Category Archives: pedagogy

The Structure of Parabolas

A year ago, I first envisioned and then taught the parabola as the sum of a parabola and a line.The standard form parabola, ax2 + bx + c, is the result of a line with slope b and y-intercept c added to a parabola with a vertex at the origin, with vertical stretch a.   This insight came after my realization that a parabola is the product of two lines (although I wrote this up later than the first).

I didn’t teach algebra 2 last semester, so I’ve only now been able to try my new approach. I taught functions as described in the second link. So the students know the vertex form of the parabola. Normally, I would then move to the product of two lines, binomial multiplication, and then teach the standard form, moving back to factoring.

But I’ve been mulling this for a few months, and decided to try teaching standard form second. So first, as part of parent functions, cover vertex form. Then linear equations. As part of linear equations, I teach them how to add and subtract functions.  As an exercise, I show them that they can add and subtract parabolas and lines, too.

So after the linear equations unit, I gave them a handout:parabolastructure

I don’t do much introduction here, except to tell them that the lighter graphs are a simple parabola and a line. The darker graph is the sum of the parabola and the line. What they are to do is explore the impact of the line’s slope, the b, on the vertex of the parabola, both the x and y values. We’d do that by evaluating the rate of change (the “slope” between two points of a non-linear equation) and looking for relationships.

Now, I don’t hold much truck with kids making their own discoveries. I want them to discover a clear pattern. But this activity also gives the kids practice at finding slopes, equations of lines, and vertex forms of a parabola. That’s why I felt free to toss this activity together. Even if it didn’t work to introduce standard form, it’d be a good review.

But it did work.  Five or six students finished quickly,  found the patterns I wanted, and I sent them off to the next activity. But most finished  the seven parabolas in about 40 minutes or so and we answered the questions together.

Questions:

  1. Using your data, what is the relationship between the slope of the line added (b) and the slope (rate of change) from the y-intercept to the vertex?
    Answer: the slope (b) is twice the slope from the y-intercept to the vertex.
    b2= rate of change
  2. What is the relationship between the slope of the line added (b) and the x-value of the vertex?
    Answer: the x value of the vertex is the slope of the line divided by negative 2.
    b⁄-2= x value of the vertex
  3. What is the relationship between the y-intercept of the line and the y-intercept of the parabola?
    Answer: they are the same.

Note: I made it very clear that we were dealing only with a=1, no stretch.

The activity was very useful–even some strong kids screwed up slope calculations because they counted graph hash marks rather than looking at the numbers. Some of the graphs went by 2s.

So then, they got a second handout: parabolastructure2

Here, they will find the slope (rate of change!) from the y-intercept to the vertex and double it. That’s the slope of the line added to the parabola (b!). The y-intercept of the line is the same as the parabola.

The first example, on the left, has a -2 rate of change from vertex to y-intercept. Since a=1, that means b=-4. The y-intercept is 8. The equation in standard form is therefore
x2 -4x + 8. In vertex form, it’s (x-2)2 +4.

Tomorrow, we’ll finish up this handout and go onto the next step: no graph, just a standard form equation. So given y=x2 -8x + 1, you know that the rate of change is -4, and the x-value of the vertex is 4. Draw a vertical line at x=4, then sketch a line with a slope of -4 beginning at (0,1).

This may seem forced, but students really have no idea how b influences the position of the vertex. I’m hoping this will start them off understanding the format of the standard form. If not, well, there’s the whole value of practicing slope and vertex form I can fall back on. But so far, it’s working really well.

By late tomorrow or Monday, we’ll be formalizing these rules and determining how an increase or decrease in a changes these relationships. So I hope to have them easily graphing parabolas in standard form by Monday. Yes, I’ll show them they can just plug x to find y.

Then we’ll talk about factored form, and go to binomial multiplication.

I’ll try to report back.

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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.


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.


Teaching Transformations

One of the most important new concepts in algebra 2 and beyond is the notion of transformation. That is, given the function f(x), we  can change any function’s position and growth by using the same instructions, much like giving directions from a map.

