# Three-Act Problem Solving

My lessons never stay the same.  They’re always evolving.  Recently, I’ve taken a look at some 3-Act Tasks I created and I noticed:

• Some of the tasks are lacking an act.
• Others have resources that no students ask for (at least students that I’ve worked with).
• The quality is low (shaky camera, point of changes, etc.)

So, I finally had a minute (read 2 days) and revisited each.  Below, you’ll see the tasks I’ve chosen to revisit.  An explanation of the original, what I changed, and why I changed it follows.  If you’d like to skip this and get to the revisited tasks, click here.

## Revisited #1 – The Candy Bowl

My very first attempt at a 3-act task was the Candy Bowl task.  I was working in an elementary school at the time and Graham Fletcher had created problem to get 2nd and 3rd grade students reasoning about subtraction by removing the numbers from the problem context.  His context involved the lunchroom and numbers of students in three classes.  We talked on the phone about this for a while and though I liked the problem, I wasn’t crazy about the context.  I sat in my room trying to think of a context that would be a bit more engaging for students to think about.  And the Candy Bowl was created.

It was a good problem, but it really lacked one of the most basic parts of a 3-Act Task… The third act.  The reveal was weak, because it relied on the teacher to give students validation.  The updated version, which had to be done from scratch (apparently whoppers candies are no where to be found anywhere near Valentine’s day), can be found here with all new updated resources for Act 2 and new video including two reveals, depending on which question students decide to tackle.

## Revisited #2 – Sweet Tart Hearts

Another one of my early tasks was Sweet Tart Hearts.  I really liked this one from the beginning. There is a huge focus on estimation which allows for students to obtain solutions that are close, but not exact in most cases.  This also allows for the teacher to facilitate a discussion about why answers may not be exact for a variety of reasons.  But again, it really lacked that third act.  The task was good, but the closing of the lesson was weak due to the fact that the students were relying on the “all knowing” teacher to give them affirmation.

Apparently Sweet Tart Hearts are a hot commodity a few days before Valentine’s day.  I went out the other day for a quick run to pick up a bag.  I had to go to 4 stores and finally found a bag (the last one).  I thought it would take about 10 minutes to do this revisit.  Surely the  numbers for the colors would be similar to the last time.  Not only was that not true, but Sweet Tarts changed the orange hearts to yellow!  But, the revisit is all done and I’m very pleased with the new reveal which allows the video to reveal the answer and the teacher to focus students on the reasonableness of their solutions.

## Revisited #3 – The Penny Cube

So, this was the quickest fix.  I just updated the Penny Cube page (all of the coin specifications are still there – in case anyone wants them).

Note:  In this post I share how I changed my approach to teaching the Penny Cube task.

So, it took a few days, but I’ve revisited some tasks that have been bugging me for a while and I hope it’s for the best.  I know I’ll probably give these another look in the future.  I’ll just need to start in early January to make sure I get the candy I need.

# The Penny Cube

I just finished a 5th grade 3-Act task called Penny Cube that I created last spring.  I tried it then, but just to get some feedback from students and see what I might need to change about how the task should be presented.  Now, after completing this task with two groups of students (at two different points in the year), I’ve learned three things:

1. Students see a video and notice a bunch of things that teachers don’t even realize are there.
2. The curious questions students ask first are often “why” questions.
3. There’s no way to predict everything a group of students might wonder.

I’ll take this reflection from the beginning.  First, I let students know that I was going to show them a video clip.  I also told them that I was going to ask them what they noticed when it was finished playing.  I gave students a chance to brainstorm ideas about what they could do to make sure they would be able to share what they noticed once the clip had finished playing.  Their ideas were amazing:

• We could look for expressions (on faces – I found out later that this student was thinking about context.  A person’s facial expression can tell a lot)  Unfortunately, there were no facial expressions in this video.
• Listen carefully (they might be able to hear something that might give them a clue about what was going on – these students were already expecting a problem situation!)
• Stay focused on the clip.
• Take notes.
• Try to remember as much as you can.

I had never done this before, but after hearing their ideas, I will be using this again.

