Lesson 12

Problemas-historia y diagramas

Warm-up: Observa y pregúntate: Mancala (10 minutes)

Narrative

The purpose of this warm-up is for students to make sense of a problem before solving it by familiarizing themselves with a context. While students may notice and wonder many things about the image, what students notice about the number and position of the seeds and students’ experiences and questions about the game are the important discussion points.

During the synthesis, invite students who share experience with the mancala family of games to share what they call the game, what materials they use when they play, and how they play. If time, or if no students share experiences with the game materials, share the facts provided in the synthesis. Consider researching the rules of one the mancala games mentioned in the synthesis or one of the variants mentioned by students to play with the class.

Launch

  • Groups of 2
  • Display the image.
  • “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
  • 1 minute: quiet think time

Activity

  • “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Facing

¿Qué observas? ¿Qué te preguntas?
Mancala game board. Some holes have seeds. Some holes are empty. Seeds outside of the game board.

Student Response

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Activity Synthesis

  • “La imagen muestra un tipo de juego que se llama Mancala. Es uno de los juegos más antiguos del mundo” // “The picture shows a type of a game called mancala. It is one of the world’s oldest games.”
  • “Mancala se inventó en Africa. Tiene más de 800 nombres diferentes y se puede jugar de varias formas diferentes. La mayoría de los juegos se juegan con un tablero que tiene varios agujeros o huecos. Cada jugador usa una cierta cantidad de semillas que ubica en su lado del tablero. Los jugadores pueden usar semillas reales o conchas marinas, piedras o chaquiras” // “Mancala was created in Africa. The game has over 800 different names and can be played in many different ways. Most games are played with a board that has different pits or holes in it. Each player uses a certain amount of seeds that they place on their side of the board. Players might use real seeds or they may use shells, rocks, or beads.”
  • “Por turnos, cada jugador pone sus semillas en el tablero. En la mayoría de los juegos, uno trata de 'capturar' más semillas que el otro jugador” // “Each player takes turns placing their seeds on the board. In most games, you try to ‘capture’ more seeds than the other player.” 
  • “En Ghana y el Caribe, un juego popular de Mancala se llama Oware. El tablero tiene 12 agujeros, 6 para cada jugador, y en el juego se usan 32 semillas” // “In Ghana and the Caribbean, one popular mancala game is called Oware. The board has 12 pits, 6 for each player, and the game uses 32 seeds.”
  • “En Sudán, un juego popular de Mancala se llama Bao. El tablero para Bao tiene 28 agujeros, 14 para cada jugador, y en el juego se usan 64 semillas” // “In Sudan, one popular mancala game is called Bao. The board for Bao has 28 pits, 14 pits for each player, and the game uses 64 seeds.”
  • “El juego de Mancala más grande se llama En Gehé y se juega en Tanzanía. ¡El tablero puede tener hasta 50 agujeros y los jugadores usan 400 semillas!” // “The largest mancala game is called En Gehé and is played in Tanzania. The board can have up to 50 pits and the players use 400 seeds!”
  • “Mancala se juega en todo el mundo. Este tablero muestra un juego que se juega en India que se llama Pallanguzhi. El tablero tiene 14 agujeros y se usan 70 semillas” // “Mancala is played all over the world. This board shows a game played in India called Pallanguzhi. The board has 14 pits and uses 70 seeds.”
  • “¿Qué preguntas matemáticas podríamos hacer sobre esta imagen?” // “What math questions could we ask about this image?” (How many seeds are there in all? How many seeds are in the holes? How many more seeds are on the top than on the bottom?)

Activity 1: Interpretemos el diagrama (15 minutes)

Narrative

The purpose of this activity is for students to make sense of tape diagrams, and how they can be used to show part-part-whole relationships. Students have previously used tape diagrams to show comparisons. In this activity, they connect tape diagrams to Compare problems and Put Together/Take Apart problems (MP2). 

Students begin the activity by looking at the first problem displayed, rather than in their books. At the end of the launch, students open their books and work to find the diagram that matches the story problem. They do not solve the problems in this activity. The activity and synthesis are focused on developing student strategies for making sense of problems before solving them (MP1).

This activity uses MLR6 Three Reads. Advances: reading, listening, representing

Required Materials

Materials to Gather

Launch

  • Groups of 2
  • Give students access to base-ten blocks.
  • “Clare y Han juegan un juego de tipo Mancala con semillas” // “Clare and Han are playing a game like mancala with seeds.”

