The Grand March in the Quad: Linear vs. Angular Velocity, Part II

So my students had done a magnificent job the previous day. And I hadn’t done too badly my own self. The original Sammy problem is, I submit, a masterpiece that integrates three different concepts without tipping its hand.

But the students’ skepticism came through loud and clear. They understood the math. They grasped the significance of the radius to the speed. But ain’t no way they bought the notion that Sammy was going faster than the bird.

I mulled this over the evening, and went back to youtube looking for videos. The Grand March? Too much time to wait for a very small demo.

But wait. What if I did my own Grand March? And in the words of the great Oracle Jones of the noted Western mockumentary The Hallelujah Trail:

THERE, now I see it!”

Alas, no whiskey to be found.

But I had the idea. And it all depended on George.
George, seen here in my algebra 2 class last year, is a top student and a fantastic young man. He also possesses a battery-operated wheelchair.

The next morning, before class, I went looking for him.George1

“George. I have a really cool idea, I think, but I need your help. Can you set your wheelchair to a particular speed?”

“Sure.” George doesn’t even ask why. He’s used to me.

“Okay, and this is a weird question–but can you, like, tip over? Do I have to worry if you go round and round on a 3-foot radius circle that it will….tilt?”

He’s kind, and doesn’t mock me.

“Ms. K, it weighs a lot. It can’t tilt over.”

“Phew. I was having nightmares.”

So before class started, I got some chalk from an old-school colleague and using a tape measure, marked out a circle around the courtyard drain, with a 3′ radius.

When the bell rang, I was ready.

“So. Yesterday I noticed skepticism about the bird’s speed. You understand the evidence, but you trust your lyin’ eyes more. I came up with a way to illustrate the proof so you won’t have to take math’s word for it.”

Then for the first time in my over five years as a teacher, I took my kids outside. Very unnerving. (Yes. I’m a big weenie.)

So the basic idea: George sets a speed and follows my traced circle at a very slow pace. The rest of the kids line up on opposite sides of the quad, and one by one they join in with George. Two kids go in on each of George’s rotation, one from each side.

The class was skeptical, but game.(In fact, this trigonometry class could not have been a better guinea pig for my first time teaching the subject. Every day, they jumped right in.)

On the first day, I just did proof of concept. I wasn’t sure how to get everyone to link together, so everyone held the edge of a tape measure.

The kids did a great job and the activity just exceeded my wildest expectations—and best of all, took less than 20 minutes from start to finish. I took pictures, and showed them to anyone walking by, including the entire admin team. This is my favorite shot from the first day:


The end of the tail really captures the movement. Austin the Action Figure!

We went back in and worked basic problems on angular and linear velocity for the rest of the day.

Then I realized that I really missed an opportunity. I was so worried my idea wouldn’t work that I didn’t take advantage of the obvious real-life problem at hand. What was our Grand March angular and linear velocity? How fast was George going? What about the speed of those at the end of the chain?

So the next day, we went out and did it again. But the kids had some changes. Nuff of those idiotic tape measures, Ms. Kerr. We are all comfortable with our sexuality, and will link arms. Football players and all.

Note–some of these pictures are taken by me with my tablet, which has a pretty low quality camera for a Samsung; other stills were taken from video that two of my students filmed.

trigferrislinkingarms trigferrislinkingarms2
trigferris5 trigferris8
trigferris9a trigferris10a
trigferris13 trigferris14
ferristrig14a trigferris15

I knew we’d added as many kids as we could when I saw Alexis nearly getting creamed by the quad wall.


So I yelled at them all to go full speed for the last half for the grand finale, the picture at the top. Here they are a couple seconds later.


You can see the “whip” effect in many of the pictures. It would work even better if we weren’t running into the quad wall towards the end.

The kids had a blast. Between the two days, almost all the kids participated in a “grand march”.

Then everyone went in and learned how fast they’d been going. I measured a bunch of them shoulder to shoulder and took an average of 36″, or 3 feet for every two students.



This lesson was a stunning success, and not just because of the fun and games. I had created memories and math that students would remember—and they did, all the way through to the final. I couldn’t wait to try it again with my two trig classes in the spring semester.

But while part 1, the Ferris wheel problem, went just as well both times, the outside activity was just a bit flat. Our German exchange student, Simon, was my TA this semester after having taken my trig class last fall (he’s the first one next to George, above). He played the anchor position, since I didn’t have George, and did it very well.

trigferrissemsd trigferrissem2e
trigferrissem2a trigferrissem2b

If you notice, two students in the last picture, above, have dropped out later:


This despite the fact that Simon was holding the same pace that George was (we had timers to confirm).

It wasn’t a disaster, and we had plenty of time to do it again. In both my spring classes, I had kids drop out, which simply hadn’t occurred the first time last fall. They seemed to have fun, but there wasn’t the same joy I saw in the fall. Simon agreed that the spring students didn’t seem to be as absolutely thrilled.

However, I had one of the best “told you so” comments in my third block class. I was explaining that the first student had to move very, very slowly—around 18 seconds per cycle–so that everyone could keep up.

“Um, keep up? Eighteen seconds around?” Braxton said. “That’s not going to be a problem.”

“Okay, everyone remember he said that!” I ordered. Which made a nice little teaching point when we got back to the room after the grand march.

Still, I wish I could have made the spring classes as absolutely perfect as the fall one was.

When I finally got around to writing this up, I suddenly had a revelation. Look at the top picture again. At least 10 of the 17 students in that picture are athletes—4 of them in two sports. I suddenly remembered all the students towards the end of the whip bouncing on their toes, warming up, waiting for the line to come around.

I mentally riffed through all my students in the two classes this time round, and yep–far fewer athletes. And here their trig teacher is demanding physical activity.

While I was always calling kids randomly, I had a much higher shot at getting an athlete in my fall class.

So same activity, same lack of warning–but not the same level of absolute ready-to-go spirit I had in the fall. I’m going to have to think about how to get them prepared to enjoy themselves, get some guidelines, maybe warn them ahead of time.

But even with less absolute magic, the kids understood and enjoyed the lesson.

I just need more space! Maybe I’ll try the football field in the fall.


The Ferris Wheel, Sammy and the Bird: Linear vs. Angular Velocity, Part I

After eighteen months focusing on pre-calc, I was assigned three trigonometry classes for this year (again, over two semester cycles). In both cases, I got a single class at first, giving me a chance to get my feet wet, and then a bigger dose later, so I could really start to experiment.

I didn’t know much more than the basics of trigonometry when I began this class, and I’m not much of a planner. So I was often learning the math a few steps ahead of my students. For example, I had absolutely no idea what linear or angular velocity was until the late afternoon the day before I introduced the concept. But hey, I’m a quick study.

What the book said:

The math symbols just when whoosh over my head (figures often do) but the ice skaters, that made sense. Thanks, William McClure!

