Series Mathematical Modeling with Three-Act Tasks: Three-Act Tasks: Modeling Subtraction


Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP1:  Make sense of problems and persevere in solving them.

    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, \"Does this make sense?\" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Download Common Core State Standards (PDF 1.2 MB)


Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP4:  Model with mathematics.

    Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Download Common Core State Standards (PDF 1.2 MB)


Common core State Standards

  • Math:  Math
  • 2:  Grade 2
  • OA:  Operations & Algebraic Thinking
  • A:  Represent and solve problems involving addition and subtraction
  • 1: 
    Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

    Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)

Download Common Core State Standards (PDF 1.2 MB)

Three-Act Tasks: Modeling Subtraction

Lesson Objective: Use modeling to solve a real-world subtraction problem about cookies
Grades K-2 / Math / Modeling
Math.Practice.MP1 | Math.Practice.MP4 | Math.2.OA.A.1


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Discussion and Supporting Materials

Thought starters

  1. How does Act 1 engage students with the problem?
  2. How are students given opportunities to learn from each other?
  3. How could you use Three-Act Tasks in your classroom?


  • Private message to Berky Lugo Salcedo
Thank you for sharing your classroom and your practice with the rest of us!!! I saw this video during an administrator's all-day professional development session provided by the Algebra for All NYC Department of Education initiative.
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  • Private message to Sailaja Vittaldev
Awesome! This is what I call visible thinking! Thanks for the share:)
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  • Private message to Skylar Dolezel
I liked the video. It was a great way to capture their attention and to see their different responses and questions.
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  • Private message to Araceli Valtierra
Really enjoyed the video and how the student's explained their thinking!
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  • Private message to Sandra Brown
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External Resource Materials


  • Three-Act Tasks: Modeling Subtraction Transcript


    +++ 00:00:04 +++
    Student: Sixteen plus 16 equals 32, plus 16 is 48.

    Three-Act Tasks: Modeling Subtraction Transcript


    +++ 00:00:04 +++
    Student: Sixteen plus 16 equals 32, plus 16 is 48.
    Sarah Dietz: So how many cookies did the cookie monster east?
    Student: Twenty-two

    Three-Act Tasks:
    Modeling Subtraction
    2nd Grade Mathematics
    Graham Hill Elementary, Seattle WA

    +++ 00:00:19 +++
    Sarah Dietz: Mathematicians, thank you for coming quickly and quietly to the carpet. Miss Ellen is here today and we are going to do some mathematical thinking.

    Lower Third:
    Ellen Kleyman
    K-2nd Grade Mathematics Specialist
    Graham Hill Elementary, Seattle WA
    Sarah Dietz: We know that mathematicians solve problems without giving up.

    +++ 00:00:33 +++
    Sarah Dietz: Today's lesson was a three act task. In a three act task, students are able to think about math in a real world context and think about what it means to them to solve a problem in the real world.

    Lower Third:
    Sarah Dietz
    2nd Grade Teacher
    Graham Hill Elementary, Seattle WA
    Sarah Dietz: And they approach it in that way. Instead of thinking, "What is the equation that I need to solve in order to answer this problem?"

    Common Core State Standard
    Math Practice 4: Modeling with Mathematics.
    Students apply the mathematics they know to solve
    Problems in contextual situations.

    +++ 00:00:50 +++
    Sarah Dietz: They're able to think about, "What is this problem really asking? What do I need to do to solve this problem?" And it is a genuine problem for them to be solving, instead of a code that they're trying to break.

    Act One
    The class discusses a high interest
    Video or picture of a real-world situation

    +++ 00:01:03 +++
    Sarah Dietz: Today, I'm going to show you a video, and I want you to be thinking about, what are you wondering during that video? Eyes on the screen.
    Student: What is that?

    +++ 00:01:29 +++
    Sarah Dietz: It's high interest video for the kids. They were instantly engrossed and already formulating questions while they were watching.
    Student: He laid the table.

