# Three-Act Math: Pyramid of Pennies

We talked about problem-posing and inspiring student curiosity in math as we tried out a three-act math task created by Dan Meyer

Facilitator(s): Mark
Date of Meeting: May 8, 2015
Problem: · url

To start off the meeting, in pairs we discussed – “Real life math”: What does it mean to you? In your classrooms?

Kevin/Carla: word problems, not the test, things we would really use, practical, sometimes math problems seem like real world math but aren’t, challenge of finding relevance, activating prior knowledge

Brian/Tyler: Using math in the problems you face, interpret situation using math, when is it real and when is it just a setup for a math problem?

Moana/Alison: connecting to money, what’s familiar to students

Avril/Eric: wanting it to be relevant, investment of interest, we care about it, meaningful, concrete stuff in the real world

Ramon/Todd: (1) making sense, (2) students don’t always check, (3) When is it a real-life problem and when is it just a good math problem (in terms of getting at a specific content we want to teach) rooted in a “real-world” situation, (4) real life math for adults and kids may be different, (5) adults might not be as comfortable playing and experimenting as kids, (6) the creative process – the creation of the problem is the math, rather than relying on what others have told you

Some examples from our discussion: Reading measuring tape, equivalent fractions; budgets: percents, fractions, using fractions in cooking; solar panel installation (negative numbers, solar panels and “the grid”); sneaker shopping; carpentry; estimating how much paint will be needed; elections; nutrition: calories, losing weight; sports

One other important question that came up was, “Whose life?”

### A Math Task in Three Acts

Mark told the group that he wanted to try out a teaching structure called 3-Act math, developed by Dan Meyer. He didn’t tell us any more about what it was, but that we would experience it and he would like feedback afterwards.

#### Act One

Mark went through the following series of questions with us. The questions are from Dan Meyer’s 3-Act math. Our responses are below Mark’s questions.

“We’re going to start by watching a short video.”

We watched the video twice.

“Write down the first question that comes to your mind? Just one question.”

“Would you introduce yourself to your neighbor and share your question? See if it’s the same question, or a different question.”

“I’m really curious what questions are out there. Just toss one out. Who else finds that question interesting?” (Mark tallied the # of people who found questions interesting.)

These are the questions we came up with (and the tallies):

• How many pennies are there? (10)
• How long does it take to put these pennies in a pyramid? (9)
• How much \$ did it take to build the pyramid? (10)
• How many pennies are there an n by n based pyramid? (5)
• How do they stay together? Are they glued? (4)
• What changed from layer/level/stack to layer/level/stack? (7) Mark had us spend a while talking about layers, levels and stacks to make sure we understood the question as it was phrased.
• How many levels are there?
• If you were an ant, how many calories would you burn climbing the stairs to the top of the pyramid? (…It got a little silly…)

“It would be great to try to answer all these questions, but since we have a limited amount of time, we’ll start with the ones that go the most votes.” (Mark admitted later that he really wanted us to choose the first or third questions and was ready to steer us that way if it didn’t happen naturally.)

This is the question we decided to answer: How much \$ did it take to build the pyramid? (10)

<<Teaching Note: The other questions were all written on the board and could be used as extensions for students who finished the first question more quickly.>>

“I want you to write down on a piece of paper your best, gut-level guess for how many coins there are. I’m curious who can guess the closest.”

“Would you also write down a number you know is too high – there couldn’t possibly be that many pennies – and a number you know is too low – there couldn’t possibly be that few pennies. Share them with your neighbor.”

“I’m very curious in here who has our highest guess.“ (These next two were the real, gut-level guesses, not the guesses we knew were too low or too high.)

\$150,000 was the highest.

“What’s our lowest guess in here?”

\$26 was the lowest.

#### Act Two

At the beginning of the second act, Mark asked us what information we needed from him in order to answer the question. We got into groups of three to discuss what information we would need, then shared our questions for Mark, who wrote them all on the board:

1. How much does the whole pile weigh?
2. How much does one penny weigh?
3. What is the length and width of each layer?
4. What is the overall height?
5. What is the height of a stack?
6. Does the height of a stack change?
7. What is the height and width of a penny?
8. How many levels are there?
9. What is the angle of the pyramid wall? Is it symmetrical?
10. How many stacks are in the base layer?
11. What is the length and width in the base layer?
12. How does the length and width change from level to level?

Mark asked us what we were talking about when we used the words “layer”, “stack” and “pile”. As a group we came up with definitions for what each meant for the sake of clarity in our discussions.

