# Lesson 9

Una suma de sumandos iguales

## Warm-up: Exploración de estimación: ¿Cuántos waffles hay? (10 minutes)

### Narrative

The purpose of this Estimation Exploration is to practice the skill of making a reasonable estimate. Students consider how the arrangement of the objects helps them estimate the total number of objects (MP7). For the first image, students should keep their books closed and discuss estimates as a group. They will record their estimates after seeing the second image. In the synthesis, students discuss their confidence in their estimates of the total number of waffles on the tray even though they cannot see all the waffles. It is an opportunity to highlight how they look for and use the structure of rows and columns to justify their estimates.

### Launch

• Groups of 2
• Have students keep their books closed.
• Display the first image:
• “¿Cuántos waffles hay en una bandeja?” // “How many waffles are on one tray?”
• “¿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

### Activity

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share responses.
• Display the second image:
• “¿Quieren reconsiderar lo que pensaron?” // “Would you like to revise your thinking?”
• 1 minute: quiet think time
• Record responses.
• “¿Cómo les ayudó la segunda imagen?” // “How did the second image help you?”
• Share responses.

### Student Facing

¿Cuántos waffles hay en la bandeja?

Escribe una estimación que sea:

muy baja razonable muy alta
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Activity Synthesis

• “Aunque no podemos ver toda la bandeja, algunos estudiantes están muy seguros de sus estimaciones. ¿Por qué creen que es así?” // “Even though we can’t see the whole tray, some students are very confident about their estimates. Why do you think that is?” (If the tray is full, it has 5 rows with 4 in each row. Even though we can’t see all the waffles, we know how many fit.)

## Activity 1: Sumas de filas y sumas de columnas (10 minutes)

### Narrative

In the first section of the unit, students determined whether a group of objects had an even or odd number of members. They found the total number of objects in even groups by finding the sum of two equal addends or by skip counting by 2. In this activity, they analyze expressions that have more than two equal addends and connect these expressions to the structure of an array. The purpose of this activity is for students to recognize that an expression with equal addends can represent the sum of the number of objects in each row or the sum of the number of objects in each column. When students analyze Diego and Mai’s equations, they reason abstractly and quantitatively by relating the expressions to the array of dots (MP2).

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• “Mai y Diego representaron el número de objetos del mismo arreglo usando expresiones diferentes. Diego escribió $$2+ 2 + 2 + 2 + 2 +2$$. Mai escribió $$6 + 6$$” // “Mai and Diego represented the number of objects in the same array with different expressions. Diego wrote $$2+ 2 + 2 + 2 + 2 +2$$. Mai wrote $$6 + 6$$.”
• “¿Con quién están de acuerdo? Decidan con su compañero con quién están de acuerdo y prepárense para explicar su razonamiento” // “Who do you agree with? Work with a partner to decide who you agree with and be prepared to explain your reasoning.”
• 1 minute: quiet think time
• 4 minutes: partner work time

### Student Facing

Mai y Diego representaron el mismo arreglo usando expresiones diferentes.

Expresión de Diego

$$2 + 2 + 2 + 2 + 2 +2$$

Expresión de Mai

$$6 + 6$$

¿Con quién estás de acuerdo?

Muestra cómo pensaste. Usa diagramas, símbolos u otras representaciones.

### Activity Synthesis

• “Algunos estudiantes estuvieron de acuerdo con Mai, algunos estuvieron de acuerdo con Diego y algunos estuvieron de acuerdo con ambos. ¿Cómo pueden ambas expresiones representar el arreglo?” // “Some students agreed with Mai, some students agreed with Diego, and some students agreed with both. How can both expressions represent the array?” (One shows adding up the numbers in each row and one shows adding up the number in each column)
• “¿Cuál expresión representa la suma del número de objetos que hay en cada fila? Expliquen” // “Which expression represents the sum of the number in each row? Explain.” ($$2 + 2 + 2 + 2 + 2 + 2$$ because there are 2 in each row.)
• “La expresión de Mai es $$6 + 6$$. ¿Cuáles son las diferentes maneras en las que Mai pudo ver $$6 + 6$$ en este arreglo?” // “Mai’s expression is $$6 + 6$$. What are the different ways Mai may see $$6 + 6$$ in this array?” (She may see 6 red and 6 yellow counters. She may see 2 rows with 6 counters in each row.)
• Consider displaying and circling the counters to illustrate student thinking:
• “Podemos encontrar el número de objetos de un arreglo sumando el número de objetos que hay en cada fila o sumando el número de objetos que hay en cada columna. Podemos representar estas sumas con expresiones como las que Mai y Diego escribieron” // “We can find the total number of objects in an array by adding up the number of objects in each row, or by adding up the number of objects in each column. We can represent these sums with expressions like the ones Mai and Diego wrote.”

## Activity 2: Clasificación de tarjetas: Arreglos y expresiones (15 minutes)

### Narrative

The purpose of this activity is for students to connect expressions to the array structure. Students match arrays, 2 expressions, and the total number of objects represented. As they do so, they look for different ways of representing the number of objects in an array in terms of the columns or the rows (MP2, MP7)In the synthesis, student share how they found their matches and discuss why there was only 1 expression that represented the array that had the same number of rows as columns.

