Holding Us Back

Which method of solving a problem would you prefer to explain to a group of novice mathematicians?

What purpose does it serve to let math technology sit still? Will the class of 2020 still take the SAT on a calculator from 1996?

Stop making students rediscover fire.

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The Estimation Wall

Amid cranking up the volume on my Algebra II curriculum, I changed the way we did warm ups. Instead of some exercise related to the lesson, I posted a daily challenge from Estimation180. I organized their progression a bit. I was worried the idea would get old after a while, but every day without fail I have no trouble getting people to participate in the daily guess. Low barrier to entry is a wonderful idea.

Having made it to Day 71, I felt the students were ready to trade places with Mr. Stadel. Early on, I knew I wanted the students to design their own tasks, but I didn't have a time table. After fiddling with the curriculum progression, I found two days last week to build our Estimation Wall. We would have 140 minutes to design and build this thing (90 minute block day plus 50 minute Friday).

Expectations

On December 4 I assigned the task. By December 6 the students had to submit an idea they would post on the Estimation Wall. We discussed the general format of an Estimation180 task, how to obscure the answer, and how to develop ideas that stack on one another. Students were required to design two tasks and supply the answers. Ideally, the tasks should be linked (similar to Day 41 through 45).

Estimation Wall idea submissions.

Production took place December 11 and 13. Students had two options for getting pictures. They could take the picture at home on their phone and submit it via e-mail (to the dummy gmail account I use for the iPads), or bring the items for their task to class on December 11 and use the class iPads to take pictures. I wanted to use card stock to mount the pictures, so students brought in \$1 for the fancy paper.

Remember how I talked about hoarding office supplies and getting a printer in the room? The Estimation Wall is the payoff activity for that investment. Scissors, glue, colored paper, iPads, and the ability to print color pictures all right there in the room. The construction process was an unholy mess, and yet beautiful with how easy it all came together. Activities earlier in the year had handled all the technology hurdles. I flipped the switch on the printer and the kids were off on their own without my help.

Construction

The students did a FANTASTIC job with the activity. Lots of creative ideas and no one refused to participate. The key to the project was where we were going to install it. An enterprising Chemistry teacher gave me the answer. Upstairs they're constructing a periodic table on our neglected lockers.

A similar bank of neglected lockers sits outside my room and after asking permission, that was that. Students would mount the question on the front of the locker and put the answer inside. The lockers are in the three rows. Task 1 goes on Row 1, Task 2 on Row 2 and Row 3 is left blank.

The trash generated by this was crazy. Had you walked in during the process it would've looked like chaos. But everyone was on task from start to finish. Most students had their pictures mounted by the end of the 90 minute period. During the 50 minute period students taped their pictures to the lockers or finished up their tasks as necessary. If they finished early, they had the review for the final to start.

Final Thoughts

Here's a shot of the full installation and a close up of a couple tasks.

On the surface, this doesn't look like Algebra II material or something that high school students should be doing (a waste a time some might say). But if you think about how hard it is to design a well done math task, these students got a great lesson in it. Will they understand my question? Did I give away the answer? Should I take a better photo? Would someone argue with my answer? It's great exposure. And considering I could've punted and just had them work on the review for the final instead, I think our class time was much better spent. Plus, the school has a cool new art installation. Less than an hour after the wall was completed, I watched a dozen random students wander by and just stop, drawn in by the lure of the Estimation Wall.

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Fixing Inverse Trig War

Summer of 2012 I found a genius idea. Log War was born from a Kate Nowak post that itself was based on the work of another. I enjoyed the idea so much in Algebra II that I generated a Pre-Cal version in Inverse Trig War. Sam Shah took THAT idea and made regular Trig War. It's a winner idea, and this year was no exception.

Because I love iterating, I took what I learned from watching the game last year and made some modifications.

Mechanics

The posts I linked explain all the rules. But, in summary: take a batch of trig equations, have students record the values of the cards then shuffle and pass them out face down. Everyone plays a card at once, the card with the highest value (or lowest) wins the pile. A tie is broken with a one card face off. When you are out of cards you're out of the game. Most cards wins.