Then Mark gave us the following information, which answered questions 2, 3, 4, 5, 6, 7, 8, 10, 11, 12:

• There are 40×40 stacks in the base layer.
• There are 13 pennies in each stack.
• 1 penny = 2.5 grams
• 1 penny = 1.55mm thick, 19.05mm wide

The question came up what if we did not ask for all of the information we actually needed to solve the problem (but we thought we had). Mark said if that had happened, he was only planning on giving us what we asked for. At least until we did some work and someone realized we needed more information.

In terms of question #12 – How does the length and width change from level to level? – Mark asked, “If the base layer is 40 by 40, what could the next layer up be?”. Some of us said it could be 38, others said 39. Mark asked, “What would it look like if the next level was 38 by 38? What would it look like if it was 39 by 39?”

Then Mark gave us a close-up photograph that showed how the pennies were stacked and we saw that there was a half a penny overlap. So the layers decrease by one, starting from 40 by 40, up to a single stack at the top.

This was enough information to answer most of the questions we asked Mark in order to get information to answer the main question: How much money would it cost to build this pyramid?

Mark then assigned pairs to work on the answer to the problem.

#### Act Three

This is the big reveal, but Mark didn’t have time to do this part with us.

There were a few groups who came up with a number and a few groups who were close. One method that came out was to find the sum of all of the squares from 40-1 (40² + 39² + 38²… 3² +2² +1²) – which would give us the number of stacks. Some groups worked it out (one of them broke the squares up among the group members to add up). Other groups tried working out a formula that would allow them to calculate the sums without doing the computation. Towards that end (and in typical CAMI fashion), Mark answered a question with a question and shared a completely separate problem that suggests an approach for deriving that formula – Pile of Ping Pong Balls. <<Similar problems dealing with a cannonball pile and a stack of oranges can be found in Lessons for Algebraic Thinking.>>

Instead of 3rd Act, we discussed a couple of teaching questions.

“How is this activity different from presenting a typical word problem with all the information presented from the beginning?”

Like this, for example

• We were interested. We created our own questions.
• We wrote the problem ourselves and equipped ourselves to solve it.
• We learned how everyone else was thinking, mathematically. It was more mind expanding. We should acknowledge when people say things about the situation mathematically. Other people as an audience to the discussion recognize they are learning. It gives voices, increases participation. Showing what we’re interested in.
• Watching the video is fascinating. I could do something that cool. We also know that someone actually did this.

“Is this a real life problem?”

• Who would stack pennies like that?
• This is a real person who did this for cancer awareness.
• You have to come up with questions and think about what you would need to know (that makes this real life) – it is real life in the sense of what Todd and Ramon were saying – in real life we are not given neat packages of information. In real life, we need to ask ourselves what is important and what is not important when it comes to the math we use.

“What were the three acts? Where did one act begin and another end?”

There was some difference in which activities folks saw as part of each act, but overall we agreed the three-acts were:

• Problem-posing
• Problem solving
• Reveal, resolution, discussion

This review from CollectEdNY.org offers a brief review/introduction to Dan Meyer’s Three-Act Math Tasks and a few suggestions for ones to get started with in adult education students. It has links to various resources, including the tasks themselves and Dan Meyer’s TEDTalk, titled “Math Class Needs a Makeover“.

We’re just starting to explore this site. Eventually there will be a review of it on CollectEdNY.org. It’s a pretty simple idea. When you get to the site, you are presented with a single photo/video and asked. “What is the first question that comes to your mind?” You can either submit a question or skip it. If you submit a question, you’ll be able to see all the other questions that people have asked. If you join – it’s free – you are able to add your own photos and videos and create your own three-act math problems. You can also bookmark photos/questions that other people have posted. You can also search by user, key word or math content, and limit your searches to posts that have lessons. Some users I’d recommend – Dan Meyer, Andrew Stadel, Robert Kaplinsky.

3. This is Dan Meyer teaching the Pyramid of Pennies as a 3-Act Task. It’s fascinating to see how similar the conversation is. Many of the same questions and explanations. Highly recommended.

In attendance: Alison, Avril, Brian, Carla, Eric, Kevin, Mark, Moana, Parvoneh, Ramon, Todd, Tyler

Programs represented: Brooklyn Public Library, Lehman College, Fifth Avenue Committee, CUNY Start, CUNY Adult Literacy, The Fortune Society, DOE District 79

Location: Murphy Institute – 25 W. 43rd St., 19th Floor

Respectfully submitted by Eric & Mark

## 2 thoughts on “Three-Act Math: Pyramid of Pennies”

1. Mark Trushkowsky says:

Dan Meyer gave a talk at the 2016 NCTM conference called Beyond Relevance & Real-World: Stronger Strategies fro Student Engagement in which he says “If you can ask questions about it, it is in your real world.” and makes a point similar to the one Ramon and Todd made during the launch discussion of this CAMI meeting. Here’s a link to Dan’s talk –