Category Archives: math

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: wuquotenu2

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.

And 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.

Let me say this again: I am not a mathematician. All I’m trying to do is evolve my students’ understanding of the equal sign as they move into more advanced math. And a simple little lecture has helped. I’ll keep pushing at it.

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Teaching with Indirection

GeoTrigRep1Technology 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 twoGeoTrigRep2 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 GeoTrigRep3students 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 andGeoTrigRep4 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.

GeoTrigRep5I 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.GeoTrigRep6 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,GeoTrigRep7 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 GeoTrigRep8their 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, GeoTrigRep9defying 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 GeoTrigRep10I 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 GeoTrigRep11motion, 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.GeoTrigRep12

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.
GeoTrigRep13Is 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. GeoTrigRep14The 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

ModRatExp1

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:

ModRatExpGraph6

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.

ModRatExp2

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.

ModRatExpGraph1This 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):

ModRatExpEquation1

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).

ModRatExpGraph2

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:

ModRatExpGraph3

$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.

ModRatExpGraph5

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:

ModRatExpEquation2

“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 :

ModRatExpEquation3

…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”:

ModRatExpGraph4

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:

prodlinesonly

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:

prodlinesoutline

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.

prodlinesplot

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:

prodlinesworksheet

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)=xtransformed 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.

prodlinesthree

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:

prodlinesextensions

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” (-b2a) 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 -b2a 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 -b2a 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.

sumparabolaline

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(!).

sumparabolalineslope

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.

sumparabolathenut

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.sumparabolalinehandout

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 -b2a  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.


Great Moments In Teaching: The Third Dimension (part II)

In our last episode, the class was engaged in sense-making, thinking aloud, arguing aloud, just plain being loud, at the math behind this sketch:

“So up to now we’ve spent a lot of time in the coordinate plane thinking about lines. In the two-dimensional plane, x is an input and y is an output. A line can be formed by any two points on the coordinate plane. We’ve been working with systems of equations, which you think of as algebraic representations of the intersections of two lines. We can also define distance in the coordinate plane, using the Pythagorean theorem. All in two dimensions. So now we’re seeing how this plays out in three dimensions.”

I drew another point, showing them how the prisms were formed, how you could see a negative or positive value:

3dplot1112colored

More students began to see how it worked, as I’d call on a kid at random to take me to the next step. When I finished a second point, the chaos was manageable, but still loud.

“Yo, you want me to be honest with you?” Dwayne shouted over everyone.

NO!” I bellowed. “I want you to be QUIET!” Dwayne subsided, a little hurt, as I go on, “Look, this is a great discussion! I love watching you all argue about whether or not I’m making sense. But let’s stay on point! I got Wendy questioning whether or not I know what I’m doing, Dwayne howling every time he loses attention for a nanosecond….”

Teddy jumped in.

“Here’s what I don’t understand. How come you have to draw that whole diagram? We don’t have to do that with the usual graph….in, what, two dimensions? So why do we have to do it with three-D?”

“Great question. Here’s why. Go back to my classroom representation. According to this, the Promethean is a quarter inch from Josh, Hillary and Talika in the front row. Was anyone thinking that I’d drawn it wrong?”

“I learned this in art!” Pam, also up front and up to now watching silently, said, while comments around nearly drowned her out. I hushed everyone and told her to say it again. “It’s like…we need to draw it in a way to make our brains see it right.”

“That’s it.”

“My brain hurts to much to see anything!” Dwayne moaned.

“But we don’t have to do it with, you know, x-y points.” Natasha.

“Good! Let’s go back to two dimensions. If I want to plot the point (2,3), I’m actually plotting the lines x=2 and y=3, like this:

Alex said “Oh, hey. There’s a rectangle. I never saw that before.”

“You never seen a rectangle before?” Dylan. I ignored him.

“Right. The point is actually the intersection of the two lines, forming a rectangle with the origin.”

Alex again: “Just like this one makes a cube…”

“..prism…”

“a prism with the intersection of the three points. But how come you have to draw it? I don’t have to draw a rectangle every time I plot a point.”

“But three dimensions make a single point much more ambiguous.”

Dwayne sighed loudly and held his head. “This ain’t English class. I can’t handle the words.”

“Ambiguous–unclear, able to be interpreted multiple ways. Let’s start with this point:

3dambiguouspoint

“What are the coordinates of this point?”

“Easy,” said Wendell. “Just count along the lines.”

“Okay. How about (-1, -1, 2)?” I count along the axes to that point.

“Yeah.”

“How about (8, -10, 11)?

“They can’t be both!” protested Wendy.

“How about (2, -4, 5)?”

While I listed these points, I followed along the axes, just as Wendell suggested.

“How can you have three different descriptions of one point?” Josh asked.

“Exactly.”

“But that means there would be three different cubes…prisms?” Manuel.

“Yep. Let’s draw them.”

Something between controlled chaos and pandemonium dominanted as I drew–with class participation–three paths to that point, and the cubes. With each point, I could see again that increasing numbers were figuring out the process–start with the intercepts, create the two-dimensional planes, join up the planes.

When all three were finished, I put them on screen one after the other.

Sophie, ever the skeptic, “Those can’t be the same point.”

Wendy: “I’ve been saying that.”

Arthur stood up. “No, you can see!” He came over to the Promethean and I gave him the pen. “Look. Here’s the original. Start at the origin, go two to the left, and one up. Each one of the pictures” and he shows it “you get to the point that way. So the point is the same on all the pictures.”

Sophie was convinced.

Dylan: “So how come it’s not just (-2, 1)?”

Arthur looked at me. “It’s not (-2,1) but it is (-2, 0, 1).” I replied.

“Oh, I see it!” Wendell came up. “See, you go along the x as negative 2. Then you don’t go along the y. Then you go up 1 on z?”

