothesize that the expressions are equal and proceed to identify a specific value of x unless we determine there isn’t one. In that pursuit,  we can use the properties of equality–again, known as “doing the same thing to both sides”.

But  in this case, the question is: are these expressions equal for all values of x?

Different question.

We can’t assume equality when working a proof. That means we can’t “do the same thing to both sides” to establish equality. Which means they can’t add, subtract, square, or do other arithmetic operations. They can combine fractions, expand binomials, use equivalent expressions, multiply by 1 in various forms. The goal is to transform one side and prove that  both sides of the equation occupy the same point on a number line regardless of the value of x.

So students have a framework. These proofs aren’t systems. They can’t assume equality. They can only (as we say) “change one side”, not “do the same thing to both sides”.

I’ve been doing this for a couple years explicitly, and I do see it broadening my students’ conceptual understanding. First off, there’s the simple fact that I hold the room. I can tell when kids are interested. Done properly, you’re pushing at a symbol they took for granted and never bothered to think about. And they’ll be willing to think about it.

Then, I have seen some algebra 2 students say to each other, “remember, this is just a question. The answer may not be a number,” which is more than enough complexity for your average 16 year old.

Just the other day, in my trig class, a student said “oh, yeah, this is the equals sign where you just do things to one side.” I’ll grant you this isn’t necessarily academic language, but the awareness is more than enough for this teacher.

 

Teaching with Indirection

Technology is a great illustrator and indispensable for presentation. But as a student tool? Eh, not so much. Certainly not laptops.   I found laptops very useful in my history class, but primarily as a delivery and retrieval mechanism, or for their own presentations.  I haven’t found that a compelling reason to submit to the logistics of handing out and collecting laptops. But then, I’m a Luddite on this.  Recently, some colleagues were jazzed with several thousand dollars of cool science tools which I oohed and ahhed over politely. But….? Basically data collection. Fast data collection, which the students can analyze.  I guess. I don’t really do science.

A couple months ago, I used laptops and Desmos to teach transformations, and after two blocks that went….well, I suppose, I used whiteboards to do the same lesson in the last block. Far superior. I wouldn’t have even considered the hassle, but last year the school decided all algebra 2 teachers warranted a laptop cart and I want to occasionally acknowledge a gift intended to be useful. I would never–I mean no excuses never–book a laptop cart from the library to teach a lesson. But if it’s sitting around my classroom, I’m bound to try and find a way to use it. Still, even if I had a lesson that would be guaranteed superior to the same lesson on paper, I’d be tough to convince. Taking them out and putting them away takes up close to 15 minutes of classtime. Wasted. If all of my students had their laptops with them at every minute, waiting to be used….maybe. I’ve certainly found uses for phones on an occasional basis. But it’s not a huge gap I’m longing to fill.

Teaching is performance art. Sometimes the art lies in holding students’ attention directly, taking them point by point through a new topic. Other times, it lies in making them do the work. In both cases, the art lies in the method of revealing, of making them come along for the ride of understanding–even if it’s just in that moment.

It’s hard to do that if you put technology in the students’ hands. First, they’re too easily distracted. Second, it’s too easy to do without understanding.  A colleague of mine simply worships Dan Meyer, and loves all the Desmos activities.  They are neat. Without question or caveat. But I have limited time, and I’d rather have my students doing math directly, by hand even, than have them work on laptops or phones. Some Desmos activities do, absolutely, require the kids to work or show their math directly. Others are an interesting form of guess and check, designed (hopefully) to help kids understand patterns. The first, I like, but am unconvinced that the time and distraction suck are an improvement over handwritten work. The second, no. Not generally interested unless I have time for games, and I don’t.

This piece is only partially about technology, though. I wanted to talk about designing experiences, and for me, technology doesn’t give me the freedom to do that. Not with my kids, ability levels, and existing technology, anyway.

But how can I claim that technology is a distraction if I’m busy performing for the students?

Well, recall I said it was great for illustration and presentation. I love my smartboard, although I move pretty effortlessly between smartboards and whiteboard walls.

I have learned it’s very simple to screw up a lesson by speeding it up, but far more difficult to do slowing it down. I like introducing a topic, sometimes in a roundabout way, and having the students do the work alongside. Consider the example displayed here. These aren’t power points of my lecture. I start with a blank screen. I give the instructions, give the kids time to follow along, then use their input to make my own diagram. That way I can circle around, see that everyone’s on track, understanding the math, seeing connections.

I spend a great deal of time looking for ways to build instruction step by step, so that the vast majority of my students have no reason to refuse the effort. Draw a square. How hard is that? Besides, most of them enjoy drawing and sketching, and this beats posters.

