Lesson 5
Another Addition Algorithm
Warmup: Notice and Wonder: Another Curious Table (10 minutes)
Narrative
The purpose of this warmup is to elicit observations about patterns in sums of two and threedigit addends in an addition table. The table is partially filled out to highlight some properties of operations. For example, the sums in the table can illustrate the commutative property (\(99 + 98\) and \(98 + 99\) both give 197). The numbers also prompt students to notice patterns in sums of odd and even numbers. For example, the sum of an odd number and an even one is always odd.
While students may notice and wonder many things about the addition table, focus the discussion on the patterns in the table and possible explanations for them. When students make sense of patterns in sums and explain them in terms of the features of the addends and how they are added, they notice and use regularity in repeated reasoning (MP7).
Launch
 Groups of 2
 Display the table.
 “What do you notice? What do you wonder?”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Share and record responses.
Student Facing
What do you notice? What do you wonder?
+  98  99  100  101  102 

98  197  199  
99  197  199  201  
100  ?  ?  
101  199  201  203  
102  201  203 
Student Response
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Activity Synthesis
 “How do you think the table works?” (The table shows addition. Each number in the row at the top is added to each number in the first column on the left.)
 “What numbers would go in the cells with a question mark?” (199 and 201.) “How do you know?”
 For each of the following questions, give students a minute of quiet think time. Illustrate their responses with equations, if possible.
 “Why do the sums go up by 2 from left to right and top to bottom?” (Because of the skipping, one number being added goes up by 2, so the sum also goes up by 2.)
 “Why do the sums from upper left corner to lower right corner increase by 2?” (Each of the two numbers being added go up by 1, so the sum goes up by 2.)
 “Why are the sums that make a line from the lower left corner to the upper right corner the same number?” (Each time, the first number being added goes up by 1 and the second number goes down by 1, so the sum stays the same.)
 Consider asking: “What do you notice about whether the addends or the sums are even or odd?” (In each pair of addends, one number is even and the other number is odd. All the sums are odd.)
Activity 1: A New Addition Algorithm (15 minutes)
Narrative
The purpose of this activity is for students to learn an algorithm in which a single digit is recorded as each place value position is added. Students use an algorithm from a prior lesson to make sense of the new algorithm. They learn that single digits can be used to represent the sum in each place value position because of what we know about place value.
Supports accessibility for: Conceptual Processing
Launch
 Groups of 2
 Display the algorithms.
 “Han and Elena used two different algorithms to solve this problem. The steps they took are labeled in order (point to the different steps). We saw Han's algorithm in an earlier lesson. Think about how Elena’s algorithm works.”
 2 minutes: quiet think time
Activity
 “Now discuss with your partner how Elena’s algorithm is different from Han’s algorithm. Also discuss why both algorithms work.”
 57 minutes: partner discussion
Student Facing
Here are two algorithms for adding \(367 + 231\).
Han’s algorithm
Elena’s algorithm
Discuss with your partner:
 How is Elena’s algorithm different from Han’s algorithm?
 Why do both algorithms work?
Student Response
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Advancing Student Thinking
If students do not explain how the algorithms are different, consider asking:
 “What are the steps for each algorithm?”
 “Why does Han's algorithm have an extra step?”
Activity Synthesis
 Invite students to share their analyses of Elena and Han's algorithms.
 Display the algorithms and annotate them to illustrate students' explanations.
 Consider asking:
 “Where do you see the 8, 90, and 500 in Elena’s algorithm?”
 “Why does Han’s algorithm have a step 4?”
 Select other students to share their thinking on why both algorithms work.
Activity 2: Compose New Units (20 minutes)
Narrative
The purpose of this activity is for students to consider how the composition of new tens and hundreds are recorded in the algorithm they saw in the previous activity. Students interpret the work and thinking of others and discuss the similarities and differences in two different strategies for finding a sum (MP3). Students see that in Elena’s algorithm, when the sum of the digits in a place has more than one digit, a newly composed ten or hundred is recorded as “10” or “100” above the addends, while the remaining value is recorded as a single digit below the addends. The synthesis focuses on clarifying how to record newly composed units when adding two numbers.
Advances: Speaking
Launch
 Groups of 2
 Display Elena’s algorithm.
 “What do you notice? What do you wonder?” (Students may notice: Elena is adding from right to left. There is a 100 at the top. She adds ones to ones, tens to tens, and hundreds to hundreds. Students may wonder: Why is there a 100 at the top? Why does she only record one number in each place?)
 1 minute: quiet think time
 Share and record responses.
 “The first problem mentions '14 tens.' Where might the 14 tens come from?” (From adding 6 tens and 8 tens, or 60 and 80.)
Activity
 “Work with your partner to answer the first question. Then, pause before moving on to the second set of problems.”
 2 minutes: partner discussion
 Invite students to share how 14 tens are recorded differently in the two algorithms.
 If no students mention that the 100 at the top represents 10 of the 14 tens, or 100 of the 140, bring this to their attention.
 “Now try using Elena's algorithm to find the value of each sum.”
 710 minutes: partner work time
 Monitor for how students record multiple compositions of tens or hundreds when finding the value of \(354 + 198\).
Student Facing
Here are two algorithms for adding \(365 + 182\).
Han’s algorithm
Elena's algorithm
 How do the algorithms show the 14 tens differently?

Try Elena’s algorithm to find the value of each sum.
 \(174 + 352\)
 \(273 + 619\)
 \(354 + 198\)
 \(525 + 376\)
Student Response
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Advancing Student Thinking
If students do not record the newly composed tens or hundreds in the second and third problems, consider asking:
 “What new units did you have to compose in this problem?”
 “Where would it make the most sense to you to record the newly composed 10? Where would it make the most sense to you to record the newly composed 100?”
Activity Synthesis
 Select students to display their work for finding the value of \(354 + 198\).
 “How did you decide where to record the new ten and hundred?” (It made the most sense to me to stack the hundred on the ten since the ten was already there.)
 “In this algorithm, we typically stack the newly composed tens or hundreds in the order they happen as we add from right to left, or from the ones place to the tens place to the hundreds place.”
Lesson Synthesis
Lesson Synthesis
Display an expression from the last activity, such as, \(174 + 352\).
“In this lesson, we have been adding from right to left, starting with the ones place. Let’s look at this expression again. Let’s consider what would happen if we started adding from the left with hundreds place.”
Work with the class to find the sum, setting it up like Elena's algorithm, but start by adding the hundreds.
“What would happen next? Can we add from left to right?” (If we worked from left to right, we would have to add the hundreds, then add them again if the tens add up to make a new hundred. It's the same with the tens. If we added them before the ones, we would have to add the tens again if the sum of the ones made a new ten.)
Cooldown: Use an Algorithm for Addition (5 minutes)
CoolDown
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