Tag Archives: 3 Acts

[3 Act Lesson] Sandboxes: Volume of Cylinders (& Spheres)

In middle school we only get 47 minutes, which is not enough time for the 3-act lessons I had gotten used to (coming from 90 minute blocks!). After a 5 minute warm-up, and 5 minutes of going over HW, that leaves no time for a lesson. But today I think I was able to get all three acts in, so I wanted to share my success!

We’re doing volume of cylinders, cones, and spheres in 8th grade math right now.

Act 1

Me: Spring break is next week, and here’s my Spring Break plans.

Image result for sand box

Students: You’re going to the beach?

Me: Nope, I’m going to build a sandbox for Benji (my 2-year-old). What do you think is the most expensive part of the sandbox?

Students (in unison, surprisingly): The Sand!

Me: Right, so forget the wood for now because I have enough scrap wood that I can probably build the frame without buying any wood. My wife and I are trying to decide between a 6’x6′ and an 8’x8′ sandbox.

Student: 8’x8′ cause it’s bigger, duh!

Another Student: But that would cost more!

Me: What do you need to know to find out how much more it would cost?

Student: How much all the sand costs.

Me: Right. You don’t buy “one sandbox worth of sand” at Lowes. It comes in bags.


Eventually they get to needing (1) how much is in a bag (0.5 cubic feet worth of sand), (2) how deep is the sand in the sandbox (6 inches), (3) how much each bag costs ($4.25 is what I found at the local Lowes).

We did the comparison together because we were so short on time. If we had a block period, then I would have let them struggle instead, but I wanted them to get to calculating volume of cylinders in context instead. The work I write on the board looks something like this:


18 cubic feet

36 bags

$153 is the total cost for a 6’x6′


32 cubic feet

64 bags

$272 is the total cost for an 8’x8′

We have a brief discussion answering “why is $272 nearly double $153 but 8′ isn’t nearly double 6′?” Unfortunately I pointed this out to them and had to start the discussion but it’s something that I feel is important enough for me to “artificially” bring up.

Act 2

Now I want you to get one large (4’x2′) whiteboard for your table, one marker, and make a cost comparison between a 6′ diameter circular sand box and an 8′ diameter circular sandbox. Go!

Image result for sand box circular

The students did a really good job (we’ve been practicing finding the volume of cylinders).

Here’s some of their work (I’m sharing the more legible ones)


Act 3

As students finished, I gave them this challenge problem:

“Suppose the silo at our farm is filled with sand. How many 8’x8′ rectangular sandboxes could we fill with all that sand?”

I draw a picture of a silo (hemisphere sitting on a cylinder) with cylindrical height of 20′ and overall diameter of 10′. As you can see from the pictures above, some students did pretty well at that problem, too!


The students were really interested in my spring break plans. The “builders” of the class liked the idea of figuring out how to get ready to build a project, even if it was just buying sand. The “caretakers” of the class like that I was doing something for my 2-year-old. Get enough of the students on board and they all really take to it, and I was fortunately that this happened here. Here’s the google slides I used for the lesson.

How would you improve it? Alter it? Thanks for any and all feedback!


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3 Acts: Volume of a Box

I’ve done this activity before, but I can’t seem to remember whether I’ve blogged on it. Here’s the gist of it (with pictures!).

  1. Ask students if they can make a box out of 8.5 x 11 paper. Give them paper, scissors and stapler (or tape).Paper Box
  2. Once they’ve made the boxes, ask the students (or let them ask) “What’s the biggest possible box with an 8.5 x 11 paper?”
  3. Let them make guesses first (way too high, way too low, just right).Guesses
  4. Point out that we should get a bunch of trials and have students, as a class, decide which boxes to make (students select which box they’ll make which is different from other students. Have students make one box each to contribute to the table.Stack of Boxes
  5. Plot these on Desmos. Then put the boxes on the board so we see what each of the points represents.Boxes on the board
  6. Talk with students about equation for volume. Have some students provide the equation.Table and Equation for Boxes
  7. Plot the equation and click on Desmos for the maximum (this is Precalculus, not Calculus). Acknowledge who was the closest and have students write down what they learned.Boxes on the board with equation


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3 Acts: Participation Quiz with Stacking Cups

I had a great block day (1.5 hr classes) today with both of my Precalculus classes because, via Tina’s Goal-Setting blog post, I found Sam’s Participation quizzes from years ago. The stacking cups video was created by Andrew Stadel, so I’ll only link to the 101qs I hadn’t read that post[1], but I loved the idea. In the comments, Julie had mentioned trying this out with Google Docs, and since I like Google things way better than my Smart-board[2], I decided to try this out. I found out that it works best with Google drawings, because you can move text-boxes around anywhere you want. I don’t know if the other teacher did this, but instead of putting the names “blue group”, “red group”, etc. up on the board, I just made the rectangle match the color of the group. The best thing about using Google was I could collaborate with myself: I brought my laptop to school and so I could enter comments either from the computer at the front of the room (not very often) or from my laptop closer to the action.


