# Lesson 4

Introduction to Addition Algorithms

## Warm-up: Which One Doesn’t Belong: 247 (10 minutes)

### Narrative

This warm-up prompts students to compare three expressions and one three-digit number. During the synthesis, ask students to explain the meaning of any terminology they use, such as the value of each expression and ways that place value was used to write the number 247 in different ways.

### Launch

- Groups of 2
- Display the expressions and number.
- “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
- 1 minute: quiet think time

### Activity

- “Discuss your thinking with your partner.”
- 2-3 minutes: partner discussion
- Share and record responses.

### Student Facing

A. \(200 + 30 + 17\)

B. 247

C. \(200 + 47 + 10\)

D. \(100 + 140 + 7\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- “How did you know that A and D were equal to 247?” (In A there were 2 hundreds, 4 tens, and 7 ones, but some of the tens were with the ones. In D there were 2 hundreds, 4 tens, and 7 ones, but some of the hundreds were with the tens.)
- Consider stating: “Let’s find at least one reason why each one doesn’t belong.”

## Activity 1: What is an Algorithm? (20 minutes)

### Narrative

In this activity, students use their knowledge of base-ten representations and place value to make sense of two addition algorithms. One algorithm shows the addends in expanded form. Both algorithms show the sums of ones, tens, and hundreds separately, but display these partial sums differently. Students notice that both algorithms show hundreds added to hundreds, tens to tens, and ones to ones, regardless of order. In the synthesis, introduce the term “algorithm.”

*MLR7 Compare and Connect.*Synthesis: Invite groups to prepare a visual display that shows the strategy they used to find the value of the sums. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask students, “What did the representations have in common?”, “How were they different?”, “How did the total sum show up in each method?”

*Advances: Representing, Conversing*

### Required Materials

Materials to Gather

### Launch

- Groups of 2
- Give students access to base-ten blocks.
- “In an earlier lesson, we saw many ways to find the value of a sum. Take a minute to look at how these 3 students added \(362+354\).”
- 1 minute: quiet think time

### Activity

- “Work with your partner to explain how each method works.”
- 7-10 minutes: partner work time.

### Student Facing

Three students found the value of \(362 + 354\). Their work is shown. Explain how each method works.

- Tyler’s drawing
- Lin’s method
- Han’s method

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students don't explain one of the written algorithms, consider asking:

- “What did Lin (or Han) do to add the numbers?”
- “How is their work related to Tyler's drawing?”

### Activity Synthesis

- For each method, ask a student share their explanation. As students share, record the sequence of steps they describe in their explanation.
- Consider asking:
- “Who can restate _______ 's reasoning in a different way?”
- “Did anyone have a similar idea but would explain it differently?”
- “Did anyone explain the method in a different way?”
- “Does anyone want to add on to____’s explanation?”

- As students add on, edit the steps so the class is in agreement about how each method works.
- “Lin and Han used
**algorithms**to solve this problem. An algorithm is a set of steps that works every time as long as the steps are carried out correctly.” - “How are Lin and Han’s algorithms the same?” (They both add ones to ones, tens to tens, and hundreds to hundreds.)
- “How are the algorithms different?” (Lin writes the number in expanded form, but Han didn’t. Lin hasn't added the sums of hundreds, tens, and ones, but Han has.)
- Consider asking:
- “Can we tell which place Lin started with? Why or why not?” (We can’t really tell with Lin’s method because of how the numbers are next to each other. She might have started with the ones or the hundreds. No matter which place she starts with she would get the same sum.)

## Activity 2: Try an Algorithm (15 minutes)

### Narrative

The purpose of this activity is for students to try the algorithms they saw earlier in the lesson. The important thing is that they combine hundreds and hundreds, tens and tens, and ones and ones, which should be a familiar idea from grade 2. The synthesis provides an opportunity to show a different way of recording newly composed tens and hundreds when compositions are required, which will be discussed in more detail in subsequent lessons. Provide access to base-ten blocks for students to use to support their reasoning about the algorithms, in case requested.

Students analyze and improve a given explanation of how to find a sum, filling in details and using more precise language to explain the calculation more fully (MP3, MP6).

This activity uses *MLR3 Clarify, Critique, Correct*. *Advances: reading, writing, representing*

*Engagement: Develop Effort and Persistence.*Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk.

*Supports accessibility for: Organization, Social-Emotional Functioning*

### Required Materials

Materials to Gather

### Launch

- Groups of 2
- Give students access to base-ten blocks.
- “Now you are going to have a chance to try the algorithms that Lin and Han used in the last activity. Take a minute to think about which algorithm you want to use for each problem.”
- 1 minute: quiet think time

### Activity

- “Work with your partner to try an algorithm to find the value of each sum.”
- 5 minutes: partner work time
- Monitor for students who use Lin’s algorithm and Han's algorithm on the last problem.

### Student Facing

Try using an algorithm to find the value of each sum. Show your thinking. Organize it so it can be followed by others.

- \(475 + 231\)
- \(136 + 389\)
- \(670 + 257\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- Select previously identified students to share their work on the last expression.

**MLR3 Clarify, Critique, Correct**

- Display the following partially correct answer and explanation: I added the ones, the tens, and then hundreds.
- Read the explanation aloud.
- “What is unclear?”
- 1 minute: quiet think time
- 2 minute: partner discussion
- “With your partner, work together to write a revised explanation.”
- Display and review the following criteria:
- Explanation for each step
- Words such as: first, next, then

- 3–5 minute: partner work time
- Select 1–2 groups to share their revised explanation with the class. Record responses as students share. (Sample explanation: First, I stacked the numbers vertically. Then, I added the ones to get 7 and recorded a 7 below the ones place. Next, I added 70 and 50 and got 120. I recorded 20 below the 7 and 100 below the 20. I added 600 and 200 to get 800 and recorded that below the 100. Then I added and wrote 927 as the answer.)
- “What is the same and different about the explanations?” (Both explanations say that they added the ones, tens, and hundreds, but the revised explanation gives more detail about how to record each step.)
- “How is this way of recording this work the same or different from Han’s method in the first activity?” (Han would record 120 in the second row where we record the tens. It was 12 tens, but that’s the same as 1 hundred and 2 tens. We could record 100 on one line and 20 on the next line.)

## Lesson Synthesis

### Lesson Synthesis

Display Lin and Han’s algorithms.

“Today we learned about two different algorithms or two different sets of steps for finding the value of a sum. How are the 2 algorithms alike? How are they different?” (Alike: They give the same result at the end. They both involve using place value and stacking the numbers being added. It doesn’t matter which place value unit we add first. Different: In one algorithm the numbers being added are written in expanded form.)

## Cool-down: Choose an Algorithm (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.