“You got it.”

“Can you do it for x and y, without z?” Josh.

“Take a look. Let’s see.” And with many shouts and much pandemonium, the class decided that the point could be plotted as (-3,1,0) or (-3,1) on the two dimensional plane.

“Whoa, there’s lots of possible ways to get to the same point.”

“Can you figure out a way to count how many different ways there are to plot the point?” asked Manuel.

“That’s a great question, and I don’t know the answer. My gut says yes, but it’d be a matter of combinatorics. Outside the scope of this class.”

“Thank god,” said Wendy, and I shot her a look.

In other classes, I let them work independently on the handout at this point, but 4th block is crazy loud and easily distracted, so I brought them back “up front” to check their progress every 10 minutes. The whole time I was thinking, man, I’ve controlled the chaos and the skeptics this far, do I dare do the last step? Or should I keep it for tomorrow?

Here’s where the performance aspect kicked in: I wanted a Big Close. I wanted to bring it all together. Even if the risk was losing the class, losing the tenuous sense of understanding that the weaker kids had.

I wanted the win. After they’d all sketched three prisms, I started up again.

“I began this lesson with a reminder of two dimensional planes. The only thing we have left is distance.”

2ddistance

“How do you find the distance of a prism?” said Francisco.

“What would the distance be?”

I drew a prism:

prismdistance

“Oh, okay,” said Sanjana. “So the distance would be to the corners.”

“Yeah, the lower left is the origin, right? And the top right is the point,” said Sophie.

prismvertex

“Now, you learned this formula in geometry. Anyone remember?” I look around, and sigh. “Really, geometry is a wasted year. Does anyone see the right triangle that the distance is the hypotenuse to?”

Dwayne sighed. “God, Hypotenuse! I don’t…”

“HUSH. Tanya?”

Tanya frowned. “Would one of the legs be the….the height?”

“But where’s the other leg?” asked Jenny. “It can’t be the length or the width.”

“No. It’s across the middle,” and I drew the second hypotenuse and labeled everything.

prismdoubletriangles

“So now you can see that the x coordinate is the length, the y is the width, and z is the height. I just labeled the other distance w. So how do I find a hypotenuse length?”

Relative silence. I growled.

“Come on, it’s the mother ship of geometry. In fact, I mentioned it earlier.”

Still silence.

“Keerist. You all bring shame to your families.”

“Wait, do you mean the…the thing. a squared plus b squared?”

“The thing?”

“Pythagorean theorem!” half the class chorused.

“Oh, NOW you know it. So how can I use that here?”

“x2 plus y2 equals w2,” offered Patty.

“Yes, and the other one would be “w2 plus z2 is distance, squared,” said Dylan, who’d decided he couldn’t disrupt class, so he may as well participate.

“Great. We have two equations, right? Kind of like a system.”

“But there’s way too many variables. We only have two, right?” Sophie frowned.

“Great question again. To solve a system, we need as many equations as we have variables. But since we aren’t using any specific values at all in this discussion, we aren’t looking for a full solution.”

prism2equations

“So what are we looking for?”

“The EXIT!” shouted Dwayne.

I howled back, “Exit is in 15 minutes. BE QUIET!–Again, a good question. What we do in math sometimes is look for meaningful algorithms that can be formalized into useful tools. Like in this case. Right now, I’d need to go through several steps in order to find the distance of a prism. I want it to be simpler. How can I simplify or restate a system? Natasha?”

Natasha, tentatively, “We can add them up.”

“OK, like combination. Anything else? (SHUT UP DWAYNE!) Natasha? No? Fine. Dwayne?”

“Who, me? I don’t know anything!”

“Bull crap. I scrawled on a new page:

x=3y
2x + y = 14

“What do you do?”

“Wait. You mean where you…oh, ok, you put the 3y in for the x and then multiply by 2.”

“Try not to be shocked by your own comprehension. That’s correct.”

“Yes,” Dylan said. “You can substitute. How come you never have a substitute?”

“Excluding his extraneous crap, Dylan has also stumbled onto truth. Substitution. And looking at the equation, I see an opportunity. An isolated opportunity, even. We have two variables that aren’t length, width, and height, right?”

“Yeah. w and distance,” Alex offered. “But we’re trying to find the distance.”

“Exactly. What I’d like to get rid of, really, is that pesky w. And oh. Hey.”

I drew a circle around x2 plus y2 and pointed it at w2 in the second equation

prismsubstitution

Manuel got it first. “Holy SH**!”

Teddy, Alex, and Sophie were right behind him verbally, although Prabh and Sanjana had already figured it out and were rapidly taking notes, working ahead of me.

As I wrote down the final equation, they were shouting it along, and eventually all the class joined in: “X2 PLUS Y2 PLUS Z2 = DISTANCE SQUARED!” and as I finished it up, hand to god about a third of the class clapped madly (while the rest looked on in bemusement).

I bowed. I don’t, usually.

“But where’d the w go?” asked Josh in bewilderment.

“That’s the point, the system substitutes x and y so you only have to use the length and width and height!” shouted Manuel.

“So you never have to find w!” added Sophie.

prismfinalformula

“Pretty cool, huh?”

And the bell rang.

How to explain the adrenaline rush all this gave me? It took me a good hour to return to earth. They clapped! Not all of them, but so what?

So much of my time is spent slowing down math to be sure everyone gets it. I rarely can really engage and challenge the top kids “up front”. I give them challenges at other times, and they usually like my lectures, but I don’t often have the opportunity teach something in a way that makes sense to the less advanced but still captures intellects at the high end.

Here, it happened. The top kids understood that I’d piece by piece revealed that 3-dimensions are just an extension of the two dimensional system they all knew, but had never thought of that way. Not only did I reveal it, but I did so while using systems, something they’d just been working with. They were admiring the artistry. They got it.