I’ve just introduced functions at this point in the calendar, so I’ve designed this activity to reinforce f(x) as a rule, that once a mapping is created, the mapping holds for all subsequent calls.

So just create a random table, one that’s simpler than anything I’d do in class. (One of the incredibly irritating things about blogging is that it’s insanely time-consuming to create images for publication that take next to no time at all to do on  a smartboard, but I never think of capturing images while on a smartboard.)

x f(x)
-3 2
-1 5
1 6
3 3
5 -1

That looks like this:

transbasepoints

So then I ask if this is f(x), what would f(x+2) look like? Someone brave will always say “Two to the right”.

At that point, I always say “This is a totally logical guess and one of the most annoying things in math from this point on is that your guess is wrong.” (I originally developed the concept of a parabola as the product of two lines as another way of explaining this confusing relationship. Confusing to normal people. Mathies think it makes sense, but they’re weird.)

I add a column to the table. “We start with x. Then we add 2. Then we make the function call. Note the function call comes after the addition of the value. This is important. Now, we have three columns, but we are starting with our x and that’s still our input value. We graph it against the outer column, the output value for f(x+2).”

x x+2 f(x+2)
-3 -1 5
-1 1 6
1 3 3
3 5 -1

I’ll ask how we can bring the -3 back in, and after some mulling, they’ll suggest that I add -5 to the table. So I add:

-5 -3 2

to the bottom. But I’ve been plotting points all along, so the kids can see it’s not going as expected.

transplustwo

“Yes, indeed. I’ll be teaching this concept in many ways over the next few months, and I ask you to start wrapping your head around this now. We have many ways of envisioning this. When working with points as opposed to an entire function, it might be helpful to think of it this way: Suppose I’m standing at -3, and I want to add two. This has the effect of me reaching to the right on the number line and pulling the output value back to me–to the left, as it were.”

I go through this several times. Whether or not students remember everything I teach, I always want them to remember that at the time, they understood the concept.

“So if standing on -3 and reaching ahead is addition and move the whole function to the left, how would I move the whole function to the right?”

If I don’t get a ready chorus of “subtract?” I know that I need to try one more addition example, but I usually get a good response.

“Exactly. So let’s try that.”

x x-2 f(x-2)
-3 -5 NS
-1 -3 2
1 -1 5
3 1 6
5 3 3
7 5 -1

transtableminusn

One year, I had a doubter who noticed that I’d made up these numbers. How did we know it’d work on any numbers? I told him I’d show him more later, but for now, imagine if I had a table like this:

x f(x)
1 1
2 2
3 3

etc.

Then I told him, “Now, imagine I put decimal values in there, fractions, whatever. Imagine that no matter how I change the x, the new value has an entry in the table and thus an output. So imagine I added 50. There’d be a value 50 ahead that I could reach forward or backwards.”

“In fact, we’ll eventually do all this with equations that are functions, instead of randomly generated points. But I start with points so you won’t forget that it works with any series of values that I can commit math on. Which isn’t all functions, of course, but that’s another story.”

“But if adding makes it go left and right, how do we make a function go up and down? Discuss that among yourselves for a minute or so.”

Sometimes a student will see that we’ve been changing x so far. Otherwise I’ll point it out.

“The function call itself is key to understanding this. If you change the value before you make the function call, then you are changing the input to the function. Simpler: you’re changing x before you call the function. But once the value comes out of the function, that is, once it’s no longer the input, it’s the….” I always wait for the class to chime in again–are they paying attention?

“Output!”

“Right. But output is no longer x. Output is”

“f of x!”

At this point, I call on a mid-level student. “So, Sanjana, up to now, we’ve been changing x before making the call to the function. See how the new column is in the middle? What could I do differently?”