Ideally, I don’t want them to see where we’re going. But then, remember I’m teaching advanced high school math. At various times, I want students to understand that math discoveries don’t always go where they were expected. The best way to do that, in my experience, is give them a situation and point out obvious things that connect in not so obvious ways.

Thus, a trigonometry class is a great place to start an activity that begins as a weird way of breaking up a square into similar triangles. The sketches in the first steps are just a way to get them started, suspend their disbelief.  The real application of knowledge begins at this step, as they identify the equivalent ratios for the different triangles. A geometry-level skill, one from two years ago, and one we try to beat into their heads. Proportionality, setting up cross products, is also something students have been taught consistently.  A trig class is going to have a pretty high percentage of functional students who remember a lot of what they’ve been taught a lot.

Which is important, because this sort of activity has to be paced properly. You have to have a number of pauses while students work independently. The pauses can’t be too short–you have to have time to wander around and explain–but not explain everything to everyone, which would take too long and kill the mood. Can’t be too fast, either, or why bother?

Ideally, students should be mildly mystified, but willing to play along. As I wrote several years ago, start slow, build student trust in your wild notions. If you keep them successful and interested, they’ll follow along working “blind”, applying their existing knowledge without complaint. Don’t deliver and they won’t follow. Which is why it’s important to start slow.

So in this particular activity, the students drew a square, some triangles, and found ratios without knowing when, or if, this was going to relate to trigonometry. Now, finally, they are using class-related knowledge, although SOHCAHTOA is technically covered in geometry and only reviewed in the early months of the year. But at least it does have something to do with Trig.

I’ve only done this once, but I was surprised and fascinated to note that some students were annoyed that I reminded them about the 1 unit substitution after they’d built the proportion statements.  I liked the structured approach of two distinct moves. They didn’t. “Why you make us do this twice?” griped Jamal, who is better at math than you might expect from his pants, defying gravity far south of his pelvis, much less his perpetually red-eyed stupor and speech patterns. (“He’s a c**n,” he informed me about a friend a month ago. I stared at him. “It’s okay. I’m half c**n, so I can say  that.  Like, my family, we all light-skinned but we c**ns.” I stared at him. “OK, I ain’t no c**n in your class.” I mentioned the discussion to an admin later, suggesting perhaps Jamal needed to be told that c**n isn’t n****r , and is an insult in any vernacular. “C**n?” she said, puzzled.  “Like….raccoon?” It took me a few minutes to realize that she was a Hispanic, so it was indeed possible she had no idea what the word meant. I should have gone to our African American admin.)

It’s not obvious to all students that the ratio labeling each triangle side is the length of that side. That is, if the base is one, then the length of the secant line will be the exact value of the secant ratio, and so on. Breaking the diagram into three distinct triangles helps, but I do recommend spending some time on this point.

So, for example, say if the angle is 30 degrees, what length would the side labeled sine be? What about cotangent? They already know about sine and cosine lengths, since I introduce this after we’ve covered the basics of the unit circle. But it helps to prod them into realizing that the cosecant length would be 2 units, and so on.

My students are familiar with my term “mother ship”. I use it in a number of contexts, but none so commonly as the Pythagorean Theorem. I ask them if they’ve seen Independence Day,  or one of the other zillions of alien invasion flicks in which the little independent saucers  all go back to the big behemoth. Because aliens will centralize, else how could humans emerge victorious? Just as all these little buzzing pods lead back to the big one, so too do so many ideas lead back to Pythagorean. Even its gaps. The Pythagorean Theorem doesn’t do angles, I point out. That’s why we started using trigonometry to solve for sides of right triangles. Originally, trigonometry was developed thousands of years ago to explain planetary motion, and was defined entirely in terms of spheres and chords. Not until Copernicus, a few hundred years ago, did we start to define trigonometry primarily in terms of right triangles.

Until this activity, I’d always taught the Pythagorean identities algebraically. I start, as many do, by reminding or introducing them to the equation for a circle, then talk about a radius of one, and so on. Then I derive the secant/tangent and cosecant/cotangent versions, which is pretty simple.

But I really like the geometric representation. The three triangles are spatial, physical artifacts of what is otherwise a very abstract concept. Ultimately, of course, these identities are used for very abstract purposes, but whenever possible, links to the concrete are welcome.

Besides, isn’t it cool that the three triangles reflect what the algebra shows? I suppose the fact that the triangles are all similar plays into it, but I’m not enough of a mathie to grasp that intuitively. The students, of course, don’t yet know the algebra. The Pythagorean identities are the one new fact set this lesson delivers.

