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 site www.joyrides.com
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 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 site www.joyrides.com, 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.)
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.
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?