We did a typical bare-bones math warm-up (is that a bad idea?). After the warm-up, I put the rules (copies straight from Sam’s post) up on the power-point:

  1. Everyone in the group must participate equally. There isn’t a leader, or the same person leading the show. The voices are shared.
  2. Students should not work too quickly. If they work simply to finish the sheet, without any other consideration, they aren’t doing it right.
  3. No one moves on until everyone understands. This isn’t about everyone having the same thing written down — but everyone has to know why.
  4. Students should think out loud. Students should check in with each other. Students should ask questions of one another.

Then I assigned groups, explained “Participation Quizzes“, showed the cup-stacking video, and said “GO!”

Here’s the result of me walking around, listening to groups, and recording what I heard/saw for 1st period and 7th period Precalculus:

1st Pd Participation Quiz -- Stacking Cups (1)

7th Pd Participation Quiz -- Stacking Cups (1)

Here’s a link to my template–feel free to make a copy of it and steal it!

*The line across the 7th period groups were before and after we discussed our answers–we then did a sequel. Here are the sequel questions I used to keep them going:

  1. What if we started with 10 blue cups and 1 white cup?
  2. How many cups does it take to reach Mr. Newman’s stool?
  3. You come up with two good questions.

When I did Robert Kaplinsky’s In-n-Out burger, he left a great comment about “breadth vs depth” of sequel questions. That really encouraged me to come up with a few deeper questions. Sort of “think about this in another way” (depth) rather than “do this similar problem and let’s make sure you can apply the skills you learned again” (breadth).

I was really impressed with how well the students started doing group work. The constantly changing board up front gave them ideas for how to better work together, and yet, it wasn’t too distracting.

Thanks to all the MTBoS members who helped shape this lesson–I’ve become a 100000% better teacher from reading you guys!

[1] I should probably stop what I’m doing and go back and read all of Sam’s posts… I even stole his way of making footnotes in blog posts!

[2] And I only have a smart-board in one of the two math classrooms in which I teach Precalculus.

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Video of a Lesson: Robert Kaplinsky’s In-n-Out Burgers

First off, thanks to Dan Meyer for alerting me to the idea of 3 Acts problems. This particular problem was created Robert Kaplinsky. You can find all of the lesson materials at this page, free of charge.

I had seen Dan Meyer present the 3 Acts lessons to teachers, and that helped shape my understanding of these problems tremendously. However, I hadn’t see one “in action”, with a bunch of real students, who may or may not already be “done” with math.


We start with some bare-bones math problem for the warm-up. Unlike last year, where “participation points” were given for working on the warm-up, there’s no obvious incentive for them to work on the warm-up, other than “so Mr. Newman doesn’t get on your case”.

Watching myself, one thing I’m glad I did was ask “if you messed up, how did you mess up?” I’m really pushing this year trying to make mistakes not only acceptable, but a great source of learning.

We open with a prayer (I try to open each class that way–this is a Christian school), and jump right into the warm-up.

The rest of the power-point is right here because you can’t really see it in the video.

Act 1

This goes pretty smoothly–most students know what In-n-Out Burger is, and the images that Robert Kaplinsky found are definitely riveting and shocking, if not simply gross enough to draw student attention. In my other period, the first question (How much does it cost?) jumps out before I even ask for questions. In the past I would have discouraged the somewhat-off-topic questions, but I find they often keep the interest of the students and makes the task genuine.

One of my favorite exchanges is:

S1: “We’re getting an In-n-Out Burger”

S2: “That’s a lie.”

It doesn’t contribute to the mathematical side of things, but it draws all the students in a little more. Time well spent (maybe 15 seconds talking about whether an In-n-out burger is coming to town?) in my opinion.

It was 10 minutes before class when I realized I hadn’t printed out the sheet that I wanted. Here’s what I had seen before and wanted (from Robert Kaplinsky:

Here was what I created, which gets at most of the same ideas:

Block Day Lesson.pdf or Block Day Lesson.docx (scribd isn’t working at the moment)

Block Day Lesson, page1Block Day Lesson, page2


I want the answers to the math questions to be earnest, so I try to treat all the questions more or less equally. That’s why I go ahead and answer the questions that I can, and we later tackle the questions that they can get. My goal is to answer everyone’s question in the end–or at least leave them with a good idea of the answers (or the tools to answer all the questions).