Activity

MLR6 Three Reads

  • Display only the problem stem for the first problem, without revealing the question. 
  • “Vamos a leer este problema 3 veces” // “We are going to read this problem 3 times.”
  • 1st Read: “Clare capturó 54 semillas. Han capturó 16 semillas menos que Clare” //  “Clare captured 54 seeds. Han captured 16 fewer seeds. ”
  • “¿De qué se trata esta historia?” // “What is this story about?”
  • 1 minute: partner discussion.
  • Listen for and clarify any questions about the context.
  • 2nd Read: “Clare capturó 54 semillas. Han capturó 16 semillas menos que Clare” // “Clare captured 54 seeds. Han captured 16 fewer seeds.”
  • “¿Cuáles son todas las cosas de esta historia que podemos contar?” // “What are all the things we can count in this story?” (Clare’s seeds. Han’s seeds. The difference between their seeds.)
  • 30 seconds: quiet think time
  • 1 minute: partner discussion 
  • Share and record all quantities.
  • Reveal the question. 
  • 3rd Read: Read the entire problem, including the question aloud.
  • Ask students to open their books.
  • “¿Cuál de los diagramas muestra una forma en la que podemos representar este problema?” // "Which of the diagrams shows a way we could represent this problem?" (See Student Responses for the first problem).
  • 30 seconds: quiet think time
  • 1–2 minutes: partner discussion
  • Share responses.
  • “Lean cada historia con su pareja. Después, individualmente, escojan un diagrama que corresponda” // “Read each story with your partner. Then choose a diagram that matches on your own.” 
  • “Cuando los dos hayan escogido un diagrama, comparen sus elecciones y expliquen por qué el diagrama corresponde a la historia o por qué otros diagramas no corresponden a la historia” // “When you have both selected a match, compare your choices and explain why the diagram matches the story or why other diagrams do not match the story.”
  • 5 minutes: partner work time

Student Facing

Marca los diagramas que correspondan a cada historia. Después, explícale a tu compañero por qué corresponden.

  1. Clare capturó 54 semillas. Han capturó 16 semillas menos que Clare. ¿Cuántas semillas capturó Han?

    Diagram. Two rectangles of equal length. Rectangle on top shaded, labeled Clare, total length, 54.  Rectangle on bottom, labeled Han, partitioned into two parts. First part, shaded, total length, question mark. Second part has dashed outline, total length, 16.

    Diagram. One rectangle, total length, question mark. Rectangle partitioned into 2 parts. First part, labeled Clare, total length, 54. Second part labeled Han, total length, 16.

    Diagram. Two rectangles of equal length. Rectangle on top labeled Clare, total length, 54. Rectangle on bottom, labeled Han. Partitioned into two parts. First part, shaded, total length, 16. Second part has dashed outline, total length, question mark.
  2. Clare tiene 54 semillas en su lado del tablero. Han tiene 16 semillas en su lado. ¿Cuántas semillas hay en total en el tablero?

    Diagram. Two rectangles of equal length. Top rectangle, shaded, labeled Clare, total length, 54. Bottom rectangle labeled Han, split into two parts. First part, shaded, total length 16. Second part has a dashed outline, total length question mark.
    Diagram. One rectangle partitioned into two parts. Total length, question mark. First part, labeled Clare, total length, 54. Second part, labeled Han, total length, 16.
  3. Clare tiene 54 semillas. 16 semillas están en su mano. El resto de sus semillas están en el tablero. ¿Cuántas de sus semillas están en el tablero?

    Diagram. One rectangle split into 2 parts. Total length, question mark. 1 part, labeled board, total length, 54. Other part, labeled hands, total length, 16.
    Diagram. Two rectangles of equal length. Top rectangle labeled hand, split into two parts. First part, shaded, total length 16. Second part has a dashed outline, total length question mark. Bottom rectangle, shaded, labeled board, total length, 54.
    Diagram. One rectangle split into 2 parts. Total length, 54. 1 part, labeled hands, total length 16. Other part, labeled board, total length question mark. 
  4. Hay 54 semillas en el tablero. Algunas semillas están en el lado de Han. 16 semillas están en el lado de Clare. ¿Cuántas semillas hay en el lado de Han del tablero?

    Escoge los dos diagramas que corresponden.

    Diagram. One rectangle partitioned into two parts, total length, question mark. First part labeled Clare, total length, 54. Second part labeled Han, total length, 16.

    Diagram. One rectangle split into two parts. Total length, 54. First part, labeled Han, total length, question mark. Second part, labeled Clare, total length, 16.

    Diagram. Two rectangles of equal length. Top rectangle, shaded, labeled Han, total length 54. Bottom rectangle labeled Clare, split into two parts. First part, shaded, total length 16. Second part has a dashed outline, total length question mark.

    Diagram. One rectangle split into two parts. Total length, 54. First part, labeled Clare, total length, 16. Second part, labeled Han, total length, question mark.