And I wait until someone suggests making the column on the right, after the f(x).

x f(x) f(x)- 3
-3 2 -1
-1 5 2
1 6 3
3 3 0
5 -1 -3

transfxminusthree

I’m giving a skeletal version of this. Often the kids have whiteboards and are calculating all this along with me. I’ll give some quick learning checks in terms of moving to the right and left, up and down.

The primary learning objective for is to grasp the meaning of horizontal and vertical translations–soon to be known as h and k. But as an introduction, I define them in terms of function notation.

tranformationsshifts

 

We usually end this activity by combining vertical and horizontal shifts.

What would f(x-2)+ 3 look like? Well, you’d need another column.

x x-2 f(x-2) f(x-2)+1
-3 -5 NS
-1 -3 2 5
1 -1 5 8
3 1 6 9
5 3 3 7
7 5 -1 2

transcombined

I connect them this time just to show that one point is in both the original and the transformation.

Ultimately, this goes to transforming functions, not points. That’s the next unit, transforming parent functions. I have a colleague who teaches transformations entirely by points. I start down that path (not from his example, just because that’s how this works), but the purpose of transformations, pedagogically speaking, is for students to understand that entire equations can be changed at the unit level, without replotting points. At the same time, I want the students to know that the process begins at the point level.

Over time, the students start to understand what I often call inside and outside, or before and after. Changes to the input value affect the x, or the horizontal because they occur before the function is called. Changes to the output value affect the y, or the vertical, because they occur after the function is called. Introducing this on a point by point basis creates a memory for that.

At best, this lesson functions as more than just a graphing exercise, something to introduce vertical and horizontal shift. It should ideally give students an understanding of the algebra behind it. Later on, when they are asked to solve equations like:

Find f(a) = 32 for f(x)=3(x-2)2+5

Weaker students have trouble with understanding order of operations, and a memory of “inside” and “outside” the function can be helpful.

If I were writing algebra 1 curriculum, I’d throw out quadratics, introduce a few parent functions, and teach them function notation and simple transformations. It’s a complicated topic that they’ll see all the way through precalc, at least.

I’ll discuss stretch and its complexities in another post.


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.


Statistics of Slaves

I vowed to spend May documenting all the curriculum I’ve built that’s kept me from writing much. But writing up lesson always takes forever, so I don’t know how much I’ll get done.

I’ve revamped a lot of my history course since I first taught it in the fall of 2014,  but this lesson has remained largely unchanged. I was looking for data, originally for a lecture, on the growth of slavery after Eli Whitney went south for a visit.  I found this report with a most gruesome title. After spending an hour or four attempting to capture the information, the horror of it, in a lecture, I suddenly realized how much better it would work to let the kids capture and represent the data themselves.

So after a brief lecture on cotton ginning, before and after, the students get the second page of the report, with the slave census data from 1790 through 1860. I always assign states by group–the eleven eventual confederate, the four border, and New Jersey for contrast, so usually each group gets four states.I then go through a brief review of Percent Change (“change in value over ORIGINAL value”, please) .

The assignment: For each state, calculate the percentage change each decade. Create Create a column graph showing the real change each decade, with the percentage change shown at the top of the column.

Once all four states in the groups are graphed, compare the growth rates.

The work so far has been done on whiteboards. Some of the whiteboards are small, for personal use. In other cases, the students did the work directly on my whiteboard walls.

LASlavegrowth

Student work: Louisiana slavery growth, 1810-1860

GASlaveGrowth

Student work: Georgia slavery growth, 1790-1860

NCSlaveStats

Student work: North Carolina slavery growth, 1810-1860

ALSlaveStats

Student work: Alabama slavery growth, 1800-1860

Right about now, the students realize it’d be much easier to compare the growth rates if they’d used a common scale. Meanwhile, I’d found it difficult to group the states in such a way that each group got a representative sample of growth rates.

In prior years, I’d just lectured through some examples. But my class was much more manageable this year, and for some reason I realized Oh, hey. A teachable moment.