And I did it all with Dwayne and Dylan yipping at my heels.

Epic.

(Here’s an actual promethean shot from that day. The rest of them I rebuilt.)

3dexampledistance


Not Negatives–Subtraction

In summer school, I’m teaching what used to be known as pre-algebra and happily, my colleagues had a whole bunch of worksheets that I got on a data stick. Very nice, and the curriculum was very good, leaving me time to tweak but not spend all my time inventing.

It’s not like the curriculum was a surprise: integer operations and fractions played a big part.

Of course, when we math teachers say “integer operations”, we mean “operations with negative integers” because while we don’t really care all that much if they’ve memorized their plus nines and times sevens (sorry, Tom!), kids that don’t fundamentally understand the process of addition are usually un-included by high school.

But negative numbers are one of those “Christ, they’ll never get it” topics. I don’t reliably have an entire class of kids who answer 9-11 with -2 until pre-calculus. I’m not kidding. They say 2, of course. But not negative 2. And if you give them -3-9, they will decide it’s 12 or -6 or, god forbid, 6. But not -12. They’re actually not terrible at subtracting negatives, provided that it’s subtracted from a positive. So they know 9-(-12) is 21, but have no idea what -9-(-12) is, and wildly guess -21.

I’ve suddenly realized that negative numbers aren’t really the problem. Subtraction causes the disconnect, as a result of the tremendous bait and switch we pull when moving from basic math to the abstractions needed for advanced math.

In elementary school, kids learn addition and subtraction. They are not told that they are learning addition and subtraction of positive integers. Nor are they told that they are only learning subtraction when the subtrahend is less than the minuend and, by the way, we need new terms. Those are horrible. In fact, kids are told that they can’t subtract in these cases.

subtractionuntruths

At no point are kids told that everything they’ve been taught is temporary, and that much of it will become irrelevant if they move into advanced math. Consider the big fuss over Common Core subtraction, which is all about an operation that has next to no meaning in advanced math other than grab your calculator. (No, this isn’t an argument pro or con calculators, put your hackles down.) Or consider the ongoing drama over the aforementioned “math facts memorization” which, frankly, gets turned ass over tincups with negatives and subtraction.

Common Core requires that sixth grade math introduce negatives. Along with ratios, rates, fraction operations, and statistical analysis, all tremendously complicated concepts. In seventh grade, things get serious:

ccseventhgrade

Never mind that most non-mathies would clutch their pearls at the very thought of parsing these demands, or that these comprise one of nearly twenty standards that have to be covered in seventh grade. Leave that aside.

Focus solely on NSA1B and NSA1C which, stripped of the verbiage, define the way we math teachers reveal the bait and switch.

So first, you teach the kids about these negative numbers and how they work. Then you show them that okay, we kind of lied before when we taught you that addition always increases. Actually, the direction depends on whether the added value is positive or negative.

But that’s it! That’s all you have to know! Just this one little thing. So negative numbers allow us to move in both directions on the number line.

And subtraction? Piffle. Because it turns out that (all together now!) Subtraction is addition of the opposite. Repeat it. Embrace it. Know it. Then everything makes sense.

So we teach them these two things. Yeah, we lied about adding because we had to wait to introduce negative numbers. But there’s this one little change. That’s all you have to know! because subtraction is a non-issue. Just turn subtraction into addition and funnel it all through the same eye of the same needle. Dust your hands. Done, baby.

Well, not done. As I said, we all know that negative numbers are brutal. We build worksheets. We support the confusion. We do what we can to strengthen the understanding.

But over the years, as I started teaching more advanced math, I realized that subtraction doesn’t go away. Subtraction is essential. It’s the foundation of distance, for starters.

And what the standards don’t mention is that introducing negative numbers changes subtraction beyond all recognition. The people who “get” it are those who reorder the integer universe spatially. Everyone else just stumbles along.

Until this summer, I never addressed this issue. I’m pretty sure most math teachers don’t, but I welcome feedback.

How do we change subtraction?

For starters, we violate the rule they’ve been taught since kindergarten. Turns out you can subtract a bigger number from a smaller number. (And, when a kid asks, “Well, in that case, how come we have to borrow in subtraction?” we teachers say…..what, exactly?)

But that’s just for starters. Take a look at the integer operations, broken down by sum and difference. (Much time is spent on teaching students “sum” and “difference”. More on that in a minute.)

addopsunmatched

So first, a row of numbers like this brings home an important fact: the Commutative Property ain’t just for mathbooks. This provides a great opportunity to show students the relevance of seemingly abstract theory to the real world of math.

But notice how much simpler the addition side is. I color-coded the results to show how discombobulated the subtraction pairs are:

addoppsmatched

Middle school math teachers spend much time on words like sum and difference, but I’m not entirely sure it helps.

For example, consider the “difference” between -9 and -5, which is -4. First, -5 is greater than -9, a complicated concept to begin with–and -4 is greater than both. And–even more confusing to kids taught to limit subtraction–none of those relationships matter to the result.

So -9 – (-5) = -4. Which is the same as adding a positive 5 to -9. So the difference of -9 and -5 is the same as the sum of -9 and 5.

Meanwhile, -9 – (-5) is subtraction of a negative, which we have hardwired kids to think of as “adding”–which it is, of course, but adding in negative-land is subtracting. So what we have to do is first get kids to change it to addition, then realize that in this case, the addition is a difference.

It’s not illogical, if you follow the rules and don’t think too much. But “follow the rules and don’t think too much” works for little kid math. As we move into algebra, not so much–we discourage zombies. Math teachers are always asking students, “Does your answer make sense?” and how can a student answer if subtraction makes no sense?