    Sarah Dietz: I want everyone to stop and think right now. don't say anything yet. And I want you to be thinking, what did you notice in that video? You've got a couple noticing. Show me that you're ready. Turn and tell your partner, what did you see?

    +++ 00:02:00 +++
    Student: It looks like he was just punching the baggie and put it back up with all of them in.
    Ellen: Oh.
    Sarah Dietz: To them, it's not a math lesson. It's a puzzle that needs to be solved. It's a problem that they want to work out.
    Sarah Dietz: What did you notice about the video, Ahmed?
    Student: That the cookie monster took the bag and eat it.

    +++ 00:02:15 +++
    Sarah Dietz: cookie monster took the cookies. What are we wondering about with this cookie monster and these Oreo cookies? What are we wondering? Tashaun?
    Student: I'm wondering, what was that white hand.
    Sarah Dietz: What was the white hand? Okay. What else are we wondering? Amina?

    +++ 00:02:43 +++
    Student: I wonder how much cookies are in each row.
    Sarah Dietz: You're wondering how many cookies in each row. Lina?
    Student: I wonder how many cookies are in the box.
    Sarah Dietz: You were wondering how many were in the box to begin with? I'm going to be able to tell you that, yes.

    +++ 00:02:59 +++
    Sarah Dietz: We wrote down what we noticed about the video, and then wrote down what we were wondering about the video. From there we then pinpointed, what was the problem we wanted to solve.

    Act Two
    The class identifies the information needed
    To answer a mathematical question about
    The situation

    +++ 00:03:11 +++
    Sarah Dietz: I think that's really the big question we're wondering, isn't it? How many cookies did the cookie monster eat? How can we figure this out? What do we need to know? Aaron.
    Student: How many was there in the beginning?
    Sarah Dietz: How many there were in the beginning? Okay. Elijah?
    Student: How many cookies are in each row.
    Sarah Dietz: How many cookies are in each row. Do you guys want to see a picture of what inside of the Oreo package looks like?
    Students: Uh-huh.
    Ellen: Ready?
    Student: There's three.
    Ellen: Voila.
    Student: What?
    Student: What?
    Sarah Dietz: This is what--
    Student: He didn't eat none of them.

    +++ 00:03:43 +++
    Sarah Dietz: Family-- this is before he ate them. This is what a family sized package of Oreos looks like, and there are 16 cookies in each row.
    Student: What's after--
    Sarah Dietz: Do you need more info? Is this not enough information?
    Student: After.
    Student: Yes.
    Student: No, it's not.
    Sarah Dietz: Okay, what other information do you guys want? Tam?
    Student: How many were left after the cookie monster ate it.

    +++ 00:04:08 +++
    Sarah Dietz: You want to see what it looks like after?
    Student: Yeah.
    Student: Yeah.
    Ellen: Okay, ready?
    Student: Wow.
    Student: Are you serious?
    Student: No, this is going to be...
    Sarah Dietz: So this is how many are left after he ate his cookies. I'm going to give you this picture for you to take back with you to your seats.
    Student: Please.

    +++ 00:04:27 +++
    Sarah Dietz: So this is what you're going to do now. You're going to go work with your partner. You are going to solve our problem. How many cookies did the cookie monster eat? I want you to show your work.
    Sarah Dietz: How many cookies did the cookie monster eat? Why do we want to add three 16s?

    Common Core State Standard
    Solve addition and subtraction word problems.

    +++ 00:04:51 +++
    Student: Because that's the number of Oreos we had.
    Sarah Dietz: There were 16 Oreos in each row, so you want to add three 16s.
    Ellen: Why did you cross out six, Ahmed's asking?
    Student: I crossed out six because the cookie monster ate six.
    Sarah Dietz: What did you do?
    Student: I counted how much was left in the box.
    Sarah Dietz: Can you show me how you counted them?
    Student: One, two, three, four, five, six, seven, eight, nine. So this one's nine and this one's 11, this one's six.
    Sarah Dietz: Then what did you do?
    Student: Then I counted how much was missing.