Issues

Problem 1 is that while playing, students had issues determining the value of their card quickly. I had them divide up the cards and solve them before playing, but because the cards weren't numbered (D'OH) there wasn't an easy to read answer key. Problem 2, the game is fun and novel, but not for long. Last year I had them play for about 15 minutes and when combined with the problem of resolving answers, the fun vanished after that long. Problem 3, there were too many cards and some of the equations were funky.

Inverse Trig War Cards Word PDF (prints to Avery 8160/5160 labels)

Fixes

Removing the kinks was pretty easy. Last year the game came at the end of a 90 minute period. This year I did it during a 50 minute period. The lesson had two parts. The previous day, I introduced inverse trig via simple equations, like 2*sin x = 1. I put a few examples on the board for them to try. On game day, the deck of cards was on the tables. Solution to Problem 3: I shrunk the deck from 40 cards to 30, and stripped out the more obscure equations. Solution to Problem 1: the cards were numbered 1-30, students made an answer key in their notebook with matching numbers. We also reviewed the answers before I introduced the game.

Solving the cards and reviewing answers took 30 minutes. I had them play for 10 minutes. For 5 minutes they played where the highest value won, for another 5 the lowest value won. It was the perfect amount of time.

It was surreal witnessing the transition from relatively silent equation solving to lively competition. I love this game.

Tournament of Champions

Later in the day I had an idea. Naturally, I get faster with my presentation and so the last two classes to play had a few more minutes to spare. I had them play highest card wins for 5 minutes, and then lowest card wins for 5 minutes. The champion of the table was brought to the center for the Tournament of Champions. Six people (one from each colored table) battled for Inverse Trig War supremacy.

It was pretty dramatic. Did I mention I love this game?

Some more video of it in action.

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Self-Sufficiency

I had an office job for a few years. An office job that generated a lot of paperwork. There was lots of printing and reprinting and copying and even faxing. But it wasn't a big deal because whether I was working out of a real office or a construction trailer, the equipment was nearby. If I needed something, the supplies were right there.

The first strange adjustment to teaching is the sheer size of a school building. You ever been in one lately? Huge. Print something? It's a 100 yard walk to the printer. Did it print wrong? A 100 yard walk to your room and then back to see if it worked this time. Need to copy something? Upstairs. Forget something to copy? Walk to the stairs, down the stairs, down the hall, find the thing, down the hall, up the stairs, walk to copier.

Need to talk to a colleague? Better be worth the three flights of stairs. Some school days I may only interact with two adults just because everyone is so far apart.

My first and second year I often found myself in a situation where I ran out of copies or needed to print new copies in the middle of a lesson. The process of printing whatever it was, finding a neighbor to listen for danger, walking to the printer and back could eat 5 minutes of class time.

Then my third year I adopted the "never again" mindset and began a small supply hoard in my room.

Printing

Seriously, buy your own printer. I know you think the laser printer in the math office is great and it seems really fancy and expensive, but it's not. Like, you have no idea. \$100 gets you black and white laser with a toner cartridge that will last two years.

Even color is getting affordable. This year I went color and it's been very beneficial.

Supplies

Colored paper entered my life in a big way during my third year. I began cranking out posters and other displayed work once a grading period. I easily used a few hundred sheets a year. The department had a supply that kept up with demand, but either my ideas are spreading or it's being used in other ways because our year's supply is already gone. Posters are harder to come by as well. A paper shortage is not getting in my way. Reams are \$8/each and 500 sheets could easily last a couple school years. This is a no brainer classroom investment.

Oh, and a full backup of toner just for kicks. And tennis balls. All the tennis balls.

Student Facing Consumables

A classroom built around a student constructed notebook goes through a lot of materials. Assignments need to be glued, taped, or stapled. Things need to be colored. Straight lines need to be drawn. For that, there's the student supply table. Everything on here is intended for student use.

Rulers, markers, scissors, tape, staplers, colored paper scraps and a hole puncher. The rulers are the 0.99 variety from Office Depot. Get wooden ones. The markings on the plastic ones rub off over time. Every August I buy two packs of new markers and put them in the bucket. Throughout the year kids are free to toss markers that are no good. Next is glue sticks. I ordered 100 through bulkofficesupply.com for 0.50 each. And these are big glue sticks too. Triple the size (0.74oz) of what you normally get (or what we get through our district supplier). Scissors are department furnished though I have 30 in reserve I just bought as well. And who needs three staplers? Teachers need three staplers. Don't skimp here either. Nothing gets abused like a classroom stapler. It needs to last. Sadly even at \$20/each I see a stapler last a year school, maybe one and a half.