One of the things I’m wondering about is the end result of the operation. Any two numbers have a difference and a sum, all expressed in absolute values. 9 and 5 have a difference of 4 and a sum of 14, and no matter what combination of sign and operation used, the answer is the positive or negative of one of these two. So I ordered them by the end result.

addoppsendresult

Notice that P+P, N+N, P-N, N-P are ultimately collective sums. No matter the relative size, P+P and P-N move to the right, N+N, N-P move to the left, and result in a positive or negative sum of the two terms.

That looks promising, but I’m not sure how to work with it yet, particularly given the confusion of the actual meanings of sum and difference.

Here’s what I’ve got so far, and how I’m teaching it:

  1. What students think of as “normal” subtraction is actually “subtraction of a positive number”, where the subtracted number is smaller. Subtraction of a positive number always involves a move to the left on the numberline.
  2. In subtraction, the starting value does not change the direction of the operation–that is, -9 – 5 and 9 – 5 will both go to the left.
  3. The starting value must not change. This is a big deal. Kids see -9-5 and think oh, this is subtracting a negative so they change the -9 to 9. No. It’s subtracting a positive.
  4. Please, please PLEASE sketch it out on a numberline. Please? Pretty please?

Hey, it’s a start. I also use a handout I built six years ago, during my All Algebra All The Time year (pause for flashback) and has proved surprisingly useful, particularly this part:

I am constantly reminding kids that subtraction is complicated, that the rules changed dramatically. Confusion is normal and expected. Take your time. I am seeing “success”, with “success” defined as more right answers, less random guessing, more consistent mistakes in conception that can be addressed one by one.

I don’t know enough about elementary and middle school math to argue for change, except to observe that much more time is needed than is given. I once took a professional development class in which a math professor covered an abstruse explanation of negatives and finished up by saying “See? Explain it logically and beautifully. They’ll never forget it again.” We laughed! Such a kneeslapper, that guy.

But I’m excited to get a better sense of why kids struggle with this. It’s not the negatives. It’s subtraction.


Great Moments in Teaching: The Third Dimension (part I)

“How many other dimensions are there?”

“Well, four, according to Einstein, and five according to Madeline L’Engle, if you’ve read A Wrinkle in Time.”

“I have!” Priya’s hand shot up. “It’s a tesseract!”

I was impressed. Not many girls read that classic anymore. “But we’re going to stick to three dimensions.”

“Isn’t real life three dimensions?” asked Tess.

“Yes. But if you think of it, up to now, we’ve only been working in two. We’ve spent a lot of time in the coordinate plane thinking about lines. In two dimensions, a line can be formed by any two points on the coordinate plane. We’ve been working with systems of equations, which you think of as algebraic representations of the intersections of two lines. We can also define distance in the coordinate plane, using the Pythagorean theorem. All in two dimensions.”

Now, no mocking my terrible art skills” and I put up this sketch, the drawing of which occurred to me the night before, and was the impetus for the lesson.

3dclassroom

Everyone gasped, as they had in the previous two classes. My instincts about that clunky little sketch proved out, beautifully. No clue why.

“Holy sh**,” groaned Dwayne, the good ol’ country boy who offered to paint my ancient Honda if I gave him a passing grade. He doesn’t like math. He’s loud and foul and annoying and never shuts up. That last sentence is a pretty good description of me, so I’m very fond of him. “What the hell is that? Get it off the screen, it hurts my eyes.”

“That is awesome,” offered Talika, a senior I had last year for history. “How long did it take you to draw that?”

“What is that white stuff spread everywhere? Did someone get all excited?” asked Dylan, a sophomore whose mother once emailed me about his grade, giving me the pleasure of embarrassing him greatly by describing his behavior.

“Ask your mother,” I replied, to a gratifying “ooooo, BURN!” from the class, who knew very well what had happened.

“How did you draw that?” asked Teddy, curious. “It’s not ordinary graph paper, right?”

“No, it’s isometric paper, which allows you to draw three dimensional images. So…”

“This is really stupid,” said Dwayne. “I’ve taken algebra 2 three times and no one’s ever taught me this.”

“Best I can tell, no one’s ever taught you anything , and not just not in algebra 2,” I replied, earning another “Oooooo” from the class and an appreciative chuckle from Dwayne.

“It’s weird, though, because in two dimensions, you start in the middle,” offered Manual, who was consulting with Prabh, another bright kid who rarely speaks.

“That’s a good point! For example, if we were going to plot seating positions in this room in two dimensions, we’d start with Tanya,” I said, moving to the class center and indicating Tanya, who looked a bit confused. “So Tanya would be the origin, and Wendell would be (1,0), while Dylan would be (1,-1).”

“I’m not negative!” Dylan said instantly, talking over my attempt to continue. “You’re saying I’m negative. You don’t like me.”

“Hard to blame anyone for that,” said Wendell who is considerably more, er, urban than Teddy, with pants down to his knees and a pick that spends some time in his hair. Despite his occasional class naps, he maintains a solid C+, and could effortlessly manage a B if I could just keep him awake. “S’easy, dude. It’s like one of those x y things, like we’re all dots on the graph.”

“You’re one down and one to the right of me,” pointed out Tanya.

Dylan was interested in spite of himself. “So Talika’s, like, (0, 4)?”

“Yes,” several students chorused.

“Then I’m negative 8.” said Dwayne, unhappy with any conversation that doesn’t have him at the center.

“More like….(-2,2), yeah,” says Cal.

Ben speaks up, “But how come Tanya’s at the center for mapping the room’s people, but your sketch is, like, from the left?”

“Or right?” Sophie, from the back.

“Or is it….outside?” asks Manuel.

