## Roller Coasting through Functions

• Lesson
6-8
1

In this lesson, students determine the time it takes for a roller coaster to reach the bottom of its tallest drop. They use tables and graphs to analyze the falls of different roller coasters. Students conclude the study by creating their own roller coaster and providing an analysis of its fall.

The vertical motion of an object falling at a constant acceleration can be modeled by the equation:

h = ‑16t2 + v0t + s

 h = height after the object has been dropped (feet) s = initial height (feet) v0 = initial velocity (feet per second) t = time in motion (seconds)

This lesson uses that equation to explore the mathematics of roller coasters. In particular, the lesson has three parts:

• Coaster Track – As an introduction, students use the equation above to determine the height of the coaster at various times and use this information to calculate average velocity.
• Match the Thrill – Students then match the results obtained from the equation with possible graphs.
• Dream Scream Machine – Finally, students use what they've learned to design their own roller coaster.

To simplify the physics, we will assume that the coaster comes to a complete stop before falling and that the train is dropping from the coaster's highest point. For this activity, the equation will be

h = ‑16t2+ s

To begin the main portion of the lesson, ask students to share their personal experiences with roller coasters. Continue the momentum with some facts and figures about roller coasters. Videos can induce a lot of excitement, so you may need to be prepared to refocus the class.

Ask, "How do you think roller coasters and math are related?" [There will be a variety of student responses, such as numbers, speed, height, and formulas.] Segue this discussion into the idea that engineers use functions like h = ‑16t2+ s to determine the coaster's height above ground after a certain amount of time.

### Coaster Track

Distribute the Coaster Track Activity Sheet and read through the introduction emphasizing the variables and what they represent.

• h is the height above ground.
• t is the time in seconds that the train has been dropping.

Students can work independently or in pairs to find the height of the roller coaster at different times by substituting values of t into the function. While evaluating the function, many students multiply ‑16 by t before evaluating t2. Review the order of operations by completing the first row of the data table together. Ask, "How will you know when the coaster has reached the bottom of the drop?" [The height above ground will be 0 feet.]

As students finish the table, discuss the answers. If necessary, conduct an error analysis, having students who disagree put their work on the board. Ask students to point out errors and guide them toward the correct answer. Anticipate the most discrepancies in the second row. Typical responses are 384 (correct), and 144, which is 400 – 162.

Read the rest of the activity sheet as a class. For Question 6, review the symbol Δ (delta). Delta is a Greek letter that mathematicians use to represent difference or change. This means that Δh is the change in height from the start to the end of the coaster, and Δt represents the change in time from start to finish.

Allow students to complete the activity sheet at their own pace. Groups who understand the concept can continue as each part of the assignment gets more challenging, while you circulate to help groups that are struggling. When students finish Coaster Track, check their work and have them move on to Match the Thrill.

### Match the Thrill

Distribute the Match the Thrill Activity Sheet.

Students will call on their knowledge from the Coaster Track activity to complete the data table in the Match the Thrill activity sheet. Students are to assist the engineers by finding the graph that matches the function h = 256 – 16t2 to determine when the Hurricane reaches bottom. They will need to compare the data they collected from the function to each graph's data.

Student must estimate the coordinates in graphs B and C. They will choose the graph that is the closest representation of their data table from Question 1. If students are struggling, lead a small group discussion including the following questions.

"How long does it take the Hurricane to reach the bottom?"

[4 seconds.]
Are there any graphs that can be eliminated?
[Yes, Coaster A. It takes 5 seconds to reach the bottom.]
What is the height of the Hurricane at the start?
[256 feet.]
Which graph starts at a height closest to 256 feet?
[Coaster C.]
How do you know?
[When x = 0, y &approx; 256.]

### Dream Scream Machine

Distribute the Dream Scream Machine Activity Sheet.

This activity will be done individually so each student is held accountable for providing his/her own work. Because students had the option to work in pairs before, you may have to monitor the students by walking around to help the students who are struggling when working on their own.

Each student will create a roller coaster, specifying the highest drop. Some students will want their coasters to have exaggerated heights like 1,000,000 feet, but stress that they are creating coasters that could actually exist. Students will draw and name their roller coaster. They will also use a table and graph to determine how long it takes the coaster to reach the bottom of the drop.