Control

In the end, I invest in this stuff because I demand control. Eating 5 minutes of class time to walk to a printer is no longer ok. Having 30 kids wait for one stapler to be passed around is not ok. Scrapping a project because there are no materials is not ok. Paying attention to the logistical demands of your classroom payoff in huge ways and shouldn't be underestimated.

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If you read enough teacher sites, you might get the impression that anyone who has one creates a magical experience during any lesson they teach. It's like Pinterest guilt or something. But, this is not true. I screw up, all the time. Just a couple days ago something I thought that would be great blew up in my face within 15 minutes of handing it out. How you adapt to failure of this kind is usually a quick way to know how long someone has been in the business. My first year of teaching, we'd be talking Level 5 FREAKOUT. Is this why some people use a worksheet for 10 years? Maybe.

Good Intentions

Because I get bored easily, I have been toying with the curriculum. I have this harmonics lesson I added last year, but wanted to set it up with some talk about trigonometric transformations. And with Desmos, it should be easier than ever, right? I generated a bunch of random sine waves and altered them in different ways, the intention being to give the students the pictures and have them poke around until they could come up with a sin x AND cos x function that would reproduce the picture.

I thought it was pretty straight forward. I did a demonstration, trying to model the thought process about how to address things like amplitude, vertical translations, etc. I passed out iPads and set them loose. Everyone figured out A. Then they ran into C and the whole process ground to a halt. Classic kid move. This is hard, me stop now.

Why? Well, C is the first one with a modified period. It's possible in setting up the activity I failed to mention this as a possibility (I definitely didn't demo it). Also, in true math fashion, for whatever reason, no matter how good a student is at determining the period with the equation present, they have their own Level 5 FREAKOUT trying to read it off a picture. Or never think it's something you could even do. Remembering that for next year.

Intervention

After about 20 minutes I took the temperature of the room. Again, A wasn't a problem, but beyond that was a crapshoot. Around 10 were able to determine several of them. But not the kind of percentage I want. So quickly, after a little lecture about giving up at the first sign of trouble, we concluded the lesson together. I made up a sine function (leaving the equation visible), and they had to determine the cosine version. We discussed what properties are the same, what's different, and how to source values for horizontal transitions (most had ignored my hint about unit circle values).

This also helped to show that there isn't a single right answer. It also gave me my warm up tasks for the next groups. You know, right after I readied a new batch of functions in 6 minutes...

Pay No Attention to the Graphs Behind the Curtain

I figured it was the period changes that were screwing this up, in addition to the short introduction. I drew up four new functions (reason 347 you need a printer in your room):

This time, no period changes. And I started each successive class with the activity I used to save the sanity of the kids in my first class. I created a sine wave, they fiddled with finding the cosine wave. We discussed what should be different. We discussed how to determine amplitude, how to determine a vertical translation, and in more explicit terms what works for horizontal translations.

Greater success this time. Everyone got at least two of them, and a significant percentage were able to get three. Given more time they'd probably get them all. Lots of interesting things this highlighted: the difficulty of open-ended tasks, the evidence that students are incredibly far from being experts, and how bad students are at reversing information (deciphering the period changes).

I'm going to keep the original batch of functions. I can make it succeed next year. Plus, I laminated them and stuff.

On the whole, this is a great activity. I would recommend it. Despite the audible, it turned transformations into a puzzle and there was genuine excitement from students as they solved them. Far better than a lot of scripted drawing.

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A big pillar of Algebra II is the quadratic formula. It's not the first introduction, but quadratics get an exalted position in the course, so there you go. Quadratics as a topic is a bit of a mess. There are so many solution methods that it's no surprise a student would freeze when presented with a problem. Do I need to factor? Could I try to complete the square? Or is a square root simplification all I need?

