Volume of Spheres: Teacher Notes

Overview

In this activity students determine the volume of spheres by examining how they are related to cylinders and cones.

Important Mathematical Ideas

  • A sphere is a perfectly round 3D figure similar to a ball you might play soccer or basketball with.
  • The volume of a 3D figure tells you how much space it takes up.
  • The volume of a sphere is related to the volume of a cylinder or a cone with the same dimensions.

Prior Knowledge

  • Apply volume formula for cones and cylinders (e.g., know and substitute).
  • Appropriate use of the basic function keys on a calculator (x², x³,).

Common Misconceptions

  • Incorrect measurement units are used or are not included.
  • Unable to identify the appropriate measurement in a context (e.g., slant height vs. perpendicular height).
  • Confusing diameter and radius.

Curriculum Notes

  • Developing the formula for a sphere, V=r3, is revisited after the students completeUnit 7: Algebraic Models and Expressions.

Information to Support/ Enhance/ Extend Learning

  • Add new vocabulary and terms to theWord Wall(e.g., 2D shape, 3D figure, sphere, rectangular prism, rectangular pyramid, triangular based prism, sphere).
  • Consider a variety of formats as an alternative to journal entries (e.g., student note, pair/share, group discussion, exit card, poster, electronic posting).

Materials

  • A variety of 3D figures for students to compare 2D drawings of 3D figures. Include spheres.

Minds On

Task 1: Name and Sort 3D Figures

  • This is anopen-endedsorting activity. Be prepared for out-of-the box thinking with non-mathematical answers.
  • Alternatively, you can cut out the drawings of the figures and ask pairs of students to sort.
  • Students identify the names of six 3D figures: Arectangular prism; B. cylinder; C. rectangular or square–based pyramid; D. sphere; E. triangular-based prism; F. cone
  • Students sort the figures into two piles
  • Sample responses:
  • Sandy's answer: Pile 1: A, C, E; Pile 2: B, D, F because Pile 1's figures all have straight edges; Pile 2's figures all have some curved edges.
  • Barb's answer: Pile 1: C, E, F; Pile 2: A, B, D because in Pile 1 all have triangular shapes in them.
  • Irene's answer: Pile 1: A, B, C, E, F; Pile 2: D because D is not a prism or pyramid and all the others are.

Discussion Prompts and Sample Responses

  • Describe how your sort is similar or different.
  • Comparing Sandy's answer to Barb's and Irene's:
  • Similar: Barb and Sandy have the same number in each pile.
  • Different: Sandy looked at edges; Barb looked at shapes; Irene looked at types of 3D figures.
  • Explain how or why you agree or disagree with their sort.

Answers will vary.

  • Describe any new learning or wonderings you have.

Answers will vary.

Action

Task 2: Volume of Spheres and Cylinders

Students will:

  • predict the relationship between the volume of this sphere and the volume of this cylinder
  • students can share their predictions before viewing the video
  • if showing the video to the whole class, it can be stopped part way to give students the opportunity to rethink their predictions
  • watchVolume of Spheres and Cylinders video
  • a sphere filled with water is being poured into a cylinder with the same diameter and height
  • compare their prediction with what they see in the video

Journal Prompt and Response

  • Volume of a sphere= 2/3volume of a cylinder with the same radius and height.

Check Your Understanding

  • Students will calculate the volume of a sphere whose radius is 10 cm using the relationship between its volume and the volume of a cylinder with the same radius and height.
  • sample answer is provided

Task 3: Volume of Spheres and Cones

  • Students will:
  • predict the relationship between the volume of this sphere and the volume of this cone
  • watchRelationship between Cones, Spheres and Cylinders Part 1 video
  • a sphere filled with water is being poured into a cylinder with the same diameter and height
  • students can watch the videostarting at 3:30
  • compare their prediction with what they see in the video
  • Students may benefit from doing the pouring experiment with 3D objects to compare the volume of a cone with the volume of a hemisphere using sand instead of water.

Journal Prompts and Sample Responses

1)How does what you saw in the video compare to your prediction?

Answers will vary.

2)What other volume relationships were referred to in the video?

Cylinder to cone; cylinder to sphere

Volume of a sphere = 2 times (volume of a cone with the same radius and height)

Check Your Understanding

  • Students will calculate the volume of a sphere the radius of which is 10 cm using the relationship between its volume and the volume of a cone with the same radius and height.
  • sample answer is provided

Consolidation

Task 4: Volume of Spheres Practice

  • Students will determine the volumes of a sphere using its relationships to the volumes of a cylinder and a cone and check with the sample answer provided

Journal Prompt and Sample Response

Describe how finding the volume of a sphere using its relationship to the volume of a cylinder with the same radius and height compares to finding the volume of a sphere using its relationship to the volume of a cone with the same radius and height.

Volume of a sphere = volume of a cylinder with the same radius and height

Volume of a sphere =2 times volume of a cone with the same radius and height

Task 5: Assignment 1 Volume of a Propane Tank

  • Posted with unit.
  • See sample solution posted on the Virtual Learning Environment (vLE).

Task 6: Student Reflection

  • Students are asked to reflect on their understanding of this topic.
  • These reflections can be used as an assessment for learning to help determine next steps for individual students.

Grade 9 Applied Blended Learning: Unit 1 Activity 2 Page 1 of 3