The purpose of this activity is for students to multiply a whole number by a fraction greater than 1 in a way that makes sense to them.

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

• Explain why the area of the shaded region is or by counting the number of shaded whole square units and then counting the number of shaded third of a square units.
• Explain how to use the expression to find the area of the shaded region.
• Explain how to use the expression to find the area of the shaded region.

The approaches are sequenced from more concrete to more abstract to support students with applying what they know about equivalent fractions to multiplying a mixed number and a whole number. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For examples for the approaches, look at the Student Responses.

As students work with fractions greater than 1, they may choose to write or rewrite them as mixed numbers. Students may also relate the expressions to the diagrams in different ways. Encourage them to interpret the diagram and find the area of the shaded region in whatever way makes sense to them. During the Activity Synthesis, show the relationships between the different ways of finding and representing the area. Identifying which expressions represent the area of the rectangle requires careful analysis of the expressions and the figure and the correspondences between them (MP7).

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • "How did each way of finding the area use whole numbers? How did each way use fractions?" (I notice that and  both represent the area of the shaded region. You can count the number of shaded whole square units to get 16 and then count the number of shaded third of a square units to get . If you group the into wholes, you have for a total of , which is .)
  • “How does represent the area of the shaded region?” (There are 56 pieces shaded in and each piece has an area of of a unit square.)
• Connect students’ approaches to the learning goal by showing the equivalent expressions and asking:
  • Display: 
  • “How do we know these expressions have the same value?” (They both represent the shaded area in the diagram. I know that 3, 6, 9, 12 thirds is 4 wholes and then there are two thirds left.)

The purpose of this activity is for students to find areas of rectangles where one side is a whole number and the other side is a fraction that is greater than 1.  Students should solve the problems in a way that makes sense to them. Ask students to explain how the diagrams show the multiplication expressions.

• Ask several students to share their responses to the second problem.
• Display:
• “How do these expressions represent the area of the shaded region?” (They are the width and length of the shaded region. In one expression the length is a whole number and some fourths, and in the other, it is all fourths.)
• “How is finding the value of these two expressions different?” (For the first one, I can multiply 3 by 3 and then 3 by and add them together. For the second one, I can see how many fourths I have using multiplication.)