The purpose of this activity is for students to articulate the relationship between multiplication and division explaining how to solve two different problems using multiplication or division. Students have observed that dividing a whole number by a unit fraction gives the same result as multiplying the whole number by the denominator. They have also observed that dividing a unit fraction by a whole number gives the same result as multiplying a fraction by a unit fraction that has a whole number as the denominator. They also know from prior units and courses that the operations of multiplication and division are closely related. This activity brings these two ideas together, making explicit how one situation and one diagram modeling the situation can be interpreted using either multiplication or division (MP2).

• Invite students to share their diagrams and equations for the first problem.
• “How does the diagram help you solve the problem?” (The number of paint trays is the number of small rectangles or s.)
• “How does the diagram let you interpret the solution using multiplication?” (There are 8 paint trays because .)
• “How does the diagram let you interpret the solution using division?” (It shows how many s there are in 2, 8.)

The purpose of this activity is for students to make sense of situations, make an appropriate representation in the form of multiplication or division equations, and use that representation to answer questions about the situation (MP2). Students who have a strong understanding of the relationship between division and multiplication can reason about the situations using either operation. In some cases a representation as division goes beyond grade level which only addresses division of a whole number and a unit fraction. When partners share their work and discuss any disagreements they critique each other's reasoning (MP3).

• “How did you know what operation you needed to perform to find the answer?” (I drew a diagram to help me visualize the situation.)
• “For which problems was it difficult to tell what operation to use?” (I wasn’t sure what to do with the strawberry problem.)
• For students who used a diagram, “How did drawing the diagram help you write an equation?” (It helped me see what operation could be used to solve the problem and see the result.)
• Point out equations that correctly represent the same problem (and are equivalent) but are expressed differently by displaying the problem about water and the equation: .
• “Many of you wrote a division equation to represent this problem. What multiplication equation can represent this problem?” ( )