Remember, I don’t use these images you see here in the lesson; rather, they represent a combination of what I say and draw during the lesson, pausing as the students work things out themselves.  Could I do this with technology? Sure. Could they? In my opinion, no. But it’s debatable, certainly. BUT–I also couldn’t do this with a book.
Is it just me, or do students take an absurdly long time, over many lessons and with many reminders,  to memorize the unit circle? I mean, my god, there are five values for each ratio. They go in order–big to small, small to big. How hard could it be? But after a couple years of students looking at me blankly at the end of the term when asked what the sine of pi over 6 is, I’ve learned to beat it into their heads. Some teachers never use the unit circle to teach ratios. I do not understand this. Steve teaches it all with co-functions and trig tables; I have taught any number of his students who know vaguely what it is, but have no conceptual understanding of it. They know the values, their operational ability is no different, but where’s the fun? The unit circle is an amazing entity.

I am a big fan of Desmos. At algebra 2 and higher, I ask my students to download the Desmos app. My students learn how to graph, how to create functions, how to explore functions. I want them to know Demos as a tool when it makes sense. Really.

So eventually–although I haven’t done it yet–I’m going to show my students this puny effort to automate the concepts we explored manually in this lesson.  Hey, I can use the laptops! It will be a great example of inverse calls.

But not right away. Look, my classes do a lot of repetition.  Plenty of worked problems. It’s not all discovery or exploration–in fact, relatively little time is spent on these. My students need to know how, building capacity. Why is the glue. The better a student is at the basics of math, the more important it is to smack them around with why, occasionally.

But I’m a performer.  English teachers talk about grabbing up front with the hook. But in math, ending big, revealing the path they’ve been wandering, is my goal. So when I draw in the circle, put in the coordinates, and hear “Holy sh**!” and various stunned gasps, following by a smattering of applause, I know my planning paid off.

“The f***? Damn. This been the unit circle all along. Shee-it.” That would be Jamal.

 

 

 

Modeling Rational Expressions

As part of our Teacher Federalism agreement, I now include the reciprocal function as one of my parent functions in Algebra 2. But time constraints don’t allow me to really dig into the function–plus, the kids are on overload by the end of the term, what with exponential functions, logarithms, and inverses. I don’t really have time to switch gears. Besides, they’ll be exploring rational expressions in depth during pre-calc.

But then I noticed, during Trig, that my students really weren’t completely understanding that four of the major trig functions are rational expressions and how they differed from sine and cosine.  Meanwhile, I’m always doing a bit of algebra review…and so I decided to kick off my trig class with a rational expressions unit. A brief one, that enabled a review of quadratics and rational expression operations (aka adding and subtracting fractions with variables).

Starting Activity

Task 1 is a straightforward linear function, so almost every kid who has made it to trig, no matter how weak, is able to quickly build the function.

Task 2, of course, is the introduction of division. In function terms, we’re dividing a line by a line, while I will eventually make clear. But practically, the big hop occurs when students realize that cost per hoodie is not constant. Usually students do this incorrectly, graphing either a constant line of 20 or confusedly dividing total cost by 20. So I’ll meander by and ask:

“According to your table for L(h), we spent $520 and got one hoodie. What was the cost of that hoodie.”

“Twenty dollars.”

Silence. I wait. And inevitably, a student will gasp, “No! One hoodie costs $520!!”

It usually takes about 45 minutes for the kids to work through both tasks, including graphing the unfamiliar rational expression. Then I call them back up front for explanation and notes.

After putting the two equations on the board (linear and rational), I point out that our cost per hoodie equation is basically a line divided by a line. I point out the two asymptotes , vertical and horizontal. Why do they exist? Most students, by trig, know that you can’t divide by zero, but why doesn’t the vertical asymptote intersect y=20?

This usually prompts interesting discussions. I usually have a couple students to correct when they build the graph, as they make it linear. So now I redraw it, making clear that the drop is sudden and sharp, followed by a leveling. Why is it leveling?

Usually, a student will suggest the correct answer. If not, I ask, idly, “Can anyone tell me why the cost per hoodie isn’t $20? After all that’s the price.”

“Because you have to pay the $500, too.” and this almost always leads to a big “aha” as the students realize that the $500 is “spread out”, as many students call it, among the hoodies. The more hoodies purchased, the higher the total cost–but less of the $500 carried on each one.

At some point, I observe that certain forms of equations are much easier for modeling than for graphing. For example, when modeling linear functions, we use standard form and slope intercept form all the time–many real-life (or close to real life!) applications fall naturally into these formats. John has twice as much money as Jane. Tacos are $3, burritos are $5, Sam has $45.  But you’d never deliberately model an application in point-slope form. You might use it, given two points, to find the equation. But it’d be an operation, not a model.