Their “statistics of slavery” handout was doublesided with graph paper. After everyone had finished their group of graphs, I took pictures of any small whiteboard graphs and displayed them on the smart board.

The assignment: quickly graph a line sketch representing the slavery trends in each states using CONSISTENT AXES.  x is year, with  1790 as x=0, or the y-intercept. y is the number of slaves, using 100K chunks through 500K.  No need to capture specific percentage growth, but the graph should reveal it. Something in between “graph every single point” and “just connect the beginning and end value.”

They did really well. A few of them forgot what I said about consistent axes–and mind you, I said this some EIGHTY TIMES but no, I’m not bitter.

SDAInconsistent SDAConsistent

Happily, most compilations got the full 5 of 5, just like the kid on the right (you can see where I corrected his first two).

So these graphs really allowed for informed discussion. (A couple students said “Wow, I actually get slope now.”)  The students were able to identify states that saw tremendous growth vs states with slow or static growth.

Why would states have different growth rates? I reminded them of the national ban on slave trade. Where would slaves come from? And so to the domestic slave trade, another cheerful topic. Unlike the Caribbean slave population, slaves in North America increased their population through natural increase. States that cultivated tobacco exhausted the soil and, as Thomas Jefferson put it  in a letter to Washington, “Manure does not enter into this [soil restoration], because we can buy  an acre of new land cheaper than we can manure an old one.”  People just up and moved, or bought more land, when the productivity dropped, and so the state populations declined. Virginia, Maryland, Kentucky, and North Carolina, tobacco states all, sold their excess slaves to the cotton states.

Interesting note 1: Washington and Madison were both passionately interested in saving Virginia’s soil. Washington abandoned tobacco early, converting to wheat and other less damaging crops. He consulted with many English experts on best practices in soil management. Madison tried to spearhead agricultural reform, but ran up against the southern dislike of centralization.

Interesting note 2: Virginia was a southern agricultural powerhouse despite its reduced tobacco crop, but its primary product was wheat, produced primarily by non-slaveholders in Shenandoah Valley, not tobacco or cotton produced by slaveholders. (Remember, Jimmy Stewart’s Anderson clan wasn’t interested in fighting for the Confederacy.)

Studying slavery reminds me of how seemingly obvious goodness probably wasn’t. So, for example, the south had constraints on manumission. Slaveholders couldn’t even free their slaves if they wanted to! Slave states didn’t want them setting a bad example! Except the constraints existed in no small part because slaveholders dumped older slaves incapable of work, putting indigent elderly slaves  with no family and no means of supporting themselves out on the street. Most of the manumission laws specified age and remuneration requirements, and most didn’t ban the emancipation of young, healthy slaves. So manumission constraints were at least in part about protecting elderly ex-slaves. But would a slave  rather be free, even if impoverished, than living as property?

Or the debate about ending the slave trade, during the Constitutional Convention, when George Mason gave a fine speech, accurately laying out the arguments against slavery–it discourages free labor, gives poor people a distaste for work done by slaves, turns slaveowning men into petty tyrants.  And then General Pinckney says, yo, fine talk from a Virginian, whose huge slave population instantly gets more valuable if we stop bringing in new ones.

What was Pinckney saying? The kids were mystified.

“Why would Virginia’s slaves get more valuable?” asked Eddie.

“Well, remember, this is banning slave trade. Not slavery. The Constitution didn’t give the federal government the right to ban slavery. So if slavery still existed, but no new slaves were being imported, the only slaves being created would be here in America.”

“Yeah, but I don’t see what makes them more valuable?” Jia was confused.

I paused. “Think about supply and demand. What would banning slave trade do to supply?”

“It would go….down.” Jun.

“Right. But demand isn’t decreasing. South Carolina, Kentucky, Georgia, they need slaves.

“They won’t be able to get anymore, though, because there won’t be any more slave trade,” offered Lee.

I stopped moving, wait until eyes are on me. (Teaching’s all about the performance.)

“There will be more slaves. The slaves themselves are having children, right?” I had barely gotten the words out when Lee figured it out, and he literally gasped.