I instantly thought of John Ford’s classic Fort Apache, which may cause you to wonder if I actually understand the concept after all. Unless you’re a really big fan of the movie and also know some math, in which case you’re thinking “Oh, yeah, the Grand March”. Long ago, I’d observed that Shirley Temple had to hustle to keep up with her screen dad Henry Fonda in the Grand March (around the 57 second mark) and then saw the same catch up effect in all the subsequent quartets making the turn. At least a decade before I ever conceived of becoming a teacher, I thought “that makes sense. She has further to go around, so she has to go faster.”

I often kick off a section with a scenario that asks a question. Sometimes the question is a short, intuitively easy problem or activity that the students can do with little analysis. Other times it’s a long, extended dive into multiple concepts, drawing on a lot of previous knowledge. But the scenario is always designed to introduce the new concept. (I read about this idea in ed school, but as it was a good year or so before I began to incorporate the practice into my teaching, I can’t remember the reading or the author. Since I save everything, I hereby vow to go back into my readers and see if I can dig up the info.)

So I originally intended to do a short, intuitively easy demonstration on the different velocities, but I couldn’t find I couldn’t find any decent videos other than the Grand March itself, which was a little to subtle. Merry-go-rounds would be great, if I could just find a video of kids on this, with some standing easily in the middle and some holding on at the ends…but no luck. I tried skating videos of crack the whip, and much as I’d love to use a Winslow Homer painting, I knew I’d have to do all the explaining. I abandoned my initial idea of presenting the phenomenon and asking the kids to explain it.

Plan B: longer, more complex problem. After perusing the book and googling, at some point I found a Ferris wheel problem asking about velocity. I wish I could remember where, because I am certain I invented this problem almost entirely, and I’ve love to include the genesis. The question that sparked mine provided the total time to complete one revolution (15 minutes, I think), and the dimensions of the Ferris wheel. It then said that someone had traveled 6 minutes, or maybe 4, and asked what his linear velocity was. It gave too much away.

But from that question (or something close), I went WHOA and morphed my plan entirely. No more small illustration, but a long extended dive into–or onto–a Ferris wheel. Because, as any trig teacher can tell you, the Ferris wheel is the mother lode for application problems, a rich source of ideas that can be turned to a number of uses. My kids had already been through Ferris wheel problems calculating heights. So once I was pointed in the direction of Ferris wheels, a multi-faceted problem was an easy next step: one that combined right triangle trig, arc length, and linear velocity, the last in an intuitive way. Booyah.

So the next day, right off the bat, I projected part one of the problem:

And figure it out they did.

My students had just learned how to find the length of an arc, whereas I, who figured it out intuitively, just took the needed percentage of the circumference. I had spent no small amount of time over the past few days explaining that the algorithm for radians, which is the product of the “angle over 180” and the radius, is exactly the same thing as taking the corresponding fraction of the circumference. I was still a bit taken aback to see them multiplying 125 by three quarters pi. Oh, wait. Yeah. Okay.



I wasn’t taking pictures through the class, unfortunately, but grabbed these shots the next day. The one on the left is most complete, but for some reason they flipped the heights. The work on the right is also done well, but they did more of it on the calculator. I remember making them talk me through their thinking.

Meanwhile, a struggling group sketched aimlessly, hoping I wouldn’t notice that they weren’t working. I reminded them of the right triangle trig, helped them to find the angle measure, then asked them to think about what it meant. This group used the circumference instead of the algorithm and made progress although they didn’t finish the problem completely by the time I called everyone back together.


At forty minutes or so, all but one group had finished the entire problem. I had to help two of the eight groups significantly; the rest just needed mild reassurance. Outstanding work, a math teacher’s propaganda day.

At that point, I defined linear velocity, which they had intuitively understood as they worked the question. Once you associate arc length with the time to travel, it’s only natural to think about the speed.

Time for part 2:

The kids all began their calculations, using Sammy’s speed.

“Hold on,” I said, calling everyone’s attention. “Didn’t you use the radius to calculate Sammy’s velocity?”

“Sure, but they’re going the same speed, right?”

“Did you use the radius to calculate Sammy’s speed?”

And I had a bunch of kids looking at me like this:


“The bird and Sammy are going the same speed!”

“But did you use the….”

“Come on! They’re going the same speed! How can the bird be going a different speed? They’re both on the same Ferris wheel!!”

“What if I’d not mentioned Sammy and we started with this problem? What would you have done?”

With much skepticism, they worked the same method and realized that the bird was, indeed, going slower. (You can see some of the work on first picture of boardwork, above. That group had finished first and I gave them the problem verbally. Everyone else started it after I called everyone back together and explained linear velocity, so their work was on paper.

And so, I introduced angular velocity. Sammy and the bird were traveling a different distance in the same time, so their speeds were clearly different. Howevever, they were both completing one complete cycle, or circumference, in 16 minutes, so their angular velocity is the same. As we watch Sammy and the bird, we see them covering a circle in the same amount of time and this fools us into thinking they’re going the same speed.

I could tell they weren’t convinced.

“So how fast is the bird going?”

“The math says the bird is going 7.85 feet a minute, which is about .09 miles per hour.”

“Well, let’s be more precise: .0892 miles per hour, right? How fast is Sammy going?”

“Half a mile…”

“.5569 miles per hour.”

“How much faster is that?”

I won some breathing room from all that doubt when the kids determined that the speeds had the same ratio as the radii. But I could see doubt.

The math proved Sammy and the bird had different velocity. But how could I get them to accept the math?

I came up with an idea for the next day. Which I’ll cover in the next post.

Classroom Alchemy

“Hey, how was Philadelphia?” asked Darius*, as I checked his work (“Sketch a parabola in which b=0”).

“Pittsburgh,” I said, pleased and taken aback. It was Wednesday, first day back after our 4-day Veterans Day weekend. Sometime on the previous Thursday, I’d mentioned casually I was going back east for my uncle’s 70th birthday. Six days later, Darius remembered my plans.

“The family reunion, right?”

“Yeah. How nice of you to remember. I had a wonderful time.”

I moved back to the front, checking for universal understanding of the impact that b=0 had on the position of a parabola, and then told everyone to sketch a parabola in which c=0.

“Did a lot of people show up?” Darius asked across the room.

“They did! Over 90 people. All my uncles and aunts on my dad’s side, and several of their cousins. Eleven of my fourteen surviving cousins on that side. At least 9 of the next generation–my son’s. And even some tiny members of the generation after that—the great-great-grandchildren of my dad’s parents.”

“Wow. Did you know them all?”

“Some of them I’d never met before, because they hadn’t been born the last time I’d visited. Others I’ve known all my life, like some cousins, and my aunt and uncles. We even had someone from my grandfather’s generation. Aunt Ruth–my dad’s aunt–who is 94, looks fantastic, and just came back from a trip to Paris.”

“Was the food good?” from Harres.

“Outstanding. It was simple, nothing dramatic. They put the food on different tables throughout the room.”

“Oh, I don’t like that,” Darius again. “I always want everything, and can’t decide which table.”

“There was a table with two big haunches of meat. One roast beef, one ham, with really good bread rolls. I had no trouble deciding which table.”