“Yes, it’s kind of like you’re standing on a desk in Ms. Chan’s room and the walls are transparent,” says Ben, more certainly. Ben is repeating Algebra 2 after having taken it with me last semester. Very bright kid who clowned incessantly, confident in his ability to learn without really trying, only to learn that Algebra 2 was different from other nights, and he wasn’t finding the afikoman. I advised him to repeat. The big sophomore not only agreed, but specifically asked to repeat with me. His attitude and behavior is much improved. I ran into him while walking across the courtyard a few weeks earlier, and he said “I just realized I was Dwayne and Dylan combined last semester, and it’s so embarrassing. I’m really sorry.”

“I’m not enough of an artist to know if I could have drawn this any other way. It just seemed intuitive to me last night, when I came up with the idea.”

“See, I knew it,” trumpeted Dwayne. “You’re making this up!”

“Yeah, I know this isn’t in any algebra book” said Wendy, a sophomore whose excellence in math is often hard to discern beneath her complaints. “This is just some weird thing you’re doing to make us think about math.”

I picked up at random one of the four algebra 2 books sitting on my desk (I’m on the textbook committee) and walked over to Wendy’s desk, opening it to the “Three Dimensional Systems” chapter. She looked, and said “Ok, maybe not.”

“So just as we can plot points in two dimensions, we can plot points in three. Take Aditya here,” my TA, who was watching the circus in amusement. “How could we represent him as a point on my graph?”

Teddy said instantly, “Yeah, I’ve been working that out. I can’t figure out which the new one is, and what do we call it? Where’s x, where’s y?”

Sanjaya said, “I think the part along the ground is x. Like if you go along the bookshelf?”

“Like this?”

3dstartoutline

“Yes,” Sanjaya said, confidently. “That has to be x. So you could count to Aditya, right?”

“Count which way?”

“The bottom!” “The bottom line!” “the bottom..axis, thing. The X!” comes a chorus of voices.

I start counting, and while I do, Sophie objected. “But hang on. I still don’t see what the new thing, direction, is. What’s the third?”

“Up! said Calvin, who rarely participates and often tunes out so far he can’t keep up. But he was watching this with interest. “You know how the class map with Tanya was going north and south and east and west. But it’s all flat, like. This picture has an up.”

“Yeah!” Ben got it. “Cal’s right.”

Dwayne has begun to grasp this. “So you can’t just draw a line? You have to follow along the…things?”

“The axes.” I finish counting along the “bottom” axis and go over to my bookshelf in the furthest corner of my room. “So the sketch starts here…QUIET! One conversation at a time, and I’m the STAR here. The origin starts at this bookshelf. I am walking along the wall, hugging it, on my way to Aditya. Does everyone see how they could track my progress on the axis?”

“YES!” from various points of the room.

“What the hell are you doing?” Dwayne is watching me carefully hug the wall.

“Everyone except Dwayne?”

“YES!” much louder.

I walked along the wall to the table where Aditya sat (fourth along the wall) and stop.

“So now what?”

“You’re there.”

“No, not yet.” countered Nadine. You have to go out….” she waved me towards her. “this way.”

“Yeah, towards Aditya,” this from Talika.

I stepped out 2 steps or so. “That all?”

Josh frowned. “Yeah. You’re there. Except…”

“UP!” Sophie shouted from the back. “That’s the third axis!” General approval reigns loudly, until I wave them all quiet, or try to.

“You go up 4!” Teddy shouted.

“OK. So Aditya is about 40 units out along the wall, 2 units out towards…the door, and 4 units up. Yes?”

“Yes!”

“So let’s draw that.”
3dfinalprism

Of course, while I’m drew this, general mayhem is ongoing with my back turned. I shouted “QUIET! or “Could someone stick a sock in Dwayne?” a few times.

“Wow, so it’s a…cube?”

“A prism, yes. So here’s what we’ve done. We’ve taken the two dimensional x-y coordinate plane and extended it.”

“We extended it up,” from Sophie.

“Yes. And now, instead of a rectangle, we have a three-dimensional rectangular prism. And we can describe things now in three dimensions. But we can do more than that. So let’s step away from my classroom sketch….”

3dcoord

“Whoa. What’s that?” Dylan.

“Man, that’s f***ed up. I just started to get this, and now you’re….” Dwayne, of course.

“No, it’s fine,” Manuel said. “It’s just like the whole thing moved to the center.”

“Oh, I see. It’s like there’s four rooms, all cornered.” Wendell.

“Yes, exactly. Except now, you want to stop thinking about it as a room and think of it as a coordinate plane. As Sophie says, the new plane is the up/down one. So the old x is now here. The old y is now here. The z is the straight up and down one. I think of it as taking the 2 dimensional plane and kind of stepping back and looking down on it.”

“That’s just….”

“DWAYNE BE QUIET. One thing to remember: when you see a 3-dimensional plane, they may be ordered differently. There’s a whole bunch of rules about it that make potentially obscene finger orientations, but I promise I won’t test you on that.”

“So let’s say we’re plotting the point (8,4,5). I’m going to show you how to do it first. Then I’ll go through why. Start by plotting the intercept along each of the planes.”

“Man, does anyone else get this?”

“YES. Shut up, Dylan,” says Natasha.

3dplot845beg

“The trick to remember when you’re graphing in 3-d is to stay parallel to the axis you’re drawing along. So never cross over the lines when plotting points. Now let’s add the yz and xz planes.

“What? This is weird. Why are you drawing so many rectangles?” Patty, frowning.

3dplot845mid

“What you have to visualize is that it’s like we’re drawing sides. So far, I’ve drawn,” I look around and grab three of my small whiteboards, “the bottom and two of the sides. Hold this, Natasha, Talika.” and I build the walls. The kids in the back stand up and look over.

“Oh, I see,” Teddy again. “You’re drawing the prism again.”

“Right. It’s just looking different because the axis is in the center.”

“You do all this just to plot one point?” Sophie, ever the skeptic.