So take a look at TL(h) and what does it look like? Usually, there’s a pause until I remind them that we could have negative hoodies, and we graph that in. Then the kids recognize the reciprocal function.

“If  we take a look at the graph and think of it as a transformation of a parent function, what’s the vertical shift?”

Silence.

I draw the parent reciprocal function . “Remember this? Where are the original asymptotes?” and eventually the kids remember y=0 and x=0.

“Right, so the original parent function, the horizontal asymptote is y=0. Where is it in this function? y=20. So what’s the vertical shift?”

Now they get it, and I hear “20” from all corners.

“Right. Is there a horizontal shift?”

“No.”

“So we know that h=0, k=20….what’s a=? What’s the vertical stretch?”

Someone always remembers that it’s the vertical distance between (1,1) and the actual output value for x=1, which is….

“Right. a=500. So TL(h) could also be written as 20+500(1⁄_h). Notice that if we split the numerator into two terms and simplify, we get the same thing. But we’d never model it that way. Much more intuitive to create the linear equation for total cost and divide it by the line.”

In other words, I point out, the hoodie activity is actually the same function that they learned about last year, but instead of just graphing or solving transformed functions, they’re modeling with it.

The second part of this activity is about 20 minutes, and comes at the end of my 90-minute block. In between, I do a lecture on  the meaning of rational expressions, vertical and horizontal asymptotes, usually bringing up something like this:

But while it works in the context of the lesson, it just pulls the focus of this post so I’ll write about that some other time. Suffice it here to say that yes, I discuss what the defining criteria of rational expressions are, what asymptotes are, and so on. One of the main reasons I teach this now is so the kids will understand both as they bump into them.

Part  three comes quickly because the students see the pattern, and that alone is enough to please a lot of them.  Suddenly, they’ve added an entire model type to their repertoire.

Part 4 is where so much gets tied together. Most students get all the way to part 3 without anything more than nudges. At that point, I usually bring it up front.

Using Desmos, we graph the same system. At this point, I’m obscuring the solution value.

This sets off discussion about the shift, how one starts out cheaper but stabilizes at a more expensive base cost. And then, look, the lines intersect? What do intersections mean, again? SOLUTION!

And in this first equation, the solution is quite simple because the equations have the same denominator. (note: I’m using x instead of h because I quickly copied these from Desmos):

So it’s a quick matter to solve the system, but again, good reminder. At 40 hoodies, they are $32.50 apiece.

Right around here, I point out that it would be convenient if one equation could show us the information we needed.  How could we show the difference between the two functions?

Fortunately, a number of the kids have lived through my algebra 2 class, and call out “subtract!”. I briefly explain that functions have operations and can be combined, for the rest. So we can subtract one equation from the other. Since ultimately the Hawk’s function will be more, we perform TW(h) – LW(h) and graph it as D(h).

Note first that it’s still a rational expression, although not the same type we’re working with. See how nice and clean the break even point is displayed!

So what we want to do is move this break even point further to the right. Luvs is ahead and has no reason to bargain. Obviously, we need to talk to Hawk’s Hoodies.

The administrators want to buy the better hoodie, but 40 isn’t enough to have a fundraiser–they want over 100. How can we get a better deal?

 

Suggestions? The ideas come fast.

First up is always “Hawk’s should sell the hoodies for cheaper.”

“Right. Hawk’s could lower its asymptote and slow the rate of increase in total cost. What would be a good price?”

We try $23/hoodie:

$22 is even better, putting the break even point at 100 hoodies.

“But here’s the thing–Hawk’s has real pride in their hoodies. They know they’re charging more, but their hoodies are worth it. That’s why we want those hoodies to begin with! They’re softer, better colors, hold up to wear and tear, whatever. So cutting their price by 12% sets a precedent. There’s a whole bunch of marketing research showing that customers don’t value luxury items if they sense the vendor will cut prices at a later date. So while we might like the price of the hoodies, Hawks could be hurting its brand if it cuts the retail price per hoodie. Take this as a given, for the moment. Is there any other way we could cut the price per hoodie that still maintains the same retail price?”

This always leads to good feedback: give a school discount, cut the price of the logo conversion, and so on.

At some point, I break in (unless the solution I’m looking for has been mentioned):

“All of you are coming up with great suggestions that involve reducing the value of the numerator. How about the denominator?”

Puzzled looks.

“If I have a division problem stated as a fraction, reducing the numerator (the dividend, if you must) will reduce the result, or the quotient. But is that the only way I can reduce the quotient?”

Pause. New teachers, let the pause hang. If it still gets no response, say “What else can I change?” because that will lead to someone saying…

“The denominator. But why would reduce the denominator?”

“Why indeed. 6 divided by 3 is 2. If I reduce the numerator to 3, my answer is 1. But….”