If that wasn't enough, this giant formula gets thrown at them in a "I know we already went over 5 methods, but this could've done the job the entire time." I paired down the number of solution options in my Algebra II Pivot Project and even then I feel like I might have done too many. However, after I got a feel for the graphing unit, I actually found a way to motivate the use of the quadratic formula.

Discovery

My students' progression has been 6 weeks of algebra followed by 5 weeks of proving that algebra via graphs, and the last 2 weeks doing special topics in quadratics. Specifically, calculating a vertex, proving the validity of vertex form, and calculating a focal distance. Along the way, we were finding solutions to equations that we did not have the algebra equipment for (not yet anyway):

I used these as an example for what was possible for graphing. But then out of curiosity I wondered, how do you get those intersection points to surface? And then DUH, there was my introduction for the quadratic formula.

Thanks, Textbooks

How did I miss this for so many years? Well, textbooks do a terrible job of putting context around it:

Holt Algebra 2 2007

Every problem has the same terrible set up. Find the roots/zeros/solutions. Why so much emphasis on the zeros? Who cares? [I have another rant about the limited worth of the discriminant, but some other time] To me, this pigeon holes the quadratic formula. It is only useful for finding x-intercepts, say the textbooks. Let it be the nuclear option it was born to be. Don't force a kid to wait and never see the usefulness until doing Heat Transfer homework. The power of showing the method with the picture cannot be understated.

Complex Numbers

Why is the quadratic formula so valuable? Well, it really does work with everything. And a student should understand this. Case in point: complex numbers. There was a whole discussion panel on the topic at TMC13.  I never got the idea until I sat there looking at my graphing example and thought: what happens if you force the issue? What if we try to prove two parabolas intersect when they clearly don't?

It's this kind of experimentation that discovered complex numbers in the first place. Which is such a cool feature of math! Why restrict it? Why is it constrained to the "finding zeros" silo? These silo techniques are what kill student motivation. If it seems like a unique one-off, they will dutifully memorize the formula, answer your silly question about roots and move on. But if they see WHY you need such a thing. That it has general purpose applications and is just an extension of their prior work with equations, that has value. Math courses should have a flow and a universal theme. Too many textbooks have turned Algebra II into a mess of unrelated drive-by topics. The quadratic formula deserves better.

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Pivot Algebra II Update Two

Another six weeks, another phase in the crazy plan finished. At the start, I didn't know how the graphing components would go, but we got some real quality conceptual work done here.

Read Part 1: The Story So Far, a summary of my equations unit

Big Ideas

First and foremost, graphing contexts in modern Algebra II are heavily concentrated on transformations. You introduce the ideas of translation and scaling and repeat it for all the parent functions. But, it's done in such a way that gives zero context. I am guilty as charged with this. In the past, we teach the transformation rules, teach their descriptions and then move on to something else. There's no emphasis on why you want to know transformations are a thing.

Step 1: De-emphasize Transformations

I took what is normally done in a week and condensed it to two days. After an enlightening discussion with Nowak, I issued them a challenge. Here are some un-labeled graphs, can you manipulate a parent function such that you get the exact same picture?

This was one of their first experiences with desmos, so we spent some time discussing notation and where manipulators can be placed. After letting them play for a while, we were able to tease out the transformation rules. And yes, it is weird that (x - 3)^2 seems to shift it right instead of left. Once they found a successful match, they added the equation to the picture and we discussed the description.

On Day 2, I wanted to develop some recognition of equation types with graph types, how to spot a radical graph a mile away, etc. They were issued this challenge:

The same writing component was added. For each challenge, they also had to annotate the graph with the domain and range.

Step 2: Graphs Have a Point

Transformations out of the way, we doubled back on all the algebra we had been doing. Amazing secret kids: all those equations you were solving were really just a manipulation of transformed graphs. Take 0.5x^2 - 5 = 3 for example. We know the algebra, but did you know you can double check this? Remember how we got two answers?

Big payoff there. They felt like suckers for doing all the work by hand. Also helps justify the hand waving about multiple solutions. Going further, what if there are expressions on either side of the equal sign? Something like 0.5x^2 - 5 = -3x^2 + 7? Boom, that works too:

Now we can get nuts. At the moment the quadratic formula is unknown, so how do I determine where an un-factorable quadratic expression equals a modified linear expression?

