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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.

Standard 1 : Explain volume formulas and use them to solve problems. (Geometry - Additional Cluster)Archived
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MAFS.912.G-GMD.1.1 : Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
  • MAFS.912.G-GMD.1.2 : Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
  • MAFS.912.G-GMD.1.3 : Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★
Cluster Information
Number:
MAFS.912.G-GMD.1
Title:
Explain volume formulas and use them to solve problems. (Geometry - Additional Cluster)
Type:
Cluster
Subject:
Mathematics - Archived
Grade:
912
Domain-Subdomain
Geometry: Geometric Measurement & Dimension
Cluster Access Points

This cluster includes the following 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.
Cluster Resources

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

Original Student Tutorial
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.

  • 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.

  • 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.

  • 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.

  • 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)].

  • 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.

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

  • 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.

  • 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.

  • Do Not Spill the Water!: Students are asked to solve a problem that requires calculating the volumes of a sphere and a cylinder.

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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

    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.

  • 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.
  • Volumes about Volume: This lesson explores the formulas for calculating the volume of cylinders, cones, pyramids, and spheres.

  • 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.

  • Cylinder Volume Lesson Plan: Using volume in the real world

  • 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.
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Problem-Solving Tasks
  • 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.

Video/Audio/Animation
  • Story of Pi: This video dynamically shows how Pi works, and how it is used.