Standard #: MA.912.GR.4.5

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.T.1.2

 

Terms from the K-12 Glossary

• Cone 
• Cylinder
• Prism 
• Pyramid 
• Sphere

 

Vertical Alignment

Previous Benchmarks

• MA.6.GR.2.3 
• MA.7.GR.2.3 
• MA.912.AR.2.1

Next Benchmarks

• MA.912.C.5.7

 

Purpose and Instructional Strategies

In middle grades, students determined the volume of right rectangular prisms and right cylinders. In Geometry, students explore for the first time the volume of pyramids, cones, and spheres. In later courses, student learn more advanced methods for calculating volume. 

• Instruction includes reviewing units and conversions within and across different measurement systems (as this was done in middle grades). 
• Instruction includes discussing the convenience of answering with exact values (e.g., the simplest radical form or in terms of pi) or with approximations (e.g., rounding to the 22 nearest tenth or hundredth or using 3.14, $\frac{\text{22}}{\text{7}}$ or other approximations for pi). It is also important to explore the consequences of rounding partial answers on the accuracy or precision of the final answer, especially when working in real-world contexts. 
• Instruction includes reviewing the definition of cylinders, pyramids, prisms, cones and spheres (as this was done in grade 5), and discussing the definitions of right and oblique polyhedrons, cubes, tetrahedrons, regular prisms and regular pyramids. 
• The population or material density based on volume is calculated by the quotient of the total population or material and the volume (i.e., population density of fish in a spherical aquarium or density of salt in a bucket of water). Have students practice finding the population or material density or the total population or material amount, given the dimensions of a three-dimensional figure. That is, part of their work includes finding the volume based on the dimensions. (MTR.7.1) 
• Instruction includes the connection to two-dimensional cross-sections of three-dimensional figures to explore Cavalieri’s Principle, which states that if in two solids of equal height, the cross-sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal. (MTR.5.1) 
  • For example, have students compare the volume of two stacks of pennies of the same height, one organized in a straight column and the other one, one penny on top of the other, but in a slanted stack. Discuss the shape of their cross-sections at the same height and what happens with their volumes. 
  • For example, have students discuss how this principle is applied in the calculation of volumes of non-right (oblique) three-dimensional figures. 
  • For example, have students discuss how this principle can be used to find the volume of a non-right cylinder given a right cylinder with the same height and same cross-sections. (MTR.4.1) 
• Instruction includes exploring a variety of real-world situations where finding the volume or volume density is relevant for different purposes. Problem types include components like percentages, cost and budget, constraints, comparisons, BTUs, nutrition (e.g., calories per cup), moisture content (e.g., ounces of water in a gallon of honey) or others. 
• Problem types include finding missing dimensions given the volume of a three-dimensional figure or finding the volume of composite figures.

 

Common Misconceptions or Errors

• Students may not be careful with units of measurement involving volume, particularly when converting from one unit to another. 
  • For example, since there are approximately 25.4 millimeters in an inch, a student may incorrectly conclude that there are 25.4 cubic millimeters in a cubic inch.

 

Instructional Tasks

Instructional Task 1 (MTR.7.1) 

• When filling cylindrical silos, the top cone is not filled. However, if the silo has a bottom cone, it is filled. Three different silos are shown in the image below. 

• Part A. In silo 3, the top and bottom cones are congruent. How much more grain could silo 3 hold than silo 1? 
• Part B. The diameter of silo 1 is 80% the diameter of silo 2. Is the capacity of silo 1 80% the capacity of silo 2? 

Instructional Task 2 (MTR.4.1) 

• The radius of a sphere is 4 units so its volume is $\frac{\text{256}}{\text{3}}$π  cubic units. 
  • Part A. Discuss the value of this kind of answer for its accuracy and precision. 
  • Part B. Discuss the effect of replacing π in the formulas with 3.14, 3.1416, $\frac{\text{22}}{\text{7}}$ and other approximations. What happens with the answer, the volume of the figure, in each case?

 

Instructional Items

Instructional Item 1 

• Joshua is going to create a garden border around three sides of his backyard deck using cinder blocks. He is going to plant a flower in each hole of the cinder block. The dimensions of the cinder blocks are 8 inches by 16 inches by 8 inches. Each hole needs to be completely filled with potting soil before the flowers can be planted. Potting soil is sold in 1 cubic foot bags. 

•  Part A. What are the dimensions of a cinder block hole? 
•  Part B. The patio is a square with a side length of 8 feet. One of the sides of the square patio is adjacent to an exterior wall of the house. If Joshua puts blocks around the other three sides of the patio, how many bags will Joshua need to purchase to fill the blocks?

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.