Standard 4: Use geometric measurement and dimensions to solve problems.

General Information
Number: MA.912.GR.4
Title: Use geometric measurement and dimensions to solve problems.
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.912.GR.4.AP.1
Identify the shape of a two-dimensional cross section of a three-dimensional figure.
MA.912.GR.4.AP.2
Identify a three-dimensional object generated by the rotation of a two-dimensional figure.
MA.912.GR.4.AP.3
Select the effect of a dilation on the area of two-dimensional figures and/or surface area or volume of three-dimensional figures.
MA.912.GR.4.AP.4
Solve mathematical and/or real-world problems involving the area of triangles, squares, circles or rectangles.
MA.912.GR.4.AP.5
Solve mathematical or real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, or cones.
MA.912.GR.4.AP.6
Solve mathematical or real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, and cones.

Related Resources

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

Formative Assessments

Inside the Box:

Students are asked to identify and draw cross sections of a rectangular prism and to describe their dimensions.

Type: Formative Assessment

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

Softball Complex:

Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

Type: Formative Assessment

Slice of a Cone:

Students are asked to sketch, describe, and compare three horizontal cross sections of a cone.

Type: Formative Assessment

Slice It:

Students are asked to identify and describe two-dimensional cross sections of three-dimensional solids.

Type: Formative Assessment

How Many Trees?:

Students are asked to determine an estimate of the density of trees and the total number of trees in a forest.

Type: Formative Assessment

Estimating Volume:

Students are asked to model a tree trunk with geometric solids and to use the model to estimate the volume of the tree trunk.

Type: Formative Assessment

Population of Utah:

Students are asked to determine the population of the state of Utah given the state’s population density and a diagram of the state’s perimeter with boundary distances labeled in miles.

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

Mudslide:

Students are asked to create a model to estimate volume and mass.

Type: Formative Assessment

Estimating Area:

Students are asked to select appropriate geometric shapes to model a lake and then use the model to estimate the surface area of the lake.

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

Sugar Cone:

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

Type: Formative Assessment

Louvre Pyramid:

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

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Burning Sphere:

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

Type: Formative Assessment

Windy Pyramid:

Students are asked to use a net to find the surface area of a triangular pyramid.

Type: Formative Assessment

Skateboard Ramp:

Students are asked to draw a net of a three-dimensional figure.

Type: Formative Assessment

Square Pyramid Slices:

Students are asked to sketch and describe the two-dimensional figures that result from slicing a square pyramid.

Type: Formative Assessment

Rectangular Prism Slices:

Students are asked to sketch and describe two-dimensional figures that result from slicing a rectangular prism.

Type: Formative Assessment

Prismatic Surface Area:

Students are asked to determine the surface area of a right triangular prism and explain the procedure.

Type: Formative Assessment

Chilling Volumes:

Students are asked to solve a problem involving the volume of a composite figure.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Cylinder Slices:

Students are asked to sketch and describe the two-dimensional figures that result from slicing a cylinder.

Type: Formative Assessment

Cone Slices:

Students are asked to sketch and describe the two-dimensional figures that result from slicing a cone.

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

2D Rotations of Triangles:

Students are given the coordinates of the vertices of a right triangle and asked to describe the solid formed by rotating the triangle about a given axis.

Type: Formative Assessment

Working Backwards – 2D Rotations:

Students are given a solid and asked to determine the two-dimensional shape that will create the solid when rotated about the y-axis.

Type: Formative Assessment

2D Rotations of Rectangles:

Students are given the coordinates of the vertices of a rectangle and asked to describe the solid formed by rotating the rectangle about a given axis.

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

Three Dimensions Unfolded:

Students will use nets of prisms to find the surface area of composite 3-D figures. Students will learn to identify the faces of 3-D figures that are needed to find the surface areas, and those that are not needed.

Type: Lesson Plan

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 create 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

My Geometry Classroom:

Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

Type: Lesson Plan

Exploring Cavalieri's Principle:

Students will explore Cavalieri's Principle using technology. Students will calculate the volume of oblique solids and determine if Cavalieri's Principle applies.

Students will also 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 Principle applies and then calculate the volume of various oblique solids.

Type: Lesson Plan

Observing the Centroid:

Students will construct the medians of a triangle then investigate the intersections of the medians.

Type: Lesson Plan

The Centroid:

Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.

Type: Lesson Plan

Plane Slice:

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Students will use modeling clay to explore the cross sections that result from slicing a 3-dimensional figure.