After we finished up c=0 and they were figuring out the significance of a parabola with just one zero/solution, Darius waited again until I was checking on his work.

“Did you talk to people there?”

“Me? Oh, yes. Non-stop talking. There were so many people I hadn’t seen in years, and then others I wanted to get to know. I wish I’d had more time. I need to go back more often. If I wait as long again, I’ll be older than my uncle is now.”

“I went to a family reunion one time.”

“You did? How was it?”

“No one talked to me. I was like this.” and Darius humorously mimed standing all alone, silent, looking about for something to do.

So that’s why he remembered.

“Darius, I can tell you for certain that no one at my family reunion was sitting all by himself. I’m sorry. That probably wasn’t fun.”

“Yeah. It was weird. I didn’t know anyone there, and they were all talking to each other.”

“That would totally suck. I’m sorry. We’d have asked all about you.”

As they worked out the next task, I had a brief moment of introspection. Darius, who’s a cool cat in every sense, is far less likely to be the one sitting alone at a party than, say, me, a cranky introvert who has to brave up for crowds so she can exercise her natural garrulousness. I know that my uncles, or my dad, would have probably joked about a teenaged African American appearing at the party. Some or all of them, egged on by siblings and downstream kin, would one up each other with ribald wordplay and puns about where and who had done what when to add color to the family tree. But they’d have sought him out, gotten him some food, grilled him on his life story, likes and dislikes, found out his plans after high school. Looked for links and common interests, bring in others to get conversation going. But would I have done everything to reach out? Or would I have been too busy enjoying not being the one sitting alone?

As the bell rang, I was actually showing Darius and others some family pictures from the night, which sounds impossibly boring, but they seemed genuinely interested in seeing evidence of my stories.

“I’m really sorry you felt isolated at your own family reunion, Darius.”

“Yeah. It’s always the same. I’m like the whitest person when I’m with my black relatives, and the darkest person when I’m with my white relatives.”

“Well, you’d have been the darkest person at my family reunion, for sure. I don’t think our bloodline moves east of Aberdeen. Maybe London. We’re pretty thoroughly white folks. But even though you felt isolated because of your race, some of it could just be family dynamics. My family’s big, boisterous. Really loud.”

“Everyone here was loud. They just were loud to everyone else but me.”

Kameron* punched his arm lightly. “I hear ya.” At Darius’s look, he elaborated. “I’m half black. My mom’s white.”

“Oh, then you know.”

“Does your black family ask if you’re ‘all-black’?”

“You get that too? Isn’t that idiotic? Like they’re measuring?”

“Well, gee, I guess at least the white side of the family didn’t ask if you were ‘all-white’.” I pointed out, and they cracked up.

“There’s a lot of research and profiles on biracial kids, did you know?”

“Really?” Both Kameron and Darius looked interested.

“Yes, that feeling you both have of not being one nor the other, of being slightly separate, is not uncommon. It’s also not unique to kids with one black and one white parent. Biracial Asians have similar feelings, whether their other parent is black, white, or Hispanic.”

“Huh. Really.”

“Sure. There are some good books that you can read about other teens with the same background. You should check them out. In any case, I promise you, Darius, that you wouldn’t have been all by yourself at our family reunion.”

“So the next one you have, invite me!”

“It’s a deal. Have a good day, guys.”

Such exchanges are classroom alchemy, a magical transformation of mundane, random elements into golden moments. They spring from elixirs of personalities, events, spontaneous conversations, the incidental inspired nudge. They are occasionally unrelated to content knowledge and always irrelevant to test scores. They will never be found in MOOCs, nor in classrooms obsessed with tight transitions. They are criterion deficient; ed schools can, to a limited extent, prepare teachers for such moments only with open-ended assignments that are probably opinion-based.

I don’t confuse alchemy with the meat and potatoes of teaching. Darius and Kameron are both doing very well, improving their competency and fluency in quadratics, modeling real-life situations with algorithms and, importantly, taking on intellectual challenges that don’t immediately hold interest.

But teachers are responsible for more than content, whether we are aware of it or not. We are the first adults students interact with, the first engagement students have with the outside world. Independent of content, we can give students a feeling of competency, of capability, or of frustration and helplessness. We can communicate values both indirectly and directly. We can teach them that work is a serious business, or we can teach them that work can be fun and entertaining—or both. We teach them how to interact with a wide range of personalities, how to ask for help, how to give help. It doesn’t matter if a teacher is determined to convey nothing but content. Simply by the nature of our job, we create an environment that has its own entirely unmeasured learning outcomes.

I am a teacher who focuses primarily on conveying content, as all observers have noted over the years. Yet for a teacher who doesn’t see her job in terms of its emotional impact, I have my fair share of classroom alchemy, the moments of knowing my classroom has been a positive force in the universe, whether for one student, a group, or a class of thirty five.

I never plan these moments. As the great Terry Pratchett noted (with props to Neil Gaiman), you can’t second guess ineffability. It’s just going to come along on its own terms.

*Darius and Kameron both confirmed this exchange as written.

On Sarcasm, Irony, and Teaching

Recently, Grant Wiggins posted a heartfelt post by his daughter who was totally gobsmacked by spending two days shadowing students. Apparently, they lead a life filled with boredom and pain, tortured by constant immobility and sarcastic teachers.

I was unmoved. It was, clearly, a minority opinion; Wiggins’ post went everywhere, and all sorts of teachers posted emotional paeans to the effect that they would change their ways this very minute. Others wrote, a tad smugly, that they had come to similar insights years ago and so no longer were that kind of teacher.

I kept my mouth shut, but when Wiggins posted a followup celebrating the fact that only two commenters spoke out in favor of sarcasm, I felt moved to comment.

Actually, I would have defended sarcasm in teachers, and wrote a long comment on the last post–and then deleted it, because really, why fight the zeitgeist?

But since you made a point of mentioning that only two commenters supported sarcasm, I thought I’d add my voice after all.

… I guess the definition is changing. I am not hurtful or unpleasant to my kids. I am definitely ironic in ways that I would describe as mildly sarcastic, and the kids enjoy it. And certainly, I use paws up, but claws not out sarcasm as a form of classroom management in ways that I am perfectly content with. Now, perhaps other teachers are incapable of non-hurtful sarcasm. Or perhaps everyone’s just a little too pure.

For example, “Ernesto, you appear perfectly enthralled with Sophia’s conversation. Must be fascinating. Sophia, perhaps you’d like to share?” is sarcasm. I do not, in fact, want Sophia to share. I want her and Ernesto to pay attention. If I say this with a bright and cheery voice, I am not being hurtful. But I am being sarcastic.

And sure enough, I got three responses that proved my point.

Your example of “mild sarcasm” might be embarrassing or humiliating to certain students. Almost fits the definition of social bullying, as it is sort of making fun of and belittling Sophia. It would have served to shut me up completely in that class, leaving me fearful that any utterance on my part would open me up for more public embarrassment. Some students can let that “mild sarcasm” roll off their backs, but certainly there are those who would feel the sting.

