Standard #: MAFS.8.G.3.9 (Archived Standard)


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Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.


Remarks


Fluency Expectations or Examples of Culminating Standards

When students learn to solve problems involving volumes of cones, cylinders, and spheres — together with their previous grade 7 work in angle measure, area, surface area and volume (7.G.2.4–2.6) — they will have acquired a well-developed set of geometric measurement skills. These skills, along with proportional reasoning (7.RP) and multistep numerical problem solving (7.EE.2.3), can be combined and used in flexible ways as part of modeling during high school — not to mention after high school for college and careers.

General Information

Subject Area: Mathematics
Grade: 8
Domain-Subdomain: Geometry
Cluster: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. (Additional Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes

Test Item Specifications

    N/A

    Assessment Limits :
    Graphics of three-dimensional figures can be included. Dimensions must be given as rational numbers. Figures must not be composite
    Calculator :

    Yes

    Context :

    Allowable



Sample Test Items (4)

Test Item # Question Difficulty Type
Sample Item 1

A cylinder with a height of begin mathsize 12px style 6 1 half end style inches (in.) and a diameter of 5 inches is shown.

What is the volume of the cylinder, in cubic inches? (Use 3.14 for begin mathsize 12px style straight pi end style)

N/A EE: Equation Editor
Sample Item 2

The diameter of a sphere is 4 inches. 

What is the volume of the sphere, in cubic inches? (Use 3.14 for ??.)

N/A EE: Equation Editor
Sample Item 3

A cone has a height of 6.4 inches and a diameter of 6 inches.

What is the volume, in cubic inches, of the cone? Use 3.14 for begin mathsize 12px style straight pi end style

N/A EE: Equation Editor
Sample Item 4

A water container in the shape of a cone has a height of 5 inches and a diameter of 3.5 inches.

Enter an equation in the first blank box and a number in the second blank box to complete the statements about the water container.

A. The formula to calculate the container's volume, V, with the given measurements is ________.

B. The container can hold approximately _____ cubic inches of water. Round your answer to the nearest hundredth.

 

N/A EE: Equation Editor


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2002100: M/J Comprehensive Science 3 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002110: M/J Comprehensive Science 3, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003010: M/J Physical Science (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003020: M/J Physical Science, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
7820017: Access M/J Comprehensive Science 3 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2023, 2023 and beyond (current))
2002085: M/J Comprehensive Science 2 Accelerated Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))


Related Resources

Formative Assessments

Name Description
Sugar Cone

Students are asked to solve a problem that requires calculating the volume of a cone.

Platinum Cylinder

Students are asked to solve a problem that requires calculating the volume of a cylinder.

Louvre Pyramid

Students are asked to find the height of a square pyramid given the length of a base edge and its volume.

Cylinder Formula

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.

Cone Formula

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.

Burning Sphere

Students are asked to solve a problem that requires calculating the volume of a sphere.

Sphere Formula

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.

Pyramid Formula

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.

Lesson Plans

Name Description
What Floats Your Boat

Students will solve real-world and mathematical problems involving area, density, volume and surface area of 2 – and 3-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Students will engineer solutions to the given problem using gained scientific content knowledge as application of mathematical skills

Slope Intercept - Lesson #1

This is lesson 1 of 3 in the Slope Intercept unit. This lesson introduces graphing proportional relationships. In this lesson students will perform an experiment to find and relate density of two different materials to the constant of proportionality and unit rate.

Knight Shipping, Inc.

In this design challenge students will take what they have learned about calculating the volumes and densities of cones, cylinders, and spheres, to decide which shape would make the best shipping container. Students will calculate the volumes and densities to help select the best design and then test them to move at least 3 containers of the chosen shape across the classroom. Students will fill the shapes with marshmallows to visually confirm which shape would hold more.

How Many Cones Does It Take?

This lesson is a "hands-on" activity. Students will investigate and compare the volumes of cylinders and cones with matching radii and heights. Students will first discover the relationship between the volume of cones and cylinders and then transition into using a formula to determine the volume.

Find your Formula!

Students will investigate the formula for the volume of a pyramid and/or cone and use those formulas to calculate the volume of other solids. The students will have hands-on discovery working with hollow Geometric Solids that they fill with dry rice, popcorn, or another material.

Silly Cylinders

This is a short activity where students determine the density of the human body by considering each part of the body to be a cylinder. I use this activity during the second week of school, so students have already had some practice with measurement. In addition to providing students with practice in data collection and problem solving, it is a good activity that allows teachers to measure students' previous knowledge in these areas.

