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.

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.

In this activity, students apply their knowledge of division of fractions to find a unknown length given the area of a triangle and a fractional base or height.

The formula for the area of a triangle presents a different multiplication situation than students have seen in this unit—there are three factors at play. This means that to find a unknown factor, at least one additional step is needed. Students are likely to approach this in a number of ways. They may: 

• Draw a duplicate of the triangle and compose a parallelogram with the same base and height, double the given area (to represent the area of the parallelogram), and then divide by the known length to find the unknown length.
• Without drawing, multiply the given area by 2 to find the value of , and then divide it by the known length.
• Perform division twice, such as divide the area by and then by the known base or length, or vice versa.
• Multiply the known length and first, so there are only two factors to work with. (For example, to find in , they may write and then .)

Monitor for these or other approaches as students work. Select students who use different strategies, and ask them to share later.

Ask 2–3 previously selected students to share their solutions and reasoning. Consider starting with students who reasoned concretely (such as by duplicating the triangle to compose a parallelogram) and following with those who reasoned symbolically (such as only by manipulating expressions or equations).

Clarify that there is no single way to solve problems such as these. Point out specific points in the solving process where division of fractions enabled them to complete their reasoning. Emphasize that what we learned about fractions and operations in this unit can help us reason more effectively about problems in other areas of mathematics.