Some students may have trouble recognizing that their equation will look similar to the ones they have used in the first and second activity sheet. Explain that the number from Question 1 is s, the initial height. All they have to do is substitute the s into the equation. Ask them if they know the values of t and h. [No, students will not know the exact values, because h and t are variables in the equation.]

Because the time it takes the coaster to reach the bottom will not be an integer, students may have a negative value for the height. Guide students to understand that the coaster reaches the bottom of the drop between two consecutive integers. Ask, "What does it mean to have a negative height?" [It would mean that the roller coaster traveled below ground, which can't happen. This is a situation when the mathematical equation gives a numerical answer that does not exactly match real life.] If students do not understand that the time to reach the bottom will be a decimal, ask, "Is the amount of time to reach the bottom of the drop a whole number?" [No.] You might also ask, "How do you know what the time will be?" [It will be a decimal. It would have to be between the two last two input values from the data table.] Suggest that students try substituting a value that is between the two consecutive integers. The students should substitute decimal values for t until value of h is close to zero.

Randomly select several students to present their coasters from the Dream Scream Machine activity sheet. They should discuss the height of the tallest drop and explain how long the coaster takes to reach the bottom.

The Coaster Track and Match the Thrill Answer Keys contains answers to the questions that appear on the first two activity sheets. (An answer key is not provided for the Dream Scream Machine activity sheet since all answers are based on student input.)

### Reference

Larson, R., Boswell, L., Kanold, T., & Stiff, L. (2001). Algebra I: Concepts and Skills, Boston: Holt McDougal.

Assessment Options

1. Give students a roller coaster's facts in order to create the function using h = ‑16t2 + s. They can create a data table and graph the function.
2. Have students research other roller coasters around the world using the Internet or resources at the library. Students can use the data to create a graph that represents the coaster dropping.

Extension

Objects thrown from the top of a coaster would use the formula:

h = ‑16t2 + v0t + s

 h = height after the object has been dropped (feet) s = initial height (feet) v0 = initial velocity (feet per second) t = time in motion (seconds)

Have students brainstorm things that fall from the top of a roller coaster. For example, sunglasses might fall off someone's head, with an initial velocity of 0 feet/second. A teddy bear might be thrown down, with an initial velocity of 10 feet/second. Have students use the model equation and tables to predict when different objects will reach the ground.

Questions for Students

1. Why are all of the graphical representations in the first quadrant only?

[Time is on the x‑axis and we cannot have negative time. Height is on the y‑axis, and roller coasters do not go below ground.]

2. The equation we are using does not take into account certain things that may have an effect on the roller coaster as it drops. What are some things that could affect the drop?

[Friction, weight of people in the car, weather.]

3. If you saw the ordered pair (3, 185) in your data table, what would it mean?

[After 3 seconds, the roller coaster is 185 feet above ground.]

4. Which of the following ordered pairs would be unreasonable if it appeared in your data table? Why?

• (2,189)
• (8,-124)
• (-2,1000)
• (0,1)

[The second, third, and fourth are unreasonable ordered pairs. The second is unreasonable because roller coasters do not travel 124 feet below ground. The third is unreasonable because time should not be negative. The last is unreasonable because the initial drop is more than 1 foot tall.]

5. Let t represent the time in seconds and h represent the height above ground of the roller coaster. What is a question that could represent the ordered pair (t, 189)? What is a question that could represent the ordered pair (4.5, h)?"

[The ordered pair (t, 189) represents the question, "How long will it take the roller coaster to be 189 feet above ground?" The ordered pair (4.5, h) represents the question:"How high above ground will the coaster be after 4.5 seconds?"]

Teacher Reflection

• Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer?
• Was students' level of enthusiasm/involvement high or low? Explain why.
• How did your lesson incorporate different styles of learning?
• Were all students able to achieve the class objectives? How do you know?
• Was the group work effective? What adjustments could you make for next time?

### Learning Objectives

Students will:

• Evaluate functions using the order of operations
• Describe and represent functions using tables, graphs and rules
• Compare and relate representations of the same relation

### NCTM Standards and Expectations

• Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

• CCSS.Math.Content.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

• CCSS.Math.Content.8.F.A.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.