I've heard lots of opinions on the unit circle. Some treat it as an object to be memorized and others something that can only be derived. And then a group who hates the whole thing and wonders why it's included.

Here's what I use traditionally:

The problem of the coordinate system is I think you lose the connection to the triangles embedded in the circle. It's not readily apparent that those xy pairs represent something other than a dot. But I've seen versions of the circle with embedded triangles and their a cluttered mess. Last year I was better about referring to them as (adjacent, opposite) rather than xy but forgot to reinforce it when doing inverses.

I played around and got here:

I really don't know if it's less confusing. The triangles are there, but do you get a sense for appropriate quadrant orientation?

Ideally I don't want a student to try and memorize the thing. Every time we get here I talk about how the first quadrant establishes all the patterns. But they still just think they have to memorize. Would four sets of triangles be better? Or a set of three axes with the base 30, base 45, and base 60 triangles in each set? Does giving it a special name like Unit Circle automatically tell a student this has no connections to other things I know about trig? Should I not name it? Banish the "1" and just label the hypotenuse as "r" ?

I am going to incorporate the decimal equivalents. I don't care who you think you are, students have very little grasp for the value of root(3) / 2.

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It has ended. It was excellent. It had kittens.

A recap of some of the events.

Wednesday

Headed to the airport at 5am after little to no sleep. All in an attempt to make it to a taping of the Price is Right. We showed up with 5 minutes to spare to what we thought would be a quick day. It's a 40 minute off the cuff show, how long could it take to film? Seven lines and five hours later, I sat in the studio audience. Then there was two hours of clapping and screaming.

Finally made it to the hotel for the meet and greet at 9pm.

Thursday

Day 1 excitement! I presented a lot last year. I wanted to take a break this year. But I had a My Favorite I couldn't pass up. I gave a short speech about Varsity Math. And a Cheeto.

I was in the Desmos morning session. Lots of Pre-Cal and Calculus people in there. Glenn Waddell, Bob Lochel, Jedidiah Butler, and Michael Fenton offered four mini groups to focus on the wide range of subjects represented in the room. Lots of play time. Fenton helped me make my solar system prettier. Picked up on some stats functions that I didn't know were built in.

In the afternoon I attended a talk about Formative Assessment with Peg Cagle and how she managed rooms of 40. Helped me with some ideas I have about assessment in Calculus. Then a fun round table lead by Sadie Estrella. Thirty minutes of people telling great stories.

Friday

In true TMC fashion, I got a look at secret Desmos features that will appear in about a week. Just in time for a presentation I'm supposed to give about the subject next month. I asked Eli my annual set of two questions.

In the afternoon I sat in on a number sense session from my man Stadel and Graham Fletcher. It was an interesting experiment using Estimation 180 in which some people are shown a task that helps with a new estimate and others aren't. We also spent about 10 minutes trying to decide what number was halfway between 10 and 1000.

In the evening the California locals threw a BBQ in a park nearby. The food was delicious. John Stevens brought an adorable set of helpers.

Saturday

The energy begins to drain. More play time in the Desmos morning session. At this point it was a lot of hopping around the room looking at what people were working on. Had some fun Pre Cal discussions.

As promised one year ago, I rode in the Estimation Mobile and we got cheeseburgers. I think we invented two estimation tasks and a 3ACTS waiting for In N Out.

Fawn Nguyen gave a keynote. Nobody cared. (Everybody cared).

And finally, I spent an hour spitballing ideas about AP Calculus with some people. I have lots of work to do there before school starts.

Conclusion

TMC13 was full of wonder and being a fangirl. TMC14 was a chance to start sharing and being recognized as a contributor to the community. TMC15 was the friendly one. No more fangirl, these were friends with a common goal to be great at their job. There were all sorts of comfortable moments from the Price is Right saga, the meeting room Wednesday night, to the dozens of casual conversations I was able to have with people who are pixels in my timeline the other 360 days of the year. It is a precious 5 days and I think I succeeded in completing my checklist even it's just 5 minutes to say, no, I don't look like a cartoon robot.

Thanks to those of you that indulged my questions and thanks to those of you who said hi and recognized me and my work. I finally got a nice headshot, so I'll be easier to spot. I started documenting stuff for my own benefit but as the audience has slowly creeped up to a pretty big number, it is crazy to know that some of you benefit from it.