“Yes, but remember this is more just to illustrate, to see how you can extend the dimensions. So after you draw the three sides, joining the intercepts for xy, yz, and xz intersections, you extend those out–again, along the lines.”

3dplot845

“So the point we’re graphing is going to be at the vertex, the intersection of the three planes, the furthest point from the origin–just like in two dimensions, the point is at the intersection of the two lines.”

“That’s really complicated.” Wendy sighed.

“No, it’s not” “Don’t you see the…” Ben, Manuel, Teddy, Wendell and others jump in at the same time, while Dwayne bellowed, Wendy and Tess were asking questions of the room, and, as the writer says, pandemonium ensued. It was a shouting match, yes, but they were shouting about math. The Naysayers, the Doubters, and the Apostles were all marking their territory and this was no genteel, elegant, “turn and discuss this with your partner”, no think-pair-share nonsense. This was a scrum, a brawl, a melee conducted across the room with the volume up at 11—but just like any good fight, there was order beneath the chaos, a give and a take at the group level.

And for you gentle souls wondering about the quiet kids, the introverts, the shy ones who need time to think, they were enthralled, watching the game and making up their mind. It may not look like everyone gets time to talk, but pretty much every time you read me call on a kid, it’s a quiet one. And I shush the room. Then the quiet kid sits there in shock as he or she realizes oh god, I’ve got the mike and I can’t be a spectator anymore.

Anyway, the story goes on with a second great moment, but I’m getting better at chunking and this half had too many details I didn’t want to give up. I’ll stop here for dramatic effect. Because oh, lord, I was high as a kite in this moment, watching the room, realizing I was riding a tremendous wave of energy and excitement. Yeah. ME. On Stage. Making Drama.1

Now I just had to come up with a good ending.

*********************
1I’m not congratulating myself, saying I’m proving kids with the great moment. No, the great moment is mine. I’m standing there going oh, my god, this is a great moment in teaching, in my life. For me! The kids, hey, if they liked it, that’s good.


Making Short Math Tests

A trig student told me he was hanging out with a group of friends, some who’d had me, some who hadn’t. One was bitching about his four page test.

My students snorted. “Ed’s tests are double sided single pages. Once we had a three page test, but only for the space.”

A debate ensued, and those with the widest range of math teacher experience agreed: My tests are shortest, and hardest.

I’m not sure what this means. I don’t try to make my tests difficult. But periodically I’ve perused other teachers’ tests off the copier, and…wow. They are four or five pages. The questions are straightforward. They are typically of what I would call rote difficulty–they could have peeled off a few pages of one of these tests. When math teachers snort about regurgitating algorithms, these are the tests they have in mind.

I used to have more traditional looking tests, but even back then I wasn’t an exact match for typical. Once I started down the multiple answer path, it became even easier to wander miles off the reservation. But without question, multiple answer tests make it easier to assess understanding on multiple topics—thus shorter tests.

This semester, I finally decided to start my class with a functions unit. Regular readers know that I’ve been beefing up my functions curriculum, after initially (as a new teacher) giving it a perfunctory treatment. But I still began the year with linear equations. This last semester start, though, I went back to the textbooks. Why do they always start with functions? I finally started to grasp the logic: beginning with functions allows the teacher to work with transformations, parent functions, mapping, as well as challenging algebra (solving for x in a square root or quadratic function, etc).

So I mapped out a basic plan:

  • Function definition: domain, range, criteria
  • Function notation
  • Transforming functions
  • Four parent functions (line, quadratic, square root, absolute value). I told them we’d be introducing lines to ignore them until the next unit.
  • Transforming parent functions.
  • Solving for input and output

I originally planned to introduce inverses, but the kids were maxed out. This was a much tougher first unit than linear equations, and a good chunk of the lower ability students were struggling with the abstractions. Generally, I was pleased.

Some new questions from my first functions unit test–which was a single page, double-sided.

ftgraphquest

Notice I slipped in a couple function notation questions? That’s how I save space.

Here’s a mapping question:

ftmapquest

Again with the function notation! Am I the only math teacher whose kids simply can’t compute the difference between “f(3) = ” and “If f(a) = 3, a=”???? I do my best to beat it into their heads.

Here’s another way I use space effectively, I think:

ftsqfunct

So a graph, some free-response algebra, and conceptual understanding. (Most of them DO NOT understand how to read graphs, and missed d.) Time and again, I had to show the students how to write the equation, but they are learning how to isolate. Relatively few order of operations errors.

I didn’t ask them to graph this next one, but again, practice at setting up an equation to find the input given the output. Another plus of doing functions early is an introduction to quadratics, which is a tremendously tough Algebra 2 unit.

ftquadquest

Hands down, this next question had the weakest response. The strongest students understood it, but many of the same students who were able to graph the square root were flummoxed by this one. Go figure. But again, notice that I assess several different knowledge areas with the same question.

ftabvalfunct

A New Quiz

I don’t usually discuss my quizzes, which are often relatively straightforward compared to my multiple answer assessments. But I created a quiz on Thursday that I’m really pleased with. It’s my second quiz for linear functions. The students have learned the three different linear forms. The first quiz covers slope intercept and standard form, which are the forms for modeling situations. This one focuses on point slope and creating equations from points, as well as parallel and perpendicular points. We actually did much more modeling of real-life situations than this quiz shows. Usually, my quizzes are a very reliable guide to what the students have done in the previous week, but this was an attempt to see how well they could transfer knowledge and work several concepts in combination.

The quiz itself, I think was cool. I stole the nuggets of two ideas from textbooks, but the presentation and questions are mine own.

lmcricketsquest

I’ve seen this crickets question in both Pearson and Holt Harcourt books. I built the graph on Desmos, and was dismayed that a number of kids counted the barely visible lines, rather than use the points. But most of them didn’t.

