This lesson is a "hands-on" activity. Students will investigate and compare the volumes of cylinders and cones with matching radius and height. They will first discover the relationship between the volume of cones and cylinders and transition to using formulas to determine the volumes.
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to explain the relationship between the volumes of cones and cylinders.
Students will be able to use the formula to determine the volume of cones and cylinders.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be able to solve for the area of a circle, given either the radius or the diameter.
Students should have learned about cylinders and cones and how they are alike and different.
Students should be familiar with pi and using it correctly in formulas for area and volume. This includes using approximations for pi, and rounding answers, as well as giving answers in "exact form" including pi.
Students should know the meaning of the concept of volume, what it measures, and that it is given in cubic units.
Students should have some experience and practice with formulas for volume.
Students should have skill with using a compass and protractor.
Students should understand about angle measure for angles between 180º and 360º.
Guiding Questions: What are the guiding questions for this lesson?
How is the volume of a cone related to the volume of a cylinder if they have the same height and radius?
Compare the diameter of the cone and the cylindrical bases? What do you notice?
Compare the height of each cone to the height of the cylinders, explain the similarities and differences.
Do you think there is a connection between the height and diameter of cone and cylinders and their volume?
Compare the volume formulas for cones and cylinders. In what ways are they similar, and in what ways are they different?
Teaching Phase: How will the teacher present the concept or skill to students?
Students should fill in the KW columns of their KWL chart (see Formative Assessment). After eliciting responses on the definition of volume, the teacher should challenge the students to determine how the volume of a cone is related to the the volume of a cylinder. In order to help discover this relationship, student should be provided with this activity:
Students should be placed in groups and given a cylindrical object like a can, oatmeal container, etc.
Students will have to measure cylinder height and radius. With this data and the information in the Further Recommendations section, students will know how to cut out the circular sector to make their cone. The cone must have the same radius and height as the corresponding cylinder. They should verify this after making the cone.
Students will be given plenty of card stock, tape, and scissors to construct a cone. Students can use the following sites to learn how to create a cone with specific measurements, or the teacher can model this for the group:
Next, each student group will use a paper cup and bowl of rice to fill the cone. Students will pour the contents of the cone into the cylinder. They will record how many cones of rice are needed to fill the cylinder.
By repeating the experiment multiple times with different shapes and sizes of cylinders, students should be able to deduce that it takes 3 cones to fill one cylinder. They should notice that this holds true for any cone-cylinder pair, provided both objects have the same height and radius.
The teacher should have groups report their findings and make sure the class agrees that a cone's volume is 1/3 of a cylinders volume when the corresponding radius and height are equal.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The teacher may provide a reference sheet that is used for testing purposes and includes volume formulas on it. Alternatively, the volume formulas for cone and cylinder may be displayed on the board for all students to see.
The teacher will lead a class discussion of the similarities in the volume formula of cones and cylinders (use of pi, radius, height, and that both formulas include the area of the circular base, (pi r ^2)).
The students will be directed to identify the difference in the formulas (The formula for volume of a cone is the same as volume of a cylinder multiplied by 1/3. It should be noted that multiplying by 1/3 is equivalent to dividing by 3).
Students should calculate the volume of a cylinder using the formula with a diameter of 10 in. and a height of 7 in. and be asked "what is the volume of a cone with the same dimensions as above?"
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
The students will practice several volume formula examples with given dimensions.
Calculate the volume of a cone that has a diameter of 12cm and a height of 22cm. What is the volume of a cylinder with the same dimensions?
Calculate the height of a cylinder that has a diameter of 5cm and a volume of 55cm cubed. What is the volume of a cone with the same dimensions as above?
How are the formulas for the volume of a cone and cylinder similar and or different. Why are certain parts similar and why are certain parts different?
Change the diameter and height figures and have students practice until a clear understanding of calculating the volume formula has been mastered.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
As a conclusion to this lesson the students will complete the L portion of the KWL chart. Students should note in the L column of the chart that the cone's volume is one-third a cylinder's volume when the corresponding radius and height are equal.
Given the height of 10 ft. and a diameter of 3 ft., what is the volume of a cone and cylinder with these dimensions?
If the volume of a cone is 82 in. cubed and the diameter is 7 in., what is the height? What is the cylinder's volume with the same height and diameter?
How is the volume of a cylinder related to the volume of a cone?
Students should be asked to fill in the KW parts of a KWL chart that asks the student what they know about volume, cones and cylinders. Have students report out what they wrote or the teacher can circulate and read them as this will help the teacher understand the prior knowledge the students have.
The teacher will guide and moderate a class discussion in which students are asked to compare and contrast cylinders and cones, how their shapes are alike and different, what variables are involved, the general meaning of volume, and the formulas used for calculating volumes of cylinders and cones. Through this discussion the teacher will be able to clarify any misunderstandings and emphasize the important concepts related to the learning objective.
Feedback to Students
A direction sheet will be given to cooperative learning groups of no more then four to follow to investigate the volume of a cone and how it compares to a cylinder. The teacher will move between the groups to answer in questions, make sure students are on task and performing the steps correctly.
Feedback to Students: A direction sheet will be given to cooperative learning groups of no more then four to follow to investigate the volume of a cone and how it compares to a cylinder. The teacher will move between the groups to answer in questions, make sure students are on task and performing the steps correctly.
The students will be investigating the volumes of a cone and a cylinder that have equal radii and heights. If the students can justify with the rice that it takes three equal cones of rice to fill the cylinder and then document their conclusion that a cone's volume is one-third that of a cylinder with equal height and radius, then they have shown mastery.
Accommodations & Recommendations
This is a cooperative learning assignment. Stronger math students could be paired with those who have accommodations of extended time and less work load. The teacher could also take a small group and let the entire lesson be a guided practice and reduce the independent work.
This lesson could also be taught with pyramids and cubes using the same activities.
Calculators could be provided when evaluating formulas.
Suggested Technology: Basic Calculators
Special Materials Needed:
Card stock for the cones
cylinders of various shapes and sizes
rice, oatmeal, sand or sugar of anything to use as fillers
How to Fold a Cone with a Certain Height and DiameterThe basic idea is that every cone can be flattened into a sector of a circle.Suppose you want to build a cone with Height, H, and radius, R. Calculate the following two parameters, L and , that respectively give the size and shape of the sector:Example:Suppose you want to construct a paper cone that is 6 inches tall and 16 inches across. Thus, H = 6 and R = 16/2 = 8.First we calculate L: inchesNext we calculate the sector angle, : degrees.So, in order to fold a cone with the required dimensions, start with a circular sector that has a radius of 10 inches and an angle of 288 degrees. Using this size and shape yields a cone whose height is exactly 6 inches and diameter exactly 16 inches.
This lesson provides work using these math practices:
MAFS.K12.MP.4.1 - Model with mathematics. (Students use physical models (cones and cylinders) to explore the relationship between their volumes.
MAFS.K12.MP.6.1 - Attend to precision. (Students must build the models precisely in order for them to discover the 1:3 relationship between the volumes of cones and cylinders.)