Lesson 9
A Sum of Equal Addends
Warmup: Estimation Exploration: How Many Waffles? (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:
 “How many waffles are on one tray?”
 “What is an estimate that’s too high?” “Too low?” “About right?”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Share responses.
 Display the second image:
 “Would you like to revise your thinking?”
 1 minute: quiet think time
 Record responses.
 “How did the second image help you?”
 Share responses.
Student Facing
How many waffles are on the tray?
Record an estimate that is:
too low  about right  too high 

\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)  \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)  \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) 
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
 “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: Sums of Rows and Sums of Columns (10 minutes)
Narrative
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give students access to counters.
Activity
 “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 \(6 + 6\).”
 “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 and Diego represented the same array with different expressions.
Diego’s expression
\(2 + 2 + 2 + 2 + 2 +2\)
Mai’s expression
\(6 + 6\)
Who do you agree with?
Show your thinking using diagrams, symbols, or other representations.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
 “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)
 “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.)
 “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:
 “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: Card Sort: Arrays and Expressions (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.
Advances: Conversing, Reading
Required Materials
Materials to Copy
 Match Arrays to Expressions Card Sort
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
 “Sort the cards into 8 groups.”
 “Each group should have an array, 2 expressions, and a total number.”
 “Each group has 4 cards, but one will only have 3.”
 12 minutes: partner work time
Student Facing
Your teacher will give you a set of cards that show arrays, expressions and a total number. Find the cards that match.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
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 \(5 + 5 + 5 + 5\))
 “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).
 “What was different about this array?” (There was only 1 expression because the number of rows and columns are the same.)
Activity 3: Add It All Up (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.
Supports accessibility for: VisualSpatial Processing, Organization
Required Materials
Materials to Gather
Launch
 Give students access to counters.
Activity
 “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 skipcounting or adding on based on the number in each row or column.
Student Facing


How many counters are there in all?
 Explain how you found the total number of counters.

Circle 2 expressions that represent the array.
\(3 + 3 + 3 + 3\)
\(3 + 3 + 3\)
\(4 + 3\)
\(4 + 4 + 4\)
\(4 + 4 + 4 + 4\)



How many counters are there in all?
 Explain how you found the total number of counters.

Circle 2 expressions that represent the array.
\(2 + 2 + 2 + 2 + 2 + 2 + 2\)
\(6 + 6\)
\(7 + 7\)
\(2 + 2 + 2 + 2 + 2\)

Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
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.
 “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
“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:
“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.
Cooldown: Match Expressions with Arrays (5 minutes)
CoolDown
Teachers with a valid work email address can click here to register or sign in for free access to CoolDowns.