Until we meet again in Minneapolis.

(The song was amazing).

 

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My official verdict on the first year of Calculus is that it went ok. I invented an assessment structure that I stuck with. We had a reasonable amount of time at the end for exam review and when the kids recounted the exam experience there was no imminent sense of dread. They all gave it an honest try. At the beginning I had minimal expectations: no quitting, no being scared of the exam. Both were accomplished. Didn't translate to exam scores unfortunately.

What I did learn, is that we have a pulse:

Two weeks after score reports, you get a breakdown from the College Board with everything in quartiles. I've obscured some data just a bit. To my utter surprise, the free response went better than the multiple choice. Overall neither are what you would call good. But to build a class from nothing and be able to get them to make a dent into the very high level language of that exam is progress.

Coming out of the exam, question 4 and question 6 were cited as the hard ones. I barely scratched the surface of differential equations, so even these metrics are nice to see. Clearly I was able to make headway when it comes to f / f' / f'' relations and table-based integration/differentiation.

What's going to move us forward? I learned some hard lessons from our AP review. In summary, we spent a couple weeks going over free response scenarios clustered by topic. I grouped released questions into seven categories: Data Tables, Differential Equations, Fundamental Theorem Relations, Particle Motion, Conceptual Integration, Rate Analysis, and Volume. That subdivision happened too late.

Rather than wait until exam time to introduce those ideas, these are now my units. All of them have overlapping skills. All of them can be discussed if I spend the first six weeks laying the conceptual groundwork. Meaning, we spend the first six weeks taking glancing blows at all the topics. I'm not going to rush u-subsitution or anything, but if we can discuss the concept of integrals, even just simple ones early and often, we'll benefit from it later on. I'm almost tempted to introduce the course as Better Understand Physics with how important f / f' / f'' is to the whole thing.

Rough order (I have 5 grading periods until the exam):

1st PREP: limits, derivatives (implicit but no chain), and integrals (no u-sub, no logs) and their basic relationships
2nd POS/VEL/ACC: integrals and area, FDT, SDT, graphical relationships
3rd PARTICLES: derivatives (all rules), integrals (all rules), final value vs final position
4th RATES/DATA: calculator based f / f' / f'' relationships, average values, related rates
5th DEQ/VOL: differential equations, volume

Those are at best, extremely simple ideas for what I want to accomplish. I have a long list of standards I'm still trying to sort. Much like Pivot Algebra II, I want to hit the same conceptual notes over and over and over. I'm going to cover all the skills, don't worry.

Assessments are changing. Sarah Hagan has this very intriguing A/B/Not Yet style of grading. It seems perfect for Calculus. Implementation details remain. AP grades are painted with such broad strokes that the difference between an 81 and an 88 is pointless. Even the 0-6 scale I was using for skills is too much of a gradient. Mock AP sticks around. They'll take half an AP test in December and half an AP test in March. I'll use them to filter out who should bother with the exam. Last year I had 50% take it, this year I want 66% or better. There's going to be more AP stuff too. When doing exit interviews, it was pretty universal that they were thankful for how much AP stuff we did, but they wanted more, and sooner. I was inclined to agree.

I pushed a lot of the early limit principles to Pre-Cal last year. That group is now my new band of Calculus kids. I'm hoping that saved me 3 weeks or more.

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AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students realize that the hypotenuse of a right triangle, the magnitude of a vector, and the r-value of a polar coordinate are three different names for the same quantity. Students realize that the direction of a vector and the theta-value of a polar coordinate are the same quantity. Students understand how to use polar equations to find r-values and how those r-values are plotted. Students understand how parameters in a polar equation affect the properties of its graph.

Progression

A couple years ago I moved straight from vectors to the polar coordinate system because the math is identical. With such recent exposure to the mechanics it just becomes a vocabulary lesson.

We start simple, taking rectangular coordinates and finding their polar equivalent. We draw some right triangle systems and define the hypotenuse as the length of the radius. The theta-value of the coordinate is just the angle relative to zero. It's the same math from vectors.

Next, what makes these things unique. Graph of some high-res polar grids. Have students make a table in 15 degree increments, 0 to 360. Give them some simple polar equations like r = 3 + 3 cos theta and r = 3 + 3 sin theta. Have them complete the table for all the values of thetas. Ignore the groans. Once finished, show them how to plot a point along the circles and then connect the dots.