“Oh, I get it! Increase the denominator?”

“What would that do? Or put it this way: what would Hawks have to do to increase the denominator?”

And eventually, everyone figures out that Hawks could throw in some hoodies for free, which would also let them maintain their higher prices while still getting the sale.

“So go figure out the equation if Hawks includes 10 hoodies for free.”

Someone will always realize that this means we could get negative hoodies. So I tell them to test some negative values and remind them to think about what this might do to the asymptotes.

When they’re done, we put the whole thing on Desmos, showing that the vertical asymptote has changed, but not the horizontal.

“See, this way, Hawks is decreasing the time it takes for our purchase to get to the lower prices, getting us to just a little over $25 per hoodie with far fewer purchased, because we’re getting $10 for free.”

Now, take a look at the new equation to find the breakeven point:

“So how many of you remember being assigned these ridiculous equations with variables and fractions and thinking oh my god, none of us will ever use this? Who would ever have to add or multiple or subtract fractions? And yet, here we are. This one has them set equal to each other, but as we said above, function D(h) is the difference between the two :

…look at that! Your math homework in real life!!!

So we discuss what d(h) is doing. I point out that “solving the system” of TL(h)  and TW(h) is nothing more than “finding the zeros” for D(h).

From a curriculum standpoint, I transition pretty quickly from rational expressions to a review of binomial multiplication and factoring. So the D(h) subtraction equation gives me a great opportunity to review the procedures before I set them on their way. I work the problem–which requires the quadratic formula at the last step, ironically, but still gives us a chance to review the steps to determine whether or not a quadratic can factor. Then I show again how Desmos takes the equation and shows us how far we’ve “moved to the right”:

Then they all work out the comparison between a $3 reduction in price and ten hoodies thrown in for free. I take a moment to point out that math drives business analysis. Today, we have technology to do the work for us, but the best analysts have an understanding of the rational expressions driving the graphs.

If I had time, I’d do this in Algebra 2, but from a time perspective, I have a choice between introducing exponential equations and logs or go deep on rational expressions. That’s a nobrainer. They need to at least be introduced to logs, and there’s no opportunity in trig to bring that topic up. Rational expressions, on the other hand, forge a connection that makes sense when we get to the graphs.

And yes, it’s made a difference. I’ve been using this activity for two years, and have seen a noticeable improvement in their understanding of the four rational expression graphs. Remember, I’m not just teaching my kids, so even those who got a full dose of the rationals with other teachers are showing increased understanding. I would like to do this and more in Precalc, and will report back.

Sorry I’ve been so long without writing. We had a ridiculous heat wave and I responded by sticking to Twitter and playing Fallout Shelter, which is kind of cool.

The Product of Two Lines

I can’t remember when I realized that quadratics with real zeros were the product of two lines. It may have been this introductory assessment that started me thinking hey, that’s cool, the line goes through the zero. And hey, even cooler, the other one will, too.

And for the first time, I began to understand that “factor” is possible to explain visually as well as algebraically.

Take, for example, f(x)=(x+3) and g(x)=(x-5). Graph the lines and mark the x-and y-intercepts:

Can’t you see the outlines of the parabola? This is a great visual cue for many students.

By this time, I’ve introduced function addition. From there, I just point out that if we can add the outputs of linear functions, we can multiply them.

We can just multiply the y-intercepts together first. One’s positive and one’s negative, so the y-intercept will be [wait for the response. This activity is designed specifically to get low ability kids thinking about what they can see, right in front of their eyes. So make the strugglers see it. Wait until they see it.]

Then onto the x-intercepts, where the output of one of the lines is zero. And zero multiplied by anything is zero.

Again, I always stop around here and make them see it. All lines have an x-intercept. If you’re multiplying two lines together, each line has an x-intercept. So the product of two different lines will have two different x-intercepts–unless one line is a multiple of the other (eg. x+3 and 2x+6). Each of those x-intercepts will multiply with the other output and result in a zero.

So take a minute before we go on, I always say, and think about what that means. Two different lines will have two different x-intercepts, which mean that their product will always have two points at which the product is zero.

This doesn’t mean that all parabolas have two zeros, I usually say at this point, because some if not all the kids see where this lesson is going. But the product of two different lines will always have two different zeros.

Then we look at the two lines and think about general areas and multiplication properties. On the left, both the lines are in negative territory, and a negative times a negative is a positive. Then, the line x+3 “hits” the x-axis and zero at -3, and from that zer on, the output values are positive. So from x=-3 to the zero for x-5, one of the lines has a positive output and one has a negative. I usually move an image from Desmos to my smartboard to mark all this up:

The purpose, again, is to get kids to understand that a quadratic shape isn’t just some random thing. Thinking of it as  a product of two lines allows them to realize the action is predictable, following rules of math they already know.