“Yeah. It’s horrible. When the federal government banned slave trade, Virginia had more slaves than any other state. And thanks to lousy farming practices, its land wasn’t much good for tobacco. But as slaves met, married, and had children, lo! the Virginians had a ready made product for sale.” More kids got it and groaned.

“That’s where the phrase ‘sold me down the river’ came from. The phrase means to betray someone. But originally, it referred to a slave whose Virginia or Kentucky owner sold them to the cotton plantations in the deep south, Mississippi or Alabama.”

“So banning slave trade was done to increase the value of slaves?”

“I’m…pretty sure that’s not true. Remember that before the cotton gin came about, many of the founding fathers really did seem to think slavery would fade out, although they were fuzzy on how that would happen. But certainly, South Carolinians would be the ones to identify the market opportunity for another state.”

Anyway.

Another little data analysis activity, done earlier than the slavery stats above: read a series of Wikipedia entries to determine when Northern states freed their slaves, then create a timeline with color-coded data. “I” was banning importation, B meant banning slavery, (“g” meant ban was gradual).    All students had to color code the dates for importing bans and slavery bans. This student came up with the idea of an identifier for those states that gave blacks the vote, and those that restricted the right to vote, particularly after the fact.

Anyway, I wanted the students to realize that organizing data can lead to insights. In this case, the bulk of the Northern states banned importation and slavery in the same 20 year cluster. New York and New Jersey stand out in sharp contrast. Another oddness: Rhode Island banned slavery earlier than it did imports, for the obvious reason that Rhode Island was the epicenter of the slave trade.

20170307_110109

I never liked all the stories about slaves quarters, and jumping the broom, and so on. Not that they aren’t interesting, but they don’t carry the weight of data, of seeing the huge numbers. Of realizing that manumission might be a way to dump non-productive workers, or that ending slave trade might be a business move to increase property value.

It’s too much like Anne Frank, or the Anne Frank that her loving dad created. Whenever I hear kids say “Oh, I identified with Anne sooooooo much!” I want to smack something. She lived in an attic for two years. She was then sent to a concentration camp where she held onto life for six month and then died of typhus, her body crawling with lice, just a month or so before liberation. Identifying with that level of suffering is well-nigh impossible, so spare me your virtue signaling, you teen drama queen. Hrmph.


Teaching Elections and the Electoral College

I couldn’t be more pleased with my US History class. First, the behavior issues are far less severe. I’m not sure why. Last time I taught, two years ago, each of my two classes had ridiculous behavior challenges of the sort I hadn’t dealt with since leaving Algebra 1 behind, thus triggering PTSD attacks and flashbacks. I had been prepared for that possibility, since US History, unlike AP, has a huge range of abilities, so I thought maybe the occasional unmanageable was just the price of admission.

But this year, I have no major challenges thus far, despite having a few unmotivated students. I don’t think I’ve dramatically improved my classroom management skills. At first, I just thought it was lucky chance, but a month in, it’s clear some students could produce the same challenge of years past. Yet they’re not nearly the hassle.

Tentatively, I’m thinking that my curriculum has helped. Rather than starting with early migration patterns, I kicked off with the 2016 election, and then went back to other elections. Competitions are naturally interesting.

I revamped the Question 1 unit from my original plans. Sectionalism will get moved to Question 2, mostly, with some in Question 3. Colonial history got moved to the immigration section, which I think is now Question 4. I only lightly touched on women getting the franchise–I’m going to put that in Question 3, paying the bills.

