The purpose of this activity is for students to interpret and represent multiplicative comparison situations in which the multiplier (the factor indicating times as many)  is unknown. Students 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).

Monitor for and select to share in the Activity Synthesis students with the following approaches:

• Draw 3 books at a time for Noah, and stop at 21.
• Draw all of Noah’s books, and circle groups of 3.
• Draw a discrete tape diagram, such as in the Student Responses.
• Write a multiplication equation, with a  symbol for the unknown, .
• Write a division equation: .

The approaches are sequenced from more concrete to more abstract to connect different ways students may represent multiplicative comparison situations. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.

• Invite previously selected students to share their representations of Priya's and Noah’s books in the given order. Record or display their work for all to see.
• 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's and Noah’s books with an equation, with a symbol for the unknown?” ( or )
• Connect students’ approaches by asking:
  • “¿En qué se parecen y en qué son diferentes estas formas de representar el problema?” // “How are these ways of representing the problem the same, and how are they different?” (They all showed 21, with groups of 3. In all of them, we reasoned about how many times the number of books. They are different because some ways show drawings or diagrams, and others have equations with multiplication or division.
• Connect students’ approaches to the learning goal by asking:
  • “Para pensar en cómo resolver el problema, ¿cómo se usó la multiplicación en algunas representaciones?” // “How did some ways use multiplication to think about how to solve the problem?” (Some drawings or diagrams kept including more and more groups of 3 to get to 21.That’s like the equation .)
  • “¿Cómo se usó la división en otras representaciones?” // “How did some ways use division?” (Some drawings partitioned 21 into groups of 3. That’s like 21 divided by 3.)
  • “¿Cuál representación fue la más útil para apoyar sus ideas?” // “Which way was the most helpful to support your thinking?” Sample response: I drew a diagram, but I like the equations because 21 squares was a lot to draw.) 

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 greater 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).

Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from starting with more accessible questions.

For example, ask: “¿Quién donó más libros?” // “Who donates more books?”
Supports accessibility for: Conceptual Processing, Organization

• 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 equation 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.)