Then we go back to Desmos and plot points that are products of the two lines.

Bam! There’s the turnaround point, I say. What’s that called, in a parabola? and wait for “vertex”.

When I first introduced this idea, we’d do one or two product examples on the board and then they’d complete this worksheet:

The kids  plot the lines, mark the zeros and y-intercept based on the linear values, then find the outputs of the two individual lines and plot points, looking for the “turnaround”.

After a day or so of that, I’d talk about a parabola, which is sometimes, but not always, the product of two lines. Introduce the key points, etc. I think this would be perfect for algebra one. You could then move on to the parabolas that are the product of one line (a square) or the parabolas that don’t cross the x-intercept at all. Hey, how’s that work ?What kinds of lines are those? and so on.

That’s the basic approach as I developed it two or three years ago. Today, I would use it as just as describe above, but in algebra one, not algebra two. As written,I can’t use it anymore for my algebra two class, and therein lies a tale that validates what I first wrote three years ago, that by “dumbing things down”, I can slowly increase the breadth and depth of the curriculum while still keeping it accessible for all students.

These days, my class starts with a functions unit, covering function definition, notation, transformations, and basic parent functions (line, parabola, radical, reciprocal, absolute value).

So now, the “product of two lines” is no longer a new shape, but a familiar one. At this point, all the kids are at least somewhat familiar with f(x)=a(x-h)²+k, so even if they’ve forgotten the factored form of the quadratic, they recognize the parabola. And even better, they know how to describe it!

So when the shape emerges, the students can describe the parabola in vertex form. Up to now, a parabola has been the parent function f(x)=x² transformed by vertical and horizontal shifts and stretches. They know, then, that the product of f(x)=x+3 and g(x)=x-5 can also be described as h(x)=(x-1)²-16.

Since they already know that a parabola’s points are mirrored around a line of symmetry, most of them quickly connect this knowledge and realize that the line of symmetry will always be smack dab in between the two lines, and that they just need to find the line visually, plug it into the two lines, and that’s the vertex. (something like this).

For most of the kids, therefore, the explanatory worksheet above isn’t necessary. They’re ready to start graphing parabolas in factored form. Some students struggle with the connection, though, and I have this as a backup.

This opens up the whole topic into a series of questions so natural that even the most determined don’t give a damn student will be willing to temporarily engage in mulling them over.

For example, it’s an easy thing to transform a parabola to have no x-intercepts. But clearly, such a parabola can’t be the product of two lines. Hmm. Hold that thought.

Or I return to the idea of a factor or factoring, the process of converting from a sum to a product. If two lines are multiplied together, then each line is a factor of the quadratic. Does that mean that a quadratic with no zeros has no factors? Or is there some other way of looking at it? This will all be useful memories and connections when we move onto factoring, quadratic formula, and complex numbers.

Later, I can ask interested students to sketch (not graph) y=x(x-7)(x+4) and now they see it as a case of multiplying three lines together, where it’s going to be negative, positive, what the y-intercept will be, and so on.

At some point, I mention that we’re working exclusively with lines that have a slope of positive one, and that changing the slope will complicate (but not alter) the math. Although I’m not a big fan of horizontal stretch outside trigonometry, so I always tell the kids to factor out x’s coefficient.

But recently, I’ve realized that the applications go far beyond polynomials, which is why I’m modifying my functions unit yet again. Consider these equations:

and realize that they can all be conceived as as “committing a function on a line”. In each case, graphing the line and then performing the function on each output value will result in the correct graph–and, more importantly, provide a link to key values of the resulting graph simply by considering the line.

Then there’s the real reason I developed this concept: it really helps kids get the zeros right. Any math teacher has been driven bonkers by the flipping zeros problem.

That is, a kid looks at y=(x+3)(x-5) and says the zeros are at 3 and -5. I understand this perfectly. In one sense, it’s entirely logical. But logical or not, it’s wrong. I have gone through approximately the EIGHT HUNDRED BILLION ways of explaining factors vs. zeros, and a depressing chunk of kids still screw it up.

But understanding the factors as lines gives the students a visual check. They will, naturally, forget to use it. But when I come across them getting it backwards, I can say “graph the lines” instead of “OH FOR GOD’S SAKE HOW MANY TIMES DO I HAVE TO TELL YOU!” which makes me feel better but understandably fills them with apprehension.

