Estimating a Hemisphere
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to review how to manipulate the formulas for volume of a cylinder and cone and consider what they look like when the height and radius are the same, which will be useful when students encounter these shapes throughout the lesson.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the fact that h=r.
Launch
Arrange students in groups of 2. Display the shapes for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Student Task
Here are two shapes. What do you notice? What do you wonder?
Sample Response
Students may notice:
- If the height and radius are the same for both the cylinder and cone, then the volume of the cone is one-third the volume of the cylinder.
- When the radius is the same as the height, the cone and cylinder seem much wider than tall.
- When the height and radius are the same, the volume acts as a function of one variable.
Students may wonder:
- Why would we want a cone or cylinder where the height and radius are the same?
- Does the volume of this type of cone or cylinder change in the same way a cube’s volume does?
- Since their dimensions match, could we put the cone inside the cylinder to create a new shape?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the shapes. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the “missing” variable for height, h, does not come up during the conversation, ask students to discuss this idea. Ensure students understand that the two equations have no variable h for height since the h was replaced by r due to the height and radius being the same for both shapes.
Standards
- 8.G.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- 8.G.C.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Estimating a Hemisphere
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to review how to manipulate the formulas for volume of a cylinder and cone and consider what they look like when the height and radius are the same, which will be useful when students encounter these shapes throughout the lesson.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the fact that h=r.
Launch
Arrange students in groups of 2. Display the shapes for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Student Task
Here are two shapes. What do you notice? What do you wonder?
Sample Response
Students may notice:
- If the height and radius are the same for both the cylinder and cone, then the volume of the cone is one-third the volume of the cylinder.
- When the radius is the same as the height, the cone and cylinder seem much wider than tall.
- When the height and radius are the same, the volume acts as a function of one variable.
Students may wonder:
- Why would we want a cone or cylinder where the height and radius are the same?
- Does the volume of this type of cone or cylinder change in the same way a cube’s volume does?
- Since their dimensions match, could we put the cone inside the cylinder to create a new shape?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the shapes. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the “missing” variable for height, h, does not come up during the conversation, ask students to discuss this idea. Ensure students understand that the two equations have no variable h for height since the h was replaced by r due to the height and radius being the same for both shapes.
Standards
- 8.G.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- 8.G.C.9·Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.