The Tides of Change

SCORE Mathematics

Standards Connections

Mark Schneider, Norwalk High School, Norwalk, CA

Introduction
In this project, students use the Internet to gather information on tides and then use this information to determine the period of revolution of the moon around the earth (part 1) and the relative pull of the sun and the moon on the tides (part 2).
Prior Knowledge
Part 1
Computing the difference between two times of day
Finding Averages
Solving proportions
Part 2 (additional)
Solving proportions involving cubes
Tasks
Part 1
Each student or partners will use the Internet to find the local time of daily high tides over a 4 day span in 4 different locations. They will then use the average of the daily time differences to determine the number of degrees the moon revolves in one day. This number will then be divided into 360° to determine the number of days required for a complete orbit. An alternate worksheet is provided for students who are more self directed and can work with less structure.
Part 2
Each student or partners will use the Internet to find the masses and distances of the sun and moon from the earth. They will then compare the ratio of the sun to the moon in each instance and find the ratios of the results using the formula for gravity. An alternate worksheet is provided for students who are more self directed and can work with less structure.
Resources
Note: If not logging on to the following sites from within the math SCORE site, append the following before each of the URL page names:

http://score.kings.k12.ca.us./math/lessons/
The Tides of Change Worksheet Part 1 (html) or The Tides of Change Worksheet Part 1 (pdf) Tides_Worksheet_1.html
The Tides of Change Worksheet Part 1 Alternate (html) or The Tides of Change Worksheet Part 1 Alternate (pdf) (less structure, more exploration) Tides_Worksheet_1.html
The Tides of Change Worksheet Part 2 (html) or The Tides of Change Worksheet Part 2 (pdf) Tides_worksheet_2.html
The Tides of Change Worksheet Part 2 Alternate (html) or The Tides of Change Worksheet Part 2 Alternate (pdf) (less structure, more exploration) Tides_worksheet_2.html
Make A Tide Prediction: USA Coast http://www.opsd.nos.noaa.gov/tp4days.html
Why the Tide Cycle is Greater than 24 hrs. Tides_Support_Files/Tides_Cycle_24hrs.html
Heavenly Proportions Tides_Support_Files/Heavenly_Proportions.html
Solar vs. Lunar Tides: Tides_Support_Files/Solar_Lunar_Tides.html
Tug of War Tides_Support_Files/Tug_of_War.html
Why the Moon is Viewed in Phases Tides_Support_Files/Why_Phases.html

Process
Part 1 (Provide The Tides of Change Worksheet Part 1 or Alt 1 )
1. Access Why the Moon is Viewed in Phases for background on the phases of the moon.
2. Access Make A Tide Prediction: USA Coast on the Internet.
3. Select 4 cities from various places along the U.S. coast.
4. For each city, record the high tides for 4 days on your Tides of Change worksheet.
5. Compute the number of minutes later each high tide occurs than the day before.
6. Access Why the Tides Cycles is Greater than 24 hrs. to see why the time changes daily.
7. Compute the average of all the time differences.
8. Use the Proportion setup on the worksheet to determine the number of degrees the moon moves daily.
9. Divide this number into 360° to determine the number of days required for a complete orbit.

Part 2 (Provide The Tides of Change Worksheet Part 2 or Alt 2 )
1. Access Heavenly Proportions on the Internet to gather information on the mass and the distance from the earth of both the sun and the moon.
2. Compute the ratio of the masses sun : moon
3. Compute the ratio of the distances from the earth sun : moon
4. Compute the ratio of gravity using the formula

ratio of the masses / cube of the ratio of the distances
5. Access Solar vs. Lunar Tides to learn about spring and neap tides.
6. Compute the ratio of the pulls during spring and neap tides
7. Access Tug of War to see the cumulative effect of the sun and moon on the tides.

Learning Advice
The Science S.C.O.R.E. site has a lesson on tides which might serve as an introduction. This will be particularly useful for students with no experience or knowledge of tides. A review of finding the difference in minutes between two times and solving proportions will also be helpful.
Depending on your class, you may want to provide fewer steps, allowing students to explore and solve the problem on their own without the help of the chart and step-by-step directions.

Extensions
The Alternate Worksheets contain additional questions for explorations. Some students using the standard worksheets may be interested in these as well:
Part 1
a). The time period we found on this worksheet is called the synodic month. Discover what a sidereal month is.
b.)See what explanation you can find on the web to explain why the daily differences between the time of high tide varies so much. Since the movement of the moon around the earth is constant, shouldn't the time difference be the same each day? Why isn't it?
Part 2
a. Do you think the planet Mars has any effect on our tides? Do some research and use the proportion formula in step 5 above to compute the greatest force Mars could have on the tides compared to the force of the moon.
b. The planet Jupiter is a sea of gasses and has many moons. The four largest are Io, Europe, Ganymede, and Callisto. Compare the pull of the sun on Jupiter to the pull of its 4 largest moons if they were all aligned.

Evaluation
Part 1
Check that the computation of the daily differences is correct. The theoretical daily difference should be around 54.8 minutes, however these will be different in each location and on each day. 54.8 minutes results in a daily movement of the moon around the earth of 13.17°, or an orbit of 27.3 days. Here are some results for various results:
Minutes of Delay

40

45

50

55

60

Daily Degrees

10

11.25

12.5

13.75

15

Days for Orbit

36

32

28.8

26.18

24
Part 2
The ratio of the masses should be around 2.7x10^7 and the ratio of distances around 387.1
The ratio of forces is about .465. The ratio of spring tide to neap tide forces is about 2.74. Accuracy will differ dependent on calculator used and the number of decimal places used for rounding
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California Mathematics Academic Standards:: Grade 8-12:
Probability and Statistics
3.0 Students demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses.
5.0 Students determine the mean and standard deviation of a normally distributed random variable.
6.0 Students know the definitions of the mean, median, and mode of distribution of data and can compute each in particular situations.
8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
NCTM 9-12: Mathematics as Problem Solving; Mathematics as Communication; Mathematics as Reasoning; Mathematical Connections; Algebra; Statistics.