Lesson 3

The purpose of this Number Talk is to elicit strategies and understandings students have for finding a unknown factor and relating multiplication and division. These understandings help students develop fluency and will be helpful later in this lesson when students represent and solve multiplicative comparison problems with unknown factors.

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿Tiene sentido que \(40 \div 8\) esté en esta conversación numérica? ¿Por qué sí o por qué no?” // “Does \(40 \div 8\) belong in this number talk? Why or why not?” (It does belong because division is like finding an unknown factor.)

In this activity, students are provided a discrete tape diagram to represent the first problem in which the multiplier (the quantity indicating \(n\) times as many) is unknown. They also rely on what they know about the relationship between multiplication and division to represent and solve each problem.

When students create their representations for the books, whether a diagram or an equation, they reason abstractly and quantitatively (MP2).

• Groups of 2
• Give students access to connecting cubes.
• “¿Qué saben sobre una colecta de libros? ¿Han participado en una coleta de libros o en un intercambio de libros?” // “What do you know about a book drive? Have you participated in a book drive or book exchange?”
• “Algunas veces, en las escuelas se organiza una colecta de libros en la que los estudiantes donan libros que ya leyeron o que ya no necesitan para que otros puedan disfrutarlos” // “Sometimes schools organize a book drive where students donate books they are done reading or no longer need so others can enjoy them.”

• 3 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who represent Noah and Priya's books with:
  • 3 books at a time for Noah and stop at 21
  • all of Noah’s books and circle groups of 3
  • a tape diagram
  • a multiplication equation with a symbol for unknown
  • a division equation

Students may lose track of the group being multiplied and the factor serving as the multiplier and begin to represent, for example, 16 times as many as 4 books. Consider asking:

• “¿Qué se está comparando en este problema?” // “What is being compared in this problem?”
• “¿Cuántas veces esa cantidad de libros describe este problema?” // “How many times as many books are being described in this problem?”

• Invite selected students to share their representation of Priya and Noah's books in the order described above.
• If no students created an equation, ask, “¿Cómo podríamos representar la cantidad de libros de Priya y la cantidad de libros de Noah con una ecuación que tenga un símbolo para representar el valor desconocido?” //  “How could we represent Priya and Noah’s books with an equation with a symbol for the unknown?” (\({?} \times 3 = 21\) or \(7 \times {?} = 21\))
• “¿Cómo usaron la multiplicación para resolver el problema?” // “How did you use multiplication to solve the problem?” (I knew \(7 \times 3 = 21\). So, Noah has 7 times as much.)
• “¿Cómo usaron la división para razonar sobre el problema?” // “How did you use division to reason about the problem?” (I knew 21 divided by 3 is 7.)

The purpose of this activity is for students to make sense of and represent multiplicative comparison problems in which a factor is unknown. Students use the relationship between multiplication and division to write equations to represent multiplicative comparisons. These problems have larger numbers than in previous lessons in order to elicit the need for using more abstract diagrams, which are the focus of upcoming lessons.

When students analyze Han's and Tyler's claims they construct viable arguments (MP3).

• Groups of 2
• Give students access to connecting cubes.
• “Tómense un momento para leer el primer problema en silencio” //  “Take a moment to read the first problem to yourself.”
• “Explíquenle a un compañero qué les piden que hagan o que encuentren” //  “Explain to a partner what you are asked to do or find out.”

• 7 minutes: independent work time
• Monitor for students who:
  • draw a tape diagram with 48 rectangles and then guess and check by circling equal groups until finding an amount with no leftover.
  • attempt to draw out all books and revise to create a more abstract diagram.
  • write multiplication and division equations with a symbol for the unknown value.
• 2 minutes: partner discussion

If students write equations that represent a different situation than presented, consider asking:

• “¿Cómo muestra tu ecuación los libros de Clare (o los de Andre)?” // “How does your equation show Clare’s (or Andre’s) books?”
• “En cada caso, ¿puede una ecuación de división representar la situación?” // “Can a division equation represent each situation?”

• Focus the discussion on why Han is correct that we can use division. (Han is correct because when a factor is missing, we can use division to find out what was being multiplied.)
• “¿Por qué es posible usar una ecuación de multiplicación o una ecuación de división para representar la misma situación?” // “Why is it possible to use a multiplication or a division equation to represent the same situation?” (Because a multiplication equation with an unknown factor is the same as a division equation with an unknown quotient.)