Examples of Functions from Geometry

Examples, videos, and solutions to help Grade 8 students learn how to write rules to express functions related to geometry

New York State Common Core Math Grade 8, Module 5, Lesson 9

Worksheets for Grade 8

Lesson 9 Student Outcomes

• Students write rules to express functions related to geometry.
  
• Students review what they know about volume with respect to rectangular prisms and further develop their conceptual understanding of volume by comparing the liquid contained within a solid to the volume of a standard rectangular prism (i.e., a prism with base area equal to one).

Lesson 9 Summary

Rules can be written to describe functions by observing patterns and then generalizing those patterns using symbolic notation. There are a few basic assumptions that are made when working with volume: (a) The volume of a solid is always a number ≥ 0. (b) The volume of a unit cube (i.e., a rectangular prism whose edges all have length 1) is by definition 1 cubic unit. (c) If two solids are identical, then their volumes are equal. (d) If two solids have (at most) their boundaries in common, then their total volume can be calculated by adding the individual volumes together. (These figures are sometimes referred to as composite solids.)

Lesson 9 Classwork

1. • 3. Use the figure below to answer parts (a)–(f).
        a. What is the length of one side of the smaller, inner square?
        b. What is the area of the smaller, inner square?
        c. What is the length of one side of the larger, outer square?
        d. What is the area of the area of the larger, outer square?
        e. Use your answers in parts (b) and (d) to determine the area of the 1-inch white border of the figure.
        f. Explain your strategy for finding the area of the white border.
        
        
2. Write a function that would allow you to calculate the area of a 1-inch white border for any sized square picture measured in inches.
   a. What is the length of one side of the smaller, inner square?
   b. What is the area of the smaller, inner square?
   c. What is the length of one side of the larger, outer square?
   d. What is the area of the area of the larger, outer square?
   e. Use your answers in parts (b) and (d) to determine the area of the 1-inch white border of the figure.
   f. Explain your strategy for finding the area of the white border.
   

5. The volume of the prism shown below is 61.6 in³. What is the height of the prism?
   
   
6. Find the value of the ratio that compares the volume of the larger prism to the smaller prism.
   
   As you complete Exercises 7–10, record the information in the table below.
   
7. • 9. Use the figure below to answer parts (a)–(c).
        a. What is the area of the base?
        b. What is the height of the figure?
        c. What is the volume of the figure?
        
        
8. Use the figure to the right to answer parts (a)–(c).
   a. What is the area of the base?
   b. What is the height of the figure?
   c. Write and describe a function that will allow you to determine
   the volume of any rectangular prism that has a base area of 36 cm²

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