Lesson 11
Analyze Subtraction Algorithms
Warmup: Number Talk: Subtract within 1,000 (10 minutes)
Narrative
The purpose of this Number Talk is to elicit strategies and understandings students have for subtracting multidigit numbers. These understandings help students develop fluency and will be helpful later in a subsequent lesson when students are to use strategies flexibly to subtract within 1,000.
Launch
 Display one expression.
 “Give me a signal when you have an answer and can explain how you got it.”
 1 minute: quiet think time
Activity
 Record answers and strategy.
 Keep expressions and work displayed.
 Repeat with each expression.
Student Facing
Find the value of each expression mentally.
 \(400  200\)
 \(450  200\)
 \(450  205\)
 \(450  215\)
Student Response
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Activity Synthesis
 “How did place value help as you subtracted these numbers?” (I subtracted hundreds from hundreds, tens from tens, and ones from ones. I was able to think about each place value position separately, which helped me find the difference.)
 Consider asking:
 “Who can restate _____'s reasoning in a different way?”
 “Did anyone have the same strategy but would explain it differently?”
 “Did anyone approach the problem in a different way?”
 “Does anyone want to add on to _____’s strategy?”
Activity 1: Compare Two Subtraction Algorithms (20 minutes)
Narrative
The purpose of this activity is for students to consider two subtraction algorithms. In the first algorithm, students first look for any place value units where they need to decompose to get more units, then subtract right to left. In the second algorithm, subtraction occurs right to left, and units are decomposed as the need arises. Students try each algorithm and consider potential advantages and disadvantages of each algorithm.
In the synthesis, students carefully analyze and discuss the two algorithms, explaining the motivation behind them and how they are the same and different (MP3, MP6).
Synthesis: For each idea that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Supports accessibility for: Attention
Launch
 Groups of 2
 Display the image.
 “The first steps of two subtraction algorithms are shown. Take a minute to think about how they are different.”
 1 minute: quiet think time
Activity
 “Discuss how the steps are different in each algorithm with your partner.”
 2 minutes: partner discussion
 Share and record responses.
 “Work with your partner to finish each algorithm.”
 2–3 minutes: partner work time
 “Now work with your partner to use both algorithms to subtract 541 from 824.”
 5–7 minutes: partner work time
Student Facing

The first steps of two algorithms are shown.
Algorithm A, step 1
Algorithm B, step 1
How are the steps different?
 Use each algorithm to find the value of \(824  541\).
Student Response
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Activity Synthesis
 Select students to share how they used both algorithms to find the value of \(824  541\).
 Keep algorithms from the first problem displayed.
 “Even though the algorithms may look the same after a few steps, they started out differently. Think about advantages and disadvantages of using each algorithm.”
 1 minute: quiet think time
 “Now, discuss the advantages and disadvantages of each algorithm with your partner.”
 1 minute: partner discussion
 Invite students to share advantages and disadvantages they come up with. (In algorithm 1, I look for decompositions first, so I probably won’t mix up the order of subtracting. In algorithm 1, I could subtract left to right or right to left. In algorithm 2, I can start subtracting right away. That means I don’t have to worry about decomposing until I know I need to do it.)
Activity 2: Use an Algorithm? (15 minutes)
Narrative
The purpose of this activity is for students to make sense of an algorithm in which a number with nonzero digits is subtracted from a number with a zero in the tens place. In the given problem, it is necessary to decompose a larger unit to have enough ones to subtract. There are no tens to decompose, however, prompt students to consider whether subtraction is possible, and if so, how it could be done.
When students make sense of Elena’s reasoning, they construct viable arguments and critique the reasoning of others (MP3).
Launch
 Groups of 2
 “Take a minute and look over Noah’s work and what Elena says about it.”
 1 minute: quiet think time
Activity
 “Now, work with your partner to complete the activity.”
 5–7 minutes: partner work time
 Monitor for a student who shows how the problem could be completed with decomposing a hundred into tens, then decomposing a ten into more ones.
Student Facing
Noah wanted to find the value of \(301  167\) and wrote:
Elena said that we can’t subtract this way because we would need more ones to subtract 7 ones, but there’s a zero in the tens place of 301.
 Do you agree with Elena's statement? Explain your reasoning.
 Show how you would use an algorithm (either Noah's or another algorithm) to find the difference between 301 and 167.
Student Response
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Activity Synthesis
 Select students to share their responses.
 Display student work that shows a hundred decomposed into tens, then a ten decomposed into ones (or show the example in Student Responses).
 “How does the work here show that we could have enough ones to subtract even though there is 0 in the tens place of 301?” (Crossing out the 3 in the hundreds and writing 10 in the tens place shows a hundred decomposed to get tens. Crossing out the 10 and writing a 9 in the tens place and writing 11 in the ones place shows a ten decomposed to get ones.)
Lesson Synthesis
Lesson Synthesis
 “We've learned different algorithms for subtracting. Which subtraction algorithm is your favorite and why?” (The expanded form algorithms because we can really see all the parts of the number. The algorithm where we decompose the units as we go because I don’t like to do them all at once. The algorithms that use 1 digit for each place value because they don’t take as long to write.)
Cooldown: Subtraction Reflection (5 minutes)
CoolDown
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