Lesson 23
Solve Problems with Many Operations
Warmup: True or False: Differences (10 minutes)
Narrative
The purpose of this True or False is to elicit strategies and understandings students have for finding differences between two numbers. These understandings help students build fluency in addition and subtraction, while preparing them to think about distances between two points.
Students may use estimation or place value understanding to solve the problems (MP7).
Launch
 Display one statement.
 “Give me a signal when you know whether the statement is true and can explain how you know.”
Activity
 1 minute: quiet think time
 Share and record answers and strategy.
 Repeat with each statement.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
 \(50,\!000  999 = 49,\!001\)
 \(4,\!799 = 5,\!000  311\)
 \(3,\!005 = 4,\!000 1,\!995\)
 \(2,\!000  1,\!234 = 1,\!876\)
Student Response
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Activity Synthesis
 “How can we tell if each equation is true without calculating?”
 “We could use the standard algorithm to find each difference and then decide if the equation is true or false. Would that be a good idea? Why or why not?”
Activity 1: Back and Forth (20 minutes)
Narrative
This activity prompts students to interpret and represent situations about distances and use multiple operations to solve problems. In problem 3, a piece of information is withheld. Students will need to make sense of what’s missing and find out that information before the question could be answered. Throughout the activity, students reason abstractly and quantitatively as they interpret the diagram and use the information to solve problems (MP2).
Supports accessibility for: SocialEmotional Functioning
Launch
 Groups of 2
 Give students access to grid paper.
 Read opening paragraphs as a class. Display the diagram.
 “What mathematical questions can you ask about this situation?”
 1–2 minutes: quiet think time
 1–2 minutes: partners compare questions
 Share and record responses.
Activity
 6–8 minutes: independent work time
 3–4 minutes: partner discussion
 When requested, tell students that 1 mile is equal to 5,280 feet.
 Monitor for:
 the different equations students write for the first question (see examples in Student Responses)
 the assumptions students make when answering the last question (as noted in the narrative)
 the different ways of reasoning (as noted in the narrative)
Student Facing
Mai’s cousin is in middle school. She travels from her homeroom to math, then English, history, and science. When she finishes her science class, she takes the same path back to her homeroom.
Mai’s cousin makes the same trip 5 times each week. The distances between the classes are shown.

How far does Mai’s cousin travel each round trip—from her homeroom to the four classes and back? Write one or more expressions or equations to show your reasoning.

Each week, Mai’s cousin makes 3 round trips from her homeroom to her music class. The total distance traveled on those 3 round trips is 2,364 feet.
How far away is the music room from her homeroom? Show your reasoning.
 Mai thinks her cousin travels 2 miles each week just going between classes. Do you agree? Explain or show your reasoning.
Student Response
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Activity Synthesis
 Consider collecting and displaying the different expressions students wrote for the first question and discussing what they tell us about the way the writer reasoned about the problems. For example:
 \(2 \times (157 + 134 + 162 + 275)\) suggests that the writer found the distance one way then doubled it.
 \((2 \times 157) + (2 \times 134) + (2 \times 162) + (2 \times 275)\) suggests that the distances between classes were doubled first before being added.
 “How did you find out if Mai’s cousin traveled 2 miles each week? What did you do first? What did you do next?”
 Select students to share their responses and reasoning.
 Highlight how the strategies are alike and different.
Activity 2: Fitness Challenge (15 minutes)
Narrative
This activity gives students another opportunity to use multiple operations to model the quantities in a situation and to solve problems involving large numbers. Students interpret the quantities in context, reason about them abstractly as they perform computations, and then return to the context to interpret the results. As they do so, students are reasoning quantitatively and abstractly (MP2).
Students may choose to answer the first problem by dividing a fivedigit number by a onedigit divisor. Though finding a quotient of a fivedigit dividend is not an expectation, this particular number ends in a 0. Students can use the division strategies they’ve learned so far and what they know about the structure of numbers in base ten to find the quotient (MP7).
Advances: Speaking
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give students access to grid paper.
 “Sometimes people compete to see who can walk the most steps in a day, week, or month.”
 “These competitions are called challenges.”
 “This activity involves a fitness challenge involving steps.”
Activity
 6–8 minutes: independent work time
 Monitor for the different ways students reason about each question.
Student Facing
 Han walked 32,550 steps in the first week. He walked the same number of steps every day. How many steps did Han walk each day? Show your reasoning.

The table shows the number steps Han took each week for the first three weeks. How much did the number of steps drop from the first week to the second week?
week 1 week 2 week 3 week 4 32,550 28,098 36,249 \(\phantom{\huge{00000}}\)  If Han wants to meet the challenge, what is the fewest number of steps that he needs to take in week 4? Show your reasoning.
 How do you know your answer to problem 3 is reasonable?
Student Response
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Activity Synthesis
 Invite students to share their responses and reasoning.
 Focus the discussion on how students found the number of steps Han took each day and how they found the last number in the table.
 “How would you know if your answer to the first question is correct?” (Multiply 4,650 by 7 to see if it equals 32,550.)
 “How would you know if your answer to problem 3 is reasonable?” (Add it to the other three numbers and see if the sum is 120,000.)
Lesson Synthesis
Lesson Synthesis
“Today we solved problems that involved numbers with four or more digits. Some of those problems could be interpreted in more than one way.”
“In the fitness challenge activity, how did you think about finding Han’s steps each day? Did you think of it in terms of multiplication (what number times 7 is 32,550?) or in terms of division (what is 32,550 divided by 7?)? Is one way of thinking more convenient? Why or why not?”
To facilitate discussion, display equations such as:
\(7 \times n = 32,\!550\)
\(32,\!550 \div 7 = n\)
“How did you think about finding Han’s steps in week 4? Did you think in terms of addition (what number must be added to 96,897 to make 120,000?) or subtraction (what is the difference between 120,000 and 96,897?)?”
Display equations such as:
\(32,\!550 + 28,\!098 + 36,\!249 + f = 120,\!000\)
\(96,\!897 + f = 120,\!000\)
\(120,\!000  96,\!897 = f\)
Cooldown: Longdistance Driving (5 minutes)
CoolDown
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