The Sum of a Parabola and a Line

For the past two years, my algebra students have determined that the product of two lines is a parabola, which instantly provides a visual of the solutions and the line of symmetry.  For the past year, they’ve determined that squaring a line is likewise a parabola, and can be moved up and down the line of symmetry, which is instantly visible as the line’s x-intercept. In this way, I have been able to build understanding from lines to quadratics without just saying hey, presto! here’s a parabola. I introduce them to adding and subtracting functions, and from there, it’s a reasonable step to multiplying functions.

Typically, I’ve moved from this to binomial multiplication, introducing the third form of the quadratic we deal with in early high-level math, the standard form. (The otherwise estimable Stewart refers to the vertex form as standard form, to which I say sir! you must reconsider, except, well, he’s dead.)

At some point in teaching this, you come to the “- b over 2a” (^-b⁄_2a) issue. That is, teachers who like to build on existing knowledge towards each new step are a bit stuck when it comes to finding the vertex in a standard form equation.

(For non-mathies, the standard form of an equation is ax²+bx+c and the vertex form is a(x-h)²+k.  The parameters “a” “b”, and “c” are often just referred to by letter. Vertex form, we’re more likely to talk about the x and y values of the vertex, just like  when we talk about lines in the form y=mx+b, we don’t say “m” and “b” but rather “slope” and “y-intercept”. But teachers, at least, often talk about teaching different aspects of standard form operations by parameters: a>1, a<0, to say nothing of the quadratic formula.  So the way to find the vertex of a parabola in standard form is to take the “a” and “b” term and use the algorithm ^-b⁄_2a to find the line of symmetry,  which is the x-value of the vertex. Then”plug it in”, or evaluate, the x-value in the quadratic equation to find the y-value for the vertex.)

The only way I’ve found until now of building on existing knowledge to establish it is setting standard form equal to vertex form to establish that the “h” of vertex form is equal to the ^-b⁄_2a of standard form, something only the top kids really understand and don’t often enjoy. (they’re much more interested by pre-calc.)

Last year, I was putting together a worksheet on adding and subtracting lines, and on impulse I added a few that involved adding a simple parabola with its vertex at the origin with a line, mainly to add a bit of challenge for the top kids. I could see that adding a line and a parabola doesn’t provide the instant visual “hook” that multiplying or squaring lines does.

It’s obvious that the y-intercept of the sum will be the same as the y-intercept of the line. One can logically ascertain that in this particular case, the right side of the y-axis will only increase—adding two positives. The left side, therefore, as x approaches negative infinity is where the action is. But not too much action, since the parabola’s y is galloping towards positive infinity at a faster clip than the line’s is trotting towards negative infinity. So for a brief interval, the negative of the line will offset a bit of the positive of the parabola, but eventually the parabola’s growth will drown out the line’s decline.

All logically there to construe, but far less obvious at a glance.

This year, I decided to explore the relationship further, because deciphering standard form is where my weakest kids tend to check out. They’ve held on through binomial multiplication, to hang on, at least temporarily, to the linear term so that (x+3)² doesn’t become x² + 9. They’ve mastered factoring quadratics, to their shock. They understand how to graph parabolas in two forms. And suddenly this bizarre algorithm that has to be remembered, then calculated, then more calculations to find “y”, whatever that is. Can you say “cognitive load“, boys and girls? Before you know it, they’re using the quadratic formula for linear equations and other bad, bad things that happen when it’s all kerfluzzled in their noggins. That’s when you realize that paralysis isn’t the worst thing that can happen.

Could I break the process down into discrete steps that told a story?  Build on this notion of modifying the parent function ax² with a line to shift it left or right? Find Raylene a new kidney now that her third husband discovered her affair with the yoga instructor and will no longer give her one of his?

My  first thought was to wonder if the slope of the line had any relationship to the graph’s location. My second thought was yes, you dweeb, “b” is the slope of the added line and b’s fingerprints are all over the line of symmetry. No, no, the other half of my brain, the English major, protested. I know that. But is there some way I can get the kids to think of “b” as a slope, or to link slope to the process in a meaningful way?

(This next part is probably incredibly obvious to actual mathematicians, but in my own defense I ran it by three teachers who actually studied advanced math, and they were like hey, wow. I didn’t know that.)

What information does standard form give? The y-intercept, or “c”. What information do we want that it doesn’t readily provide? The vertex. Factors would be nice, but they aren’t guaranteed. I always want the vertex. So if I graph the resulting parabola of the sum of, say,  x² and 6x + 5, how might the slope be relevant?

The obvious relationship to wonder about first is the slope between the y-intercept, which I have, and the vertex, which I want. Start by finding the slope between these two points. And right at that point I realize hey,  by golly, that’s the rate of change(!).

The slope–that is, by golly, the rate of change(!)–is 3. The line of symmetry is -3. The vertex is exactly 9 units below the y-intercept, or the product of the rate of change and the line of symmetry. Heavens. That’s interesting. Does it always happen? Let’s assume for now a=1.

