# Cluster 1: Explain volume formulas and use them to solve problems. (Geometry - 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.

General Information
Number: MAFS.912.G-GMD.1
Title: Explain volume formulas and use them to solve problems. (Geometry - Additional Cluster)
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Geometry: Geometric Measurement & Dimension

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.G-GMD.1.AP.1a
Describe why the formulas work for a circle or cylinder (circumference of a circle, area of a circle, volume of a cylinder) based on a dissection.
MAFS.912.G-GMD.1.AP.3a
Use appropriate formulas to calculate volume for cylinders, pyramids, and cones.

## Related Resources

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

## Assessments

Sample 1 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 3 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 2 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

## Formative Assessments

Volume of a Cylinder:

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.

Type: Formative Assessment

Area and Circumference – 1:

This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.

Type: Formative Assessment

Volume of a Cone:

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.

Type: Formative Assessment

Area and Circumference - 3:

This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.

Type: Formative Assessment

Area and Circumference - 2:

This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r2 · A(1)].

Type: Formative Assessment

Volume of a Pyramid:

Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.

Type: Formative Assessment

Snow Cones:

Students are asked to solve a problem that requires calculating the volumes of a cone and a cylinder.

Type: Formative Assessment

Sports Drinks:

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.

Type: Formative Assessment

The Great Pyramid:

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.

Type: Formative Assessment

Do Not Spill the Water!:

Students are asked to solve a problem that requires calculating the volumes of a sphere and a cylinder.

Type: Formative Assessment

## Lesson Plans

Filled to Capacity!:

This is a lesson where students investigate, compare, dissect, and use the relationship between volume of a cone and cylinder with equal corresponding dimensions.

Type: Lesson Plan

The Relationship Between Cones and Cylinders:

Students are guided through the creation of a cone and a cylinder with the same height and base. At the conclusion of the lesson, the students will know that the volume ratio between the cone and cylinder is 1:3.

Type: Lesson Plan

Exploring Cavalieri's Principle:

In this 90 minute lesson/exploration of Cavalieri's Principle using technology (GeoGebra 5.0), students calculate volume of oblique solids and determine if Cavalieri's applies to given scenarios.

Students will perform transformations of a base figure in a 3 dimensional coordinate system to observe the creation of right and oblique solid figures. After these observations, students will create a conjecture about calculating the volume of the oblique solids. Students will use the conjecture to determine situations in which Cavalieri's applies and calculate volumes of oblique solids.

The teacher may choose to split lesson into two 45 minute sessions.

Variable representation of volume formulas V = BH and V = 1/3BH:

• B = Base Area
• H = Height of Solid
• b = length of base figure
• h = height of base figure

The differentiation is used so that the height of the solid is not confused with the height of the base.

Type: Lesson Plan

Discovering the Formulas for Circumference and Area of a Circle:

Using reasoning skills, students will understand how the formulas for circumference and area of a circle are derived. Students will use a wide array of skills such as deductive reasoning, finding patterns, using algebra, modeling and transformation of an object. The teacher ensures student success through direct instruction, investigation and collaborative group work.

Type: Lesson Plan

Cape Florida Lighthouse: Lore and Calculations:

The historic Cape Florida Lighthouse, often described as a conical tower, teems with mathematical applications. This lesson focuses on the change in volume and lateral surface area throughout its storied existence.

Type: Lesson Plan

Yogurt Land Container:

The student will assist Yogurt Land on choosing a new size container to offer their customers. The choice of containers are different three dimensional figures. Students will revisit the concepts of volume, surface area, and profit in order to make a decision.

Type: Lesson Plan

Evaluating Statements About Enlargements (2D and 3D):

This lesson is intended to help you assess how well students are able to solve problems involving area and volume, and in particular, to help you identify and assist students who have difficulties with the following:

• Computing perimeters, areas and volumes using formulas.
• Finding the relationships between perimeters, areas, and volumes of shapes after scaling.

Type: Lesson Plan

This lesson explores the formulas for calculating the volume of cylinders, cones, pyramids, and spheres.

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

Cylinder Volume Lesson Plan:

Using volume in the real world

Type: Lesson Plan

Calculating Volumes of Compound Objects:

This lesson unit is intended to help you assess how well students solve problems involving measurement, and in particular, to identify and help students who have the following difficulties:

• Computing measurements using formulas.
• Decomposing compound shapes into simpler ones.
• Using right triangles and their properties to solve real-world problems.

Type: Lesson Plan

## Original Student Tutorial

I Scream! You Scream! We All Scream for... Volume!:

Have you ever ordered a scoop of ice cream in a cone and wondered how much ice cream actually fits inside the cone? By the end of this tutorial, you should be able to answer this question and solve other real-world problems by using the formula for the volume of a cone.

Type: Original Student Tutorial

## Perspectives Video: Expert

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.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

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.

Type: Perspectives Video: Professional/Enthusiast

Mathematically Optimizing 3D Printing:

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

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

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

Using Geometry for Interior Design and Architecture:

An architect discusses how he uses circumference and area calculations to accurately create designs and plans.

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]

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

Robot Mathematics: Gearing Ratio Calculations for Performance:

A science teacher demonstrates stepwise calculations involving multiple variables for designing robots with desired characteristics.

Type: Perspectives Video: Teaching Idea

Doctor's Appointment:

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.

Centerpiece:

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).

Use Cavalieri’s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

## 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

## Video/Audio/Animation

Story of Pi:

This video dynamically shows how Pi works, and how it is used.

Type: Video/Audio/Animation

## Virtual Manipulative

Which Holds More? :

This interactive manipulative will let you compare and calculate volumes of different solids.

Type: Virtual Manipulative

## Student Resources

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

## Original Student Tutorial

I Scream! You Scream! We All Scream for... Volume!:

Have you ever ordered a scoop of ice cream in a cone and wondered how much ice cream actually fits inside the cone? By the end of this tutorial, you should be able to answer this question and solve other real-world problems by using the formula for the volume of a cone.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

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

Doctor's Appointment:

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.

Centerpiece:

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).

Use Cavalieri’s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

## Virtual Manipulative

Which Holds More? :

This interactive manipulative will let you compare and calculate volumes of different solids.

Type: Virtual Manipulative

## Parent Resources

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

## Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

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

Doctor's Appointment:

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.

Centerpiece:

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).

Use Cavalieri’s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.