The possibilities become endless. Absolute values intersecting with radicals, quadratics and linears and students eye-balling the solutions like it's no big deal. How fun would those be on a TI? Second question, how much time would you waste trying to graph all this by hand?

Step 3: The Meaning of Inequality

And lastly, what about these inequality things? Ok, so the algebra isn't much different, but what does is really mean when I get x < 3 rather than x = 3?

Previously, a single point on the graph was considered the solution. Now, what's going on here? So an inequality opens it up to various sections of the graph? How do we define that? And now we have some justification for domain. For these, we redrew the graphs to better represent what was valid. If it's contained in the shaded region, that section should be solid, the rest not so much. There were some struggles with domain and range here, but it was nothing a little extra explanation couldn't fix.

In the final phase of our graphing unit, we got into complex solution regions bounded by functions of all types. I discussed the details last week. In this age of beautiful, modern graphing tools, the complexity is only limited by your imagination:

It just blows my mind how casually we can do this. It took 30 years for me to appreciate math like this, why do my students have to wait that long?

Conclusion

In summary, the goals of the graphing unit:

• spontaneous recall of graphs of four parent functions
• justification of algebra mechanics
• make transformations just a footnote
• lower the entry level on very complex mathematics

When fall semester draws to a close, I will neatly package this material as much as possible. There were four tests during this unit. Students had access to Desmos while working on them.

I never want to graph on a TI ever again.

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More Inequality Explorations

Over a year ago I hit upon a simple idea related to linear inequality explorations: mapping locations. In this version, with limited iPads at my disposal, I put a list of predetermined locations on the board "demonstrate west of Toyota Center" for example. This year, in my crazy Algebra II experiment, we entered graphing with knowledge of more than just linear functions. In fact, at this point, the idea of quadratic or absolute value inequality was no big deal to any of the students.

Anyway, while discussing the idea of single condition inequalities of all types I retreated back to this mapping activity as a way to discuss multiple conditions in a realer sense. What does it mean to be a valid point within an inequality?  Multiple inequalities?

I pulled up my (not actual) house in Google Maps and discussed how I would describe the location of a Halloween party they weren't invited to:

It wasn't hard to grasp the idea that discussing the house this way helped narrow down its location. Given a driver with decent knowledge of the city, this might be enough to place the address.

Idea in hand, I passed out iPads, styluses (stylii?) and gave them their instructions:

• Given six locations: your house, your middle school, our high school, the US Capitol, Empire State Building, and free choice
• Locate four of these items on a Map
• Screenshot the map
• Find two ways to describe the location

I gave a brief tutorial on Google Drive export, but since they had to submit four items, they got it down after a bit.

Results were pretty good. This took 40 minutes or so of class time. I used to jump off on their task for the next day.

The Extension

The next class day we explored multiple inequality conditions created by all sorts of functions. I prepared and array of points and had them test where they were valid. There were a set of 5 condition groups they tested.

Going further, later they were given other conditions and we created sketches of the entire solution region and attempted to describe the domain and range of them.

The most enlightening thing for me about this task is that it would be much more cumbersome without a modern graphing utility. The plots could be done by hand or with a TI, yes. But what is my goal here? To be able to discuss the nature of meeting a condition, or watch kids struggle to even get there because they took 20 minutes to make one pair of graphs?

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More Frictionless Sharing

Classroom technology should have two functions: let you do something not possible with or better than paper and not be the center of the attention. If you want students to create something with technology (presentation, drawing, etc), the PROCESS of creating it should not overshadow the POINT of creating it.

Tablets have to be treated differently from computers, though. EdTech marketing material is clueless on this front, not even mentioning the ways you would do this. Even the official iPad Education site skips ALL of the details of how teachers collect all these great things they talk about.

Tablets will start working like magic if you do a couple things:

• sign every device into the same iCloud/Apple ID account
• enable Photo Stream
• create 1 new gmail account
• sign this new gmail account in to Google Drive on every iPad

The idea is to clone 1 device however many times you need in a class set. The students does not engage with the tablet as an individual, they engage with it as a terminal to a larger system that doesn't care who they are.

Photo Stream

Here's the idea. Take a picture on a device. Moments later that photo copies itself to ALL devices in the room. It's incredibly useful for two reasons.