After showing the video clip for Act 1, I immediately had them talk about what they noticed with their groups.  Then, they were asked to share with the whole group.  Here is what they noticed:

What’s missing from this picture is the wonderful reasoning given for some of these.  For the last bullet, “container is open in the front,” the student told the class that it was open in front so the pennies could be placed in the container more easily (I never thought they’d see or think about that).  They even began to wonder a bit here – “it might be an expression or it might be counting.”  My favorite, though, is the estimation by the girl who said “it looks like 100 pennies in the stack (\$1.00).”  This was particularly interesting to me because of what happened when they were asked to estimate for the focus question.

The wonders were typical from what I usually get from students new to 3-Act tasks, but I handled it a bit differently this time.  Here are their wonders (click here for a typed version of Penny Cube Notices&Wonders):

In my limited (yet growing) experience with teaching using 3-Act tasks, I’ve noticed that the wonders are initially “why” questions (as stated in number 2 above).  I told the class that I noticed that the questions they were asking were mostly “why” questions.  I asked them what other words could be used to begin questions.  Rather than trying to steer students to a particular question, I decided to focus the students’ attention on the kinds of questions they were already asking, and guide them to other types of questions.  It didn’t take long!  Within about 5 minutes, students had gone from “why” questions to “how many . . .” and “how much . . .” questions which are much easier to answer mathematically.

The students were then asked to figure out what they needed to solve the problem.  From experience with this task, I knew that most students would want pennies, so I had some ready.  I didn’t give them the Coin Specifications sheet, because no one asked for it.  I did have it ready, just in case.  Every group asked for pennies and rulers.  I wasn’t sure how they would use them, but I was pleasantly surprised.

Here’s what they did:

How many pennies in 2 inches

How many pennies in an inch

How many pennies fit on a 6 inch edge of the base

How many pennies cover base

How many stacked pennies in 2 inches?

The students all started in a place that made sense to them.  Some wanted to figure out how many in the stack, so they stacked pennies and quickly realized (as I did when filling the cube) that you can’t stack pennies very high before they start to wobble and fall.  So, they measured smaller stacks and used that info to solve the problem.  Others wanted to find number of pennies along an edge to find how many cover the base, then work on the stacks.  Students were thoroughly engaged.

After three 1 hour classes, students were wrapping up their solutions.  Some groups were still grappling with the number of pennies in a stack.  Others were finished.  A few were unsure about what to do with some of the numbers they generated.  All of this told the classroom teacher and me that there were some misconceptions out there that needed to be addressed.  Many of the misconceptions had to do with students disengaging from the context, rather than integrating their numbers into the context:

• One group was unsure of whether to multiply  the number of pennies in a stack by 12 (6 inches + 6 inches) or to use 64.
• Another group found the number of pennies to cover the base and multiplied it by itself to get their solution.
• A third group found 37 pennies in 2 1/2 inches and was having a difficult time handling that information.
• A fourth group had come up with two different solutions and both thought they were correct.  Only one could defend her solution.

Eventually, several groups arrived a solution that made sense to them.

Time to share!

I chose one group to share.  This group had a reasonable solution, but their method and numbers were different from many of the other groups, so this is where we were hoping for some light bulbs to begin to glow a bit.

This group shared their work:

I asked the class what they liked about the work.  The responses:

• The math (computations) are written neatly and they’re easy to follow.
• I know what their answer is because it has a bubble around it.
• The question is on it.
• It’s colorful.

All good.  Now, for the best part:

What questions do you have for this group?  The responses:

• Where did you get 34?
• What does the 102 mean?

Any suggestions for this group to help them clarify their work to answer some of your questions?

• Maybe they could label their numbers so we know what the numbers mean.
• Maybe they could tell what the answer means too.  Like put it in a sentence so it says something like “6,528 pennies will fit in the container.”
• Maybe they could have a diagram to show how they got a number like 64 or 34.  I know that would help me (this student had a diagram on his work and thought it was useful).

The light bulbs really started to glow as students began making suggestions.  As soon as a suggestion was made, students began to check their own work to see if it was on their work.  If it wasn’t, they added it.  All of the suggestions were written on the board so they could modify their work one final time.  The best part about this whole exchange was that students were suggesting to their peers to be more precise in their mathematics (SMP 6 – Attend to precision).  And, they really wanted to know what 34 was because they didn’t have that number on their boards (which is why I chose this group).

Now for the reveal!  When I asked the class if they wanted to know how many pennies were in the cube, they were surprised when I pulled up the reveal the video.  I guess they thought I’d just tell them (that’s so 1980’s).  They watched to see how close they were and when the total came up on the screen, many cheered because they were so close!

