The purpose of this activity is to review and illustrate the idea that the area of a rectangle with fractional side lengths can be found by multiplying the two side lengths, just as in the case of whole numbers.

Students are to justify that the area of a rectangle that is inches by inches is square inches. As they make their case and listen to others’ cases, students practice constructing a logical argument and critiquing the reasoning of others (MP3).

Monitor for students who use different approaches to reason about the area of the rectangle in the last question. Here are some likely approaches, from less direct to more direct:

• Find the number of -inch squares in the rectangle, and multiply it by , which is the area of one -inch square.
• Tile the rectangle with 1-inch squares and parts of 1-inch squares, find the area of each part, and add the areas.
• Decompose each side length into a whole number and a fraction (3 and on one side, 2 and on the other), partition the rectangle accordingly, find the area of each part, and add the areas.
• Multiply the side lengths of the rectangle.

The goal of the discussion is to make explicit that even when the side lengths of a rectangle are not whole numbers, its area is still the product of the two numbers.

Invite previously selected students to share their responses to the last question. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

If no students reasoned about the area by decomposing the rectangle and finding the sum of partial areas, display one or both diagrams that are shown in the Student Response. Ask students to explain how each diagram allows us to find the area. If no students calculated the product of and , ask students to find that value and compare it to the area found using their way.

Connect the different responses to the learning goals by asking questions such as:

• “Which strategies are more elaborate or involve more steps? Which ones are quicker or more straightforward?”
• “What are some benefits of reasoning about the areas of smaller squares or rectangles in the -by- rectangle? What are some disadvantages?”
• “What are some benefits of multiplying the side lengths? What are some disadvantages?”

This activity gives students another opportunity to reason about multiplication and the area of a rectangle with fractional side lengths. Specifically, it helps students see how the product of two mixed numbers (or two fractions that are greater than 1) can be found by decomposing the factors into whole numbers and fractions and finding partial products.

Students are first presented with multiplication expressions and diagrams of rectangles with a shaded region. They match each diagram with an expression that can represent the area of the rectangle. Then students use their understanding of the connections across representations to show that the value of one expression they matched, , is .

Because the diagrams are unlabeled, students need to look for and make use of the structure in the expressions and diagrams to make a match (MP7). To justify that a multiplication has a certain value, they need to construct a logical argument (MP3).

Display the expressions and diagrams for all to see. Invite one or more students to match each expression to a rectangle whose area the expression can represent. Ask students to explain their reasoning. If possible, record students’ reasoning on or near the representation referred to.

To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Does anyone want to add on to ’s reasoning?”
• “Do you agree or disagree? Why?”

Next, invite other students to share how they used Diagram B to show that , starting with how they knew the side lengths of each sub-rectangle. As students explain, label each sub-rectangle with its side lengths and its area.

If not articulated by students, highlight that combining all the partial areas gives us a sum of , which is the area of the entire rectangle.

In this activity, students determine how many tiles with fractional side lengths are needed to completely cover another rectangular region that also has fractional side lengths. To do so, students need to apply their understanding of the area of rectangles and of division of fractions. They also need to plan their approach, think about how the orientation of the tiles affects their calculation and solution, and attend carefully to the different measurements and steps in their calculation. The task engages students in aspects of mathematical modeling (MP4) and prompts them to attend to precision (MP6).

As students work, monitor for those whose diagrams or solutions show different tile orientations.

Students might find only the number of tiles needed to line the four sides of the tray. Suggest that they refer to their drawing of the tray and check whether their calculations include tiles that cover the entire tray.

Students might try to reason about a division such as by drawing a diagram or another likely time-consuming way (given the numbers involved). Remind them, as needed, that they have one or more efficient strategies at their disposal.

Invite students who chose different tile orientations to show their diagrams and explain their reasoning. Display the two diagrams shown in the Student Response, if needed.

If time permits, point out how in this problem, the two different tile orientations do not matter because the length and the width of the tiles are factors of both the length and the width of the tray. This means that we can fit a whole number of tiles in either direction, and we can use the same number of tiles to cover the tray regardless of orientation.

But if the side lengths of the tiles do not both fit into and evenly, then the orientation of the tiles does matter. In that case, we may need more or fewer tiles, or we may not be able to tile the entire tray without gaps if the tiles are oriented a certain way.

Use the opportunity to point out that a diagram does not have to show all the details (such as every single tile) to be useful.