In the first section of this unit, students determined whether a group of objects had an even or odd number of members. Students 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, students 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’s and Mai’s equations, they reason abstractly and quantitatively by relating the expressions to the array of counters (MP2).

• “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 number of counters in each row, and one shows adding up the number of counters in each column.)
• “Which expression represents the sum of the number in each row? Explain.” ( because there are 2 in each row.)
• “Mai’s expression is . What are the different ways Mai might see in this array?” (She may see 6 red and 6 yellow counters. She may see 2 columns with 6 counters in each column.)
• Consider displaying and circling the counters to illustrate student thinking:

  

  

• “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 (MP7). In the Synthesis, students 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.

• Display Card A and Card F.
• Invite previously selected students to share the expressions that match Card A and Card F. ( and )
• “How is it possible that each expression can represent these 2 arrays?” ( shows the sum of the columns for the basketballs, but it shows the sum of the rows for notebooks. shows the sum of the rows for the basketballs, but it shows the sum of the columns for the notebooks.)
• Display Card C (popcorn in 3 rows of 3 bags).
• “What was different about this array?” (There was only 1 expression that matched it 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 .

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.

• 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.
• “How are these expressions the same? How are they different?” (All the expressions show the total number of counters. They all use addition. 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.)