Tag Archives: Desmos

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.

 

 

 

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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.