## Unit Overview

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Plan C.9: Volume of Cylinders, Cones, and Spheres

**Download Seeds, Plans, and Resources (zip)**

**Content Emphasis By Clusters in Grade 8**

**Progressions from Common Core State Standards in Mathematics**

Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.

### Unit Overview

This unit builds on the properties of circles, and computation with rational and irrational numbers. The unit requires students to apply prior knowledge about the properties and formulas for circles and the Pythagorean Theorem to accurately approximate the volume of cylinders, cones, and spheres.

**Teacher Notes:**

Students should:

- be well-grounded in their knowledge of square units as a measure of area.
- understand concept of area and relate area to multiplication
- be able to determine the volume of rectangular prisms, using the dimensions of length, width, and height.
- solve real-life and mathematical problems involving area of two-dimensional objects composed of triangles.

**Enduring Understandings:**

At the completion of this unit on problem-solving with the volume of cylinders, cones, and spheres, the student will understand that:

- an approximation of Pi is integral to volumes of solid figures that are related to circles
- relationships exist between dimensions and volumes of cylinders, cones, spheres. Changes in the dimensions affect volume in specific ways
- volume and surface area depend on many of the same dimensions, but result in different calculations.

**Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):
**

- PARCC has not provided examples of opportunities for in-depth focus related to solving real-world and mathematical problems involving volume of cylinders, cones, and spheres.

**Possible Student Outcomes:**

Students will be able to:

- identify the differences between formulas for volume of cylinders, cones, and spheres.
- choose the appropriate formula to closely approximate volume of each type of figure and to solve authentic problems
- create models that show why and how the volume formulas for cylinders, cones, and spheres "work."

**Evidence of Student Learning:**

*The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.*

**Fluency Expectations and Examples of Culminating Standards:**

**Common Misconceptions:**

Students may

- View volume merely as a number that results from substituting other numbers into a formula, rather than recognizing volume as a measure related to an amount of space occupied.

**Interdisciplinary Connections:**

Interdisciplinary connections fall into a number of related categories:

*Literacy standards within the Maryland Common Core State Curriculum*

*Science, Technology, Engineering, and Mathematics standards*

*Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.*

**Sample Assessment Items: ***The items included in this component will be aligned to the standards in the unit and will include:*

*Items purchased from vendors*

*PARCC prototype items*

*PARCC public release items*

*Maryland Public release items*

*Formative Assessment*

**Interventions/Enrichments/PD: ***(Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)*

**Vocabulary/Terminology/Concepts: ***This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.*

**
Part I – Focus Cluster
Understand and Apply the Pythagorean Theorem**

** volume of cones: **The formula for the volume of a cone can be determined from the volume formula for a cylinder. We must start with a cylinder and a cone that have equal heights and radii, as in the diagram below.

Imagine copying the cone so that we had three congruent cones, all having the same height and radii of a cylinder. Next, we could fill the cones with water. As our last step in this demonstration, we could then dump the water from the cones into the cylinder. If such an experiment were to be performed, we would find that the water level of the cylinder would perfectly fill the cylinder.

This means it takes the volume of three cones to equal one cylinder. Looking at this in reverse, each cone is one-third the volume of a cylinder. Since a cylinder's volume formula is V = Bh, then the volume of a cone is one-third that formula, or V = Bh/3. Specifically, the cylinder's volume formula is and the cone's volume formula is .

** volume of cylinders: **The process for understanding and calculating the volume of cylinders is identical to that of prisms, even though cylinders are curved. Here is a general cylinder.

**General Cylinder**

Here is a specific cylinder with a radius 3 units and height 4 units.

**Specific Cylinder**

We fill the bottom of the cylinder with unit cubes. This means the bottom of the prism will act as a container and will hold as many cubes as possible without stacking them on top of each other. This is what it would look like.

The diagram above is strange looking because we are trying to stack cubes within a curved space. Some cubes have to be shaved so as to allow them to fit inside. Also, the cubes do not yet represent the total volume. It only represents a partial volume, but we need to count these cubes to arrive at the total volume. To count these full and partial cubes, we will use the formula for the area of a circle.

The radius of the circular base (bottom) is 3 units and the formula for area of a circle is A = . So, the number of cubes is (3.14)(3) , which to the nearest tenth, is equal to 28.3.

If we imagine the cylinder like a building (like we did for prisms above), we could stack cubes on top of each other until the cylinder is completely filled. it would be filled so that all cubes are touching each other such that no space existed between cubes. it would look like this.

To count all the cubes above, we will use the consistency of the solid to our advantage. We already know there are 28.3 cubes on the bottom level and all levels contain the exact number of cubes. Therefore, we need only take that bottom total of 28.3 and multiply it by 4 because there are four levels to the cylinder. 28.3 x 4 = 113.2 total cubes to our original cylinder.

If we review our calculations, we find that the total bottom layer of cubes was found by using the area of a circle, . Then, we took the result and multiplied it by the cylinder's height. So the volume of a cylinder is π times the square of its radius times its height. Sometimes geometers refer to the volume as the area of the bottom times the height. Symbolically, it is written as V = Bh, where the capitol B stands for the area of the solid's base (bottom).

** volume of spheres: **A sphere is the locus of all points in a region that are equidistant from a point. The two-dimensional rendition of the solid is represented below.

**General Sphere**

To calculate the surface area of a sphere, we must imagine the sphere as an infinite number of pyramids whose bases rest on the surface of the sphere and extend to the sphere's center. Therefore, the radius of the sphere would be the height of each pyramid. One such pyramid is depicted below.

The volume of the sphere would then be the sum of the volumes of all the pyramids. To calculate this, we would use the formula for volume of a pyramid, namely V = Bh/3. we would take the sum of all the pyramid bases, multiply by their height, and divide by 3.

**Part II – Instructional Connections outside the Focus Cluster**

** proof of the Pythagorean Theorem and it converse: **In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse . The figure below shows the parts of a right triangle.

** distance formula: **The distance

*d*between the points A:(x1, y1) and B:(x2, y2) is given by the formula:

The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points. The formula below shows an application of the Pythagorean Theorem for right triangles: