# Standard 2: Solve problems involving three-dimensional figures, including right circular cylinders.

General Information
Number: MA.7.GR.2
Title: Solve problems involving three-dimensional figures, including right circular cylinders.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.7.GR.2.AP.1
Match the parts of a given formula to the right circular cylinder using the figure’s net.
MA.7.GR.2.AP.2
Given the formula, use tools to find the surface area of a right circular cylinder using the figure’s net.
MA.7.GR.2.AP.3
Given a formula, use tools to calculate the volume of right circular cylinders.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessment

Platinum Cylinder:

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

Type: Formative Assessment

## Lesson Plans

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.

Type: Lesson Plan

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.

Type: Lesson Plan

Students will apply skills from the Geometry Domain to build graduation caps for themselves using heavyweight poster paper. They will also apply some basic mathematical skills to determine dimensions and to determine minimum cost. Some of the Geometric skills reinforced in Building Graduation Caps: Cooperative Assignment are finding area, applying the concept of similarity, and the application of the properties of parallelograms. Other skills also involved in this application are measuring, and statistical calculations, such as finding the mean and the range. In addition to the hands-on group project that takes place during the lesson, there is the Prerequisite Skills Assessment: Area that should be administered before the group activity and a home-learning activity. Building Graduation Caps: Individual Assignment is the home-learning assignment; it is designed to reinforce the skills learned in the group activity.

Type: Lesson Plan

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.

Type: Lesson Plan

Netty People and Pets:

Students will learn what the "net" of a three-dimensional figure is, draw nets of right-rectangular prisms and right-rectangular prisms, accurately calculate the surface area of nets, and put nets together to create an original “person” or “pet.”

Type: Lesson Plan

The Cost of Keeping Cool:

Students will find the volumes of objects. After decomposing a model of a house into basic objects students will determine the cost of running the air conditioning.

Type: Lesson Plan

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.

Type: Lesson Plan

Can You Cut It? Slicing Three-Dimensional Figures:

In this lesson, students will sketch, model, and describe cross-sections formed by a plane passing through a three-dimensional figure. Students will create a cube, right rectangular prism, and right rectangular pyramid using modeling clay dough, and then slice the model using parallel, perpendicular, and intersecting lines. Students will describe the two-dimensional figure resulting from slicing the three-dimensional model.

Type: Lesson Plan

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.

Type: Lesson Plan

A Pi-ece of Florida History:

A Pi-ece of Florida History discovers significant dates in Florida History in the first 8 digits after the decimal of the number Pi. Historical people associated with those dates are identified and described. Students then use body measurements to approximate volume.

Type: Lesson Plan

Boxing Candles:

In this MEA, students select jars for candles based on a variety of factors and then design boxes to contain the jars.

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.

Type: Lesson Plan

How Much Surface Area Does Your Skin Take Up?:

Students relate their body parts to six cylinders. They adjust the surface area of a cylinder formula, and then measure, record, and compute the approximate surface area of their skin. Students will develop a better understanding of how much skin covers up their body when they calculate how much surface area their skin takes up.

Type: Lesson Plan

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.

Type: Lesson Plan

## Perspectives Video: Professional/Enthusiasts

Volume and Surface Area of Pizza Dough:

<p>Michael&nbsp;McKinnon&nbsp;of Gaines Street Pies explains how when making pizza the volume is conserved but the&nbsp;surface area changes.</p>

Type: Perspectives Video: Professional/Enthusiast

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.

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

Surface Area Misconception:

Unlock an effective teaching strategy for identifying the base and height of figures in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

How thick is a soda can? (Variation II):

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.

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.

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.

## Unit/Lesson Sequence

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.

Type: Unit/Lesson Sequence

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

How thick is a soda can? (Variation II):

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.

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.

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.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

How thick is a soda can? (Variation II):

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.

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.