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 “¿Esto tiene sentido?” // “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.

• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

• “¿Qué partes de la foto usaron para hacer su estimación?” // “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.)
• “¿Qué estimaciones serían poco razonables?” // “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
• “Con su pareja, escojan un tipo de mesa que permita sentar a diferentes grupos de estudiantes” // “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.
• “¿La clase de Noah se puede acomodar exactamente en mesas de 6 sin que sobren puestos? ¿Cómo lo saben?” //“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.)
• “¿La clase de Noah se puede acomodar exactamente en mesas de 8 sin que sobren puestos? ¿Cómo lo saben?” // “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.)
• “¿Cómo nos ayudan los hechos de multiplicación o la idea de los múltiplos a pensar en este problema?” // “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.)
• “En el problema de ambas clases y los profesores, ¿podrían acomodarse todos en mesas de 6? ¿Por qué sí o por qué no?” //“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.)
• “¿Podrían acomodarse todos en mesas de 8? ¿Por qué sí o por qué no?” // “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.
• “¿Qué preguntas pueden hacer sobre la situación?” // “What questions could you ask about the situation?”
• 1 minute: quiet think time
• Share responses.
• “Resolvamos problemas sobre salchichas y panes de perros calientes que vienen en paquetes de tamaños diferentes” // “Let's solve problems about hot dogs and hot dog buns, which come in different-size packages.”
• “En cada problema, busquen posibles números de paquetes que se necesiten para servir perros calientes en un pícnic” // “For each problem, look for possible numbers of packages that are needed to serve hot dogs at a picnic.”
• “Dediquen un minuto a leer los problemas y a pensar en cómo los van a resolver” // “Take a minute to read the problems and think about how you would solve them.”
• 1 minute: quiet think time

• “Ahora trabajen con su grupo para resolver estos problemas” // “Now work with your group to solve these problems.”
• 8–10 minutes: group work time
• Monitor for students who:
  • use the term “múltiplo” // “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: “¿Cómo estás escogiendo los números que ensayas?” // “How are you choosing which numbers to try?” or “¿Cómo te puede ayudar pensar en múltiplos a resolver el problema?” // “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.
• “¿Por qué hay diferentes respuestas a esta pregunta?” // “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.)
• “¿Es posible comprar exactamente el mismo número de salchichas y de panes? Si sí, ¿cuántos de cada uno?” // “Is it possible to buy exactly the same number of hot dogs and buns? If so, how many of each?” (Yes, 40, 80.)
• “¿Por qué es posible comprar exactamente el mismo número de salchichas y de panes aunque las cantidades en los paquetes sean diferentes?” // “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.)