So what was left in Question 1:

  1. What is the electoral college?
  2. Why does the electoral college exist? Move from Articles to Constitution, small states vs large states, slavery representation.
  3. Political systems: America is a two party country. To oversimplify, one party has historically represented economic and business concerns, the other representing the rights and interests of the individual. (So, class, which party would you think championed abolition, the end of slavery? WRONG! But thanks for playing. They’re very interested in seeing how that came about.) So periodically, we’ll cover the political systems in effect at the time. This go-round, I covered Federalist-Democrat Republican split purely in terms of its existence, and Democrat-Republican split on tariffs. I’ll bring up additional details each cycle.
  4. Expansion of the franchise in the 19th century–specifically, white men without property and black men, as well as the failure to expand it to women.
  5. Elections: The elections with a EV/PV split, while perfectly legit constitutionally, have all been unusual. Two probably involved fraud or intimidation. I also threw in two early elections and Tyler’s assumption of the presidency.

This isn’t a government class, but we looked at Amendments 12-15.
Elections covered in this order:

  • I used the 2016 election to illustrate, which was a romping success and set the tone for the class.
  • After covering the Constitutional Compromises, we looked at the 1796 and 1800 elections–the first to show the unexpected consequences of “first place, second place” in light of the development of parties, the second to show the results of the 3/5ths Compromise.
  • The Corrupt Bargain of 1824
  • Tyler and the Vice Presidency: I included this because it seemed the right place. It wasn’t obvious that the Vice President would assume the presidential role until he said well, yeah, it is.
  • The End of Reconstruction, Compromise of 1877.
  • Election of 2000–out of order, because I had to take a day off for algebra 2 planning. So they watched (and mostly didn’t understand) Recount and then I finished up with that one.
  • Election of 1888

Methods: varied. I did more lecturing than last time, but still a lot of variety. So for the 1800 election, they read about the 3/5ths compromise and used census data to calculate how many extra votes were handed to Thomas Jefferson and what would have happened otherwise (thanks, Garry Wills!). For the Corrupt Bargain, they evaluated Henry Clay’s predictions and compared them to the actual electoral votes–and then compared actual electoral votes to the House vote. They did a lot of reading, usually in class-led situations (otherwise, they won’t do it), including longer pieces on Reconstruction and the Whisky Rebellion. I covered the 2000 election with a CNN documentary and a NY Times retrospective. Then I jigsawed the election of 1888, giving eight groups a different aspect to cover.

One of my favorite activities was on the 15th amendment–I created a group of profiles–black sharecropper, white female abolitionist in Ohio, white former slave owner, Hispanic Texan, married female ex-slave in Mississippi, etc. Then they considered how each person might feel about expanding the franchise. Would a black woman in Kentucky want women to get the franchise, or would that just increase the number of whites voting against issues that mattered to her? Would a black man in New York have different opinions about giving white women the vote than one in South Carolina? Why might a white woman in Kentucky have different opinions about abolition than a white woman in Ohio? And so on. We focused on the franchise not as a right, but rather as a pragmatic consideration–who’s going to vote for the things I want? Then we revisited the issue in 2000 and 2016 with the different views on voter identification.

Onwards. If you’re interested in the test questions, here you go.

I love building history tests with multiple answers–which, as I’ve mentioned, become True/False.

They did pretty well. This was a tough test. Highest score: 91%. Student performance showed a clear pattern: they knew some topics better than others, but the topics varied. Which means it wasn’t my teaching or the curriculum that determined the variance.

I figured out a way to weight half the results at 2 points, the other half at one point, giving each student more credit for the questions they knew well. This boosted everyone’s grade 10-20% over what a straight percentage calculation would have done (except for the high scorer, who I calculated as a percentage).

I also corrected the tests without actually writing on them, just putting the total correct at the bottom of each page. Students will be able to research the actual answers, write up a brief analysis, and turn in the corrected test. I won’t boost their first grade, but count it as a second 10-point test, which will give another boost.