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

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

Propensity for Density:

Students apply concepts of density to situations that involve area (2-D) and volume (3-D).

Type: Lesson Plan

Wrapping Up Geometry (Surface Area of Triangular Prisms) :

This lesson is designed to take students from recognizing nets of triangular prisms and finding areas of their individual faces, to finding the surface area of triangular prisms.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

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

The Grass is Always Greener:

The lesson introduces area of sectors of circles then uses the areas of circles and sectors to approximate area of 2-D figures. The lesson culminates in using the area of circles and sectors of circles as spray patterns in the design of a sprinkler system between a house and the perimeter of the yard (2-D figure).

Type: Lesson Plan

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.

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

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

St. Pi Day construction with a compass & ruler:

St. Pi Day construction with compass

This activity uses a compass and straight-edge(ruler) to construct a design. The design is then used to complete a worksheet involving perimeter, circumference, area and dimensional changes which affect the scale factor ratio.

Type: Lesson Plan

Turning Tires Model Eliciting Activity:

The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

Type: Lesson Plan

Original Student Tutorials

Volume of Spherical Bubble Tea:

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

Type: Original Student Tutorial

Ninja Nancy Slices:

Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial.

Type: Original Student Tutorial

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

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: Experts

Velocity of the Aucilla River:

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

Type: Perspectives Video: Expert

MicroGravity Sensors & Statistics:

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Type: 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

Implications of the Spherical Earth:

To understand atmospheric and oceanic currents, one needs a well-rounded understanding of geometry and the shape of the Earth.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiasts

Unit Rate and Florida Cave Formation:

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

Type: Perspectives Video: Professional/Enthusiast

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

Reflections, Rotations, and Translations with Additive Printing:

Transform your understanding of 3D modeling when you learn about how shapes are manipulated to arrive at a final 3D printed form!

Type: Perspectives Video: Professional/Enthusiast

3D Modeling with 3D Shapes:

Complex 3D shapes are often created using simple 3D primitives! Tune in and shape up as you learn about this application of geometry!

Type: Perspectives Video: Professional/Enthusiast

Scale and Proportion for Bird Photography:

Mathematics plays a role in what we perceive as beautiful! Learn more about it while you learn about bird photography! Produced with funding from the Florida Division of Cultural Affairs.

Type: Perspectives Video: Professional/Enthusiast

Design Process for a Science Museum Exhibit:

Go behind the scenes and learn about science museum exhibits, design constraints, and engineering workflow! Produced with funding from the Florida Division of Cultural Affairs.

Type: Perspectives Video: Professional/Enthusiast

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

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

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

NASA Space Flight Hardware Geometry:

If you want to take things to space, you have to have a place to put them. Just make sure they fit before you send them up.

Type: Perspectives Video: Professional/Enthusiast

Perspectives Video: Teaching Ideas

Ecological Sampling Methods and Population Density:

Dr. David McNutt explains how a simple do-it-yourself quadrat and a transect can be used for ecological sampling to estimate population density in a given area.

Type: Perspectives Video: Teaching Idea

KROS Pacific Ocean Kayak Journey: Kites, Geometry, and Vectors:

Set sail with this math teacher as he explains how kites were used for lessons in the classroom.

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: Teaching Idea

Toilet Roll:

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Ice Cream Cone:

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

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.

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

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Global Positioning System II:

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Archimedes and the King's Crown:

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.

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.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

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.

Tennis Balls in a Can:

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

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.

Tutorial

Find the Volume of a Triangular Prism and Cube:

This video will show to find the volume of a triangular prism, and a cube by applying the formula for volume.

Type: Tutorial

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.

Original Student Tutorials

Volume of Spherical Bubble Tea:

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

Type: Original Student Tutorial

Ninja Nancy Slices:

Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial.

Type: Original Student Tutorial

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

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: Expert

MicroGravity Sensors & Statistics:

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiasts

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

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

Toilet Roll:

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Ice Cream Cone:

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

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.

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

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Global Positioning System II:

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Archimedes and the King's Crown:

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.

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.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

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.

Tennis Balls in a Can:

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

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.

Tutorial

Find the Volume of a Triangular Prism and Cube:

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

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

Perspectives Video: Professional/Enthusiasts

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

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

Toilet Roll:

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Ice Cream Cone:

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

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.

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

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Global Positioning System II:

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Archimedes and the King's Crown:

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.

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.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

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.

Tennis Balls in a Can:

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

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.