    +++ 00:05:24 +++
    Sarah Dietz: So you figured out each row individually. You know that ten is missing from here, five is missing from here and seven is missing from here. What now are you going to do?
    Student: I want to see how much the cookie he ate, how much he ate.
    Sarah Dietz: Yeah, okay.
    Student: So ten plus five, 15.
    Sarah Dietz: Mm-hmm.
    Student: Fifteen plus five is 20, plus two is 22.

    +++ 00:05:51 +++
    Sarah Dietz: So how many of the cookies did the cookie monster eat?
    Student: 2; Twenty-two.

    Common Core State Standard
    Math Practice 1: Make sense of problems
    And persevere in solving them.
    Sarah Dietz: You kind of ran into a little bit of a problem here, and you didn't go, "Oh, this is too hard. I'm giving up." You just kept at it and you kept solving it until you figured it out. Nice work.

    +++ 00:06:04 +++
    Sarah Dietz: So once the students were done solving their problems, the third act would be that they come back to the carpet and share their thinking with one another.

    Act Three
    The problem is resolved and students share
    Solution strategies.
    Sarah Dietz: This one was interesting because there was two different ways to do it. They could subtract the remaining number from the total, or they could find the difference.
    Sarah Dietz: All right, Luis, can you tell us what you were thinking?
    Student: What I was thinking is that there was 48.
    Sarah Dietz: There was 48 cookies at the beginning, okay.
    Student: And then I counted back to see if there was 26. So I counted 22 back.

    +++ 00:06:33 +++
    Sarah Dietz: So you made a big hop of ten and landed on 38. Made another big hop of ten and landed on 28, and then made two little hops and landed on 26. So where is the answer on Luis's number line? Penny?
    Student: At the top, where the circle is in the ten and the 20.
    Sarah Dietz: So this ten, this ten and this two? And so we need to add these up together? So how many cookies did the cookie monster eat?
    Student: Twenty-two.

    +++ 00:07:04 +++
    Sarah Dietz: Twenty-two. All right, thank you for sharing, Luis.
    Sarah Dietz: There's a range of strategies. Some of them figured out how many were left and were subtracting that from the total. Some of them were adding up to get to the total.
    Sarah Dietz: Notice what's the same and notice what's different. Okay, Aaron, what did you do first?
    Student: I did 16 plus 16 plus 16 equals question? But then I decomposed it to ten and six and the other 16s to ten and six. And then ten plus ten plus ten equals 30. Then six plus six plus six equals 18. So 18 plus 30 equals 48.

    +++ 00:07:45 +++
    Sarah Dietz: So what does this 48 represent? Forty-eight what?
    Student: Cookies.
    Sarah Dietz: At first, before the cookie monster started eating. Okay. Then what did you do?
    Student: I did 48 minus 26. I decomposed the 48 to 40 and 8 and 26 into 20 and 6. Forty minus 20 equals 20 and eight minus six equals two. So 20 plus two equals 22.
    Sarah Dietz: And what does this represent?
    Student: The answer.

    +++ 00:08:17 +++
    Sarah Dietz: Yeah, the cookie monster ate 22 cookies. What do you see that is the same, and what did you see that is different between Aaron's way and Luis's way? Elijah?
    Student: I noticed that Luis subtracted. He minused until he got to the answer of how much are left, and Aaron subtracted till he got to how much he ate. Twenty-two is how much he ate, and 26 is how much is left

    +++ 00:08:46 +++
    Sarah Dietz: So I think what I'm hearing you say, Elijah, is that Luis started at 48 and he subtracted back until he got to 26.
    Sarah Dietz: And Aaron started at 48, subtracted 26 and then found out, got the 22. Two different ways to think about it.

    +++ 00:09:07 +++
    Sarah Dietz: It's really exciting to see the kids have such high interest in these problems. My hope and my desire is to just look around their world and making those real world connections for them, highlighting for them that math is an everyday thing. They're solving math problems all the time without knowing it.
    Ellen: So how many did the cookie monster eat?
    All: Twenty-two.


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