Act 2, part 1

I’m not sure where Act 1 ends and Act 2 begins, but I decided to cut the video where I said “Go” to the students. I was less than thrilled with students’ creative thinking, so I had a divergent thinking interlude.

Divergent Thinking interlude

My goal of this task was getting them to be more creative[1]. Part of that is helping them realize that they are more capable of being creative than they think they are.

Act 2, part 2

Now the students are more oriented to the task, and come up with a little better information. I give them the necessary information (the menu) and they take off.

Act 2, part 3

Here the students are working in groups at different rates. I do an okay job giving the group that was done first the longest/most difficult task. I really wanted one of each of the following from different groups: (1) a graph, (2) an equation, (3) a table, or a solution in another way. Some groups “thought it out” and used words, which was great.

Act 3

We (I) talked about what the graph means, what the axes mean, and what the equation “y=mx+b” means in this situation. After that, I showed them the “answer” (the receipt) and they thought it was cool. I mean, I got several of them to clap–that’s always fun when that happens in math class.

Unfortunately we ran out of time and I didn’t really get to explore “How many calories is that?” in this class, although one group in my other class (which I didn’t film) did. They found the information online by themselves (identical to the numbers that Robert Kaplinsky provides!), answered it, and even answered the question of “How much does that weigh?” They put their answer in terms of Chromebooks so we could compare to what was right in front of us.

The last part of class (which I cut from the film) just involved me teaching the students how to log into ActiveGrade. Mostly just classroom-administrative stuff that’s not nearly as interesting to watch.

Things to Improve On

Others talk about having students make approximations or estimates to increase “buy-in”. I didn’t do that because I forgot about that aspect of it, however, I think there was sufficient “buy-in” (this is pretty close to the start of the year). That’s definitely something I’ll need to do in the future, though, because it also increases their estimation abilities and allows us to discuss afterwards “Does this answer make sense?”

One thing I struggle with is finding a balance between giving students the distance they need to be creative and think on their own, yet being close enough to make sure they’re focused on the task at hand. Ideally the 3Acts is a great hook and I don’t need to be hovering over students to get them to work. But this hasn’t been my experience.

For the first half of class, the student in the front had his head down, and I had to go over and talk to him to make sure he was participating.

Watching myself teach, I talk way too much. And I answer my own questions way too much. I’ll walk away from a class thinking “that was a good class”, probably because I understood everything that happened. I need to do more formative assessment to make sure that they understand everything that’s happening.

I also haven’t made my Popsicle sticks in this class yet, so I also had the problem of the same 2-3 students answering all of my questions. Oh, and my wait time is awful. Every now and then I consciously think “just wait”, but not often when I’m excited about something. And these 3-act lessons always make me excited.

I didn’t end the lesson with a “summary activity” to make sure the students learned something. I even have a box at the bottom of my sheet “what did you learn?”, yet I didn’t take the time to fill it out. Now another day of school and a weekend will have passed, and I’m left asking myself whether it would be worth it to return just to answer that question.

One other thing I think I do is that I teach as if I’m in a rush. Yes, it feels like we have a lot to go through, and we do, but if I could slow down, I wouldn’t lose so many students, and I could take time to do things like acknowledge when a student is brave and submits his or her own mistake for review by the class.

Other Notes

I used Doceri to write on the iPad and have it show up on the projector.

I introduced Desmos to students, and they got a small glimpse of how awesome I think it is as a tool. I don’t they understand just how truly awesome it is yet, but that’ll come.

I know that this is a ridiculously long post, and I don’t expect many people (anyone?) to read it all the way through, though I hope that the video of the lesson was at least helpful. That is one thing that I would like to see more of on the MTBoS: video of teachers in real-time. I understand that many teachers (mostly public school, but some private as well) have tons of red tape to walk through to put videos online of them teaching since it usually involves students’ faces. Teacher blogs are a great window into a teacher’s classrooms, but I want to see how other teachers handle behavior problems, or keep students excited about a lesson when it starts to turn sour. This is like getting to observe other teachers in the MTBoS, which would be an awesome experience!

Another note: while this reflection/sharing was good, it took way too long–especially the video editing on my 6-year-old computer!