Notice that this is a relatively easy question. I didn’t want to focus on the algebra needed to find the y-intercept. I wanted them instead to look at the patterns (the 120 chirps is exactly halfway between 0 and 240), and think about what graphs say vs. what they mean. Most kids confused question c and d, explaining that the temperature was too cold for chirping, or that the crickets died. But after a few pushes, they go…”negative chirps?” which is fun.

Here, I’m just testing their fluency:

lmfluencyquest

Lots of room for self-correction. One student asked me why all her solutions were “Neither”, and I suggested that perhaps she should check her algebra, where she’d handled a negative value incorrectly. Other students plotted the points incorrectly and, because they were only able to find slopes from the graphs, couldn’t catch their mistake–thus giving me an opportunity to reiterate the importance of using different methods to validate and self-correct.

As part of the work leading up to this quiz, they’d derived the Celsius to Fahrenheit conversion algorithm, given two points. I decided to give them the formula to see if they could recognize the errors in verbal description and work a solution using fractions.

lmcelsiusquest

And then my favorite:

lmhelicopterquest

I got the basic idea from my new favorite textbook series, Big Ideas Math, then played with the goals a bit. Big Ideas has wonderful scenarios.

As always, if you spot any errors or ambiguities, let me know.


Jake’s Guest Lecture

Our well-regarded local junior college is the top destination for my high school’s graduates, a number of whom are more than bright enough to go to a four-year university but lack the money or the immediate desire to do so. Case in point: Jake, my best case for the hope that subsequent generations of Asian immigrants will adopt properly American values towards education, now at the local community college with a 4.0 GPA. He earned it entirely in math classes, having taken every course in the catalog–and nothing else. This from a kid who failed honors Algebra/Trig for not doing homework, and didn’t bother with any honors courses after that.

Jake visits four or five times a year, usually coming during class to see what’s up, working with other students as needed, then staying afterwards to chat. This last week he showed up to my first block trig class, with the surly kids who mouth off. We were in the process of proving the cosine addition formula.

The day before, I started with the question: “cos(a+b) = cos(a) + cos(b)?” and let them chew on this for a bit before I introduce remind them of proof by counterexample. A few test cases leads to the conclusion that no, they are not equal for all cases.

Then we went through this sketch that sets up the premise. I like the unit circle proof, because the right triangle proofs just hurt my head. So here we can see the original angle A, the original angle B, and the angle of the sum. Moreover, the unit circle proof includes a reminder of even and odd functions, a quick refresher as to why we know that cos(-B) = cos(B), but sin(-B) = -sin(b).

cosineadditiondiag

Math teachers often forget to point out and explain the seemingly random nature of some common proof steps. For example, proving that a triangle’s degrees sum up to 180 involves adding a parallel line to the top of the triangle and using transversal relationships and the straight angle.

Didn’t I make that sound obvious? You have this triangle, see, and you wonder geewhiz, how many degrees does it have? Hmm. Hey, I know! I’ll draw a parallel line through one vertex point! Who thinks like that? The illustration of a triangle’s 180 degrees is much more compelling than any proof.

So when introducing a proof, I try to make the transition from question to equation….observable. Answering the question requires that we define the question in known terms. What is the objective? How does the diagram and the lines drawn get us further to an answer?

Point 1 in the diagram defines the objective. Points 2 and 4 allow us to represent the same value in known terms–that is, cos(A) and cos(b). And thanks to some geometry that is intuitively obvious even if they’ve forgotten the theorem, we know that the distance between Point 1 and Point 3 [(1,0)] is equal to the distance between Point 2 and Point 4.

So I’d done this all the day before in first block, setting up the equation and doing the proof algebra myself, and the kids were lost. In my second block class, I turned the problem over to the kids at this point.

cosineadditionmath

The solution involves coordinate geometry, algebra, and one Pythagorean identity. No new process, nothing to “discover”. Familiar math, unfamiliar objective. Perfect.

I grouped the second block kids by 5 or 6 instead of the usual 3 or 4 (always roughly by ability), giving each team one distance to simplify (P1P3 or P2P4). Once they were done, they joined up with kids who’d found the other distance, set the two expressions equal and solve for cos(A+B). The group with the strongest kids were tasked with solving the entire equation, no double teaming.

Block Two kids worked enthusiastically and quickly. I decided to retrace steps and do the same activity with block 1 the next day. Which is when Jake—remember Jake? This is a story about Jake—showed up.

“Hey, Jake! You here for the duration? Good. I’m giving you a group.”

Jake got those who had either been absent or were too weak at the math to be comfortable doing the work. I kept a watchful eye on the rest, who tussled with the algebra. I tried not to yell at them for thinking (cos(A) + cos(B))2 = cos(A)2 + cos(B)2, even though they all passed algebra 2 (often in my class), even though I’ve stressed binomial multiplication constantly throughout the year but no, I’m not bitter. Meanwhile, Jake carefully broke down the concept and made sure the other six understood, while they paid much more attention to him than they ever did to me but no, I’m not bitter.

Result: much better understanding of how and why cos(A+B) = cos(A)cos(B) – sin(A)sin(B). One of my most hostile students even thanked me for “making us do the math ourselves” because now, to her great surprise, she grasped how we had proved and thus derived the formula.

And then she went on to ask “But we have calculators now. Do we need to know this?” She looked at me warily, as I’m prone to snarl at this. But I decided to use my helper elf.

“Jake?”

Jake, mind you, gave exactly the same answer I would have, but he’s just twenty years old, so they listened as he ran through the process for cosine 75 (degrees. 75 degrees. Jake’s a stickler for niceties.)

“But why is this better?” persisted my skeptic.

“It’s exact,” Jake explained. “Precise. When we use a calculator, it rounds numbers. Besides, who programs computers to make the calculations? You have to know the most accurate method to better understand the math.”