To better understand the weird shapes, they do an exploration. Given a set of polar graphs and some equation frames (a+/-b cos/sin theta; m sin/cos k theta) students determine the equation associated with a graph and we have a discussion about properties. They have iPads to do the graphing.

Questions to ask: why does 5 + 3 sin theta appear the same as -5 + 3 sin theta? why do graphs have a favored orientation? what determines petal count in a rose curve? What induces a loop? is there anything odd about petal counts? how would you make a rose curve with 10 petals?

Next we get a little more technical. Students are given a second set of polar graphs. This time, no iPads to help. Come up with the equation. Using the equation you came up with, find r-values at 0º, 90º, 180º, and 270º. Use this information to make a connection between negative r-values and the appearance of loops. Discuss the angle separation of rose curve petals. Discuss how far off-axis a sin based rose curve is rotated from its cos counterpart.

One last set of polar equations. Having their fill of cardiods, limaçons, and roses, we discuss ellipses and circles. Pass out the iPads again and send them to a polar ellipse generator. Give them a set of ellipses and have them determine the appropriate parameters. Have a brief discussion about any effects they notice and give them some definitions: semi-major axis, eccentricity, and focus. The key part of this study is to get an idea for eccentricity and how it relates to the dimensions of a shape.

If you really want to keep going (and at this point it's mid-April and probably have the itch to start Calculus), you can explore parabolas and hyperbolas.

Activities

Great stuff here. After the initial discussion of equations that match graphs, the students get a chance to design their own. Then we draw them outside on the sidewalk.

When discussing polar conics, you can have some really fun discussion about the solar system. Polar ellipses are plotted using Kepler's equations, which nicely map the orbits of any celestial body you've heard about. After the exploration of making equations of arbitrary ellipses, send them to Wikipedia and lookup the semi-major axis and eccentricity for Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, Eris, and Halley's Comet. Sticking those values into a Kepler equation gets you a lovely solar system, complete with Pluto's encroachment of Neptune.

There's a great video to show as a followup that discusses how bodies interact with one another in space.

Assessment

I assessed the mechanics: coordinate conversion and appropriate equations for a graph. Then wandered into the conceptual with discussion questions about eccentricity and written descriptions of rose curves. It spanned three tests and 4 SBG topics.

Test 16, Test 17, Test 18

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AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students can define a vector as an arrow of varying length pointed in a particular direction. Student can compose a right triangle from a vector by finding the horizontal and vertical components of the vector. Students can combine vectors in component form or with given magnitudes and directions and finding a resulting magnitude and direction. Student can correct the results of an inverse tangent function based on the quadrant of the vector.

Progression

Vectors goes pretty quickly because all the math is a new spin on right triangle stuff. If you spent some time relating cos with horizontal measurements and sin with vertical measurements the breakdown of a vector into components is pretty easy. Some years I've talked about coming up with vector components to describe how to travel from one coordinate point to another. I skipped that this year.

I don't have any particularly interesting intros. It's a lot of drawing arrows with a known magnitude and direction and turning them into right triangles, then just showing them the mag * cos theta or mag * sin theta thing. You could have them draw some arbitrary arrows, do some measuring, and then work back to solving right triangles if you want.

They practice component breakdowns and combining vectors that are already in component form (I use the i, j, and k physics conventions). They figure that out on their own, it's easy.

Show some videos of airplanes landing in crosswinds.

Next we work backwards on finding magnitude and direction given only the components. I spend a moment talking about solving right triangles which will lead them to the use of an inverse tangent.  I always discuss angles as relative to zero (getting answers that range from 0-360), and we always work in degree mode. Calculator output needs to be interpreted here, because in quadrant II, III, and IV the calculator is solving a triangle, and the answer will always be less than 90. The answers are also relative to the horizontal, not zero. I emphasize drawing vector systems quite heavily to help with this idea. If the vector is located in quadrant III, but the calculator tells me the direction is 75º, what is that 75º really describing?

Through this discussion we come up with corrections: quadrant I is fine, quad II is ans+180, quad III is ans+180 and quad IV is ans+360. Students will usually fail to correct an angle for a quad III vector because the calculator outputs a positive number. Kids always seem to think positive = success.

