Lesson 5

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it. For example, students may reason that there are 10 chairs at each table and then make an estimate for the number of tables (MP3). Asking yourself “Does this make sense?” is a component of making sense of problems, and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4). 

• Groups of 2
• Display the image.

• “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

• “What part of the picture did you use to make your estimate?” (The number of seats at one table. The number of rows and tables in each row.)
• “Which estimates would have been unreasonable?” (353, 22, or 198, and other numbers not ending in a zero. Because there are ten chairs at a table, the total number has to be a multiple of 10 and end in a 0.)

The purpose of this activity is to find multiples of two different numbers in context. Students decide possible table sizes for a party based on whether or not a given number of people is a multiple of 6, 8, both, or neither. In situations where a given number is not a multiple of 6 or 8, they reason about what it means in context (MP2).

Focus the synthesis on how the number of seats at the two table sizes relate to multiplication and what it means when the number of people is not a multiple of the number of seats at a table.

• Groups of 2
• “Work with your partner to choose a type of table to seat different groups of students.”

• 8–10 minutes: partner work time
• Monitor for students who:
  • draw pictures to highlight why the number of students at tables of 6 is a multiple of 6 and that at tables of 8 is a multiple of 8
  • list the multiples of 6 and multiples of 8 to determine which table to use
  • use multiplication facts to explain their reasoning

• Invite students who used drawings, the idea of multiples, or multiplication facts to share while discussing the following questions.
• “Can Noah’s class fit exactly in tables for 6? How do you know?” (Yes. \(30 = 5 \times 6\), so Noah’s class will fill 5 tables exactly.)
• “Can Noah’s class fit exactly in tables for 8? How do you know?” (No, 30 is not a multiple of 8. They would need at least 4 tables but that’s too many seats, so 2 of the seats would be empty.)
• “How do multiplication facts or the idea of multiples help us think about this problem?” (We know that 24 and 32 are products of 8 and another number, but 30 is not.)
• “For the problem with both classes and the teachers, could they all fit at tables of 6? Why or why not?” (Yes, they could get 9 tables of 6, but there will be 4 empty seats. If they don’t want empty seats, tables of 6 won’t work as 50 is not a multiple of 6.)
• “Could they all fit at tables of 8? Why or why not?” (Yes, they could get 7 tables of 8, but there will be 6 empty seats. If they don't want empty seats, tables of 8 won't work as 50 is not a multiple of 8.)

In this activity, students solve problems that involve finding multiples that are shared by two different numbers: the number of hot dogs in a package and the number of hot dog buns in a package. They reason about how many of each package to get to make a certain number of servings of hot dogs. As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).

To solve the problems, students may decontextualize the situation and reason about factors and multiples, and then recontextualize the solutions in terms of servings of hot dogs. As they do so, they practice reasoning quantitatively and abstractly (MP2). During the synthesis, analyze different solutions and discuss why numbers that are multiples both of 8 and 10 are useful in this situation.

• Groups of 2–4
• Read the opening paragraph as a class.
• “What questions could you ask about the situation?”
• 1 minute: quiet think time
• Share responses.
• “Let’s solve problems about hot dogs and hot dog buns, which come in different-size packages.”
• “For each problem, look for possible numbers of packages that are needed to serve hot dogs at a picnic.”
• “Take a minute to read the problems and think about how you would solve them.”
• 1 minute: quiet think time

• “Now work with your group to solve these problems.”
• 8–10 minutes: group work time
• Monitor for students who:
  • use the term “multiple” in their explanations
  • try different possibilities for the number of packages to buy
  • try to find numbers of packages that would give the exact (or close to the exact) numbers of servings

If students are using guess and check, consider asking: “How are you choosing which numbers to try?” or “How can thinking about multiples help you solve the problem?”

• Invite students to share their responses to the problem about 72 hot dogs, including those who get 70 hot dogs and 72 buns or 80 hot dogs and 80 buns. Encourage students to highlight how they are using multiples in their reasoning.
• “Why are there different answers to this question?” (The numbers do not work out exactly to get 72 hot dogs and 72 buns, so we need to look at different alternatives.)
• “Is it possible to buy exactly the same number of hot dogs and buns? If so, how many of each?” (Yes, 40, 80.)
• “Why is it possible to buy exactly the same number of hot dogs and buns when the packaging is different?” (Some multiples of 10 are also multiples of 8, such as 40 and 80.)