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.

In this activity, students find the volume of a prism with fractional edge lengths. In an earlier grade, students learned that the volume of a rectangular prism with whole-number edge lengths can be found by computing the number of unit cubes that can be packed into the prism. Here, the cubes have a unit fraction (such as or ) for their edge lengths. Students calculate the number of such cubes that can be packed into a rectangular prism and use that number to find the volume in a standard unit of volume measurement (cubic inches, in this case).

By reasoning repeatedly with small cubes (with -inch edge lengths), students notice regularity (MP8): that the volume of a rectangular prism with fractional edge lengths can also be found by directly multiplying the edge lengths in inches.

This activity works best when each student has access to physical cubes because students will benefit from manipulating the cubes and building prisms by hand.

If physical manipulatives are not available, consider using the digital version of the activity. In the digital version, students use an applet to build rectangular prisms from cubes with edge lengths of unit. The applet allows students to measure the edge lengths of the resulting prisms.

The digital version is adapted from an applet made in GeoGebra by Susan Addington.

The purpose of the discussion is to highlight that the volume of a rectangular prism can be found by:

• Packing it with cubes with no gaps and multiplying the number of cubes by the volume of each cube.
• Multiplying the edge lengths of the prism.

Display a completed table for all to see. Give students a minute to check their responses. Invite a few students to share how they determined the volume of the prisms.

If no students mention using the number and volume of the -inch cubes, ask them to observe the relationship between the values in the last two columns of the table (the number of cubes with -inch edge lengths and the volume of prisms in cubic inches). Highlight that the volume can be found by calculating the number of cubes and multiplying it by , because each small cube has a volume of cubic inch.

If no students mention finding the product of the edge lengths in inches, draw students’ attention to the values in the first three columns of the table and in the last column. (Looking at the whole-number values in the second, fourth, and sixth rows of the table may be particularly helpful.) Make sure students see that the volume of each prism can also be found by multiplying its side lengths in inches.