[1] I know that divergent thinking and creative thinking aren’t identical, but for the purposes of this activity, I used the two phrases interchangeably. Sorry.


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[3 Acts] The Coefficient of Friction of My Son

Last year in Physics, I used the PhET simulation Ramp: Forces and Motion to teach about calculating the coefficient of friction in the situation of an object on a ramp (in this case, a box).

This has worked well enough for me in the past. However, as I was travelling home with my son, I had an idea to create a 3 act lesson using my experiences.

Perhaps I talked too much in the video, so I might just show them the final part where I’m sitting up with Benji.

The idea really did just hit me as I was sitting there in the train, thankful that I didn’t have to keep my arm under him, propping him up. So often, ideas will come to me, but I’m just not in the right location to jot them down to save for later.

Another thing I thought of pertaining to this lesson is how it relates better to the girls in my class than most of the lessons. Sure, some girls like talking about drag racing or shooting a basketball, but this connects better to the majority of girls than those other lessons[1].

One downside to this activity is that on the train & plane, I’m not concerned about the coefficient of friction–I’m concerned about what angle I can sit up with him without him falling down. And that’s what we’re measuring at the start, so finding the coefficient of friction has become a superfluous academic exercise despite the “real-world-ness” of the problem.

And then there are the other things that I hope students will consider: things like my chest isn’t actually flat and we both have shirts on, so I guess we’re actually finding the coefficient of friction between his shirt and my shirt. But those are great things for students to consider on their own, so I don’t want to spoil it by pointing out all the inconsistencies in the video from the start.

[1] If you think I’m being sexist here, you should try walking down an airport terminal with a 3-week old. At least 10 times more women than men will stop you and say something.

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[Productive Struggle] Precalculus Logarithms

I’m trying to keep a positive spin on my Precal lesson from Wednesday (a few weeks ago, now), but it really, really flopped.  To be fair, it wasn’t entirely my fault: I had lesson plans using the internet, and the internet was totally out.  So it was only mostly my fault because I didn’t have backup plans

So I decided to try a 3-acts on the spot.  Note to self: do not try this unless you’ve done the specific 3-acts before and you’re very, very experienced at 3-acts.  I have been doing about a 3-act lesson every week or every other week or so, but this little amount of “experience” does not make up for the lack of planning and preparation.

I saw this chart shooting around the Twitterverse:

TECHNOLOGY Price of 1gb of storage over time:
1981 $300000
1987 $50000
1990 $10000
1994 $1000
1997 $100
2000 $10
2004 $1
2012 $0.10

After asking “What questions do you have?”, and discussing “how much a GB is”, they got to work plotting this on a whiteboard.

My first goal for students was to graph this and quickly realize that a “normal” scale wouldn’t work here because over half the points just sit on the x-axis, not really telling you anything.  A few creative students decided to make the “squiggles” and represent a significant change on the scale of the y-axis, but these students did not realize that (a) you really shouldn’t do that between data points and (b) you really, really shouldn’t do that multiple times on the same scale.  So they saw the need for a logarithmic scale, but even after they graphed the points on Desmos, they had no way of making the data scale that way.  Mistake #1.

The next mistake that I made was thinking that “because the data merits a logarithmic scale, then the best-fitting function must be logarithmic”.  I didn’t tell students to choose a specific function, but I hinted that since we had been working with “logarithmic functions recently, it’s probably a good place to look”.  I need to get better at Dan Meyer’s slogan of “be less helpful”.  I might as well have required them to use logarithmic functions with that kind of hint.  Even 2 minutes of playing around with the data before class and I would have realized that it is definitely not a logarithmic function.  Instead the students struggled for a good 10-15 minutes before I realized what was going on.  Mistake #2 (at least).

So I decided to give students a “break” while I regrouped and gathered together my thoughts.  Since my school has regular classes on Monday, Tuesday, and Friday, and block periods on Wednesday and Thursday, most teachers give students a break partway through the long periods for students to use the bathroom, get water, and just regroup mentally.  Until this class, I hadn’t given my Precal students a break because they’d been busy with the 3 Acts lesson we were doing.  However, my own fumbles demanded a break.

When the students got back, I explained to them my mistake and pointed out what kind of function they should have been looking for.  They jumped back into groups and started working on Desmos to find an exponential function that fits the data.  Once groups started getting an appropriate equation, I asked them more probing questions about the domain, range, and other specific questions (“How much will a GB cost in 2020?”).  However, I didn’t have one specific goal for all the groups to come back together and discuss, so I lost their focus unless I was standing over their group shooting questions at them.  Mistake # too-many-I-lost-count.