Second: I have to respond to the sarcasm. I have a middle school daughter. She is a high honor roll student. She finds pride and accomplish in her academics, and places far more pressure on herself than I ever would. Last week she forgot her homework, not because she chose not to do it, but because like adults, she made a mistake. As she got into class and realized this she panicked. Her nervous reaction is to cover her mouth, 12 year old age appropriate. Her teacher gave her some comment that crushed her. Upon picking her up even her explaining to me her “humiliation and embarrassment” brought tears to her eyes. “I am not that student that just doesn’t do my homework or that student that thinks it is not serious” she stressed over even going into class the next day. What was accomplished by that??? On a much brighter note my daughter to the initiative to email the teacher and explain she wanted to apologize at the end of class for the misunderstanding but was “scared” to approach him! I was very thankful to see an email back from the teacher thanking her the email and encouraging her to never fear approaching him. It was not the lack of missing homework that had her that upset. She had already figured out with no grade under a 93 and HW only 20% of her grade she was in the clear for high honors. It was the comment and the weight of how that teacher made her feel. Please understand this teacher at parent teacher night seemed great. I feel he really does care about the students he teaches everyday. This is not meant to be a teacher bashing rather hopefully another perspective on what some may see as harmless in a classroom.

I had a Spanish teacher for 3 semesters that had a reputation as the worst teacher in our high school. (Of course, I do not compare you to her, but I want to give an example of how disruptive sarcasm can be to a learning environment.) Sarcasm was her only classroom management strategy. All the students were terrified of her. If you asked a question, she sarcastically said you should have paid better attention to the lecture. If she asked a question and you answered wrong, she’d simply raise her eyebrows at you and not help or offer suggestions as you struggled to find the right conjugation of a verb. There was no kindness, no empathy, no humanity. I knew only two students that liked her. They claimed you had to understand her sarcasm, but most students couldn’t. Many begged guidance counselors to be switched to another teacher. She was the only 4th year teacher the last year I had her and so few students took 4th year (because it was not required to graduate) that they had to make her the only 3rd year teacher this year so that 4th year Spanish could continue to be a class. The only reason I stuck through it was because I wanted to minor in Spanish. Classmates that before loved Spanish hated it with a passion after having this sarcastic teacher.

So. (In my response, please note that I teach high school students.)

The first commenter wants me to feel badly about Sophia, who is talking in class. Sophia may never open her mouth in class again! she says dramatically. Well, if Sophia is constantly talking in class when she shouldn’t be, then this would not be the worst thing. My experience suggests that the kids who talk a lot in class when they should be listening are not easily discouraged.

But in any event, why should Sophia be any less intimidated by a stern request to stop talking? Or would the commenter prefer that the teacher simply say “Everyone stop talking, please!”, without identifying any particular person?

The latter issue, whether or not a teacher should “call out” offenders, is a subject of considerable debate in the teaching community. I am unsympathetic on this point—if you’re the one talking, you’re the one I’m going to tell to hush, one way or another. But in either case, the issue in that event is not the sarcasm, but the calling out.

The second commenter, the mom, doesn’t even describe the comment made, merely conveying that her daughter was devastated because she’s extremely sensitive. So I can’t really tell if the teacher used sarcasm or not. If the teacher wasn’t saying anything deliberately hurtful, then it’s hard to argue that the teacher should change. Sometimes you have to tell girls to get a grip and stop thinking it’s all about them. And kindly note the name of this blog before you tell me I don’t understand.

Notice that I am speaking here mostly of suburban kids with plenty of privileges—the ones who post on blogs or have moms who do. High poverty kids who are offended by an unintentionally hurtful comment are a different issue. But ultimately, all kids who are sensitive to basically normal interactions should be supported sympathetically but led to see that they need to take a breath and realize no harm was meant.

Third commenter is specific! And none of that is sarcasm!

Therefore, for starters, let’s define terms.



I am uncertain as to what “irony” is without sarcasm. We Americans definitely conflate the two terms. So let me try to define: ironic utterances intended as humorously warning rebuke.

Some people find “Sophia, perhaps you’d like to share?” to be a comment beyond the bounds of acceptable teacher discourse because it’s hurtful and unfair to speak in such a fashion to a student who is rattling away cheerfully while the teacher is addressing the class.

Those who oppose “sarcasm” (whatever that is) also tend to be bothered by exchanges like this:

“Fredo, are you intentionally trying to be irritating?”

“What!?! I wasn’t doing anything!”

“Exactly! Pick up your pencil and at least pretend to work!”

“Jimmy, either stop tapping that pencil of your own accord or sit on your hands.”


“Sandy, the sign says no grooming. Put the mirror away. You are naturally fabulous.”

None of these are sarcastic comments. None of them are respectful. All of them are comments that, in my class, get a laugh from the students (including Jimmy, Fredo, and Sandy). All of them identify a student who is in some way violating the rules of my classroom—or simply driving me crazy, in the case of Jimmy’s tapping.

I have run into two sorts of sarcasm opposers. One sort really opposes meanness and is only thinking of mean, hurtful sarcasm. I think all teachers would agree that hurtful language towards students is never appropriate. They have only the “mean” type in mind, and readily agree that much verbal irony does not qualify.

The second sort really opposes not just sarcasm, but all verbal expression that isn’t polite, respectful, and utterly indifferent to the student behavior. When you give them an example, they respond with an alternative. Emilio should be gently asked to go back on task. Jimmy should be allowed to tap. Sandy should be counseled on feminist empowerment. Or something.

This may be an issue of class. I do not come from a polite ring of the social strata, which may be why I don’t consider the occasional verbal riposte a source of endless psychological damage.

So to those people who simply think teachers shouldn’t be mean, we agree. Very few teachers would disagree.

To those people who think teachers should be intensely conscious that their every word might scar a super-sensitive teenager, we need to remember those super-sensitive teenagers will grow up. Grant teachers some leeway, and tell the kids to toughen up a bit, with sympathy and understanding. And most teachers will get a bit milder with the sensitive ones.

To those who believe all teachers should be respectful, firm, sincere in every utterance and unfailingly polite, I certainly agree that teachers who want to should operate in such a manner. But requiring that behavior from all teachers would suck the joy out of my classroom and I’d leave teaching if we were all expected to act like pedantic ninnies with high school students. I secretly wonder if such people are evidence that much of what makes America great is in the process of being castrated.

Yeah, don’t mind that last sentence. It was just me being sarcastic ironic.

The Panda Problem

Our school was having visitors that week, and I needed a show-off problem in case they dropped by my room. The circle square problem caught my interest. As posted by Dan Meyer,

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

The math behind this problem was perfect. I was in the middle of systems and just been through transformations with my algebra II/trig classes. The math was a bit challenging for A2/Trig, but it would yield so much insight into the math they were working on. Then I figured my pre-calc classes would have fun with it, and I’d get more insight into the lesson if I ran through it more than once. So I did the same problem for all my classes that day, but with a different approach.

