standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
The motion of a falling object is commonly modeled with a quadratic function. This activity prompts students to build a very simple quadratic model using given time-distance data of a free-falling rock. By reasoning repeatedly about the values in the data, students notice regularity in the relationship between time and the vertical distance the object travels, which they then generalize as an expression with a squared variable (MP8). The work here prepares students to make sense of more complex quadratic functions later (that is, to model the motion of an object that is launched up and then returns to the ground).
Launch
Display the image of the falling object for all to see. Students will recognize the numbers from the warm-up. Invite students to make some other observations about the information. Ask questions such as:
“What do you think the numbers tell us?”
“Does the object fall the same distance every successive second? How do you know?”
Arrange students in groups of 2. Tell students to think quietly about the first question and share their thinking with a partner. Afterward, consider pausing for a brief discussion before proceeding to the second question.
Speaking, Reading: MLR5 Co-Craft Questions.
Begin the launch by displaying only the context and the diagram of the building. Give students 1–2 minutes to write their own mathematical questions about the situation before inviting 3–4 students to share their questions with the whole-class. Listen for and amplify any questions involving the relationship between elapsed time and the distance that a falling object travels.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
A rock is dropped from the top floor of a 500-foot tall building. A camera captures the distance the rock traveled, in feet, after each second.
[IMG]
How far will the rock have fallen after 6 seconds? Show your reasoning.
Jada noticed that the distances fallen are all multiples of 16.
She wrote down:
\(\displaystyle \begin {align}16 &= 16 \boldcdot 1\\64 &= 16 \boldcdot 4\\144 &= 16 \boldcdot 9\\256 &= 16 \boldcdot 16\\400 &=16 \boldcdot 25 \end {align}\)
Then, she noticed that 1, 4, 9, 16, and 25 are
\(1^2, 2^2, 3^2, 4^2\)
and
\(5^2\)
.
Use Jada’s observations to predict the distance fallen after 7 seconds. (Assume the building is tall enough that an object dropped from the top of it will continue falling for at least 7 seconds.) Show your reasoning.
Write an equation for the function, with
\(d\)
representing the distance dropped after
\(t\)
seconds.
Student Response
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Anticipated Misconceptions
Some students may question why the distances are positive when the rock is falling. In earlier grades, negative numbers represented on a vertical number line may have been associated with an arrow pointing down. Emphasize that the values shown in the picture measure how far the rock fell and not the direction it is falling.
Activity Synthesis
Discuss the equation students wrote for the last question. If not already mentioned by students, point out that the
\(t^2\)
suggests a quadratic relationship between elapsed time and the distance that a falling object travels. Ask students:
“How do you know that the equation
\(d=16t^2\)
represents a function?” (For every input of time, there is a particular output.)
“Suppose we want to know if the rock will travel 600 feet before 6 seconds have elapsed. How can we find out?” (Find the value of
\(d\)
when
\(t\)
is 6, which is
\(16 \boldcdot 6^2\)
or 576 feet.)
Explain to students that we only have a few data points to go by in this case, and the quadratic expression
\(16t^2\)
is a simplified model, but quadratic functions are generally used to model the movement of falling objects. We will see this expression appearing in some other contexts where gravity affects the quantities being studied.


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