Lesson 8

The purpose of this warm-up is to allow students to think about how organizing objects can help them find the total number more quickly. Each image shows 20 counters. Image 1 makes it difficult to count quickly due to a lack of organization, Image 2 makes it more clear and allows for the conversation about even and odd, and Image 3 connects a familiar structure (10-frame) to the new structure being introduced (an array).

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking oneself “Does this make sense?” is a component of making sense of problems (MP1). Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

• Groups of 2
• Display the image or arrange 20 counters (10 red and 10 yellow) in a random arrangement:
• “¿Cuántas fichas ven?” // “How many counters do you see?”
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Arrange counters to show 2 circular arrangements of 10 red and 10 yellow counters:

• “Estas son las mismas fichas, pero organizadas de otra manera. ¿Quieren reconsiderar lo que pensaron antes?” // “Here are the same counters in a different arrangement. Would you like to revise your thinking?” (I counted the red and there were 10. I think there’s the same amount of yellow, so I said 20.)
• 1 minute: quiet think time
• Share responses.
• After Image 2 ask, “¿Cómo les ayuda esta imagen a pensar si el total es par o es impar?” // “How does this image help you think about whether or not the total is even or odd?” (It helps me see the number of counters as 2 equal groups. So it must be an even number.)
• Arrange the counters into an array of 2 rows of 5 red counters and 2 rows of 5 yellow counters with some space to separate the two colors:

• “Esta imagen muestra el mismo número de fichas, pero organizadas de otra manera. ¿Quieren reconsiderar lo que pensaron antes?” // “This image shows the same number of counters, but in a different arrangement. Would you like to revise your thinking?” (It’s easy now to see that it’s 20. There’s 10 red and 10 yellow.)
• 30 seconds: quiet think time
• Share responses.

• “Vimos distintas maneras de organizar el mismo número de fichas. ¿Cuál hace que sea más fácil saber cuántas fichas hay en total? Expliquen” // “We saw different arrangements of the same number of counters. Which one makes it easier to tell how many there are altogether? Explain.” (In the last way, it is easy to see it is 10 and another 10. It looks like 2 10-frames.)
• Refer to the counters in an array.
• “Organizar los círculos en arreglos nos ayuda a ver formas de encontrar más rápido el total. ¿Cómo puedo usar el conteo a saltos para encontrar el número total de fichas?” // “Organizing the circles into arrays can help us see ways to find the total more quickly. How could I use skip counting to find the total number of counters?” (We could count by 5 or 2.)

The purpose of this activity is for students to describe the structure of an array by identifying the number of columns, the number of objects in each column, and the total number of objects. They learn that the columns of an array go up and down. Students begin to reason about how they can use the structure of the array to find the total number of objects without counting by 1.

This activity uses MLR8 Discussion Supports. Advances: speaking, conversing

• Groups of 2
• Give students counters.
• “Usen sus fichas para hacer un arreglo que tenga 3 filas con 4 fichas cada una. Asegúrense de que las fichas de cada fila estén alineadas con las fichas de las otras filas” // “Use your counters to make an array with 3 rows with 4 in each row. Make sure the counters in each row line up with the counters in the other rows.”
• Display counters in an array as shown:

• “Los arreglos están formados por filas, pero también tienen columnas que van de arriba hacia abajo. ¿Cuántas columnas hay en este arreglo?” // “Arrays are made up of rows, but they also have columns that go up and down. How many columns are in this array?” (4)
• 30 seconds: quiet think time
• Share responses.
• “¿Cuántas fichas hay en cada columna del arreglo que hicieron?” // “How many counters are in each column of the array you made?” (3)
• 30 seconds: quiet think time
• Share responses.

• “Ahora van a examinar unos arreglos para decidir cuántas columnas tienen, cuántas fichas hay en cada columna y cuál es el número total de fichas. Después, van a usar fichas para hacer arreglos y a responder unas preguntas” // “Now you will look at a few arrays to decide how many columns they have, how many in each column, and find the total number of counters. Then you will make arrays using counters and answer a few questions.”

MLR8 Discussion Supports

• Display sentence frames to support students when they describe the structure of the array:
  • “Hay _____ columnas en el arreglo” // “There are _____ columns in the array.”
  • “Hay _____ fichas en cada columna” // “There are _____ counters in each column.”
  • “Hay _____ fichas en total” // “There are _____ counters in all.”
• 8 minutes: partner work time

• Display student work for an array with 10 counters.
• “Describan el arreglo de _____” // “Describe _____’s array.” (It has 10 counters, it has 2 columns and 5 counters in each column. It has 5 rows with 2 counters in each row.)
• “¿Cómo podemos mostrar que hay 10 fichas en este arreglo?” // “How can we prove there are 10 counters in this array?” (We can skip count by 2. We can add \(2 + 2 + 2 + 2 + 2\). We can add \(5 + 5\).)

The purpose of this activity is for students to distinguish between rows and columns of arrays based on given clues. The activity synthesis invites students to use the vocabulary they have learned in this lesson and the previous lesson to describe and create arrays (MP6). When making arrays based on clues, it might be helpful to have partners sit back to back.

• Each student needs 25 counters.

• Groups of 2
• “4 estudiantes usaron el mismo número de fichas para hacer arreglos. Marquen cada arreglo con el nombre del estudiante y encuentren el número total de fichas” // “4 students made arrays using the same number of counters. Label each array with the student’s name and find the total number of counters.”
• 6 minutes: independent work time
• Share and record responses.
• Give each student 25 counters.

• “Ahora van a trabajar con un compañero para ver si pueden hacer arreglos que coincidan” // “Now you will work with a partner to see if you can make matching arrays.”
• “El compañero A va a hacer un arreglo con fichas. Puede usar hasta 25 fichas. No va a dejar que su compañero vea el arreglo” // “Partner A will make an array using up to 25 counters. Don’t let your partner see.”
• “Después, le da pistas sobre su arreglo a su compañero para que intente hacerlo” // “Give your partner clues about your array, so they can try to make it.”
• “Comparen sus arreglos para ver si hicieron el mismo arreglo. Después, intercambien roles” // “Compare to see if your partner made the same array, and then switch.”
• 10 minutes: partner work time
• Monitor for students who gave clues using odd or even and used the rows and columns language correctly to share in the synthesis.

If students give clues that are not related to the rows and columns of their array, consider asking or displaying:

• “¿Cuántas filas tiene tu arreglo?” // “How many rows does your array have?”
• “¿Cuántas columnas tiene tu arreglo?” // “How many columns does your array have?”
• “¿Tu arreglo tiene más filas, tiene más columnas o tiene el mismo número de filas y de columnas?” // “Does your array have more rows, more columns, or the same number of rows and columns?”

• Invite previously identified students to share their clues.
• Students can use their counters to see if they make an array that matches the chosen student’s.
• “¿Cuáles fueron las pistas que más les ayudaron a hacer sus arreglos?” // “Which clues helped you most when making your arrays?” (It was easy when they said how many rows and columns.)