# Lesson 14

Fractional Lengths in Triangles and Prisms

### Lesson Narrative

In this transitional lesson, students conclude their work with area and begin to explore volume of rectangular prisms. First, they extend their work on area to include triangles, using division to find the length of a base or a height in a triangle when the area is known. Second, they undertake a key activity for extending their understanding of how to find the volume of a prism.

In previous grades, students learned that the volume of a prism with whole-number edge lengths is the product of the edge lengths. Now they consider the volume of a prism with dimensions \(1 \frac 12\) inch by 2 inches by \(2 \frac 12\) inches. They picture it as being packed with cubes whose edge length is \(\frac 12\) inch, making it a prism that is 3 cubes by 4 cubes by 5 cubes, for a total of 60 cubes, because \(3 \boldcdot 4 \boldcdot 5 = 60\). At the same time, they see that each of these \(\frac12\)-inch cubes has a volume of \(\frac18\) cubic inches, because we can fit 8 of them into a unit cube. They conclude that the volume of the prism is \(60 \boldcdot \frac18 = 7 \frac12\) cubic inches.

In the next lesson, by repeating this reasoning and generalizing (MP8), students see that the volume of a rectangular prism with fractional edge lengths can also be found by multiplying its edge lengths directly (e.g., \(\left(1 \frac12 \right) \boldcdot 2 \boldcdot \left( 2\frac12 \right) = 7 \frac12\)).

### Learning Goals

Teacher Facing

- Apply dividing by fractions to calculate the base or height of a triangle, given its area and the other measurement.
- Determine the volume of a rectangular prism by counting how many 12-inch or 13-inch cubes it takes to build, and explain (orally and in writing) the solution method.
- Generalize that the volume of a rectangular prism with fractional edge lengths can be found by multiplying the edge lengths.

### Student Facing

Let’s explore area and volume when fractions are involved.

### Required Materials

### Required Preparation

For the Volumes of Cubes and Prisms activity, prepare 20 half-inch cubes for every group of 3–4 students. Wooden ones are available inexpensively at craft stores. If you have access to centimeter cubes, you could use those instead. Tell students that we will consider them half-inch cubes for the purposes of that activity.

### Learning Targets

### Student Facing

- I can explain how to find the volume of a rectangular prism using cubes that have a unit fraction as their edge length.
- I can use division and multiplication to solve problems involving areas of triangles with fractional bases and heights.
- I know how to find the volume of a rectangular prism even when the edge lengths are not whole numbers.

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