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Unit 7, Lesson 5

Building Quadratic Functions to Describe Situations (Part 1)

Algebra 1

5.1

Warm-up

5 mins

Notice and Wonder: An Interesting Numerical Pattern

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

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 that all the values are multiples of 16 and perfect squares. Some may notice that the pattern is not linear and wonder whether it is quadratic.

Launch

Arrange students in groups of 2. Display the table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things that they notice and wonder about.

Activity

None

Student Task Statement

Study the table. What do you notice? What do you wonder?

	0	1	2	3	4	5
	0	16	64	144	256	400

Activity Synthesis

Ask students to share the things they noticed and wondered about. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the table. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the patterns for the values do not come up during the conversation, ask students to discuss this idea.

5.2

Activity

15 mins

Falling from the Sky

Instructional Routines

MLR5: Co-Craft Questions

Materials

None

Activity Narrative

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).

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

Launch

Arrange students in groups of 2. Introduce the context of a rock falling off of a building. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.

Display only the problem stem and related image, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation. After they have done so, ask the partners to compare their questions.
Invite several partners to share one question with the class. Record the questions that they share. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “gravity,” “distance,” “multiple of 16,” and “pattern.”
Reveal the questions, and give students 1–2 minutes to compare them to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
- “Which of your questions is most similar to or different from the ones provided? Why?”
- “Is there a main mathematical concept that is present in both your questions and those provided? If so, describe it.”

5.3

Activity

15 mins

Galileo and Gravity

Instructional Routines

MLR2: Collect and Display

Materials

None

Activity Narrative

In this activity, students continue to explore how quadratic functions can model the movement of a falling object. They evaluate the function seen earlier () for a non-integer input, and then build a new function to represent the distance from the ground of a falling object 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 the ground) record distances measured from opposite ends. (Any value of at a whole-number could work. In this case is selected.)

Lesson Synthesis

To highlight the key ideas from this lesson and the connections to earlier lessons, discuss questions such as:

“We used two different functions to describe the movement of a falling object. One function measured the distance that the object traveled from its starting point, and the other measured its distance from the ground. How are the representations of these functions alike and different?” (The equations both have , but one is positive and the other negative. Their graphs are both curves, but one graph curves upward and the other downward. One table shows increasing values and the other shows decreasing values, but the values change by the same amounts from row to row.)
“How are these functions like or unlike those representing visual patterns in earlier lessons?” (They can all be represented by quadratic expressions. The relationships between the step number and the number of squares or dots are easier to see. The relationships between time and distance are not as obvious.)
“How are the graphs representing falling objects like or unlike those representing visual patterns?” (The graphs representing the patterns curve upward. They show plotted points at whole-number inputs because non-whole-number steps do not make sense. The graphs of falling objects curve upward or downward. The graphs can be continuous, because we can measure the distances even when the number of seconds is fractional.)

to access Cool-down.

Student Lesson Summary

The distance traveled by a falling object in a given amount of time is an example of a quadratic function. Galileo is said to have dropped balls of different mass from the Leaning Tower of Pisa, which is about 190 feet tall, to show that they travel the same distance in the same time. In fact the equation models the distance , in feet, that a metal ball falls after seconds, no matter what its mass.

Because , and the tower is only 190 feet tall, a metal ball hits the ground before 4 seconds.

Here is a table showing how far a metal ball has fallen over the first few seconds.

time (seconds)	distance fallen (feet)
0	0
1	16
2	64
3	144

Here are the time and distance pairs plotted on a coordinate plane:

<p>Graph of the quadratic function <span class="math">\(y=16t^2\)</span> on a coordinate plane, origin <span class="math">\(O\)</span>.</p>

Notice that the distance fallen is increasing each second. The average rate of change is increasing each second, which means that the metal ball is speeding up over time. This comes from the influence of gravity, which is represented by the quadratic expression . It is the exponent 2 in that expression that makes it increase by larger and larger amounts.

Another way to study the change in the position of the metal ball is to look at its distance from the ground as a function of time.

Here is a table showing the distance from the ground in feet at 0, 1, 2, and 3 seconds.

time (seconds)	distance from the ground (feet)
0	190
1	174
2	126
3	46

Here are those time and distance pairs plotted on a coordinate plane:

<p>Graph of the quadratic function <span class="math">\(y = 190 - 16 t^2 \)</span>on a coordinate plane, origin <span class="math">\(O\)</span>.</p>

The expression that defines the distance from the ground as a function of time is . It tells us that the metal ball's distance from the ground is 190 feet before it is dropped and has decreased by when seconds have passed.

Launch

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?”

MLR2 Collect and Display. Collect the language that students use to identify and describe what is the same and different between the tables. Display words and phrases such as “increasing,” “decreasing,” and “difference.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading

Activity

None

Student Task Statement

Galileo Galilei, an Italian scientist, and other medieval scholars studied the motion of free-falling objects. The law they discovered can be expressed by the equation , which gives the distance fallen in feet, , as a function of time, , in seconds.

An object is dropped from a height of 576 feet.

How far does it fall in 0.5 seconds?
To find out where the object is after the first few seconds after it was dropped, Elena and Diego created different tables.

Elena’s table:

time (seconds) distance fallen
(feet)

0 0

1 16

2 64

3

4

Diego’s table:

time (seconds) distance from the ground (feet)

0 576

1 560

2 512

3

4
1. How are the two tables alike? How are they different?
2. Complete Elena’s and Diego’s tables. Be prepared to explain your reasoning.

time (seconds)	distance from the ground (feet)
0	576
1	560
2	512
3
4

Student Response

to access Student Response.

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

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 in 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 and 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 , and the other shows . 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 that students can interpret them. For example, they should see that the -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.)

<p>Graph with x-axis labeled time in seconds and y-axis labeled distance fallen in feet.</p>

<p>Graph with x-axis labeled with time in seconds and y-axis with distance above the ground in feet.</p>

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.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code how the values in each table correspond to the values in each graph.
Supports accessibility for: Visual-Spatial Processing

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Lesson 5

Building Quadratic Functions to Describe Situations (Part 1)

Warm-up

Notice and Wonder: An Interesting Numerical Pattern

Standards Alignment

Building On

Addressing

Building Toward

HSF-BF.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Falling from the Sky

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Galileo and Gravity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Are You Ready for More?

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.A.2

Building Toward