The students in this class were engaged in multiple content standards over the course of 3 days.  They reasoned, critiqued, made sense, and persevered.  It’s almost difficult to believe that this class was a “remedial” class!

Below, I’ve included a picture of each group’s final work.

Finally, one of the conversations witnessed in a group was between a girl and a boy and should have been caught on video, but wasn’t.  This group had an incorrect solution, but they were convinced they were correct, so to keep them thinking about the problem, I asked them how many dollars would be equal to the number of pennies in their answer (3,616).

• Girl:  There are 100 pennies in a dollar.  So 600 pennies is . . .
• Boy in group: \$6.00
• Girl (after a long pause):  1,000 pennies equals \$10.00
• Boy:  So that’s . . . um. . .
• Girl:  We have 3,000 pennies, so that’s \$30.00.
• Boy:  \$36.00
• Me:  Share with your group how you know it’s \$36.00
• Boy:  Because \$30.00 and \$6.00 is \$36.00
• Girl:  And the rest (16) are cents.  \$36.16!

And they didn’t even need a calculator!

Math really does make sense!

The short answer:  It’s what’s best for kids!

If you want more, read on:

The need for students to make sense of problems can be addressed through tasks like these.  The challenge for teachers is, to quote Dan Meyer, “be less helpful.”  (To clarify, being less helpful means to first allow students to generate questions they have about the picture or video they see in the first act, then give them information as they ask for it in act 2.)  Less helpful does not mean give these tasks to students blindly, without support of any kind!

This entire process will likely cause some anxiety (for all).  When jumping into 3-Act tasks for the first (second, third, . . .) time, students may not generate the suggested question.  As a matter of fact, in this task about proportions and scale, students may ask many questions that are curious questions, but have nothing to do with the mathematics you want them to investigate.  One question might be “How is that ball moving by itself?”  It’s important to record these and all other questions generated by students.  This validates students’ ideas.  Over time, students will become accustomed to the routine of 3-act tasks and come to appreciate that there are certain kinds of mathematically answerable questions – most often related to quantity or measurement.

These kinds of tasks take time, practice and patience.  When presented with options to use problems like this with students, the easy thing for teachers to do is to set them aside for any number of “reasons.”  I’ve highlighted a few common “reasons” below with my commentary (in blue):

• This will take too long.  I have a lot of content to cover.  (Teaching students to think and reason is embedded in mathematical content at all levels – how can you not take this time)
• They need to be taught the skills first, then maybe I’ll try it.  (An important part of learning mathematics lies in productive struggle and learning to persevere [SMP 1].  What better way to discern what students know and are able to do than with a mathematical context [problem] that lets them show you, based on the knowledge they already have – prior to any new information. To quote John Van de Walle, “Believe in kids and they will, flat out, amaze you!”)
• My students can’t do this.  (Remember, whether you think they can or they can’t, you’re right!)  (Also, this expectation of students persevering and solving problems is in every state’s standards – and was there even before common core!)
• I’m giving up some control.  (Yes, and this is a bit scary.  You’re empowering students to think and take charge of their learning.  So, what can you do to make this less scary?  Do what we expect students to do:
• Persevere.  Keep trying these  and other open problems.  Take note of what’s working and focus on it!
• Talk with a colleague (work with a partner).  Find that critical friend at school, another school, online. . .
• Write a comment below. 🙂

The benefits of students learning to question, persevere, problem solve, and reason mathematically far outweigh any of the reasons (read excuses) above.  The time spent up front, teaching through tasks such as these and other open problems creates a huge pay-off later on.  However, it is important to note, that the problems themselves are worth nothing without teachers setting the expectation that students:  question, persevere, problem solve, and reason mathematically on a daily basis.  Expecting these from students, and facilitating the training of how to do this consistently and with fidelity is principal to success for both students and teachers.

Yes, all of this takes time.  For most of my classes, mid to late September (we start school at the beginning of August) is when students start to become comfortable with what problem solving really is.  It’s not word problems – mostly. It’s not the problem set you do after the skill practice in the textbook.  Problem solving is what you do when you don’t know what to do!  This is difficult to teach kids and it does take time.  But it is worth it!  More on this in a future blog!

### Tips:

One strategy I’ve found that really helps students generate questions is to allow them to talk to their peers about what they notice and wonder first (Act 1).  Students of all ages will be more likely to share once they have shared and tested their ideas with their peers.  This does take time.  As you do more of these types of problems, students will become familiar with the format and their comfort level may allow you to cut the amount of peer sharing time down before group sharing.