Next we do the whole thing in three dimensions. It's limited though. They'll describe the appropriate octant for a 3D vector, and its magnitude. I don't have time to get into the subtleties of angles relative to the three axes. In the future when they're learning with holograms, sure.

Last, we talk about the angles between two given vectors. And no, I don't use the hokey formula with dot products. To start they draw the two vectors and literally measure it. Going further we talk about how to find relative angle measurements. If you're feeling really adventurous you can have them solve the arbitrary triangle created or find its area.

Vector problems are easy enough that a lot of the intro stuff I'll just make things up on the board. I have two typed up bits. Vector Intro, 3D Vectors

Activities

Three activities that crop up. A couple of them are going to change in the future due to changing time constraints.

To stress how a vector system is arranged, I have them create a few in three of the four quadrants. They demonstrate the components and the vector itself using straws cut to scale. An error I've made is rushing this activity before the students have a good idea what I'm talking about. It's also a bit messy and there's nothing necessary about the straws other than giving a physical component.

Since students are bad at visualizing 3D coordinates (they really are, I know you don't believe me), we do some walking around in mathematically defined 3D space. Take some yarn and make an x-y-z axis set in the room. Label the octants. Show the students around and discuss sign conventions. Then have them locate a big set of coordinates printed on labels (set of 60, Word, PDF) inside your 3D coordinate box.

Last, I have them construct their own 3D coordinate system using a bunch of straws and electrical tape. They label each octant and then give me an example of a point that belongs there. Only problem with this one is it is very time consuming (solid 80 minutes start to finish) and might be a little light on learning. I have some ideas about how to improve it next year.

Due to the nature of the components these will fall apart very quickly, some within hours of finishing. Students or are more dutiful with their taping have much better results.

Assessment

The concepts in vectors span a long period of time even though the material is pretty simple. There are a lot of conceptual things to talk about here. The foundations of this unit come back when discussing polar coordinate systems, so the more time you spend on it the better.

The topics covered span four tests and strike a 50/50 balance between mechanics and concepts.

Test 13, Test 14, Test 15, Test 16

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AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students understand how values of the unit circle are periodic and are connected with the appearance of a sin and cos graph. Students understand the ideas of amplitude and period. Students understand that it is possible for an identical graph to be produced by a sin and cos function with appropriate use of transformations. Students understand that tan has a different period by default.

Progression

There's a lot to do here and I find this to be one of my favorite topics. We move from values on the unit circle to more generic methods for sketching the graph of sin and cos functions. We make some science connections with sound waves.

First grab some trig graph paper and have students get out a unit circle and some markers. Have them systematically plot the values of sin for the whole circle and connect the dots. Do the same for cos and then tan.

Define the terms amplitude and period. Some of this will be familiar from prior science classes. I interchange period with wavelength and frequency. As a further connection, we talk about the "speed" of the function and define this base form as "normal" speed.

Together, I show them how you can plot the graph of modified sin/cos/tan functions if you know something about how amplitude and period frame the picture. We plot a few examples together. Students often find it weird that sin 2x has a shorter period. There is a tendency to start talking about arbitrary opposites a la "oh well, we do the opposite of the modification, therefore 2x implies x / 2" hand waving. I found it really useful to continue that metaphor of speed. The function sin 2x completes its run two times "faster" than the "normal" we defined previously. For something to be faster, the elapsed time must be shorter. Therefore, one wave takes up less distance along the x-axis. This becomes really handy when you want to talk about sin 2x/3, where the whole opposite operation thing gets ugly to describe. Rather, at 2/3 the speed, we should expect waves to be longer. Throwing in stuff like -3 sin x or sin x + 2 doesn't cause much confusion. I don't do horizontal transformations just yet.

Next they work through two activities to get more practice. It's easy to go fast through this topic and wind up with lots of confusion. I made a concerted effort to slow down and offer lots of opportunities for students to discuss this with each other.

After the activities, they completed some individual practice.

Activities

First I have to make some connections between pictures and the equations associated with them. I hand a out a set of 16 unidentified graphs (Answer Key). Students are given an iPad and desmos to determine how the pictures were made.

Next students play a little game drawing randomly generated trig functions. Students are given whiteboards, markers, cards, and 1 die. Students flip a card which describes something about the amplitude, period, and/or transformation of a function. Students roll the die. A 1/2 means sin, 3/4 cos, and 5/6 tan. Students must draw the appropriate function that has the properties of the card. I wrote a more extensive description at the time.

