Series: Mathematical Modeling with Three-Act Tasks

Math.Practice.MP1

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)

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Math.Practice.MP4

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)

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Math.K.OA.A.1

Common core State Standards

  • Math:  Math
  • K:  Kindergarten
  • OA:  Operations & Algebraic Thinking
  • A:  Understand addition, and understand subtraction
  • 1: 
    Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.


    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)

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Math.K.OA.A.2

Common core State Standards

  • Math:  Math
  • K:  Kindergarten
  • OA:  Operations & Algebraic Thinking
  • A:  Understand addition, and understand subtraction
  • 2: 
    Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings 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 Addition
Lesson Objective: Use modeling to solve a real-world addition problem about birthday candles
Grades K-2 / Math / Modeling
Math.Practice.MP1 | Math.Practice.MP4 | Math.K.OA.A.1 | Math.K.OA.A.2

Thought starters

  1. What is the purpose of Act 1?
  2. In Act 2, how do you see students approaching the problem? How are their strategies similar and different?
  3. How does Ms. Alfonzo reflect on her students' learning?
17 Comments
Love the lesson! I added it to my workspace and set a reminder to use later. But, where can I access the video to of the birthday party to show my students? It wasn't included in the supporting materials.
Recommended (0)
Love the lesson and I will be introducing Three Act tasks to my Foundation children this week. I took a screen shot of the birthday video to use with my children.
Recommended (0)
Three act tasks are great to teach perseverance. I will definitely be using them to help build stamina and to teach that there is more than one way to solve a problem.
Recommended (0)
This is my first introduction to three act tasks. I thought it was great that the students were able to use their language skills to discuss all of the things they notice about the video. I found it very interesting and love that the students had to go explore at their tables with manipulatives to find the answer. They found numerous ways to model their thinking and were able to justify their answer as well. Even though the teacher had not taught them about estimation, this could be a good introduction into it and one of those "teachable moments."
Recommended (0)
I love how the students experienced estimation for the first time organically. For them to think/look back on their initial number of candles and learn that they "estimated" was pretty interesting. They had already estimated by the time they found out what it was called. It was also interesting how flexible the students were in mixing different colors of cubes, rather than identifying just the yellow and blue cubes. Thank you so much for sharing!
Recommended (0)

Transcripts

  • Three-Act Tasks: Modeling Addition Transcript
    GFX:
    Tch
    TeachingChannel

    +++ 00:00:04 +++
    Sophia: One, two, three, four, five, six--
    Student: I wrote

    Three-Act Tasks: Modeling Addition Transcript
    GFX:
    Tch
    TeachingChannel

    +++ 00:00:04 +++
    Sophia: One, two, three, four, five, six--
    Student: I wrote the letter six and then there’s nine.
    Kristen Alfonzo: The thing I enjoy most about three-act tasks is it makes them interested in something right away and makes them personally invested in thinking about math in the real world.
    Card:
    Three-Act Tasks:
    Modeling Addition - Kindergarten

    +++ 00:00:26 +++
    Kristen Alfonzo: We are gonna do a three-act task today like we have done before. And we know in three-act tasks, Satchel, that we start with watching a video. Our video today is about a birthday party.
    Card:
    Kristen Alfonzo
    Kindergarten Teacher
    South Shore PK-8, Seattle, WA

    +++ 00:00:36 +++
    Kristen Alfonzo: We start off with a video or a picture, an image, something that the kids can get interested in and then we ask them what do they notice about the video, what do they wonder about the video. In kindergarten I usually try and remind them that we are thinking about math.
    Card:
    Act One
    The class discusses a high interest video or picture of a real-world situation.
    Kristen Alfonzo: Are your eyes on the screen?
    Student: Yeah!
    Kristen Alfonzo: Here we go.
    Children: Happy birthday to you, happy birthday, dear Abby [ph?], happy birthday to you!

    +++ 00:01:15 +++
    Kristen Alfonzo: Okay, give us a private thumb if you think you have something you notice, something you’re wondering about the video? I want you to quickly turn knee to knee, tell your partner what do you notice or wonder about this video.
    Student: There’s so many candles. I wonder how many are candles. How many stuff are in the bag?
    Kristen Alfonzo: Modeling’s important because that’s how we use math in the real world. It’s in life. They’re not gonna be able to reason with their partner at the grocery store in the future how much their budget can afford if they don’t know how to explain their thinking.

    +++ 00:01:49 +++
    Kristen Alfonzo: Amira, what did you notice?
    Amira: I wonder how many candles.

