Given a mathematical or real-world context, find the surface area of a right circular cylinder using the figure’s net.

### Clarifications

*Clarification 1:*Instruction focuses on representing a right circular cylinder with its net and on the connection between surface area of a figure and its net.

*Clarification 2:* Within this benchmark, the expectation is to find the surface area when given a net or when given a three-dimensional figure.

*Clarification 3:* Within this benchmark, the expectation is not to memorize the surface area formula for a right circular cylinder.
Clarification 4: Solutions may be represented in terms of pi (π) or approximately.

General Information

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**7

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Cylinder (Circular)
- Net
- Pi (π)
- Surface Area

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students found the area of quadrilaterals and composite figures by decomposing them into triangles or rectangles, which developed into finding the surface area of right rectangular prisms and right rectangular pyramids using a figure’s net. In grade 7, students find the area of a circle as well as the surface area of a right circular cylinder using the figure’s net. In high school, students will solve mathematical and real-world problems involving the surface area of cylinders, pyramids, prisms, cones and spheres.- Instruction includes problems that make connections to the understanding of a formula for the surface area of a right circular cylinder and its net (soup can and finding the area of the paper around the can). Show how different parts of the formula correspond to different parts of the net
*(MTR.7.1).* - Instruction allows students the opportunity to develop the formula for the surface area of a right circular cylinder.
- For example, provide students cans of various sizes, paper, scissors, and tape
*(MTR.2.1).*Ask students to cover the can completely without overlapping the paper and only using as little tape as possible. Ask students to explain their method if they feel it was successful in covering the can completely without overlapping any paper, until someone presents the idea of cutting two circles, taping them to the base and then one rectangle to cover the curved surface*(MTR.4.1, MTR.5.1).*

- For example, provide students cans of various sizes, paper, scissors, and tape
- Instruction allows for students to use various nets for cylinders which they can cut out and form the three-dimensional figure or use virtual simulations which show the “unrolling” of the cylinder into its net
*(**MTR.2.1*).

### Common Misconceptions or Errors

- Students often confuse the vocabulary base, length, height and “$B$” (base area), when moving between two-and three-dimensional figures. To address this misconception, continue to use the parts of the net to calculate the surface area, rather than focusing on the formula.
- Students may incorrectly believe that the part of the cylinder that is lying flat is the base of the figure. To address this misconception, remind students that while a cylinder may lay on its side, the bases are the circles with the height being the perpendicular distance between them. Provide multiple orientations of objects and continue to break them down to their nets.
- Students often forget or confuse the formulas for area, surface area and volume. To address this misconception, use these concepts in context, or a manner in which they understand the meaning behind the terms, will be important in fostering their conceptual development
*(**MTR.7.1*).

### Strategies to Support Tiered Instruction

- Instruction includes the use of geometric software to allow students to explore the difference between base, length, height and “
*B*” (base area). - Instruction includes co-creating a graphic organizer to define the dimensions of rectangles, circles and right circular cylinders.
- Teacher provides students with an example of a three-dimensional figure in its original position then provides multiple orientations to discuss how the location of the figure’s base changes, but the dimensions of the figure do not change.
- For example, two right circular cylinders are shown below with the same dimensions but in different orientations. The base is highlighted in each.

- For example, two right circular cylinders are shown below with the same dimensions but in different orientations. The base is highlighted in each.
- Teacher instructs students to draw a visual of a three-dimensional figure and its dimensions in the context of a real-world problem.
- Teacher directs students to find the exact area of a given circle in terms of pi before replacing the value of pi with an approximation to determine the estimated area.
- Instruction includes color-coding and labeling the dimensions of rectangles, circles and right circular cylinder.
- Teacher provides instruction focused on manipulatives or geometric software for students to understand the difference between the formulas for area, surface area and volume.
- Teacher encourages students to continue to use the parts of the net to calculate the surface area, rather than focusing on the formula.
- Teacher reminds students that while a cylinder may lay on its side, the bases are the circles with the height being the perpendicular distance between them. Provide multiple orientations of objects and continue to break them down to their nets.

### Instructional Tasks

*Instructional Task 1*

**(***MTR.7.1*)An ocean resort decided to build a large room in the shape of a cylinder to host events. The room is 34 feet in diameter with a height of 9 feet. They are going to paint the floor, wall and ceiling blue to make attendees feel like they are floating in the sky. Determine the surface area to be painted so they may order the needed supplies.

*Instructional Task 2*

**(***MTR.7.1*)The reviews from several events in the blue room have come in and attendees are reporting feeling trapped in the enclosed room, rather than floating. So the resort has decided to replace the solid wall with windows for a 360° view of the ocean and surrounding area.

- Part A. How many square feet of windows will need to be ordered to do so?
- Part B. If the resort decided to make a glass ceiling instead of replacing the curved wall, how much glass would be needed?

### Instructional Items

*Instructional Item 1*

Determine the surface area of the cylinder below. Round to the nearest tenth. Note: Figure is not drawn to scale.

*Instructional Item 2*

Determine the surface area of the cylinder below. Write your answer as the exact surface area.

*Instructional Item 3*

Lyndon is making a nylon case for his new snare drum which measures 14 inches in diameter and is 6 inches deep. If the case fits snugly around the drum, how much nylon will Lyndon need?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

This benchmark is part of these courses.

1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))

1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

MA.7.GR.2.AP.1: Match the parts of a given formula to the right circular cylinder using the figure’s net.

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