Lesson 5

The purpose of this warm-up is to elicit the idea that the values in the table have a predictable pattern, which will be useful when students consider the context of a falling object in a later activity. While students may notice and wonder many things about this table, the patterns are the important discussion points, rather than trying to find a rule for the function. Because the rule is not easy to uncover, studying the numbers ahead of time should prove helpful as students analyze the function later.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is all the \(y\) values are multiples of 16 and perfect squares. Some may notice the pattern is not linear and wonder whether it is quadratic.

Invite students to share their observations and questions. Record and display them for all to see.

After all responses are recorded, tell students that they will investigate these values more closely in upcoming activities.

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).

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.

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.

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.

In this activity, students continue to explore how quadratic functions can model the movement of a falling object. They evaluate the function seen earlier (\(d=16t^2\)) at a non-integer input, and then build a new function to represent the distance from the ground of a falling object \(t\) seconds after it is dropped. To find a new expression that describes the height of the object, students reason repeatedly about the height of the object at different times and look for regularity in their reasoning (MP8).

The number 576 is chosen as the height (in feet) from which the object is dropped to make it more apparent for students that the values in the two tables (distance fallen and distance from ground) record distances measured from opposite ends. (Any value of \(16t^2\) at a whole-number \(t\) could work. In this case \(t=6\) is selected.)

Arrange students in groups of 2, and suggest that they check in with each other after trying each question. To facilitate peer discussion, consider displaying sentence stems or questions that students could use, such as:

• “Why do you think the object will have fallen that amount in 0.5 seconds?”
• “How do you think the values in the first table are changing? What about in the second table?”
• “How are the two tables alike? How are they different?”

To help students make sense of the two functions, compare and contrast their representations (tables, equations, and graphs) and discuss the connections between them. Ask questions such as:

• “How did you complete the missing values in the first table?” (Substituting 3 and 4 for \(t\) in \(16t^2\) gives the distances fallen after 3 and 4 seconds.)
• “What about those in the second table?” (The distance from the ground is 576 minus the distance fallen, so we can use the values for \(t=3\) and \(t=4\) from the first table to calculate the values in the second table.)
• “Why do the values in the first table increase and those in the other table decrease?” (The distance from the top of the building increases as the object falls farther and farther away. The distance from the ground decreases as the object falls closer and closer to it.)
• “The expression representing the distance fallen shows \(16t^2\) and the other shows \(576 - 16t^2\). Why is that?” (In the first function, the distance fallen, measured from where the object is dropped, will always be positive. In the second function, what’s measured is the height from the ground, so the distance fallen needs to be subtracted from the height of the building.)
• “If we graph the two equations that represent distance fallen and distance from the ground over time, what would the graphs look like? Try sketching the graphs.”

Display graphs that represent the two functions and make sure students can interpret them. For example, they should see that the \(y\)-intercept of each graph corresponds to the starting value of each function before the object is dropped.

They should also notice that the difference in distance between successive seconds gets larger in both cases, hence the curving graphs. (If the differences were constant, the graphs would have been straight lines.)

Display the embedded applet for all to see. Ask students how the graph of the height of the object is related to the path that the object takes as it falls.