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 Activity Synthesis, introduce the term “algorithm.”

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

• For each method, ask a student to share their explanation. As students share, record the sequence of steps they describe in their explanations.
• 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 to the explanations, 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?” (Both add ones to ones, tens to tens, and hundreds to hundreds.)
• “How are the algorithms different?” (Lin wrote the number in expanded form, but Han didn’t. Lin didn’t add the sums of hundreds, tens, and ones, but Han has.)
• Consider asking:
  • “Can we tell at which place Lin started? 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 started with, she would get the same sum.)

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 Activity 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 Critique, Correct, Clarify. Advances: reading, writing, representing.

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

MLR3 Critique, Correct, Clarify

• Display the response for students to consider:

  

  I added the ones, the tens, and then the hundreds.
• Read the explanation aloud.
• “What parts of this response are unclear, incorrect, or incomplete?” (It tells the order of places they added, but it's missing a lot of details. It doesn't explain what all the numbers under 670 and 257 mean. It doesn't explain why there's 100 and 800.)
• 1 minute: quiet think time
• 2 minute: partner discussion
• Invite 2 or 3 groups to share what they discussed. Record for all to see.
• “With your partner, work together to write a revised explanation.”
• Display and review the following criteria:
  • An explanation for each step.
  • Words such as “first,” “next,” and “then.”
• 3–5 minute: partner work time
• Select 1 or 2 groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each group shares, and then invite the whole class to contribute additional language and edits to make the final draft even clearer and more convincing. (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 what is different about the explanations?” (Both explanations add the ones, the tens, and the hundreds, but the revised explanation gives more detail about how to record each step.)
• “How is this way of recording this work the same as 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.)