Sum	Slope from y-int to vertex	Line of Symmetry	units from y-int to y-value of vertex	Vertex
x2 – 4x – 12	-2	x=2	-4	(2,-16)
x2 – 10x + 9	-5	x=5	-25	(5,-16)
x2 – 2x – 3	-1	x=1	-1	(-1,-4)
x2 +6x + 8	3	x=-3	-9	(-3,-1)

Hmm. So according to this, if I were trying to get the vertex for x² +12x + 15, then I should assume that the slope–that is, by golly, the rate of change(!)– from the vertex to the y-intercept is 6. That would make the line of symmetry is x=-6. The y-value of the vertex should be 36 units down from 15, or -21. So the vertex should be at (-6,-21). And indeed it is. How about that?

So what happens if a is some other value than 1? I know the line of symmetry will change, of course, but what about the slope–that is, by golly, the rate of change(!). Is it affected by changes in a?

Sum	Slope from y-int to vertex	Line of Symmetry	units from y-int to y-value of vertex	Vertex
2x2 – 8x – 5	-4	x=2 (-4/2)	-8	(2,-3)
-x2 +2x + 4	1	x=1 (-1/-1)	1	(1,5)
-2x2 +14x +7	7	x=3.5 (-7/-2)	24.5 (49/2)	(3.5,31.5)
4x2 +8x -15	4	x=-1 (-4/4)	-4	(-1,-19)

Here’s a Desmos application that I created to demonstrate it.  The slope–that is, by golly, the rate of change(!)–from the vertex to the y-intercept is always half of the slope of the line added to the parabola–that is, half of “b”. The rate of change is not affected by the stretch factor, or a. The line of symmetry, however, is affected by the stretch, which makes sense once you realize that what we’re really calculating is the horizontal distance (the run) from the vertex to the y-axis. The stretch would affect how quickly the vertex is reached. So the vertex y-value is always going to be the rise for the number of iterations the run went through to get from the y-axis to the line of symmetry, or the rate of change multiplied by the line of symmetry x-value.

Mathematically, these are the exact steps used to complete the square but considerably less abstract. You’re finding the “run” to the line of symmetry and the “rise” up or down to the vertex.

Up to now, I’ve been describing my own discovery? How to explain this to the kids? As is always the case in a new lesson, I keep it pretty flexible and don’t overplan. I created a quick activity sheet.

The goal here was just to get things started. Notice the last question on the back: “Do you notice any patterns?” I was fully prepared for the answer to be “No”, which is good, because it was. We then developed the table similar to the first one above, and they quickly caught on to the pattern when a=1.

I was a bit worried about moving to other a values. However,  the class eventually grasped the basic relationship. The slope from the vertex to the y-intercept was always related to the slope of the line added  to the parabola. But the line of symmetry, the distance from the y-axis, would be influenced by the stretch. This made intuitive sense to most of the kids. They certainly screwed up negatives now and again, but who doesn’t.

Good math thinking throughout. I heard a lot of discussions, saw graphs where kids were clearly thinking through the spatial relationship. Many kids realized that when a=1, a negative b means the slope of the line from the y-intercept to the vertex is also negative, which means the vertex must be to the right of the y-intercept. A positive “b” means the slope is positive which means the vertex is to the left. Then they realize that the sign of “a” will flip that relationship around. he students start to see the “b” value as an indicator. That is, by making bx+c its own unit, they realize how important the slope of the added line is, and how essential it is to the end result.

All that and, you might have noticed, they get an early peek at rate of change concepts.

Definitely no worse than my usual ^-b⁄_2a  lesson and the weak kids did much, much better. This was just the first run; the next time I teach algebra 2 I’ll get more ambitious.

So I can now build on students’ existing knowledge to decipher and graph a standard form equation rather than just provide an algorithm or go through the algebra. On the other hand, the last tether holding my quadratics unit to the earth of typical algebra 2 practice has been severed; it’s now wandering around in the stratosphere.

I don’t mean the basics aren’t covered. I teach binomial multiplication, factoring, projectile motion, the quadratic formula, complex numbers, and so on. But the framework differs considerably from my colleagues’.

But if anyone is thinking that I’m dumbing this down, recall that my students are learning that functions can be combined, added, subtracted, multiplied. They’re learning that rate of change is linked directly to the slope of the line added to  the parabola, and that the original parabola’s stretch doesn’t influence the rate of change–but does impact the line of symmetry. And the weaker kids aren’t getting lost in algorithms that have no meaning.

I could argue about this, but maybe another day. For now, I’m interested in what to call this method, and who else is using it.