MLR2 Collect and Display. Collect the language students use as they complete the card sort. Display words and phrases such as: “arreglo”, “filas”, “columnas”, “grupos”, “ecuación”, “total”, “suma repetida” y “contar a saltos” // “array,” “rows,” “columns,” “groups,” “equation,” “total,” “repeated addition,” and “skip count.” During the synthesis, invite students to suggest ways to update the display: “¿Qué otras palabras o frases deberíamos incluir?” // “What are some other words or phrases we should include?” and so on. Invite students to borrow language from the display as needed.

### Required Materials

Materials to Copy

• Match Arrays to Expressions Card Sort, Spanish

### Required Preparation

• Create a set of cards for each group of 2–3.

### Launch

• Groups of 2–3
• Give each group a set of cards.

### Activity

• “Clasifiquen las tarjetas en 8 grupos” // “Sort the cards into 8 groups.”
• “Cada grupo debe tener un arreglo, dos expresiones y un número total” // “Each group should have an array, 2 expressions, and a total number.”
• “Cada grupo tendrá 4 tarjetas, menos uno de ellos que solo tendrá 3” // “Each group has 4 cards, but one will only have 3.”
• 12 minutes: partner work time

### Student Facing

Tu profesor te va a dar varias tarjetas que muestran arreglos, expresiones y un número total. Encuentra las tarjetas que corresponden entre sí.

### Activity Synthesis

• Display Card A and Card F.
• Select students to share the expressions that match Card A and Card F ($$4 + 4 + 4 + 4 + 4$$ and $$5 + 5 + 5 + 5$$)
• “¿Cómo es posible que estos 2 arreglos se puedan representar con las mismas expresiones?” // “How is it possible that these 2 arrays have the same expressions?” (Sample response: $$4 + 4 + 4 + 4 + 4$$ shows the sum of the columns for the basketballs, but it shows the sum of the rows for notebooks.)
• Display Card C (popcorn in 3 rows of 3 bags).
• “¿En qué se diferencia este arreglo del resto?” // “What was different about this array?” (There was only 1 expression because the number of rows and columns are the same.)

## Activity 3: Sumémoslo todo (10 minutes)

### Narrative

The purpose of this activity is for students to determine the total number of objects in an array and match expressions to arrays by paying attention to the number of objects in each row and the number of objects in each column. For example, students recognize that 3 rows with 4 in each row would be $$4 + 4 + 4$$. The arrays in this task provide students opportunities to compare different ways an array could be decomposed to find the total number of objects. In the synthesis, students compare the different ways they find the total number of objects in the array to expressions that use equal addends to represent the sums of rows or sums of columns.

Representation: Develop Language and Symbols. Support understanding of the problem by inviting students to use a blank piece of paper to cover counters and show each row in one expression, and then each column in the other expression. For example, in task 1, students will use a piece of paper to cover all counters except the first row of 4 counters, slide it down to reveal the second row of 4 counters, and slide it down one last time to show the last row of 4 counters ($$4 + 4 + 4$$). Then, in a similar fashion, cover all counters but the first column of three counters, slide right to reveal the second column of three columns, and so on. ($$3 + 3 + 3 + 3$$)
Supports accessibility for: Visual-Spatial Processing, Organization

### Required Materials

Materials to Gather

### Activity

• “Ahora van a encontrar el número total de fichas de los arreglos usando un método que tenga sentido para ustedes. Después, asocien cada arreglo con expresiones” // “Now you will find the total number of counters in arrays using a method that makes sense to you. Then match each array to expressions.”
• 6 minutes: independent work time
• Monitor for the array that generates the largest variety of different ways students find the total number of counters in each array, including skip-counting or adding on based on the number in each row or column.

### Student Facing

1. ¿Cuántas fichas hay en total?

2. Explica cómo encontraste el número total de fichas.
3. Marca 2 expresiones que representan el arreglo.

$$3 + 3 + 3 + 3$$

$$3 + 3 + 3$$

$$4 + 3$$

$$4 + 4 + 4$$

$$4 + 4 + 4 + 4$$

1. ¿Cuántas fichas hay en total?

2. Explica cómo encontraste el número total de fichas.
3. Marca 2 expresiones que representan el arreglo.

$$2 + 2 + 2 + 2 + 2 + 2 + 2$$

$$6 + 6$$

$$7 + 7$$

$$2 + 2 + 2 + 2 + 2$$

### Activity Synthesis

• Invite previously identified students to share how they found the total for the same array. Record each method using an expression.
• If no student counted by adding the sum of each row or sum of each column, select a student to share the expressions they circled to match the array.
• “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “How are these expressions the same? How are they different?” (All the expressions show the total number of counters. They have different numbers of addends. They use different numbers. Some show a way to find the total, but they don’t match the number of counters in each row or each column.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy aprendieron que el número de objetos que hay en un arreglo se puede representar usando expresiones que muestran la suma del número de objetos que tiene cada fila o la suma del número de objetos que tiene cada columna” // “Today you learned that the number of objects in an array can be represented using expressions that show the sum of the number of objects in each row or the sum of the number of objects in each column.”

Display counters to show:

“¿Qué me pueden decir sobre este arreglo?” // “What can you tell me about this array?” (It has 5 rows with 4 in each. It has 4 columns with 5 in each. An expression that represents the sum of the rows is $$4 + 4 + 4 + 4 + 4$$. An expression that represents the columns is $$5 + 5 + 5 + 5$$. There are 20 counters in all.)

Share and record responses.