There were two tests, of course. So in one, Matthew Quay helped Benjamin Harrison (true), whereas in this one below, Quay helped Cleveland (false).

matest1800election

I thought they’d do well on this one. Love the cartoon. But performance on this really lagged.

mattylertariff

The Tyler question saw good performance overall, although some kids clearly had no clue and had blanked it out of memory. The tariff question was designed so that if a student didn’t remember Dem/Rep position, he was guaranteed to get hurt.

matcompof1877

Worst overall performance. Almost all the information for this came from a reading, and clearly a lot of them didn’t retain and didn’t study.

mat2016
mat2000recount

No student got fewer than 8 of these correct. Great performance.

matvotingrights

Generally good performance. This alerted me to students who simply weren’t paying any attention at all, since the questions covered topics throughout the unit.

matcorruptbargain

Here’s something I struggle with on grading: what do I do with students who mark D, E, and F as True? Some of them clearly weren’t random guessers. So were they not sure and hedging their bets? Or did they think each one was true, and in that case, what are they failing to put together? Also sad: the number of students who got everything right EXCEPT they thought Clay wrote this in 1888. I’m doing a lesson on how to order events mentally–I don’t think kids always think this through.
matfrances

In the “jigsaw” lesson, I needed an extra, simple topic for some weaker students and other than Hillary, no woman had been mentioned since the semester began. Frances Townsend was an interesting, off-beat topic to explore, plus she was relatively close to my students’ age when she became First Lady. But this quote was great, because it allowed me to test them on the election results as well.

Dig that Matthew Quay pickup! Hat tip and thanks to Richard Brookhiser, who put me onto the Tammany Hall connection. We covered different perspectives on what, exactly, Quay did. I really had no idea that the election of 1888 involved so much fraud. The AP US History test hasn’t covered it, at least in the years I prepped for it.

matfeddr

Really good performance on both, which surprised me because I taught this early in the semester–at the same time as the first question. I would have thought they’d remember people and results better than political parties and outcomes. Moreover, the Whiskey Rebellion was also a reading, something we only covered in a day. Weird.

It was a tough test, with relatively few clues and a great deal of reading. I also do brief essay questions, but we didn’t cover any issue in enough depth to warrant one. That will come later.


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.


Teaching US History in the Trump Era

So the first semester is coming to an end, with its three different preps and an ELL class. Up next: three trig classes. Normally, I kvetch at the idea of teaching three classes in a row. By time three, I’m improvising just to relieve the sense of deja vu (which isn’t as bad as it sounds, since it usually leads to insights into the next day). But I’m unlikely to complain anyway, since this semester I came perilously close to burning out. I managed my Thanksgiving break effectively, getting in sleep, grading, gardening, and holidaying in equal measure. I welcomed Christmas in the normal fashion, without the sense of needing it as I did going into Thanksgiving. So apart from the tedium of grading a hundred plus tests at a time (as opposed to 35 each time now), there’ll be no complaints from this quarter.

And! I’m teaching US History again. Whoo and hoo. I never thought I’d see another year when I’d use all my credentials.

When I last taught it, the big challenge was balancing content. I like teaching history in a semi-linear fashion, but there’s always something interesting in the past to bring up, and I forget all about the time. (Ha, ha.) I forgave my failings because we don’t have state tests and all evidence shows kids never remember the details anyway. You know how all the curriculum folk like E.D. Hirsch, Robert Pondiscio, Dan Willingham all say “Teachers today don’t teach knowledge?” They’re goofy. We do. Trust me. We do. But they tend not to remember. That’s another story.

Anyway. I wanted to get past World War II while still teaching my favorite topics of the past, and have been mulling possibilities in my copious spare time without much progress until The Election Happened. That, coupled with some breathing room over Thanksgiving, gave me a framework.

Five Questions:

  1. Wait–the Candidate With the Most Votes Didn’t Win?
  2. Why Black Lives Matter?
  3. What does “American” Mean?
  4. How Will You Contribute to the US Economy–aka, How Will You Pay Your Bills?
  5. What do Fidel and Putin Have to Do With Us?

I’ll continue to wordsmith the questions, but I do want them to be instantly relevant to a high school junior.

