standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Optional activity
In the previous activity, students relate the distance a wheel travels to the number of rotations the wheel has made. This activity introduces a new quantity, the
speed
the wheel travels. As long as the rate the wheel rotates does not change, there is a nice relationship between the distance the wheel travels,
\(d\)
, and the amount of time,
\(t\)
. With appropriate units,
\(d = rt\)
: here
\(r\)
is the speed the wheel travels, which can be calculated in terms of the rate at which the wheel spins. The goal of this activity is to develop and explore this relationship.
As they work on this activity, students will observe repeatedly that the distance the wheel travels is determined by the amount of time elapsed and the rate at which the wheel spins.
Watch for students who use the calculations that they have made in earlier problems as a scaffold for answering later questions. For example, when the car wheels rotate once per second, the car travels about 3.7 mph. When the wheels rotate 5 times per second, that means that the car will travel 5 times as fast or about 18.5 miles per hour. Ask these students to share their work during the discussion.
Launch
Remind students to be careful with units as they work through the problems. Also remind them that there are 5,280 feet in a mile and 12 inches in a foot.
Student Facing
The circumference of a car wheel is about 65 inches.
If the car wheel rotates once per second, how far does the car travel in one minute?
If the car wheel rotates once per second, about how many miles does the car travel in one hour?
If the car wheel rotates 5 times per second, about how many miles does the car travel in one hour?
If the car is traveling 65 miles per hour, about how many times per second does the wheel rotate?
Student Response
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Anticipated Misconceptions
Some students may do the calculations in feet but not know how to convert their answers to miles. Remind them that there are 5,280 feet in 1 mile.
Activity Synthesis
Have students share their solutions to the first question and emphasize the different ratios and rates that come up when solving this problem. There are
65 inches of distance traveled
per
rotation of the wheels
1 rotation of the wheel
per
second
60 seconds
per
minute
Finally, there are 12 inches
per
foot if students convert the answer to feet. While this is not essential, it is useful in the next problem as feet are a good unit for expressing both inches and miles.
For the last problem, different solutions are possible including:
In the second question, students have found that the car travels 19,500 feet in an hour if the wheels rotate once per second. This is about 3.7 miles. The number of rotations per second is proportional to the speed. So one way to find how many times the wheels rotate at 65 miles per hour would be to calculate
\(65 \div 3.7\)
.
Students could similarly use their answer to the third question, checking what multiple of 18.5 miles (the distance traveled in an hour at 5 rotations per second) gives 65.
A calculation can be done from scratch, checking how many inches are in 65 miles, how many seconds are in an hour, and then finding how many rotations of the wheel per second result in traveling 65 miles.
The first of these approaches should be stressed in the discussion as it uses the previous calculations students have made in an efficient way.
Conversing: MLR8 Discussion Supports.
To help students to compare and justify their methods for answering the problem about the car traveling 65 mph, ask them to compare their responses with a partner. Listen for and amplify any language that describes different methods for calculations. Invite students to zero in on “why” there are more differences here. Provide students with question stems to help them compare and contrast, such as: “Do you have the same answer? If not, then why?” (Students may have estimated or converted differently or incorrectly.), “Did you use the same conversions for units?”, and “Why do your answers differ?”
Design Principle(s): Maximize meta-awareness; Cultivate conversation


aligns: no