Unit 4, Lesson 14: Fractional Lengths in Triangles and Prisms Lesson Goals Use multiplication and division to solve problems involving fractional areas and lengths in triangles. cubes with fractional edge lengths and finding the number of cubes. Understand that volume of a rectangular prism can be found by packing it with unit Know that the volume of a rectangular prism can also be found by multiplying its edge lengths. Required Materials -inch cubes geometry toolkits 14.1: Area of Triangle (5 minutes) Setup: Students in groups of 2. 2 minutes of quiet work time, followed by 1 minute of partner discussion. Review the formula for the area of a triangle before students begin. Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 1
Student task statement Find the area of Triangle A in square centimeters. Show your reasoning. Possible responses cm 2 Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 2
14.2: Bases and Heights of Triangles (10 minutes) Setup: Students in groups of 2. 4 5 minutes of quiet work time, followed by 2 minutes of partner discussion. Student task statement 1. The area of Triangle B is 8 square units. Find the length of. Show your reasoning. Possible responses 1. 6 units 2. 6 units Anticipated misconceptions 2. The area of Triangle C is square units. What is the length of? Show your reasoning. 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. Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 3
14.3: Volumes of Cubes and Prisms (20 minutes) Setup: A brief discussion of 1-inch and 2-inch cubes. Students in groups of 3 4. Twenty -inch cubes per group. 5 minutes of work time for the first set of questions, followed by a brief whole-class discussion. 7 8 minutes on the remaining questions. Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 4
Student task statement 1. Your teacher will give you a set of cubes with an edge length of inch. Use them to help you answer the following questions. a. Here is a drawing of a cube with an edge length of 1 inch. How many cubes with an edge length of inch are needed to fill this cube? Possible responses 1. a. 8 b. in 3 c. in 3 or in 3 2. a. See lesson plan for completed table. b. Answers vary. Sample explanation: I noticed that the volume is equal to. b. What is the volume, in cubic inches, of a cube with an edge length of inch? Explain or show your reasoning. c. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of inch. Sketch the prism, and find its volume in cubic inches. 2. Use cubes with an edge length of inch to build prisms with the lengths, widths, and heights shown in the table. 3. in 3 Anticipated misconceptions Students may need a reminder to label volume in cubic units. a. For each prism, record in the table how many -inch cubes can be packed into the prism and the volume of the prism. Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 5
prism prism prism number of -inch volume of length (in) width (in) height (in) cubes in prism prism (cu in) 1 1 2 1 2 2 1 4 2 5 4 2 5 4 b. Analyze the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume? 3. What is the volume of a rectangular prism that is inches by inches by 4 inches? Show your reasoning. Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 6
Are you ready for more? A unit fraction has a 1 in the numerator. These are unit fractions:. 1. Find three unit fractions whose sum is. An example is:. These are not unit fractions: Possible Responses Answers vary. Sample response: 1. How many examples like this can you find? 2. Find a box whose surface area in square units equals its volume in cubic units. How many like this can you find? 2. The denominators of the fractions that work for the first question can be used as the length, width, and height of the box. Lesson Synthesis (5 minutes) How do we find an unknown base or height measurement in a triangle when other known measurements involve fractions? How do we find the volume of a rectangular prism with fractional edge lengths? 14.4: Triangles and Cubes (Cool-down, 5 minutes) Setup: None. Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 7
Student task statement 1. A triangle has a base of inches and an area of square inches. Find the height of the triangle. Show your reasoning. 2. Answer each of the following questions and show your reasoning. Possible responses 1., so is and the height is inches. a. How many cubes with an edge length of inch are needed to build a cube with an edge length of 1 inch? b. What is the volume, in cubic inches, of one cube with an edge length of inch? 2. a. 27 b. in 3 Unit 4: Dividing Fractions, Lesson 14: Fractional Lengths in Triangles and Prisms 8