But what problem? I decided to show my students both the abstract, general version of the problem, and the one I’d doctored up to make more accessible. In my A2/Trig class, I showed the abstract form first; in precalc, the doctored one first. All my students saw both problems and given the opportunity (which most took) to recognize the difference.

Lesson objectives:

  1. Give students an insight into the general case, by comparing it to an applied case.
  2. Gain a deeper understanding into geometric relationships.
  3. Understand that “systems of equations” were more than just two lines (A2/Trig).
  4. Improve modeling skills, “making math out of words”.
  5. Gain a deeper understanding of transformations (A2/Trig) as well as the link between graphic and algebraic representations.
  6. Solve the problem and understand what the solution means.

I read through all the comments and links in the original blog post, finding two of them extremely helpful in formulating my thoughts. First, Dan Anderson’s work on Desmos was invaluable, not because I understood the sliders (I don’t) but because I hadn’t considered the problem in terms of function transformations until I saw his work, even though he didn’t address it directly. Then I found this essay by Tim Erickson to be the most helpful as a framework for considering the problem, even though I didn’t use any of his suggested approaches. He also provided me with the observation I used to close the lesson (and hadn’t thought of on my own). I wouldn’t even know how to solve this using calculus. I combined Erickson’s Queen idea with the cowboy ranch thing mentioned here and came up with the Empress of China and pandas.


You’re thinking oh, god, a cutesy problem. Well, yes, but it’s an ironic cutesy! Plus, come on, the pandas are adorable.

Whenever I’m doing a complicated project for the first time, I overwrite the handout. That way if I forget something before the release I can point the kids to text. So the Algebra II/Trig handout goes into considerable detail because it’s covering the actual lecture. I was also moving a little more slowly with the A2 kids, as I wanted them to understand the process of understanding an abstraction. I made my teaching literally visible to help them see what they, too, could do to unlock a difficult problem.

By contrast, in the PreCalc class, I tossed them in with only one hint: there were three equations.

I wanted a manipulative, both to literally help kids make concrete sense of the problem, but also to demonstrate what manipulatives do. It’s just a line. It’s just the word “webbing”. But now, look. It’s the actual thing. See how this thing, even if you don’t really use it, clarifies the next step?

So I needed some gold webbing, and it can’t be a coincidence that I settled on paper clips. (The big question is why didn’t Dan Meyer think of it?)

I wanted something I could reuse, and that the kids could reuse, unclipping the chain for one test and then putting it together again, creating several different versions if needede. In truth, I didn’t think they’d use the manipulatives much, but the idea of them, as well as one walkthrough, would probably get the more concrete thinkers moving. Plus, the kids got a kick out of the chains.

I spent $4 in paper clips, which would have only been $3, but I kept losing track of how many I’d chained together, so I went back to the dollar store and bought some colored clips to use every 20th clip. I did one at school for proof of concept, then went home and did five more while watching TV. Which probably wasn’t a good idea, because the couch ate several chains before I got wise and put them on the floor.


Then I had to figure out how to bring them back into school without getting them all snagged. A bamboo skewer worked as a rod and the tiny ponytail bands that can’t handle my hair kept them from sliding around.

Intro to the problem in both cases took no more than ten minutes; the kids were up and working by 15 minutes in. I have whiteboards all around the room, so the kids are up sketching and thinking in groups of 4.

I wish I’d captured the Algebra II/Trig work, but I forgot. Several of these groups actually read my handout, and selected 60-40 as their first modeling scenario. The precalc kids had the same suggestion, but everyone jumped right to 50-50. It didn’t make any difference, but I would have liked to have shown the different thought processes.


All the kids unclipped the chains, made squares or circles on desks or with hands and giggles and quickly grasped the algorithm for moving from perimeter to area of a square. To my pleased surprise, all but a couple groups also uncovered the process from circumference to area of a circle. Those who didn’t had the first step down, but were hung up on dividing by pi. I told them to get over it; they’d be taking the square root of pi before they were through.

When all the groups had identified the three equations needed, I stopped and went to Desmos.


This is a recreation of the conversation as it went in A2/T; the precalc conversations were the same except transformations (we haven’t covered those yet).

[See note at bottom]

“So before we go on to treat these equations as a system, let’s consider each equation. The graph above shows all three of the equations, without treating them as a system. Mandy, what do you see?”

“Um. Two parabolas, and one line?”

“Fair enough. Now, everyone look at the graphs and think of transformations: vertical shifts, horizontal shifts, stretches, compressions. You have your parent graph notes. Take a look at the parent graphs for lines and quadratics.”

Serge: “Hey, I see. So the quadratics are stretched.”

“Vertically or horizontally?”

“Horizontal,” says Dani. “The denominator changed and got bigger, from 1 to 16.”

“Good. What about the line?”

Shane: “It’s reflected over the x-axis, and shifted….vertically? Up?”

“Or horizontal to the right.”

“Good! So now, back to the stretch. Are they stretched the same amount?”

“No.” Judy. “The…which one is red, the circle? No, the square. The square is stretched…less, vertically. It’s going up slower.”

“Very nice. What’s that mean, Julian?”

“I don’t know.”

“Oh, stop. Think. Take a look at the equations, everyone. Right now, we’re using x in both, right? So the graphs show what happen to each if we cut the webbing exactly in half. If we look at our equations, we take the perimeter, square it, and take one-sixteenth of that value, right?”

“We divide it by sixteen,” said Ann. “Oh, that’s the same thing.”

“So what do we do with the circumference? Remember, what’s pi again?”


“So if I multiply that by 4, I get around…”


“Right. So if I start with equal values and square them, which is going to be larger? The one I divide by 16, or by 12? I’m taking a sixteenth of one squared perimeter, and a twelfth of the other. Which is bigger?”

Pause. “Come on, think. I divide a number by 16, or I divide it by 12.”

Ah, now they get it. “Twelve!”

“So what does this mean? How can we use this knowledge? What does it mean about the relationship of perimeter to area?”

Pause again.

“So if I have a circle and square with the same perimeter, which shape will have the bigger area?”

“Circle.” This they’re sure of.

“Okay. So let’s think about our task at hand. We need to make our areas equal. What does that tell us about their perimeters?”

Now they get it, and I hear the knowledge in their voices as they talk over another. The square has to have the larger perimeter. That means it has to have the bigger piece of the chain.

And then I send them off to solve the algebra. I remembered to capture pre-calc work—in a few cases there are errors that I later corrected.

If I were half as orderly as this student, I wouldn’t ever lose my glasses, car keys, or Promethean pen. Except–alas–they’d made a mistake on the first attempt. I pointed it out, they went back and fixed it, but I forgot to take another picture.

I gave a fair amount of algebra assistance in the A2 class. For the pre-calc kids, I just said “Don’t distribute. Ever. Get a calculator when you realize you need a co-efficient.” From that, most of them called me over to confirm that they were ready to use the calculator. I corrected algebra and encouraged, but the kids weren’t sitting around passively. Active working and thinking the whole time..