Next students are given a chance to design some trig functions of their own, labeling the graphs appropriately. They design 3 sin and 3 cos functions. They must demonstrate amplitude changes, slower periods, and faster periods in whatever combination they choose. Students create the graphs with desmos on an iPad, print them out and annotate.

Last, we make some connections with sound waves. How much I spend with this depends on where I am in the semester. Last year we were right up against the end, so the science stuff was just a couple side discussions. You could flesh it out more as I have in the past, or start with this. I use a tone generator to talk about how we perceive amplitude and period with sound. I also have a set of blinking lights that are fun to talk about too. You can go really deep into harmonics and springs and things if you have time to fill in the physics background.

Assessment

This traditionally hits at the end of the first semester. Assessment was pretty straightforward. I'm looking at their ability to sketch sin/cos/tan functions by interpreting amplitude and period correctly spread across two SBG topics. I ask some conceptual questions as a third topic.

Test 9, Test 10

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Last year I rattled through some frequently asked questions.

People point to me as a user of interactive notebooks. I would not call my notebooks interactive in the same category as what the internet considers an interactive notebook. If you're looking for foldables, tables of contents, and scripted notebook pages, please go check out Sarah Hagan. She even had a whole day celebrating her.

Anyway, this is what I do to get students to do stuff. Time for bullet points!

Gallery

Here's some pictures. I'll do a little explaining after.

Structure

  • composition books preferred, hold up better over time
  • students can keep them in the room, I use a 30 qt. tub for each class
  • colored duct tape identifies the class period
  • a manilla folder fragment stapled to the back holds old tests
  • students track their SBG scores in the front, it's not a table of contents

Procedures

  • students use it every day, no questions
  • all classwork and notes are contained in the book
  • from time to time work is pre-typed and handed out on 1/3 or 1/2 sheets
  • students do a lot of work copied from the board
  • students don't have homework, they work when they are in class
  • students can use their books on tests, tests are designed to be about thinking, not parroting, we don't do review days
  • class time is intended to provide as much student time as possible, stop talking
  • books are checked every 3 weeks for updated SBG charts and cleanliness
  • students can put things wherever they want
  • assignments are graded a day or so after completion, I walk around with a clipboard while students are working on something newer
  • it takes time to cut things out and glue, I mean, just accept it
  • I never take them home
  • kids forget them, it happens, usually not a chronic issue

Critics of the notebook thing dislike the time required for the cutting, taping, gluing, and preparation. They'd prefer a binder full of daily worksheets or something. Mine don't suffer those issues. I don't find myself spending an inordinate amount of time waiting on kids to do stuff. Of course, I've set up class to be for them. You don't see me talking much. Assignments are short and sweet so they can do all the cutting and still accomplish something. I'm allergic to full page handouts. Over time it's just so much paper. That volume of copying is just unnecessary to me. Same with packets. Yuck. One year I'm going to count how little I copy as proof.

After five years with these things I'd never do binders or folders, ever. Student productivity is at ridiculous levels since I made it a requirement.

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This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students define sin, cos, tan, csc, sec, and cot as the ratio of sides in a right triangle. Students identify certain special ratios for 30º, 45º, and 60º angles and organize them in the unit circle. Students can determine unknown sides and angles in a right triangle. Students should see that the size of an angle has a direct or inverse relationship to the value of certain side ratios.

Progression

Other good resources on the topic come from Sam Shah (with a follow up) and Kate Nowak.

My approach and theirs have some similar messages. Students should understand that trig ratios mean something and that while 30/45/60 ratios are considered special, there's nothing particularly special about them other than reducing to nice-ish numbers.

First, I start with the "special" triangles. Have them draw a set of right triangles (4-5) of arbitrary size for 30/60/90, 60/30/90, and 45/45/90. Establish the definition of the opposite, adjacent, and hypotenuse. Measure the ratio of opposite/hypotenuse and adjacent/hypotenuse for your triangles and record them in a table.

They should see the ratios are the same for an angle regardless of triangle size. We use the three triangles and build a collection we call the Unit Circle where the hypotenuse is 1. I offer them things like root(2)/2 as the exact value of the ~0.71 number they see. We define opposite/hypotenuse as the sine of an angle, adjacent/hypotenuse as the cosine. It's important to take a moment and talk about opposite and adjacent as relative definitions. Some will catch that, others might not. They spend a week finding ratios on the unit circle and defining them as sin/cos/tan/csc/sec/cot.