    Kristen Alfonzo: How many candles are there? If you are also wondering how many candles are there, would you do this? Say, “Me, too. I was thinking the same thing.” Solaiya, what’s your notice or wonder about the video?
    Solaiya: I wonder who’s gonna blow the candle off the fire.
    Kristen Alfonzo: Who will blow the candles out?
    Solaiya: Yeah.
    Kristen Alfonzo: Sophia, what were you noticing?
    Sophia: How many napkins?
    Kristen Alfonzo: How many napkins? Orelio [ph?], what were you wondering?
    Orelio: How many birthday bags?
    Student: There are five.

    +++ 00:02:22 +++
    Kristen Alfonzo: How many birthday bags. There are so many things to notice about this birthday.

    +++ 00:02:26 +++
    Kristen Alfonzo: The math can be anything the kids truly want to solve. So, tomorrow, if my kids say, “You know, I was really wondering how many cups there were,” we can go back and say, “Well, what information do we need? How can we find that information and solve it from there?” So it’s taking, like, a real-world scenario and helping them notice where the math is and then not only being able to find an answer, but also having to model their thinking, how they solved it.
    Card:
    COMMON CORE STATE STANDARD
    Math Practice 4: Modeling with Mathematics.
    Students apply the mathematics they know to solve problems in contextual situations.

    +++ 00:02:50 +++
    Kristen Alfonzo: We are all wondering how many are there of all of those things. I was really wondering about the candles. It’s kind of hard to tell exactly how many candles there are. I see some people counting. You have some guesses? I tried counting, but it was hard for me to see.
    Student: Twelve. Twelve. Twelve.
    Kristen Alfonzo: How many candles do you think there were? Minyette?
    Minyette: Twelve.
    Kristen Alfonzo: Minyette believes there were twelve candles. Do you have a different number you want to ask? Russo [ph?], what do you think? What’s your guess?
    Russo: Thirteen.
    Kristen Alfonzo: Thirteen candles.
    Card:
    Act Two
    The class identifies the information needed to answer a mathematical question about the situation

    +++ 00:03:26 +++
    Kristen Alfonzo: Should we find out exactly accurately how many candles there really are?
    Student: Yeah.
    Kristen Alfonzo: Okay, so we have some information to help you with that. Everyone say out loud, how many blue candles are there?
    Children: Six!
    Kristen Alfonzo: There are six blue candles? Can we find out exactly how many candles there are now? What other information do we need to find out? How many candles all together? Rose?
    Rose: Twelve-- yellow.

    +++ 00:03:51 +++
    Kristen Alfonzo: How many yellow candles? Do you agree with Rose? We need to know how many yellow? Solaiya agrees, Louie agrees, okay. How many yellow candles are there? Can you say it with me all together? How many yellow candles are there?
    Children: Nine!
    Kristen Alfonzo: Nine. Okay, we know there are six blue candles and nine yellow candles. Do you have the information to find out exactly how many candles there are on the cake now?
    Card:
    COMMON CORE STATE STANDARD
    Represent addition and solve word problems.
    Student: Yeah.

    +++ 00:04:19 +++
    Kristen Alfonzo: The three-act tasks engage every single kid, not just by having a video, but also, especially I think for my English language learning students, they’re able to have a constant visual up there and something to attach it to, so they’re not trying to puzzle through the language and the words I’m giving them.
    Kristen Alfonzo: Think you can find out? We have our guesses, but we have so many different numbers. We want to find out exactly how many. Are you ready to go to your tables and find out?
    Children: Yes.
    Kristen Alfonzo: Then they go off to their desks and solve the problem or model the math thinking that they’ve been doing. Sometimes they work on their own, sometimes they work with a partner.

    +++ 00:04:51 +++
    Kristen Alfonzo: So how many all together?
    Student: Eleven?
    Kristen Alfonzo: You don’t sound sure. How could you be sure?
    Kristen Alfonzo: Use cubes.
    Kristen Alfonzo: Cubes. Okay, how about if I come back. He’s showing you six. What is that from the video? What is that number six?
    Orelio: Six and nine. I think I’ll need four hands.
    Kristen Alfonzo: You need four hands.
    Orelio: ‘Cause Minga has four hands. I mean, Minga has two and I have--
    Kristen Alfonzo: So you have four hands to use. So Orelio, you said you were showing six blue. What do you need your partner to show for your idea?

    +++ 00:05:23 +++
    Orelio: Nine.
    Kristen Alfonzo: The nine yellow. How can we find out now how many candles all together?
    Orelio: One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen.
    Kristen Alfonzo: So you think there are how many candles all together?