In A2/Trig, I ensured everyone started by solving the linear equation for C, then solving the entire equation for P (the perimeter). In Precalc a number of groups figured out the substitution before I got there, solving the line for P and the entire equation for C.

Once everyone had solved for P or C, or I’d talked them through it in a few cases, back to Desmos.

“We looked at the three graphs. Now let’s consider it as a system.”


In all cases, the classes literally gasped. Some context—most of my kids don’t have graphing calculators. Many students had just realized now, for the first time, that “solving graphically” meant something other than graphing two lines manually and finding the point of intersection, that it meant saving a hell of a lot of time and algebra.

Did I point this out? No. I missed it. Only yesterday, when typing this up, did I suddenly flash back to the gasps and realize what they meant. Arggggghh. I don’t think graphing calculators and solutions should make an appearance until the algebra is beyond the kids’ abilities, because most of my kids need strong algebra skills far more than they need advanced math analysis. But I still should have made the point.

For homework, the pre-calc kids had to take a whack at the abstract version; the A2/T kids had to answer some questions.

The next day, I closed the lesson out first by restating the objectives. Systems come in all forms. Transformations can help us make sense of relationships.

And then, the geometry, which the kids clearly found fascinating. In each class, I pointed out that circles are more efficient at closing space than squares are, and before I could ask the question, students—many of them strugglers—pointed out that squares could be grouped more efficiently. Even better, several strong students pointed out that we could probably calculate the difference between the individual square’s “lost” area and the space you always see between four circles. (And yes, that’s going to another fun problem in the future!)

Finally, the perplexing reality: giving Ping and Sing their own private space lost them a huge amount of space! If they’d left it as one big piece of webbing, the two would have had a palace.


“You’d think the carpenters could just make one big circle and give Sing a bunch of picture frames or something.”

At the various blogs I read, many math teachers sniffed at the idea of turning this into simple problem-solving. Well, I like problem solving; I want my kids to be strong at the algebra and modeling necessary to solve this problem. But I also found many links to important concepts through the simple “problem solving”. This activity provides connections a seemingly simple problem, which they grasp, to transformations, systems, and geometric figures, which they often just treat as “things to be learned to survive math class”.

From idea to execution was just over 24 hours elapsed. Actual time spent building the lesson, about 6 hours—3 of them on paper clips. Next time, I’ll do it later in the year for A2/Trig, and haven’t yet decided when to do it for Pre-Calc. The handouts need some tweaking, and I need to start working more graphing calculator work into my pre-calc classes. Definitely a keeper. Thanks, everyone, for the mind food.

[Note: Dan Meyer pointed out something in the comments that led me to re-write this slightly. I didn’t make it clear in the first write up that the first graph display was of the three equations–NOT a system. I added what I actually said in class, that these equations assumed that the two strands were equal. The purpose here was to show the kids why, in fact, the areas would NOT be equal if the strands were. I did cover that in all three classes, and completely forgot to write it up, since I was focused on the transformation aspect.

The second point Dan raises, that the line is not using the same data is a good point. I was focusing on showing the equations together, but the x and y aren’t the same and in the second image, that’s not correct. I’ll do it differently next time.]

Halloween Candy

One notable gap in my curricular resume is any sort of performance work based on real-life data. Like most math teachers who work with struggling populations, I think performance task projects waste hours of time for limited learning outcomes. So how could I create a performance-based task that didn’t waste time, had a number of easily achievable goals with real-life data that yielded observations worth the time and effort?

What I Borrowed:

I got the germ of an idea from Dan Meyer, particularly this picture:

Meyer surveyed about 100 people with some metadata categories and loaded all this information into an Excel spreadsheet. His students generated the a list of questions:

Paired off with laptops, the students downloaded the data and were given Excel to analyze it. Meyer looked for the students who didn’t know how to use Excel (manually sorting data, manually calculating differences, and so on).

Meyer’s primary teaching goal was data analysis and tech transfer. I was more interested in getting my kids to think about data organization and description.

What I Changed

Collect the data myself? Nuts to that. I made the kids do the surveying. I created a Survey Monkey poll, which I wouldn’t do again–too much hassle, and even though all the kids have smart phones, they were much happier just filling out a paper form. I ended up doing an hour of data entry. Nothing terrible.

Dan spoke ill of the maltballs he used. I used Lemonheads, which are awesome, as well as a few of Lemonhead’s friends. Plus, a Halloween Theme.

My photo:


The flip side of the form I created to

Thirty kids were assigned to collect ten guesses each. I was hoping for at least a hundred responses, got two hundred.

Learning Objective

I have traditionally given pretty short shrift to absolute value, because the formula (split it up into two equations/inequalities, solve) has relatively little to do with the concept itself. So I give them some practice but don’t focus much on it. Like performance work, one of my goals for the year was beefing up my instruction, to give kids a good sense of what absolute value is.

I redesigned my AV unit introduction, which I really should write up. It went well, and over the next week the kids did well with the algorithms. It’s still difficult to keep both the concepts and the algorithms unified; they aren’t intuitive and I don’t yet have a meaningful model for a multi-step absolute value equation or inequality. But I’ve been pleased with my additional curriculum work this year and have a good basis to build on.

The Halloween unit had two objectives. First, it would reveal another use of the concept—or, if nothing else, reveal why contestants on The Price is Right have be come the closest without going over, since Bob Barker deals with retail price, not absolute value. But a second, linked learning objective was the meaning of central tendencies. Half the class is sophomores, so a little refresher course on mode, median, mean, and range wouldn’t hurt a few months before the CAHSEE.

No Technology

Doing an about face from the Meyer lesson, I eschewed all technology short of calculators. I mean, seriously, who wants to spend the time getting all the students up and running on Excel? Besides, I wanted the kids to think about the ordering the data, finding the mode, literally counting to find the median. I like the occasional tactile exercise and it doesn’t hurt to remind kids that way, way back in the day, computer programmers had to write their own data sorting routines, from bubble sorts on up. Sneer if you like at menial tasks, but I’m not a big fan of just pushing buttons. No Excel utilities until you can do the same tasks manually, dammit.

I had eight groups of four students, so I broke the list up into groups of twenty five. To ensure a random sampling, I added a new column, filled it with randomly generated numbers, and then sorted the responses by the random values. (Yes, I used the Excel utilities. I can do the same tasks manually, dammit.)


So each group of four got a list of twenty five numbers something like this (not an actual list, which I didn’t save. Argggh.):






I wrote the following instructions on the board:
I described “measures of central tendency”, and prompted the class for the definitions of mean, median, and mode. I pointed out that task 1 was pretty straightforward, that task 2 required some thought and (hint hint) required a concept they’d been working with recently.

Each group was given their own mid-sized whiteboard, which they could lay across their desks or lean against the wall. One group used a full-size whiteboard.

And off they went.


