## Competing Coasters

- Lesson

In these activities, students look at attributes that vary from coaster to coaster, attributes that make each scream machine uniquely thrilling. Students use a Web site to view photos of roller coasters from around the country. Based on the pictures, students predict which coaster is faster, which is higher, which goes farther, and which takes longer. They look up data on another Web site to check their predictions. Finally, students evaluate their estimates of speed, height, etc., to see whether their estimation skills improve with experience.

Begin the lesson by assessing students’ prior knowledge of the lesson’s mathematical concepts. Use short activities, problems, or other appropriate means for students to discuss and demonstrate their current knowledge. This information enables you to build upon the understanding of students.

Have students explain their roller coaster experiences. Encourage students to describe:

- How fast they went
- How high they went
- How far they went
- How long the ride lasted

Ask questions, such as the ones below, to stimulate discussion. Make a list of phrases that describe students’ experiences and responses.

- What do you remember most about your first roller coaster ride? What do you like (dislike) about roller coaster rides? (Give students time to share their personal experiences. List student responses in two columns labeled Likes and Dislikes.
- How fast do you think roller coasters go? Why would one roller coaster go faster than another? (Help students see that the coaster’s height, especially the height of the first drop, is the main factor affecting maximum speed.)
- About how long do you think a roller coaster ride lasts? Why would some roller coaster rides last longer than others? (Guide students to see that the height of the coaster and the length and steepness of the track may all influence the duration of the ride.

### Activity: Comparing Coasters

Tell the students they are going to compare and estimate the speed, height, track length, and duration of ride for two roller coasters just by looking at photos. Direct students to the web to choose two pictures of roller coasters (Vortex and King Cobra, for example). Have students list the names of the coasters on their Recording Sheet under Coaster.

As students study the two photos, ask questions such as:

- Which of the two coasters do you think is faster? Why? (Sample response: The King Cobra because it looks higher and it looks like a pretty smooth ride.)
- About how fast do you think each coaster can go? Faster than a school bus? Faster than a car on a highway? Record your estimates in the first two rows of the chart on your Recording Sheet. (Estimates will vary. See sample chart below for actual data.)
- Which of the two coasters is higher? Why do you think so? (Sample response: The King Cobra looks higher with the circular piece of track.)
- About how high do you think each coaster is? Record your estimates. (Estimates will vary. See sample chart below for actual data.)
- Which coaster appears to have the longer track? Why do you think so? (Sample response: The Vortex looks longer because it has lots of dips and twists.)
- About how long do you think each coaster is? Record your estimates. (Estimates will vary. See sample chart below for actual data.)
- Which ride do you think lasts longer? Explain why you think so. (Sample response: The Vortex, because it has to slow down as it tries to climb steep tracks.)
- About how long do you think each ride lasts? Record your estimates. (Estimates will vary. See sample chart below for actual data.)

Once students have made their comparisons and shared their estimates, direct students to the Roller Coaster Database (perform a simple search on the web to find this site) to get the actual data on each roller coaster. Have students record this information on their Recording Sheet as shown below. Give students time to compare their estimates with the actual data for the two roller coasters.

Estimate | Actual | ||||||||

Coaster | Speed | Height | Length | Duration | Speed | Height | Length | Duration | |

King Cobra | 75 mph | 150 ft. | 500 | 1 min. | 50 mph | 95 ft. | 2219 ft. | 2:00 | |

Vortex | 90 mph | 200 ft. | 1000 ft. | 45 sec. | 55 mph | 148 ft | 3800 ft. | 2:30 |

### Activity: Coaster Contest

Organize the class into pairs. Tell students that partners in each pair will take turns:

- Choosing a picture of a coaster
- Estimating the speed, height, length and duration of the ride, and
- Recording those estimates.

Then partners work together to find and record the actual data. Students earn a point for each category (speed, height, etc.) in which their coaster is better than their partner’s.

From the web, Player A chooses a picture of a coaster that he or she thinks is fast, high, etc.

