Lesson 19
Compose and Decompose to Add and Subtract
Warmup: Number Talk: Subtract Fractions (10 minutes)
Narrative
The purpose of this Number Talk is to elicit strategies and understandings students have for subtracting fractions and mixed numbers, particularly cases where it is necessary to rewrite a whole number as a fraction, or decompose it into a different whole number and fraction, in order to perform subtraction. These understandings will be helpful later in this lesson when students subtract multidigit numbers that involve decomposing units.
Launch
 Display one expression.
 “Give me a signal when you have an answer and can explain how you got it.”
Activity
 1 minute: quiet think time
 Record answers and strategy.
 Keep expressions and work displayed.
 Repeat with each expression.
Student Facing
Find the value of each expression mentally.
 \(2\frac{3}{4}  1\frac{1}{4}\)
 \(1\frac{1}{4}  \frac{3}{4}\)
 \(5\frac{1}{8}  2\frac{3}{8}\)
 \(3\frac{2}{10}  2\frac{7}{10}\)
Student Response
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Activity Synthesis
 Highlight strategies in which students decomposed the first mixed number in each expression.
 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 expression in a different way?”
 “Does anyone want to add on to____’s strategy?”
Activity 1: Find and Check Sums (15 minutes)
Narrative
The purpose of this activity is to use the standard algorithm to add multidigit numbers, taking care to compose a new unit and record it accurately. They also analyze common errors and critique the given reasoning when composing a new unit (MP3).
Required Materials
Materials to Gather
Launch
 Groups of 2
 “Complete the first 2 problem and then talk to your partner about any patterns you notice.”
Activity
 3 minutes: independent work time
 2 minutes: partner discussion
 Share and record responses.
 If needed, use the expanded form to help students make connections between the way composing a larger unit is recorded when using the standard algorithm and what is happening in each place. In the bottom line of the example here, we see 10 in the ones place and 100 in the tens place. Both partial sums do not match the assigned values of their places.
 3–4 minutes: independent work time for the last problem.
Student Facing
 Find the value of each sum.
 Use the expanded form of both 8,299 and 1,111 to check the value you found for the last sum.

Each computation shown has at least one error. Find the errors and show the correct calculation.
Student Response
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Advancing Student Thinking
Activity Synthesis
 Ask students to share the errors they identified in the last set of questions.
 “What are some common errors when adding large numbers?”
Activity 2: Priya’s Family Heirlooms (20 minutes)
Narrative
This activity revisits the idea of decomposing a unit in one place into 10 units of the place value to its right when subtracting multidigit numbers using the standard algorithm. Students recall how this is done as they subtract numbers in which decomposition is necessary.
To support students with understanding the context, the activity launch introduces a saree (traditional wedding attire for women in India) and the idea of family heirlooms, or gifts passed down from generation to generation.
When students create a subtraction problem that does not require decomposition of a unit when using the standard algorithm, they make use of structure and their understanding of subtraction as they choose the digits for the numbers in their difference (MP7).
This activity uses MLR7 Compare and Connect. Advances: representing, conversing
Supports accessibility for: Conceptual Processing, Memory, Language
Launch
 Groups of 2
 “What do you notice and wonder about these pictures?”
 Collect student ideas.
 “In this activity, Priya is researching her family history. Let’s see what she discovers.”
 “The women in the picture are dressed in a traditional Indian garment called a ‘saree’. Sarees are made of colorful fabric and often have intricate embroidery or patterned print designs.”
 “The bracelets in the picture are also from India. Sometimes jewelry like this is used as heirlooms, or gifts that are passed down from one generation to the next.”
Activity
 Groups of 2
 5–6 minutes: independent work time
 3 minutes: partner work time
 As students work, listen for student discourse that includes language about the place value of the digits and when to decompose a unit.
Student Facing
Priya’s mom wore an heirloom bracelet at her wedding in 1996. The bracelet was made in 1947.
Priya subtracted to find out how old the bracelet was when her parents were married.
Priya learned that her grandmother had also worn the bracelet at her wedding 24 years earlier.
Priya subtracted to find out when her grandparents were married.

Are both calculations correct? Why does one calculation have some numbers crossed out and some new numbers, but the other one does not? Explain your reasoning.

Priya’s grandmother wore an heirloom necklace and earring set that was 63 years old when she was married in 1972.
 If Priya uses the standard algorithm to subtract \(1972  63\) will she need to decompose a unit? Explain your reasoning.
 Use the standard algorithm to subtract \(1972  63\) and find the year the necklace was made.

Create a subtraction problem that would not require decomposing a unit to subtract. Then solve the problem.
Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
Students may not remember when to decompose a unit or how to record regrouping. Urge students to begin to subtract by place value, either by using expanded form or lining up the digits. Ask: “What issue comes up when you subtract the ones in \(1972  63\)?” Allow students to explain that they don't have enough ones in the ones place to subtract 3, but they can decompose a ten to get 10 ones and add it to the 2 already there. Then consider asking:
 “How will you record all of the ones you have after you decompose a ten?”
 “How will you know if you need to decompose a unit when subtracting one number from another?”
Activity Synthesis
 “Let’s do a gallery walk to see what problems you created.”
 “As you walk, discuss with a partner what you notice about the value of the digits in the numbers that were chosen.”
 3 minutes: gallery walk
 Collect 1–2 responses from student discussions during the gallery walk.
 Share 1–2 of the responses you collected.
 1 minute: partner discussion
 “What is the same about each of the problems you created?” (In each problem each digit is greater in the first number than in the second number)
 2 minutes: wholegroup discussion
Lesson Synthesis
Lesson Synthesis
Write \(1972  63\) for all students to see.
“When we look at a problem, how do we know if we will need to decompose a unit?” (If the digit we are subtracting is larger than the digit we are subtracting from, we will need to decompose a unit and regroup.)
Display for all to see:
“When we look at an addition problem, how do we know when we will need to compose a new unit?” (If the sum of the digits in one place is greater than 9, we will compose a new unit, and record 1 more for the place to the left.)
Cooldown: Difference and then Sum (5 minutes)
CoolDown
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