    +++ 00:05:39 +++
    Orelio: Fifteen.
    Kristen Alfonzo: Fifteen.
    Kristen Alfonzo: I noticed that because we chose numbers higher than ten they couldn’t be successful with just their fingers alone. Two students got creative and used each other’s fingers, which was an excellent strategy. But I also saw that kids went directly to direct modelling. So they either used a drawing or tally marks or cubes to show exactly how many candles there were.
    Student: I put six here and I put nine right here.
    Kristen Alfonzo: And how does that match the video again? Six blue candles, right? Where was the nine in the video?

    +++ 00:06:17 +++
    Student: Is yellow!
    Kristen Alfonzo: Nine yellow candles. You have--
    Student: Nine, ten, eleven, twelve, thirteen, fourteen, fifteen.
    Card:
    COMMON CORE STATE STANDARD
    Math Practice 1: Make sense
    and persevere in solving them.
    Kristen Alfonzo: Fifteen. Fifteen candles on the cake.
    Card:
    Act Three
    The problem is resolved and students share solutions strategies.

    +++ 00:06:33 +++
    Kristen Alfonzo: Right now is our time for us to be sharing out the strategies we use to find out how many candles are on the cake and it’s also time for us, Lily, to hear and listen to how other people did it. That’s how we learn new ways of doing math and new ways of thinking is by hearing other people’s strategies. So Amira offered to share her good thinking with us. Will you come on up, Amira? She’s gonna teach us how she solved the problem. Can you tell us a little but about your first drawing?
    Amira: I used six blue candles.
    Kristen Alfonzo: Where are those? Can you show us the six blue in your drawing?
    Amira: And there’s nine yellow candles.

    +++ 00:07:12 +++
    Kristen Alfonzo: So how did you find out how many candles were on the whole cake?
    Amira: Because I looked on the screen.

    +++ 00:07:19 +++
    Kristen Alfonzo: How did-- did you use your picture to help you? How did you use your picture? Did you count them all? One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve-- will you count with me? Thirteen, fourteen, fifteen. If you also came up with fifteen candles, please touch your head. But just like Amira, many of you had a different number to start with. Amira, can we share what happened with
    +++ 00:07:52 +++
    you? At the beginning didn’t you guess there were ten? And I see you wrote “ten” and I also saw other people do this. Did anyone else look and say, “Well, I guess there were eleven or fourteen. And so there has to be that many!”

    +++ 00:08:04 +++
    Kristen Alfonzo: Kids were thrown off by their estimate. We had never done an estimate in my class before. So at the beginning of the task they all said, “Oh, let’s-- when we find out how many candles there are,” they immediately, of course, started counting and then they really took ownership of that number that they really thought they saw on the screen.

    +++ 00:08:21 +++
    Kristen Alfonzo: So Amira drew the ten that she knew she thought there were, but then she thought, “Wait a minute. That doesn’t actually match the video.” So she revised her thinking, did a different picture and found out the number fifteen. And that’s okay. It’s okay if our first guess isn’t the correct one.
    Kristen Alfonzo: So I think my next step is to talk a little bit about estimation and why it’s okay to revise thinking if our first guesses aren’t correct in math.

    +++ 00:08:49 +++
    Kristen Alfonzo: Satchel, would you be willing to share your strategy with us? Satchel used cubes. How did you use cubes? First you had six. How did that match the video?
    Satchel: Six blue.
    Kristen Alfonzo: Six blue candles. And then--?
    Satchel: Nine yellow.
    Kristen Alfonzo: Okay, then what?
    Satchel: I counted them.
    Kristen Alfonzo: How many candles did you decide were on the cake?
    Satchel: Fifteen.
    Kristen Alfonzo: Fifteen all together.

    +++ 00:09:15 +++
    Kristen Alfonzo: I love that three-act tasks are usually just difficult enough that even if kids can figure out really quickly on the carpet, they still have to go back to their tables and show us and be able to prove their thinking to us.

    +++ 00:09:27 +++
    Kristen Alfonzo: You worked hard today to watch a video, notice what was happening in it, think about the math you saw happening at that birthday party, and then you solved a problem. You made a model with a picture or with your fingers or with cubes to show how many candles were on the birthday cake.

    +++ 00:09:47 +++
    Kristen Alfonzo: They did an amazing job of solving the number story correctly, first of all, and also of showing their ways of thinking.
    Kristen Alfonzo: Everybody give yourself a pat of the back. That was a lot of great thinking today.
    GFX:
    Tch
    TeachingChannel

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