Now we take a known angle and find the missing side of a right triangle. This is an exercise in vocabulary. Given the location of angle, what is the relative definition of the three sides? What trig ratio is relevant to the given information? This is the first time they compute sin/cos/tan values with a calculator. You could choose not to and save it for later.

Next, using a calculator I have them complete a table of values for sin/cos/tan of the angles 0 to 90 in increments of 5 (this is similar to Sam's large packet of triangles to build a database of ratio values). I give them a set of right triangles of arbitrary size and they use their table of ratios to approximate the angles. I avoid using the inverse trig functions on the calculator until we start talking about trig equations later on. I want them to make a connection between ratio value and angle value. I want them to have an idea of sin and cos trends to help when we graph those functions later.

Lastly, they create a set of right triangles, one in each quadrant, of arbitrary size where they demonstrate the values of the six trig functions for one interior angle and use those ratios to approximate the value of the interior angle.

They have an assignment with a set of right triangles where sides and angles are missing and they have to find the missing information using any of the methods we discussed.

Activities

Most of their work through this unit is a series of activities. Here's a picture of the arbitrary special right triangles and the table of ratios:

Here's a picture of the ratio table they complete with the calculator:

Here's a picture of the final assignment, an arbitrary set of right triangles with all ratios computed and angles approximated:

Another option is the classic "how tall is the tall thing?" activity. Take them somewhere where you have some tall objects. Have them measure 1 side and 1 angle. Use a relevant trig ratio to approximate the height of the object and compare results.

Assessment

Triangles cover 5 topics in my SBG system. Student demonstrate knowledge similar to classwork and in a couple cases are asked to draw triangles oriented correctly in a quadrant, label sides based on a defined trig ratio, and determine the rest of the information.

Test 7, Test 8, Test 9

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There are rumors big scale Chromebook deployments could come to the district. There were some pilot programs last year in English IV and Government. I was given a Dell Chromebook 11 to see what possibilities there were for high school math.

Hardware

My primary point of comparison is an iPad. The Chromebook I used sells for $249. You can get refurbished iPad minis for that price, but not with retina screens. Definite price advantage. The keyboard is fine. The case and hinge feel sturdy. Battery lasts for quite a while, with infrequent use would only need a charge once a week. But man, the trackpad is no good. Plugging in a mouse was necessary.

Software

Google Drive and its companion apps are definitely meant for a desktop browser. If given a set of these, I could have students save a lot more of their work. I'm still not sold on handing things in digitally. I like physical products. If anything being able to save things like custom trig graphs or piecewise functions would let students spend time on them outside the classroom if they don't finish. I could add some more elaborate write up portions perhaps?

I couldn't try out Google Classroom. It requires inviting accounts enabled for it through Google Apps for Education. And I have no children at the moment. It doesn't seem to be much more than a central point for students to grab things from you. The add-on Doctopus appears to offer similar management features. I didn't explore add-ons much.

Printing was among my major questions. Google handles this through Cloud Print, a work in progress. In theory you tie a wireless printer or a printer tethered to a desktop you control to your Google account and any subsequent device you sign into can print to that device. I successfully sent stuff to the printer many times, but some OS X weirdness prevented it from working. I don't blame Google here. Everything on their end seemed to work.

Products

I set out to reproduce things I've done with iPads. Regardless of hardware, when it comes to technology the application has got to really trump the quickness of pencil/marker and paper. Remaking things you can do in Desmos on an iPad was simple. My experience with Desmos classroom activities was no different than with an iPad.

Clockwise from top left: Identify a region between curves; determine linear equations for a street map; import data and perform a regression; prepping Desmos output for print

The Chromebook excelled in two scenarios. For a couple years I had Pre-Cal kids wander around and take pictures of random right triangles, dimension them, and then give me a the set of trig ratios associated with an angle in the triangle. The drawing tools in Drive make for some better results. A student submission from that version is on the left, my recreation is on the right.

The desktop implementation of Drive is what wins here. You have do a lot of import/export steps on the iPad to get it sent to the right place, part of the reason I abandoned this idea.