Area to Volume Exploration

In this student-centered lesson, the formulas for the volume of a cylinder, cone, and a sphere are examined and practiced. The relationship between the volume of a cone and a cylinder with the same radius and height is explored. Students will also solve real-world problems involving these three-dimensional figures.

Pack It Up

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.

Victorious with Volume

In this lesson, the students will explore and use the relationship of volume for cylinders and cones that have equal heights and radii.

M&M Soup

This is the informative part of a two-lesson sequence. Students explore how to find the volume of a cylinder by making connections with circles and various real-world items.

Cylinder Volume Lesson Plan

Using volume in the real world

Relating Surface Area and Volume

Students will recognize that while the surface area may change, the volume can remain the same. This lesson is enhanced through the multimedia CPALMS Perspectives Video, which introduces students to the relationship between surface area and volume.

Original Student Tutorial

Name Description
Volume of Spherical Bubble Tea

Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.

Perspectives Video: Experts

Name Description
Velocity of the Aucilla River

Harley Means discusses the mathematical methods hydrologists use to calculate the velocity of rivers.

Download the CPALMS Perspectives video student note taking guide.

Carbon Foam and Geometry

Carbon can take many forms, including foam! Learn more about how geometry and the Monte Carlo Method is important in understanding it.

Perspectives Video: Professional/Enthusiasts

Name Description
Unit Rate and Florida Cave Formation

How long does it take to form speleothems in the caves at Florida Caverns State Parks?

Download the CPALMS Perspectives video student note taking guide.

Volume and Surface Area of Pizza Dough

Michael McKinnon of Gaines Street Pies explains how when making pizza the volume is conserved but the surface area changes.

Mathematically Optimizing 3D Printing

Did you know that altering computer code can increase 3D printing efficiency? Check it out!

KROS Pacific Ocean Kayak Journey: Food Storage Mass and Volume

What do you do if you don't have room for all your gear on a solo ocean trek? You're gonna need a bigger boat...or pack smarter with math.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Download the CPALMS Perspectives video student note taking guide.

Problem-Solving Tasks

Name Description
Glasses

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.

Comparing Snow Cones

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.

Flower Vases

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.

Shipping Rolled Oats

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.

Shamu Stadium Geometry-SeaWorld Classroom Activity In this problem solving task, students investigate Shamu Stadium at Sea World. They will use knowledge of geometric shapes to solve problems involving area and volume and examine as well as analyze a diagram making calculations. Students will also be challenged to design an advertising poster using the measurements they mind.

Student Center Activity

Name Description
Edcite: Mathematics Grade 8

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Teaching Idea

Name Description
Modeling: Making Matchsticks This lesson unit is intended to help you assess how well students are able to:
  • Interpret a situation and represent the variables mathematically.
  • Select appropriate mathematical methods.
  • Interpret and evaluate the data generated.
  • Communicate their reasoning clearly.
The context is estimating how many matchsticks (rectangular prisms) can be made from this tree (conic).

Tutorials

Name Description
Cylinder Volume and Surface Area

This video demonstrates finding the volume and surface area of a cylinder.

Volume of a Sphere

This video shows how to calculate the volume of a sphere.

Volume of a Cone

This video explains the formula for volume of a cone and applies the formula to solve a problem.

Unit/Lesson Sequence

Name Description
Three Dimensional Shapes

In this interactive, self-guided unit on 3-dimensional shape, students (and teachers) explore 3-dimensional shapes, determine surface area and volume, derive Euler's formula, and investigate Platonic solids. Interactive quizzes and animations are included throughout, including a 15 question quiz for student completion.

Student Resources

Original Student Tutorial

Name Description
Volume of Spherical Bubble Tea:

Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.

Problem-Solving Tasks

Name Description
Glasses:

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.

Comparing Snow Cones:

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.

Flower Vases:

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.

Shipping Rolled Oats:

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.

Student Center Activity

Name Description
Edcite: Mathematics Grade 8:

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Tutorials

Name Description
Cylinder Volume and Surface Area:

This video demonstrates finding the volume and surface area of a cylinder.

Volume of a Sphere:

This video shows how to calculate the volume of a sphere.

Volume of a Cone:

This video explains the formula for volume of a cone and applies the formula to solve a problem.



Parent Resources

Problem-Solving Tasks

Name Description
Glasses:

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.

Comparing Snow Cones:

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.

Flower Vases:

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.

Shipping Rolled Oats:

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.



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