options: Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
problem_activity: Activity
The purpose of this activity is to give students opportunities to find the volumes of some cylinders. By finding the area of the base before finding the volume, students are encouraged to compute the volume by multiplying the area of its base by its height. This way of thinking about volume might be more intuitive for students than the formula
\(V=\pi r^2 h\)
. Notice students who plug the radius into a formula for the volume and students who find the area of the base and multiply that by the cylinder’s height.
The second problem of this activity focuses on exploring cylinders in a context. Generally, the volume of a container is the amount of space inside, but in this context, that also signifies the amount of material that fits into the space. Students are not asked to find the area of the base of the silo. Notice students that find the area before computing the volume and those that use the volume formula and solve for it directly. Notice at what step in the computations students approximate
\(\pi\)
.
When working with problems in a given context, it is sometimes convenient or practical to use an approximation of
\(\pi\)
. An example of this is given in the question regarding the volume of a grain silo and interpretations of the answer.
Launch
Provide access to colored pencils to shade the cylinder’s base. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Here is a cylinder with height 4 units and diameter 10 units.
[IMG]
Shade the cylinder’s base.
What is the area of the cylinder’s base? Express your answer in terms of
\(\pi\)
.
What is the volume of this cylinder? Express your answer in terms of
\(\pi\)
.
A silo is a cylindrical container that is used on farms to hold large amounts of goods, such as grain. On a particular farm, a silo has a height of 18 feet and diameter of 6 feet. Make a sketch of this silo and label its height and radius. How many cubic feet of grain can this silo hold? Use 3.14 as an approximation for
\(\pi\)
.
Student Response
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Student Facing
Are you ready for more?
One way to construct a cylinder is to take a rectangle (for example, a piece of paper), curl two opposite edges together, and glue them in place.
Which would give the cylinder with the greater volume: Gluing the two dashed edges together, or gluing the two solid edges together?
[IMG]
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students might use the silo’s diameter instead of the radius to find the volume. Remind them of the formula for area of a circle and the discussion about diameter and radius earlier in this lesson.
Activity Synthesis
The goal of this discussion is to ensure students understand how to use the area of the cylinder’s base to calculate its volume. Consider asking the following questions:
“How does knowing the area of a circular base help determine the volume of a cylinder?” (The volume is this area multiplied by the height of the cylinder.)
“If the cylinder were on its side, how do you know which measurements to use for the volume?” (Since we need area of the base first, the radius or diameter of the circle will always be the measurement used for
\(r\)
, and the height is always the distance between the bases, or the measurement perpendicular to the bases. It doesn’t matter which direction the cylinder is turned.)
“When is it better to use approximations of pi instead of leaving it exact?” (Approximating pi helps us interpret an answer that has pi as a factor, like the area of a circular region or the volume of a cylindrical container.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement.This will help students produce and make sense of the language needed to communicate about different strategies for calculating volume of cylinders.
Design Principle(s): Support sense-making; Optimize output (for explanation)


correct_option_index: 6