The second improvement came with Geogebra. I don't have much experience with it, but I've known it to be a little fiddly and it works better with a mouse. Were I geometry teacher, I'd be very happy at the thought of student computers capable of using it. Geogebra installs a Chrome app (nothing but a bookmark really) version of itself.

The 3D graphing features seem worthwhile. It could offer a lot of enhancement to the modest discussion of 3D I'm able to do today.

Conclusion

At first pass, a Chromebook would allow me to do the same things I can do with my iPads. As stated before, the best thing I've found for technology is off loading the heavy lifting parts of graphing. That's roughly 10 to 12 class days (once every 4-5 weeks) throughout the course of the year. Putting a desktop experience in a cheap package gives me a chance to do those activities and a little more. They still wouldn't come out everyday.

There are easy ways to modify some of my established practices. Students could do more writing. Students could submit projects to me instead of printing them out. Students have all the benefits of the internet and we could find some research angles. Students could do a lot more self-paced learning (like watching tutorial videos! just seeing who's still with me...) There's tons of unexplored potential with Geogebra and I just teach the wrong subject to exploit it. But, the whole experience doesn't do much to make the pencil and paper needs of math class better or obsolete.

If the testing conglomerate every modernizes I could see using these things all the time in Calculus. A notebook, pencil, and Chromebook/iPad is not a bad way to do math.

So are 1:1 Chromebooks a good idea for math? As part of a larger deployment where ELA/SS have gone paperless, sure. Exclusively for math? Just give me a cart.

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This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students should be comfortable with function notation. Students should see the division of two functions as a ratio. Students should be able to discuss the difference between an asymptote and removable discontinuity. Students should be able to understand why they occur and how they appear on a graph.

Progression

There is a lot here. I talk a lot about algebra, a lot about graphs, and try to cycle between them frequently.

Start where we left off in composite functions. We discussed addition, subtraction, and multiplication but have ignored division. I specify division as the ratio of two independent function and coin the term rational as shorthand. Define some functions f(x) and g(x), both factorable quadratics, and such that the ratio will generate some errors. Have the students create a table of x values from -10 to 10. Have them evaluate f(x) and g(x) for the length of the table. Then evaluate the ratio of f(x) / g(x). They should see a g(x) value of 0 causes a problem. Glenn Waddell has a prepackaged alternative version of this in Desmos.

Graph the ratio of f(x) / g(x) with Desmos (either individually or just you) to provide some explanation for the errors. At this point I'll define asymptote and removable discontinuity.

Now we get into why these errors happen, why some are considered removable, and how you could find their location without a graph. I prepare a number of factorable quadratics as ratios and have them simplify them through factoring to see if any share a binomial in the denominator and numerator. It is tempting to say that if you get something like (x - 3)(x + 4) / (x + 4)(x - 7) that the ratio will fail at -4 and 7 because those are the opposites. It is much better to phrase it as "what value of x will make the binomial zero?" I'm careful to say divide out and not cancel out as well when defining x = -4 as a hole. 

Activities

Graph a Rational without a Calculator: present them with a ratio made of factorable quadratics, have them identify the location of any asymptotes or removable discontinuities. Use these locations as boundaries. Mark the asymptote with a dotted line and notch a spot on the axis where the hole should appear. Use test points on either side of the asymptote and hole to see if the graph is generally positive or negative. Draw curves that follow that behavior.

Identify a ratio from a graph: Provide some graphs of rational functions. Have them work backwards from the location of the asymptotes/holes to write out the possible denominator. Fill in what's possible based on the presence of a hole. Use the idea of a placeholder in the numerator to signify the lack of additional information. Ex: unknown A / (x + 3)(x - 9) for something with two asymptotes but no holes.

Design a Rational: make up two ratios that demonstrate the characteristics we have seen. Demonstrate a function with multiple asymptotes and one of them should have at least one asymptote and a hole. Label the graph with the corresponding ratio and explain why the features appear where they do.

Assessment

Students are tested on mechanics: identifying the location of an asymptote or removable discontinuity from a ratio of factorable quadratics/cubics. Students are tested on context by being given a ratio and possible graph of that ratio and have to develop an argument as to whether or not that graph could represent the ratio. A few students did a fantastic job with those questions. This is tested as two SBG topics.

Test 5, Test 6

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AuthorJonathan Claydon
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