Example 1: locate a picture of [thing]

Students search for whatever you requested. These are either images on the internet that they save to the local Camera Roll, or through creating new pictures with the iPad camera. How do we discuss these as a class? Downloading them to a computer would take forever. Connecting individual devices to the projector or what have you would take equally as long. With Photo Stream, as the students go about their business collecting photos, the 16 iPads operating as one silently dump the results in one location. You the teacher, plugged into this hive mind, can pull it up with no need to take a device from anyone. I discussed it last school year, here's what the collection looks like in real time:

Example 2: annotate atop picture of [thing]

Even with 16 iPads, that's not enough to go around. And if we go wandering around the school for a picture collection activity, we don't need to take them all with us. Just a few days ago, I wanted each student to annotate some things about a right triangle they found in the school. Each student needed their own picture. I sent them into the wild with 12, but we were able to create with all 16. How? Every snapped picture was on the photo stream. The 4 iPads sitting in the classroom were quietly receiving every picture being collected. When it came time to create, I passed out the extras and students were able to find their photos without a problem. The device that took the photo is irrelevant.

In the last year, Google Drive for iPad has made great improvements. It's a native interface to Google Docs but is a location that can be sent items from other apps on the iPad. In my annotation example, I wanted each student to complete a drawing. Using Adobe Ideas the student brought in their triangle of choice and made their annotations.

When complete, we had some options. They could send the drawing back to the camera roll (where it would be sucked back into the Photo Stream), print the drawing, e-mail it to me (bwahahaha), or send the photo off device for viewing by me on the computer. I went with option three because I wanted to quickly see who was complete (by naming the file with their first name).

Having signed all devices into Google Drive with the same account, the annotated triangles could be dumped in the same location and monitored by me in a desktop browser. The process to send a completed drawing out of Ideas and into Google Drive takes less than a minute:

Students could verify their work was received by looking at the big screens in the room and watching for their name. I could also glance at this and get a feel for the progress of the room. Here's 10 minutes worth of Google Drive activity in my desktop browser sped up by 10x:

Collection is quick. Students figured out the procedure quickly. There was a bit of a start up cost to teach the saving process. But it will save me time in the future if I want future work to be submitted this way. Not a single kid ran into a technology hurdle with this. They drew a picture, they sent it to me, and we got back to work.

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Textbook Makeover Four

At the opening of trig, most Pre-Cal texts will start with radian measure. You do some converting of angles between the two measuring systems and boom, a random digression about linear and angular speed. Most of the problems are like this:

Precalculus with Limits 4th Edition

This unit sticks out a bit, and the problems are quite random in the midst of all the triangle talk. For years it's been a throwaway for me and I didn't have a great way to look at these concepts. About a year ago, I handed out a really lame assignment for this. The issues I had with that assignment are the same as this saw blade problem: information given up front, explicit instruction, no opportunity for curiosity.

Challenge

In our math department office we have some measuring tapes. They're 100' long and can be rolled up with a handle.  They come in handy for lots of things and are especially useful here.

Instructions were simple. One person got our their phone and opened the stopwatch. The other person manned the tape. The timer stops when all 100' are rolled up. The person doing the rolling had to keep track of their rotation count.

Goal: Can you spin the handle at 1 mph?

Most students experience miles per hour in a car. The challenge would imply "I should spin pretty slow." Yet, the handle isn't very long. And there's the curiosity. What DOES 1 mph look like at this small scale?

So they spun.

Real data in hand, we walked through turning their rotation counts and times into a rotations per minute number. Then we discussed distance. What distance did the handle cover? Most were able to see that as circumference. A few unit conversions and we were able to see how everyone did. Some were pretty close. Many overshot.

Second Challenge

Now with some reference about what 1 mph feels like, I gave a second task and had them do all the work to determine their speed. So, you think you can spin the handle at 3 mph?

I collected the data publicly. For the initial test, I had them calculate RPM on their own with a formula reveal to check work. Same for the final speed. They told me, then ran a formula to verify.

Fun task. Lots of interest from the kids from start to finish. Cut all the stuff that lacks real context (looking at you, radians per minute and πft/s). This unit is still a bit of an oddball, but it was more interesting this time.

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