There are tons of other mistakes with this lesson that I’d like to point out:

  • I didn’t have a good “hook”, or even a good idea where to take the students after they got their graph.  If I had spent some planning time before to come up with that, then it would have vastly improved their experience.
  • I didn’t have any good ideas for how to view the data-that-should-be-on-a-logarithmic-scale.  I’ve never learned how to put data onto a logarithmic scale accurately, so I wouldn’t feel comfortable showing students how to do it.
  • A “hook” isn’t just a bunch of questions, but you do need questions before you get a good hook, and I had neither.  I didn’t record the students’ questions beforehand, like I almost always do, and therefore I certainly didn’t come back to them at the end of the lessons, which I also almost always try to do.  In short, I just killed some of my students’ trust in asking me questions in the future.
  • I didn’t have any sequels ready for students.
  • The information, by itself, wasn’t particularly compelling.  I can imagine making a slide-show of the cost and amount of data on a slide with a picture of an object with that storage capacity.  To actually see it go from several buildings down to the size of less than a thumbnail would leave an impression and provide some other sequel questions.  Missed opportunities.

There were a few positives: students felt the need to create and use a logarithmic scale (however fleeting that feeling was), students practiced fitting an exponential curve to data (they’re getting quite good at fitting all kinds of functions lately), and they learned what a GB is (super-important in today’s world, in my opinion).  However, it felt worse than a wasted class period–it felt like a wasted block period.  Even though this is my 4th year teaching, I’ve got to have back-up plans for technology failures and I’ve got to get better at putting time into these kinds of tricky lessons.

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Entire Lesson Videotaped: Intro to Parametrics

On a whim, I decided to videotape an entire 45 minute lesson in one of my classes. The only thing I had on hand was an iPad (2nd Gen), so the video quality is low, but I’m also kinda banking on that so I don’t have to blur out students’ faces.

Other information possibly of importance:

  1. There are 15 students in this Precalculus class.
  2. This class is the last period of the day. [1]
  3. Yes, that is my principal who strolls in at [29:43], and yes he picks up a student’s guitar and starts playing in the last 10 minutes of class.   (Oh, and it was his birthday.) Don’t you wish you had a principal as cool as mine?

Outline of the Video

[0:00] to [3:46] Waiting for class to start. (I should have edited it out, but this is why the video starts at [3:46].

[3:46] to [5:40] Waiting for student let out of choir to get to my room (they’re late most of the time because our choir teachers let them out late).  Because I have all these students for Chemistry, we discuss a little Chemistry while waiting for everyone else to get here.

[5:40] to [8:36] I show students how they can find all the standards from the class on my website.  We also discuss Inverses of functions and they convince me to give the quiz on Tuesday instead of Monday of next week.

[8:36] to [12:30] 1st Act: I develop a reason for Parametric Equations and we do a really rough experiment of a student walking into the room.

[12:30] to [31:47] 2nd Act: Through questions I build an intuition for Parametric Equations through graphs and the measurement of our “experiment”.

[31:47] to [40:04] 3rd Act: We answer the questions I provided for them at the start of the lesson (although I didn’t vocalize them, and I didn’t have a “hook” for them like your typical good 3 Acts lesson. We look at two ways to use technology to graph this equation and I give them two of these types problems for homework (see the sheet below). I use the TI Calculator because they’ll need to know how to use that for most standardized tests. I use Desmos because it is awesome and easy to use (and way cooler than the TI Calculators).

[40:04] to [50:03] I promised them from the day that I would do the Fibonacci Magic trick.  If you haven’t seen it, here’s a video of it being done, long with an explanation of how to do it (however, I don’t like his explanation of how to multiply a 3-digit number by 11).  If you prefer to read, here’s an good explanation of how to do it.

Supplemental Materials

Since the board is hard to see in the video, here are two pictures of what I put on there.  For the most part, the black is what I had up before class started and green is what I added during class.

Picture of Classroom

Four Representations of a Parametric Equation

Here’s the worksheet I gave for homework.  Notice how I set up the board so all 4 parts match the worksheet.

Please leave feedback on my lesson and on my teaching style: both constructive and destructive comments are welcome, so please let me know what you think!


[1] The students recognize their tiredness at this point and frequently complain when I ask them to do a particularly mentally challenging problem or task.  I suppose I am fortunate that they recognize this, though I wish they had a bit more motivation and didn’t use it quite so often as an excuse.  Fortunately these are all good kids who try despite their tiredness, though they have vocalized to me that they wish this class was taught earlier in the day.

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