“Class, one thing I’d add to Jake’s answer is that depending on circumstances, you might want to factor the numerator, particularly if you are in the middle of a process.” and I added that in:

cosinefactor

“Yeah, that’s right,” Jake confirmed. “like if you were multiplying this, I can think of all sorts of reasons a square root of two might be in the denominator. But other times you need to expand.”

I suddenly had another idea. “Hey. How about if we use right triangles?”

“Like how?”

I sketched out two triangles.

“Oh, good idea. Except you forgot the right triangle mark.”

I sighed. “Class, you see how Jake is insanely nitpicky? Like he’s always making me write in degrees? He’s right. I’m wrong. I’ve told you that before; I’m not a real mathematician and they have conniptions at my sloppiness. But…” I’m struck by an idea. “I don’t need to mark it here! These have to be right triangles. Neener.” (I nonetheless added them in, although I left them off here out of defiance.)

cosinepythagexamp

“This is good. So suppose you want to add the two angles here. These right triangles have integer sides, but their angle measures are approximations. Let’s find those values using the inverse.”

Ahmed has his calculator out already. “Angle A is…53 degrees, rounded down. Angle B is 67.38 degrees.”

Me: “Just checking–does everyone understand what Ahhmed did?” I wrote out cos-1(35). “He used the inverse function on the calculator; it’s just a reverse lookup.”

” Let’s keep them rounded to integers. So 53 + 67 is 120 degrees, which has a cosine of ….what?” Jake paused, waiting for a response. Born teacher, he is.

By golly, my efforts on memorization have paid off. Several kids chimed in with “negative one half.”

“Meanwhile, if we multiply all these values using the cosine addition formula…” he worked through the math with the students, “we get -3365“.

Dewayne punched some numbers and snorted. “-0.507692307692. That’s practically the same thing!” .

I had another idea. “You know how I said you should look at things graphically? Let’s graph this out on the unit circle.”

cosdesmoscircle

Jake was pleased. “This is excellent. So where would cosine(A+B) show up? We need to find the sine of each to plot it on the circle.” We worked through that and I entered the points.

Isaac: “Yeah, Dewayne is right. The two points are the same on the graph!”

“But this is a unit circle,” Jake said. “Just a single unit. As the values get bigger….I wish we could show it on this graph. Could we make a bigger circle? Or that probably wouldn’t scale.”

“How about if we just show all the values for every x? We could plot the line through that point? From the origin?”

“What would the slope be?” Gianna asked.

“Yeah, what would the slope be? Rise over run. And in the unit circle, the rise is sine, the run is cosine, so…”

“Tangent!” everyone chorused.

Jake was impressed. “See, this is why I should have taken trigonometry. I never thought about that.”

“OK, so I’m going to graph two lines. One’s slope is the tangent of 120, the other’s is the tan(cos-1(-3365))), which is just using the inverse to find the degree measure and taking the tangent in one step. Shazam.”

cosdemotangentline

We then looked more closely at different points on the graph and agreed that yes, this piddling difference became visible over time.

“So the lines show how far apart the points would be for 120 and the addition formula number if you made the circle to that radius?” Katie asked.

“Yep. And that’s just what we can see,” Jake added. “The difference matters long before that point.”

When second block started, after brunch, Abdul rushed in, “Ahmed said we had a genius guest lecturer? Where is he?”

I faced a cranky crowd when I told them the genius had to go to class, so Jake will have to come back sometime soon.

*************************************************************************

Two months ago, Jake stopped by for a chat and I asked him about his transfer plans.

“Oh, I don’t know. Four year universities, I’ll have to take other classes, instead of what interests me.”

“You can’t be serious.”

“Well, maybe in a few years. But I have to wait a while for the computer programming classes I need to take, and the math classes are more fun.”

“Computer programming?”

“Yeah. That’s what I want to….what. Why are you laughing.”

“Do you know anything about computers?”

“No, but it’s a good field, right?”

“I think you’re one of the most gifted math students I’ve bumped into, and you’ve never shown the slightest interest in technology or programming.”

Jake sat up. “My professor told me that, too. He said I should think about applied math. Is that what you mean?”

“Eventually, probably, but let’s go back to why the hell you don’t have a transfer plan.”

“Well, should I go to [name of a local decent state university]?”

I brought up his school website, keyed in “transfer to [name of elite state university system]”.

Jake looked on. “Wait. There’s a procedure to apply to [schools much better than local decent state university]?”

“You will go to your counselor, tell her or him you want to put together a transfer plan. Report back to me with the results in no less than 2 weeks. Is that clear?”

“OK,” meekly.

Just five days later, Jake’s cousin, Joey, my best algebra 2 student, reported that Jake had a transfer plan started and was getting the paperwork ready.

So after this class, I asked him about transfer plans.

“Oh, yeah. I’m scheduled to transfer to [extremely elite public university] in fall of 2017. I’ve been taking all math classes, so I have a bunch of GE to take. But it’s all in place.” He grinned wryly. “I didn’t think I’d be eligible for a school that good.”

“And that’s just the guarantee, right?”

“Yes, I want to look at [another very highly regarded public]. Do you think that’s a good idea?”

“I do. You should also apply to a few private universities, just for the experience. It’s worth learning if they give transfer students money.” I named a few possibilities. “And ask your professors, too.”

“Okay. And you don’t think I should major in computer programming?”

“Do you know anything about programming right now? If not, why commit?”

“I don’t know. I never knew about applied math possibilities. It sounds interesting.”

“Or pure math, even. So you’ve got some research to do, right? And keep your GPA excellent with all that GE.”

“Right.”

“And at some point, you’re going to think wow, I never would have done any of this without my teacher’s fabulous support and advice.”

“I already think that. Really. Thanks.”

Just in case you think his visits pay dividends in only one direction.