What do you do if they don’t generate the question suggested?  Well, there are several ways that this can be handled.  If students generate a similar question, use it.  Allowing students to struggle through their question and ask for information is one of the big ideas here.  Sometimes, students realize that they may need to solve a different problem before they can actually find what they want.  If students are way off, in their questions, teachers can direct students, carefully, by saying something like:  “You all have generated some interesting questions.  I’m not sure how many we can answer in this class.  Do you think there’s a question we could find that would allow us to use our knowledge of mathematics to find the answer to (insert quantity or measurement)?”  Or, if they are really struggling, you can, again carefully, say “You know, I gave this problem to a class last year (or class, period, etc) and they asked (insert something similar to the suggested question here).  What do you think about that?”  Be sure to allow students to share their thoughts.

After solving the main question, if there are other questions that have been generated by students, it’s important to allow students to investigate these as well.  Investigating these additional questions validates students’ ideas and questions and builds a trusting, collaborative learning relationship between students and the teacher.

Overall, we’re trying to help our students mathematize their world.  We’re best able to do that when we use situations that are relevant (no dog bandanas, please), engaging (create an intellectual need to know), and perplexing .  If we continue to use textbook type problems that are too helpful, uninteresting, and let’s face it, perplexing in all the wrong ways, we’re not doing what’s best for kids; we’re training them to not be curious, not think, and worst of all . . . dislike math.

# Real Math Homework and Real Learning

Had a great night the other night with my (almost) 14 year old son. Connor had some math homework (Pythagorean Theorem worksheet from an outdated math series) and I was just looking over his work, making sure he understood the concept. He was coming along ok, I guess, so he finished up and sat on the couch to veg. for a bit.

Now, at first I thought what happened next was fate, but the more I think about it my subconscious probably took over. I checked my email and saw one from earlier in the week that I wanted to look at. It was from Dan Meyer’s blog and had a couple of links that I wanted to check out. After about 20 minutes of looking at some stuff I hadn’t seen before– including Estimation 180 (great site by the way), I stumbled upon Dan Meyer’s Taco Cart Problem again and began to grin.

Since I was on the couch with Connor by now, I showed him the video. When it ended abruptly, he said, “THAT’s IT!” I asked what was wrong. He said, “I want to know who gets there first.” We started to talk about it and maybe 20 minutes passed.  After realizing this was difficult to do with the limited resources we had on the couch, he asked, “Can we go sit at the table and work this out?  These numbers are too big.” I said, “SURE!” (but in a subdued voice so as not to sound “giddy” in front of my teen-aged son).

We sat there for a while talking about what he needed to know.  He knew he needed to know how far each person needed to travel, but didn’t make a connection about what he knew about the Pythagorean Theorem to solve the problem.  Yes, problem solving should be at the heart of every lesson!  He hadn’t been introduced to any ideas about distance and rate, but he knew he needed to know how fast they walked.  We talked about the relationship of distance, rate, and time and how these relationships can be use to find solutions to problems like this one.

After a little discussion and a lot of questions, Connor got to work.  He stumbled with some of the fraction “mechanics,” but with a little questioning, came through just fine.  Connor did more thinking during this task than I’ve witnessed him doing in a long time.  He was engaged from the start and he would not stop until he figured it out.  This is what students need to do all day in math class!

At the end, it was beautiful! He not only solved the problem, but when I asked where the cart should be for both people to get there at the same time, he was ready to go. He marked a new spot, and figured out the new distance. We had to set it aside, though, because it was getting late. He wants to see how close his placement of the taco cart is for the two guys to get there at the same time. We’ll be looking at it again over the weekend!

Kim, my beautiful bride, stopped at the table and asked what he was doing.  He told her, then she asked him if he finished his math extra credit (he doesn’t really need it – it’s just improving his grade, not really improving his understanding of mathematics). I found it a bit humorous because he was learning more doing this problem, than by doing the extra credit sheet of 19 naked equations.  Context and comprehension mean everything in mathematics!

Just before he went to bed, I asked him what he thought about the taco cart problem. He said he wished he got to do those kinds of problems at school instead of the “stupid problems he gets in class.”

Connor just recreated his thinking through the Taco Cart problem below using the Educreations App.  Enjoy!