Lesson 14

This warm-up allows students to review calculation of triangular area and prepares them to use division of fractions to solve area problems involving triangles later. Students calculate the area of a triangle given fractional base and height measurements.

Arrange students in groups of 2. Give students 2 minutes of quiet work time, followed by 1 minute of partner discussion. Before students begin, review the formula for the area of a triangle. Consider displaying a drawing of a triangle with one side labeled as a base and a corresponding height shown and labeled as such. 

Ask a student to share their solution and reasoning. Record it for all to see. Poll the class to see whether students agree or disagree. Ask if others had alternative ways of reasoning or calculating.

Tell students that they will solve more problems involving the area of triangles in this lesson.

In this activity, students apply their knowledge of division of fractions to answer questions about bases and heights of triangles, which they studied earlier in the year. In the warm-up, students recalled how to find the area of a triangle given a pair of base and height. Here, they find a missing length given the area of a triangle and a fractional base or height.

The formula for the area of a triangle \(A = \frac12 \boldcdot b \boldcdot h\) presents a different multiplication situation than students have seen in this unit—there are three factors at play. Students are likely to approach 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 \(b \boldcdot h\), and then divide it by the known length
• Perform division twice, i.e., dividing the area by \(\frac12\) and then by the known base or length, or vice versa
• Multiply the known length and \(\frac12\) first, so there are only two factors to work with (e.g., for \(\frac12 \boldcdot b \boldcdot \frac83 = 8\), they may apply the commutative and associative properties of operations and write \(\frac12 \boldcdot \frac83 \boldcdot b = 8\) and then \(\frac43 \boldcdot b = 8\))

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

Keep students in groups of 2. Give students 4–5 minutes of quiet work time and 2 minutes to discuss their responses and complete the activity with their partner. Keep the formula for the area of a triangle and a labeled drawing of a triangle displayed.

Some students may not know how to find a missing factor when three—instead of two—factors are involved. Ask if there is any step to take or any calculation to make first so they are only working with two factors.

Ask previously selected students to share their solutions and reasoning. Sequence their presentations so that students who reasoned concretely (e.g., by duplicating the triangle to compose a parallelogram) present first and those who reasoned symbolically (e.g., only by manipulating expressions or equations) present last. After each student shares, ask all students to indicate whether they reasoned the same way.

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.

This activity extends students’ understanding about the volume of rectangular prisms from earlier grades. Previously, 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, they draw on the same idea to find the volume of a prism with fractional edge lengths. The edge length of the cubes used as units of measurement are not 1 unit long, however. Instead, they have a unit fraction (\(\frac 12\), \(\frac 14\), etc.) for their edge length. Students calculate the number of these smaller cubes in a prism and use it to find the volume in a standard unit of volume measurement (cubic inches, in this case).

By reasoning repeatedly with small cubes (with \(\frac12\)-inch edge lengths), students notice that the volume of a rectangular prism with fractional edge lengths can also be found by directly multiplying the edge lengths in inches.

 Students may need a reminder to label volume in cubic units.  

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, and their observations about the relationships between the number of cubes (with \(\frac12\)-inch edge lengths) and the volume of prism in cubic inches. Highlight that the volume can be found by calculating the number of cubes and multiplying it by \(\frac 18\), because each small cube has a volume of \(\frac 18\) cubic inch. 

If students do not notice a pattern in their table, ask them what they notice about the edge lengths of the prisms in the second, fourth, and sixth rows of the table and their corresponding volumes. Make sure they see that the volume of each prism can also be found by multiplying its side lengths. In other words, we can find the volume of a rectangular prism with fractional edge lengths the same way we find that of a prism with whole-number edge lengths.