This group had a whiteboard leaning against the wall, despite my exhortations, and were working in pairs because of spatial constraints. So I made them put the desks in a tight group instead of an L, which you can see in the background of this group:



This last guy wasn’t working alone; his teammates just didn’t make it in the picture. Here’s their final product:


Once two groups of four students finished, I paired them up to join forces as a new group of eight and find the information for the list of fifty numbers.


Here’s a sample of boardwork from group 4, a group of girls who typically struggle a bit and had a tough time figuring out the difference between ordering by number and ordering by difference from the winning amount. But they finally had their “aha” moment when they were placing the number 1500.


Four of the original eight groups—by and large, the strongest kids—struggled a bit with the organization of data and the final task of finding the absolute value. They all finished in time to form two new groups compiling fifty numbers.

The other four groups, including group 4 above, finished their first analysis quickly,paired up to produce new lists of fifty and were done with twenty minutes to spare.

Then I said, “Now, guys, put it all together.” and they asked me if I was kidding.

Nope. And thus was born the Mega-Group. (I did let them off the hook for median.) Fifteen kids putting all the data together, and I missed getting a picture of it. Sigh. There wasn’t one board with all the data on it, but here’s the final working board I used to talk to the class, with some markup:


I really need to make sure my boardwork is understandable after the fact. I know the kids get it at the time, but it’s a bit like architectural layers.

In brown–two groups’ work. The other eight’s work is on a board to the right. In green–the kids rework their average by finding the sums for the two averages and then adding them to find the total, which they then divide by 100. (They were greatly cheered to discover that they didn’t have to add up all the terms one more time.) In blue–the mean, mode, and range for the hundred numbers.

Then I used the board as the discussion pointer for the whole class. To the right, with a red square around it (on the whiteboard, not my markup), the overall mean, mode, and median for the entire set. In darker brown outline, my additional instructions on using the average formula to recreate the sum. I wasn’t teaching so much as demonstrating a concept (it was about a week to the PSAT) to show how the mega-group had been able to recalculate their averages without going through all the work of adding up the numbers individually.

See? If they’d been working in Excel, they never would have learned a useful algebra short cut, the explanation of which is difficult to see in my admittedly disorganized boardwork, so here you go:



So where’s the “real life”?

Time worked out perfectly; the kids came to a natural stopping point with fifteen minutes left in class. I could have done an entire class on the data discussion, but it’s an algebra II class, not stats. What I wanted was some “aha!” moments, as the kids realized what the data revealed about the contestants. Here’s the unmarked up discussion board:


But then we looked at all the groupwork data, looking for patterns. In almost every case, the median value was somewhere in the high 600s, low 700s. The mode of 480 quickly emerged as the groups combined. Averages consistently lowered as groups merged.


I put up a picture of the original poster and suddenly I’m looking at a sea of faces that get it.

“So what relationship does the mode of 480 have to the picture you used to collect data?”

“It’s four times the sample size!” said Brad.

“Which means, Tracy?”

“People thought that the jack o’lantern was four times as big as the owl jar.”

“All people?”


“So, Sharon?”

“More people guessed exactly four times than any other guess.”

“There you go. So how is that different from the median? Remember, the median reflects the middle value. But if I look around the room, I see that all the groups had middle values in the 600s or so. What does that mean, Mark?”

“That’s 6 times, right?”

“Good. But exactly six times would be 690, and I don’t see 690 as a mode, surprisingly.”

“Yeah, it’s like people said it was about six times, and then, you know, went up or down.”

“Oh, so the people who thought it was 6 times were more likely to estimate, figure ‘a little over or under 690’. So you get a lot of numbers in the high 600s, low 700s. But the people who guessed four times….”

Kevin jumped in, “It’s like the people who guessed four times as big were…not as realistic about what the numbers could be?”

“Oh, interesting. So someone who guessed four times might not have had as much experience thinking about weird questions like how many teeny tiny Lemonheads would fit in and around, so they went with a straight 4:1. The people who guessed six times as much were more versed in estimates, and fudged.”

“Yeah, but it wasn’t six times. It’s more like eight times bigger.”

“Which brings up the biggest point of all, for me! I was really worried. I thought everyone would ballpark a thousand Lemonheads and we’d have 20 people tied for the win. Instead, just three people guessed a thousand, and they tied for second. One of them is in this room–yay, George!”(George is in purple on the left in the first picture.)

Jose said, “Some people guessed really, really high. Did that change the average?”

“Great question. Look around at all the numbers. What’s the lowest you see?”

“We had it,” Manuel said. “115–someone guessed the big and little jars had the same amount, which is crazy.”

“Exactly. So the lowest number was about one thousand less than the actual count, and it was totally ridiculous. What about the highest?”

“9999!” everyone chorused.

“that’s nearly NINE THOUSAND more. So the biggest guess is nine times as far off as the lowest guess.”

“We had a three thousand guess.”

“So did we.”

“Those kind of guesses are called outliers, data that is significantly off the average. Many times, statisticians will discard this data to see if it is distorting the outcome. Here’s a question—if you get rid of data that is really large or really small, which measure is it most likely to affect, if any: the mean, the median, or the mode.”

“Not the mode,” Khan said with certainty.

“Right. Unless the outlier data was a repeated number—like, maybe someone bored just kept typing in 9999—the mode is unlikely to be affected.”

“It might change the median, but not a lot.”

“Not if your center is robust, right. And if the center isn’t robust–if taking a few numbers off the top or bottom really changes the center, then maybe the data aren’t outliers.”

“It could really change the mean.”

“Yep. Since the average uses all the numbers in a sum, removing really big ones—even though we’re also reducing the number of terms–can have an impact on the mean. I checked that out, though.”


“Since a few numbers skewed really high, I decided I’d pull out the guesses that were greater than 1.5 times the actual count. That left 6 guesses out: 9999, 6000, 4000, 3000, 3000, 3000. Those six guesses comprised nearly 17% of the total sum that made up the average.”

“Just six numbers?”

“Yep. Just 3% of the guesses made up 17% of the total. So what would removing them do to the average? I would be dividing by 194 instead of 200, remember.”

“Make the average smaller.”

“Right. How much smaller?”


“Before you answer, I want you all to think about how much your GPA changes just because you get an A instead of a B. A lot or a little?”

“A little!” Jun complained.

“Right. Averages are pretty robust—they don’t change easily. On the other hand, this is a whole bunch of data. So I’ll put it this way. If I remove the big numbers, do you think the average would be higher than the median, or lower?”

“Higher” says Dante, with certainty.

“Good. How much higher?”


“Not bad. The new average is 744.”

“You took out 9000 and 6000 and the average only went down one hundred and twenty some points?” asked Raman.

“That’s a lot, actually. It’s proof that the big guesses were skewing the mean. But either mean–867 or 744—tells us what about the people guessing?

“They guessed low.”

“Most people guessed low. Once we remove the outliers, we see that the average guess was just over six times the number in the hint, when the actual amount was nine times. So each central tendency–the mean, the median, and the mode—gives us an insight into our guessers and their decisions.”

The kids could see actual human behavior in each description and yes, my performance task skeptic brethren, it did matter that the behavior came from data my students collected themselves.

