### Clarifications

**Fluency Expectations or Examples of Culminating Standards**

When students learn to solve problems involving volumes of cones, cylinders, and spheres — together with their previous grade 7 work in angle measure, area, surface area and volume (7.G.2.4–2.6) — they will have acquired a well-developed set of geometric measurement skills. These skills, along with proportional reasoning (7.RP) and multistep numerical problem solving (7.EE.2.3), can be combined and used in flexible ways as part of modeling during high school — not to mention after high school for college and careers.

### General Information

**Subject Area:**Mathematics

**Grade:**8

**Domain-Subdomain:**Geometry

**Cluster:**Level 2: Basic Application of Skills & Concepts

**Cluster:**Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. (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.

**Date Adopted or Revised:**02/14

**Date of Last Rating:**02/14

**Status:**State Board Approved

**Assessed:**Yes

### Test Item Specifications

**Assessment Limits :**

Graphics of three-dimensional figures can be included. Dimensions must be given as rational numbers. Figures must not be composite**Calculator :**Yes

**Context :**Allowable

### Sample Test Items (4)

**Test Item #:**Sample Item 1**Question:**A cylinder with a height of inches (in.) and a diameter of 5 inches is shown.

What is the volume of the cylinder, in cubic inches? (Use 3.14 for )

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 2**Question:**The diameter of a sphere is 4 inches.

What is the volume of the sphere, in cubic inches? (Use 3.14 for ??.)

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 3**Question:**A cone has a height of 6.4 inches and a diameter of 6 inches.

What is the volume, in cubic inches, of the cone? Use 3.14 for

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 4**Question:**A water container in the shape of a cone has a height of 5 inches and a diameter of 3.5 inches.

Enter an equation in the first blank box and a number in the second blank box to complete the statements about the water container.

A. The formula to calculate the container's volume, V, with the given measurements is ________.

B. The container can hold approximately _____ cubic inches of water. Round your answer to the nearest hundredth.

**Difficulty:**N/A**Type:**EE: Equation Editor

## Related Courses

## Related Access Points

## Related Resources

## Assessments

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Experts

## Perspectives Video: Professional/Enthusiasts

## Problem-Solving Tasks

## Student Center Activity

## Teaching Idea

## Tutorials

## Unit/Lesson Sequence

## STEM Lessons - Model Eliciting Activity

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.

## MFAS Formative Assessments

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

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.

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.

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

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

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.

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.

## Original Student Tutorials Mathematics - Grades 6-8

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

## Student Resources

## Original Student Tutorial

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

Type: Original Student Tutorial

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

This video demonstrates finding the volume and surface area of a cylinder.

Type: Tutorial

This video explains the formula for volume of a cone and applies the formula to solve a problem.

Type: Tutorial

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

## Parent Resources

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task