Word (.docx)
Acrobat (.pdf)
Examples
Example: A cylindrical swimming pool is filled with water and has a diameter of 10 feet and height of 4 feet. If water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank to the nearest pound?Clarifications
Clarification 1: Instruction includes concepts of density based on volume.Clarification 2: Instruction includes using Cavalieri’s Principle to give informal arguments about the formulas for the volumes of right and non-right cylinders, pyramids, prisms and cones.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Cone
- Cylinder
- Prism
- Pyramid
- Sphere
Vertical Alignment
Previous Benchmarks
Next Benchmarks
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, 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 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, 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?
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Experts
Perspectives Video: Professional/Enthusiasts
Problem-Solving Tasks
Tutorial
Unit/Lesson Sequence
STEM Lessons - Model Eliciting Activity
Students use geometry formulas to solve a fruit growing company's dilemma of packing fruit into crates of varying dimensions. Students calculate the volume of the crates and the volume of the given fruit when given certain numerical facts about the fruit and the crates.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
In this activity, students will utilize measurement data provided in a chart to calculate areas, volumes, and densities of cookies. They will then analyze their data and determine how these values can be used to market a fictitious brand of chocolate chip cookie. Finally, they will integrate cost and taste into their analyses and generate a marketing campaign for a cookie brand of their choosing based upon a set sample data which has been provided to them.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
Students are asked to solve a problem that requires calculating the volume of a sphere.
Students are asked to write the formula for the volume of a cone, explain what each variable represents, and label the variables on a diagram.
Students are asked to write the formula for the volume of a cylinder, explain what each variable represents, and label the variables on a diagram.
Students are asked to solve a problem that requires calculating the volumes of a sphere and a cylinder.
Students are asked to model a tree trunk with geometric solids and to use the model to estimate the volume of the tree trunk.
Students are asked to find the height of a square pyramid given the length of a base edge and its volume.
Students are asked to write the formula for the volume of a pyramid, explain what each variable represents, and label the variables on a diagram.
Students are asked to solve a problem that requires calculating the volumes of a cone and a cylinder.
Students are asked to write the formula for the volume of a sphere, explain what each variable represents, and label the variables on a diagram.
Students are asked to solve a problem that requires calculating the volume of a large cylindrical sports drink container and comparing it to the combined volumes of 24 individual containers.
Students are asked to find the height of the Great Pyramid of Giza given its volume and the length of the edge of its square base.
Students are asked to derive and explain a formula for the volume of a cone given a pyramid with the same height and the same cross-sectional area at every height.
Students are asked to derive and explain a formula for the volume of a cylinder given a prism with the same height and the same cross-sectional area at every height.
Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.
Original Student Tutorials Mathematics - Grades 6-8
Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.
Original Student Tutorials Mathematics - Grades 9-12
Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.
Student Resources
Original Student Tutorials
Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.
Type: Original Student Tutorial
Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.
Type: Original Student Tutorial
Perspectives Video: Professional/Enthusiast
You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!
Type: Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.
Type: Problem-Solving Task
This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.
Type: Problem-Solving Task
The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.
Type: Problem-Solving Task
The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).
Type: Problem-Solving Task
This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.
Type: Problem-Solving Task
This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder
Type: Problem-Solving Task
In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.
Type: Problem-Solving Task
Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.
Type: Problem-Solving Task
The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.
Type: Problem-Solving Task
Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.
Type: Problem-Solving Task
Tutorial
This video will show to find the volume of a triangular prism, and a cube by applying the formula for volume.
Type: Tutorial
Parent Resources
Perspectives Video: Professional/Enthusiast
You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!
Type: Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.
Type: Problem-Solving Task
This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.
Type: Problem-Solving Task
The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.
Type: Problem-Solving Task
The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).
Type: Problem-Solving Task
This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.
Type: Problem-Solving Task
This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder
Type: Problem-Solving Task
In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.
Type: Problem-Solving Task
Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.
Type: Problem-Solving Task
The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.
Type: Problem-Solving Task
Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.
Type: Problem-Solving Task
