Lesson 9

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.

• 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

• “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.

• “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.)

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).

• Groups of 2
• Give students access to counters.

• “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

• “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.”

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.

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

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

• “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

• 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.)

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.

• Give students access to counters.

• “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.

• 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.)