### Clarifications

*Clarification 1:*Within this benchmark, the expectation is not to memorize the volume formula for a right circular cylinder or to find radius as a missing dimension.

*Clarification 2:* Solutions may be represented in terms of pi (π) or approximately.

**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)
- Pi (π)
- Volume

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

Students progress through solving problems involving volumes with a focus of right rectangular prisms in grade 6, right circular cylinders in grade 7 and cylinders, pyramids, prisms, cones and spheres in high school.- Instruction builds on students’ knowledge of finding the volume of a rectangular prism, which also includes the area of the base multiplied by the height. Ask students to make conjectures about how to calculate the volume of a right circular cylinder. Provide the radius and height of several cylinders for students to verify or revise their conjecture
*(MTR.6.1).* - Instruction includes physical or virtual representations for the volume of a right circular cylinder
*(MTR.2.1).*- For example, stack quarters one at a time to show repeated addition (the height) on the area of the base (the first quarter used).

- Instruction focuses on real-world situations to reinforce conceptual understanding of volume versus surface area
*(MTR.7.1).*

### 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 whatever 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 may incorrectly apply the formulas for area, surface area and volume.

### 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).
- Teacher creates and posts an anchor chart with visual representations of a right circular cylinder to assist in correct academic vocabulary when solving real-world problems.
- 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.
- Instruction includes opportunities for students to solve for the volume of a given right circular cylinder in terms of pi before replacing the value of pi with an approximation to determine the estimated volume.
- Instruction includes co-constructing a graphic organizer of a right circular cylinder and color-coding and labeling the dimensions.
- Teacher provides instruction focused on manipulatives or geometric software for students to develop understanding of the difference between the formulas for area, surface area and volume.
- Teacher provides opportunities for students to comprehend the context or situation by engaging in questions (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
- What do you know from the problem?
- What is the problem asking you to find?
- Can you create a visual model to help you understand or see patterns in your problem?

- 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.1.1)**Coffee2Go wants to build a record breaking giant coffee cup for a promotional celebration at a convention for coffee drinkers. In 2020, the Guinness Book of World Records reported that the World’s Largest Cup of Coffee contained 2,010 gallons of fresh-brewed coffee brewed.

- Part A. What questions would need to be answered to approach this problem? Do you have all the information you need to solve the problem? Why or why not?
- Part B. The Guinness Book of World Records, in 2020, documented the Largest Coffee Cup with a height of 8 feet and a diameter of 8 feet. How many cubic feet of coffee can be held in the World’s Largest Coffee Cup?
- Part C. There are approximately 1.25 gallons of coffee in a cubic foot. If Coffee2Go has a goal of creating a coffee cup that holds 2,410 gallons of coffee, what could be the height and diameter of the coffee cup if they are the same size?

*Instructional Task 2*

**(***MTR.7.1*)Karim is setting up an inflatable pool in the yard for the little kids to play. The pool measures 3$\frac{\text{1}}{\text{2}}$ feet across and 16 inches deep. What is the volume of water needed to fill the pool all the way to the top?

### Instructional Items

*Instructional Item 1*

What is the height of a cylinder, with a radius of 3.4 meters (m), whose volume is 198 m³? Round to the nearest hundredth.

*Instructional Item 2*

Determine the exact volume of each of the following cylinders.

*Instructional Item 3*

A candle mold has 24π in³ of space in which wax can be poured to make a candle. If the radius of the mold is 2 inches (in), what is its height?

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

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessment

## Lesson Plans

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## Unit/Lesson Sequence

## STEM Lessons - Model Eliciting Activity

In this MEA, students select jars for candles based on a variety of factors and then design boxes to contain the jars.

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.

## MFAS Formative Assessments

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

## Student Resources

## Problem-Solving Tasks

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.

Type: Problem-Solving Task

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

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

## Parent Resources

## Problem-Solving Tasks

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.

Type: Problem-Solving Task

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

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