Player A estimates the chosen coaster’s speed, height, length, and duration, and writes the estimates on the Roller Coaster Data Activity Sheet. (Students will evaluate these estimates at the conclusion of the activity to see if their estimating skills improved during the game.)

Roller Coaster Data Activity Sheet

Player B repeats steps 1 and 2, trying to pick a coaster that appears to be faster, higher, etc. than Player A’s estimates. These estimates are also written on the same activity sheet as Player A.

Together, the players obtain actual data for their coaster’s speed, height, etc. from the Roller Coaster Database and record this information on the Roller Coaster Data Activity Sheet.

Students compare the actual data for their coasters, noting which coaster is faster, higher, etc. by circling the greater measurement in each category. A point is awarded to the player whose coaster is faster, a point to the player whose coaster is higher, and so on. Players total their scores for the round.

Estimate | Actual | Points | ||||||||

Coaster | Speed | Height | Length | Duration | Speed | Height | Length | Duration | ||

King Cobra | 75 mph | 150 ft. | 500 | 1 min. | 50 mph | 95 ft. | 2219 ft. | 2:00 | 0 | |

Vortex | 90 mph | 200 ft. | 1000 ft. | 45 sec. | 55 mph | 148 ft | 3800 ft. | 2:30 | 4 |

On the next round, Player B chooses a picture first, Player A chooses second. Players must always choose a picture that has not been used by either player in the current round or in previous rounds.

The player with the most points from all rounds is the winner. After the game has finished, pose the following questions to students:

- What features in a roller coaster will you look for that might contribute to a faster speed? (Sample responses: Height, a smoother track, not having too many dips and twists.)
- How many coaster pictures do you want to review before choosing one you think will score a lot of points?
- To the player who goes second: What will you look for in your roller coaster picture now that you have seen what your partner has chosen? Why do you think the roller coaster you’re choosing goes faster (is higher, is longer, lasts longer) than your partner’s?)

Use the following guiding questions about students' estimates:

- How can you use the information from previous rounds to estimate the speed (height, length, duration) of this coaster from its picture?
- Why is your estimate for this roller coaster’s speed (height, length, duration) greater (less) than your previous roller coaster?
- To the player who goes second: About how much faster do you think this roller coaster goes than your partner’s? Why do you think so?

### Activity: Are Your Estimates Getting Better?

After students have played a game of two or more rounds, ask them if they think their ability to estimate from pictures is improving. Tell them they are going to see if their estimates have been getting better, but they need a way to judge how good an estimate is.

Tell the students that Kim and Zach have made estimates. Kim’s estimate is off by 20 feet. Zach’s is off by 3 feet. Ask them whose estimate is better. (They will probably say Zach’s.) Then explain that Kim and Zach estimated two different things. Kim estimated the distance from her home to school. Zach estimated the length of his school desk. Once again, ask whose estimate is better. (You might draw a diagram, such as the one below, to help students see that Zach’s estimate is relatively closer to the actual amount.)

Explain to the class that there is a way to calculate how good an estimate is. Show students the formula given below and have them use calculators to find the Estimation Score for each estimate they made. Make sure students understand that the lower the Estimation Score, the better the estimate.

Estimation Score = | Difference between estimated and actual amounts | x 100 |

Actual amount |

For example, suppose the actual speed is 50 mph and the estimate is 90 mph:

Estimation Score = | 90 - 50 | × 100 = | 40 | × 100 = 80 |

50 | 50 |

If students get stuck while using this formula with estimates of duration, suggest that they try changing each time into seconds only. For example, suppose the ride actually lasts 1:40 and the estimate is 45 seconds:

1:40 = 1 min. + 40 sec. = 60 sec. + 40 sec. = 100 sec.45 sec. = 45 sec |

Estimation Score = | 100 - 45 | × 100 = | 55 | × 100 = 55 |

100 | 100 |

Tell students to round each score to the nearest whole number. (If they haven’t done decimal rounding yet, simply tell them to use just the whole numbers in their answers.)