Student Response

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Advancing Student Thinking

Students may say Diagram A matches the second story problem because the diagram shows the amount of seeds both students have in the story. Consider asking:

  • “¿Qué representa el signo de interrogación en cada diagrama?” // “What does the question mark in each diagram represent?”
  • “¿Cuál diagrama corresponde mejor a las semillas de Han y de Clare y a la pregunta que se hace en la historia?” // “Which diagram best matches Han and Clare’s seeds and the question the story asks?”

Activity Synthesis

  • Invite students to share correct diagrams for the last three problems.
  • Display each correct choice. Consider asking:
    • “¿Cómo corresponde el diagrama al problema-historia?” // “How does the diagram match the story problem?”
    • “¿Cómo muestra el diagrama lo que se conoce y lo desconocido?” // “How does the diagram show what is known and what is unknown?”
    • “¿Dónde ven sumas o restas?” // “How do you see addition or subtraction?”

Activity 2: Clasificación de tarjetas: Problemas-historia y diagramas (20 minutes)

Narrative

The purpose of this activity is for students to connect different types of story problems to tape diagrams and solve the problems. Students identify representations that match a story problem and justify their decisions by describing how the diagrams represent the relationships between the quantities or the actions in the story problem (MP4). When students analyze and connect the quantities and structures in the story problems and tape diagrams, they think abstractly and quantitatively (MP2) and make use of structure (MP7). Students will use the story problems from this card sort again in a future lesson.

Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, make sure students are discussing amongst themselves and highlight groups that are following the protocol, so that students know what is being expected of them. 
Supports accessibility for: Social-Emotional Functioning, Organization

Required Materials

Materials to Gather

Materials to Copy

  • Story Problem and Diagram Cards, Spanish

Required Preparation

  • Create a set of cards from the blackline master for each group of 2.
  • The sets of cards will be used again in the next lesson.

Launch

  • Groups of 2
  • Give each group one set of cards from the blackline master.
  • Give students access to base-ten blocks.
  • “Por turnos, van a leer un problema-historia. Después de que una persona lee, trabajen juntos para encontrar el diagrama que corresponda. Cuando piensen que encontraron una correspondencia, explíquenle a su pareja por qué las tarjetas corresponden” // “You are going to take turns reading a story problem. After one person reads, work together to find the diagram that matches. When you think you have found a match, explain to your group why the cards match.”
  • “Si les ayuda, pueden escribir sobre los diagramas para explicar sus correspondencias” // “If it helps, you may label the diagrams to explain your matches.” 
  • As needed, demonstrate the activity with a student volunteer. 
  • “Cuando su grupo termine, escojan 2 problemas-historia de las tarjetas A, B, C o D y resuélvanlos” // “When your group finishes, choose 2 story problems from Cards A, B, C, or D and solve them.”

Activity

  • 8 minutes: small-group work time
  • Monitor for students who explain how each diagram matches the quantities in the context of the story problem.
  • 4 minutes: independent work time
  • Monitor for students who solve Card B or C using methods based on place value for sharing in the synthesis.

Student Facing

  1. Empareja cada problema-historia con un diagrama. Explica por qué las tarjetas corresponden.
  2. Escoge dos problemas-historia y resuélvelos. Muestra cómo pensaste.

Student Response

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Advancing Student Thinking

Students may mismatch stories and diagrams. Consider asking:

  • “¿Qué sabes? ¿Qué te pide el problema-historia que encuentres?” // “What do you know? What does the story problem ask you to find?”
  • “¿Cómo muestra el diagrama los números que ya conoces? ¿Cómo muestra el diagrama lo que tienes que encontrar?” // “How does the diagram show the numbers that you know? How does the diagram show what you need to find?”

Activity Synthesis

  • Invite students to share and explain their matches for cards A, B, and C.
  • Consider asking:
    • “¿Cómo saben que el diagrama corresponde al problema-historia?” // “How do you know the diagram matches the story problem?”
    • “¿Qué hicieron para resolver el problema?” // “What did you do to solve the problem?”
  • If time, select previously identified student(s) to share their method for solving Cards B or C.

Lesson Synthesis

Lesson Synthesis

“Hoy leímos muchos tipos diferentes de problemas-historia y los asociamos con diagramas” // “Today we read many different kinds of story problems and matched them with diagrams.”

Display Card L and Card N from the card sort.
 

Diagram. Two rectangles of equal length. Rectangle on top not labeled, shaded, total length, 65. Rectangle on bottom not labeled, partitioned into 2 parts. First part, shaded, total length, 28. Second part has dashed outline, total length, question mark.
Diagram. One rectangle split into two parts, total length, question mark. First part, label blank, total length, 65. Second part, label blank, total length, 28.

“¿En qué se parecen estos diagramas? ¿En qué son diferentes?” // “How are these diagrams the same? How are they different?”

“Piensen en una historia que pueda corresponder a la Tarjeta N” // “Think of a story that could match Card N.”

Cool-down: Encuentra cuál corresponde (5 minutes)

Cool-Down

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