Question One

Main Idea: The Electoral College plays an important role in balancing regional tensions, a role that’s remained constant even as we’ve dramatically expanded the voting pool.
I.   History of colonial development
II. Brief (I said BRIEF, Ed!) history of Revolutionary Era
III.Constitutional Convention
IV. Rise of sectionalism and the role the electoral college played in balancing power (Hartford Convention, Missouri Compromise, Nullification Crisis, Compromise of 1850).
V. Expansion of franchise: all property holders, all men (technically, all women (technically), all citizens (really).
VI. Popular Vote/EC Splits a) Jefferson-Adams (Jefferson only won EV because of slave headcount) b) The Corrupt Bargain; c) Compromise of 1876; d) Cleveland-Harrison e) Gore-Bush f) Trump-Clinton, which I’ll probably defer until later.

Question Two:
Main Idea: “Black Lives” matter because the US violated its fundamental values to achieve and maintain unity, and our African American citizens paid the price.

I.   Development of slavery (I go way back to Portugal and kidnapping, the Papal Bull and so on)
II.  The evitable roots of American slavery and its development: Jamestown, South Carolina, Bacon’s Rebellion.
II.  The rise of Cotton
III. Deeper look at sectionalism from slavery standpoint: rise of abolition, range of reasons for opposition, free black role in movement, etc.
IV.  Civil War, Reconstruction
V.    Rise of black intellectual debate (Booker T, WEB, Garvey, MLK,).
VI.  Post-Civil Rights era–I see history past the Voting Rights as rather gloomy. Maybe examine riots in 60s/70s and compare to today?

Question Three:

Main Idea: From the first Beringian wanderers to the desperate migrants hoping for a miracle in Turbo, everyone wants to find a home here. At some point, the United States imposed its will on the process. What does that mean to the world? What does the expanding definition of “American” mean to its citizens?

I.    Early Americans and Corn Cultivation (one of my favorite topics!)
II.  Age of Exploration (again, brief, Ed!)
III.  Immigration Waves and Westward Expansion
IV.   Restriction: 1888, 1924
V.   Expansion: 1965
VI.  I’m still figuring out how to organize this.

Question Four
Main Idea: The United States’ economy has changed in many ways over the years. Many people think Trump’s victory was due in part to regional dissatisfaction with those changes. How do the transformations in the past help us understand the future–or do they?

This is a big section and I’ll have to chop it down. But it’s my favorite, so I’m listing everything to see if I can find any synergies to improve coverage.

I.    Colonial Mercantilism
II.   Hamilton vs. Jefferson (again, a favorite of mine)
III.  Rise of Industry (Eli Whitney! McCormack! Industrial espionage! and so on)
IV.  The “Worker” as opposed to the farmer or merchant (Jackson Kills the Bank will make an appearance)
V.    The Rise of Mechanization and the Industrial Era (immigration will show up again here)
VII. America as Industry Giant (Ford, impact of WWI/WWII on our dominance, the automated cotton picker & Great Migration, etc), including the rise of unions (thanks to Wagner Act)
VIII. Early Computing through the WWW and Information Age
IX.   Globalization and Automation, coupled with the fall of unions.
X.   Growing–and reducing–the work force

Question Five

Main Idea: How has the United States interacted with its neighbors near and far?

As I’ve written before, I’m a big fan of Walter Russell Mead’s Special Providence, and will use that as a sort of syllabus to outline key events in American foreign policy: neutrality, acquisitions, native American screwovers, world wars, and cold wars. I don’t have this one fleshed out, but the topic will definitely include the important international alliances that occurred before and during the Revolution, Founding Fathers, John Quincy Adams (you can get a hint of my thoughts here ). Then I’ll pick key events of interest in the 19th century, limiting my scope. Again, some talk of America’s position post-WWI/WWII, but bulk of time will be spent on Cold War and beyond, is my hope.

So.

I have a lot of these lessons done already. I didn’t like to lecture the last time I did the class because it was too tempting to just lecture the entire time. But with this structure, I think I’ll be able to give lectures as well as do a lot of readings and analysis. That’s the hope, anyway.