If the students are familiar with percent, you can tell them that the Estimation Score is commonly called percent error.

Tell students to record each Estimation Score in one of the charts on the Estimation Score Charts Activity Sheet. The other chart is for the other player.

Estimation Score Charts Activity Sheet

Below is a sample Estimation Score Chart for the first round example:

Estimate | Actual | Points | ||||||||

Coaster | Speed | Height | Length | Duration | Speed | Height | Length | Duration | ||

King Cobra | 75 mph | 150 ft. | 500 | 1 min. | 50 mph | 95 ft. | 2219 ft. | 2:00 | 0 | |

Vortex | 90 mph | 200 ft. | 1000 ft. | 45 sec. | 55 mph | 148 ft | 3800 ft. | 2:30 | 4 |

Ask questions such as the following to help students evaluate their improvement.

- As you look down your Estimation Score Chart, are your Estimation Scores for speed (height, length, duration) getting lower or higher?
- Are you getting better at estimating the speed (height, length, duration) of coasters? How can you tell?

**Assessment Option**

Pose questions, such as the ones in the

Questions for Studentssection, to assess students' understanding of the mathematical concepts involved in this lesson. You may ask students to provide their responses in writing and collect their written responses.

**Extension**

Have students make line graphs of their data. They should use one line for each category (speed, height, length, duration). They will need to decide how to calibrate the vertical axis.

Ask questions, such as the following:

How does the graph help you see whether your estimates are improving?

Both the graph and your score card contain information about your scores. How are these two representations alike and how are they different? What are the advantages of each?

**Questions for Students**

1. Did your ability to estimate the height of roller coasters improve as you looked at more coaster pictures and data? How can you tell?

[Student responses will depend upon their experiences.]

2. Can an Estimation Score be greater than 100? How?

[Yes, if the estimate is more than double the actual amount.]

3. Can an Estimation Score be greater than 100 if the estimate is lower than the actual amount? Explain.

[No, because the difference cannot be more than the actual amount, unless the estimate is less than zero.]

4. Did you notice a relationship between the speed of the coasters you chose and their height? For example, does a taller coaster go faster? Explain what relationship, if any, exists with the coaster data you gathered on speed and height.

[Student responses will depend upon their experiences.]

5. Do you notice a relationship between speed and the duration of the ride for the roller coasters you chose? If so, explain the relationship. If not, tell why you think a relationship does not exist.

[Student responses will depend upon their experiences.]

6. Would the length of a roller coaster track affect its speed? Why or why not?

[No, the length of the coaster track would not affect its maximum speed.]

7. What other factors could be affected by the length of the coaster track?

[The duration of the ride, for example.]

**Teacher Reflection**

- Did students achieve the objectives for this lesson? What evidence supports this claim? What changes should I make to create a more effective lesson?
- What additional experiences do students need to be successful with this activity?
- Were students able to explain their reasoning in a clear and logical manner?
- Have students gone beyond simply stating procedures by justifying and defending their reasoning?
- What are the indicators that students were able to work together and share responsibilities?
- What is the evidence that students have assumed individual responsibility for understanding the mathematics content shared in their group?
- Have individual students asked questions to clarify and extend their understanding of the mathematics content?
- Were students able to quantify, organize, and/or record information?
- What new vocabulary did students use that might need to be reinforced in the next lesson?
- Were directions in the lessons clear and usable by students? If not what adjustments would be appropriate for me to make?
- What additional extensions/experiences would be appropriate?

### Learning Objectives

Students will:

- Make estimations on the quantifiable aspects of a roller coaster’s structure (height and length) and ride (speed and duration).
- Collect data about roller coasters.
- Draw conclusions about the relationships among variables.
- Analyze estimation skills by using a percent error formula.
- Examine patterns on score cards.
- Represent personal scores data on a line graph for the extension.

### NCTM Standards and Expectations

- Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate attribute.

- Select and use benchmarks to estimate measurements.