Designing lessons like this is, I think, one of the great thrills of teaching. It’s not enough to define a task. You have to give careful thought to how you will use the information. Most teachers err by giving the students too much responsibility—discuss the data in groups, and decide what it reveals. But most kids, given this information, would never have come up with the same insights. They’d have just shrugged and given up. You have to give them plenty of success, plenty of faith, before you give them analysis, and even then you need to guide your strugglers.

I read lots of lessons in which the teachers say “Most kids gave up, but my top kids were able to figure out a couple of the tasks.” Or they group kids heterogeneously and the weaker kids don’t do much work except watch the stronger ones. In this case, my strugglers got the charge of being in the top performance group on their own merits.

I don’t mind struggle, but I want the struggle to either pay off for 90% of the class OR be a brief struggle for my weaker students while my top students get a challenge. But far too often, I see or hear of lessons in which the teacher ends up going from group to group or pair to pair explaining what the kids were supposed to figure out. I’d rather do that in a class discussion, or let kids move things along with my guidance. That way I know all the kids are paying attention, rather than talking about the Homecoming rally.


Two people guessed 1035—off just by one. A student’s father won half the candy, but a sophomore in Algebra II/Trig got the other half, along with a cute trick or treat pillowcase. Three runners up guessed 1000, and got a pack of Halloween minipencils.

The learning objectives were met. The class discussion involved the central tendencies, but all the groups quickly realized the relevance of absolute value, and every single group mentioned game shows.

I didn’t put on any weight, despite sneaking an ungodly number of Lemonheads while filling the jack o’lantern. Counting makes me hungry.

So all ended well.

Ambiguous Factoring

Fun with factoring, part I–Difference of Cubes

I wish I could remember the teacher’s blog that alerted me to this. I can see the page in my memory, but I can’t find it. I will happily include a link if anyone can tell me the blog. It was written in February, I think, a year ago.

The blog entry involved an interesting order of operations ambiguity when factoring a difference of two sixths as a difference of two cubes. I note that Wikipedia gives patterns for factoring differences of 4, 5, 6, and 7th powers, but in precalc and earlier we limit ourselves to squares and cubes.

Here’s an example of the same polynomial factored as a difference of two cubes and a difference of two squares:

So factoring it as a difference of squares allows for a second factoring round as a sum and difference of cubes, which fully factors the polynomial. Factoring it as a difference of cubes first allows for one factor to be further broken down, but not the other.

The additional factors on the squares side are indeed factors of the unfactorable term on the cubes side.

Most instructions for sum and difference of cubes explain that the second term is “unfactorable”> Clearly, this isn’t the case. We may not be able to factor it with our usual methods, but the term is factorable. We can only find the factors if we take the right road? Is that how it works?

I thought it was interesting, anyway, and gave the exercise to my precalc students, telling them I had no idea why the difference, but I’d find out if I could. Meanwhile, wasn’t it kind of cool? And the neat thing about precalc was that most of the kids were able to tangle with the math, realize the ambiguity and likewise be interested.

The blog entry I read and can’t find didn’t have an answer, either. In googling, I could only find one site that specifically mentioned order of operations when the cubes were also squares. Most didn’t mention it at all.

Interlude: Khan Academy Rant

While wandering round the web looking for sites that might address this ambiguity, I stumbled onto Khan Academy’s explanation for factoring the sum of cubes. While I think the fuss about Khan is little more than a reminder that great riches come to those who never need it and really don’t deserve it, I’ve never looked at his pedestrian videos closely enough to be flatly offended until now. Please understand that I am not the sort who goes through a blow by blow of the faults with the video, so it’s best to go see it yourself.

Khan starts with a sum of two cubes. Then he says, in order to do this, “this is really just something you need to know…I’d argue whether you REALLY need to know this, but to actually do this problem, it’s something you just need to know…”

Now, I want to be clear that math teachers will occasionally say “here’s something you need to know”–give a formula or fact that’s needed to explain or prove some other concept, when the students aren’t quite ready to grasp that fact. If you’re the type of teacher who just says “Sum of cubes, here’s the pattern”, that’s fine, too, if the students don’t have the math ready to understand it. However, Khan is doing the “here’s something you need to know” routine about the subject at hand. That’s not only weird, it’s bad. If you’re doing direct instruction without the why, then you say “Here’s what a sum of cubes looks like, here’s the pattern you use” and work some examples, Khan’s saying “well, in order to do this problem, here’s this little thing you need to know.” It’s unclear, and of course, Khan then undercuts it, saying that as far as he’s concerned, it’s debatable whether the student will need it at all. Simply dreadful lecture technique, and he doesn’t make it at all clear what he’s doing.

Worse, instead of just giving the pattern, Khan decides to “explain it”. You know, he’s not one of those schlubs that tells math without showing it.

It gets worse. Instead of just giving the pattern—because Khan, he don’t want to be one of those teachers, the ones that just show how but not why—the guy sets out to explain the pattern. Not prove it, god forbid, just explain where it comes from.


He just says “…it’s something you need to know. Let’s take the term” and he comes up with a2 – ab + b2 OUT OF THE BLUE. The student has absolutely no idea where this term comes from. Then he comes up with (a+b), again, not giving any reason. Then he multiples them and, in effect, says “Wow when we multiply these two terms that I seemed to just have pulled right out of my posterior, the product looks very similar to the problem we started with, the one I’m supposed to be showing you how to factor.”

Notice that, in the sequence of lessons, Khan has already covered division. But instead of doing a simple algebraic proof, he seemingly at random comes up with two factors. There’s no logical progression, no use of concepts already covered. Perhaps Khan does a better job when he thinks the students should actually know how to do the math.

As if that’s not bad enough, Khan finishes the multiplication of (a+b)(a2 – ab + b2) to get a3+b3 and then, I kid you not, says “Another way to look at this is to say that (a+b)(a2 – ab + b2) = a3+b3

Um, that’s not another way. That’s the same way.

Anyway. Speaking of ways. Rant over. But it’s truly dreadful lecture, and yet more proof that Khan is a fad, not a teacher.

I am hoping to have my students prove the pattern as part of our polynomial division section. This is the logic I’m planning on having them discover—let me know if I’ve missed something.

Fun with Factoring, Part II: Quadratic with Irrational Coefficients

On Friday, I set my precalc students an interesting problem, as part of a lesson on solving difficult equations:

I used the area model approach:


But two different groups of students came up with a solution that looked different on the face of it, yet had the same solution values:


So presumably one of them isn’t factored completely? But which one?

Here’s the solution using the quadratic formula, which is the method the math book used.


I was really quite surprised when the students came up with the second method, which they did with guess and check. Everyone agrees that the area model solution is “cleaner”, but I readily admitted that I didn’t know why we could get two solutions.

I don’t have any ego about appearing uncertain in front of my students. I know more than enough math, particularly for someone who isn’t a mathematician, and I”m a good problem solver. But I’d love to find out more about this. How were we able to get the same solutions but different factors?