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Everest Math
Teacher's Notes
Unit Summary
These lessons are designed to augment existing unit planning. Use to add color to textbook exercises! They can be taught separately as one-off lessons. Teach one - or teach them all! Extension exercises are offered for each lesson AND other unit ideas are provided as well.
Teaching the Lessons
Monday: Lesson One
Incorporate into the following possible units:
Decimal computation
Discrete mathematics
Distance problem solving
The Maneuvering through Kathmandu work sheet, using the Kathmandu Map, asks students to add and multiply decimal numbers as they determine the shortest route between several locations on a map of Kathmandu. The cost for traveling the route via rickshaw must also be determined. A great problem solving exercise!
Extending the learning:
Create new problems by changing the distance to each of the three locations. For example, what happens if the park is closed and they are forced to go to another lunch vendor that is 1.5 miles from the park, 1.24 miles from the hospital, 2.3 miles from the hostel and 21.809 miles from Durbar Square? Assume all other distances remain the same. Which route will be the shortest OR cheapest?
Make a new map that shows FOUR places Ben and Nancy must stop before they go to Durbar Square. How many routes are possible now? Which is the shortest route?
Have students think of other real-life situations where they need to determine the shortest route possible (home, school, hockey practice, friends' houses)? How would they solve these new problems?
The Measuring Mountains work sheet is a backgrounder on mountain heights. Students are asked to find the difference between the greatest possible height and the least possible height of two mountains.
A logical extension would be to give multiplication equations from a textbook.
For older students, you may actually want to get them to attempt to MEASURE the height of Mt. Everest for a trigonometry or angle of elevation lesson. Here's a "possible" suggestion:
The height of Everest can be measured using the law of gravity, that is: W = G(m1 m2)/(r*r) where G is a constant, m1 and m2 are the masses of two objects and is the distance between these objects. Let m1 be the mass of the earth, and m2 the mass of another object. Assuming we know the mass of the earth we can measure the height of Mount Everest by first weighing the object at the sea level, to getW1, and then take the object to the top of the mountain, and weigh again (using a spring scale) to get W2. In these two cases the value would be different. In the first case it's R, the radius of the earth, and in the second case, R + H, with H the height of the mountain. Therefore we have two equations W1 = G(m1 m2)/( R*R )W2 = G(m1 m2)/( (R+H)(R+H) ). Since mass is constant, W1 and W2 are known, we can solve the above two equations to get the height of the mountain.
SUGGESTED VIEWING:Twin Peaks, a nine-minute video demonstrates surveying techniques used to measure the length of India and the relative heights of Everest and K2. Discusses triangulation, global positioning satellites, trig, and 3-D mapping. Includes questions. Supplier: B.C. Learning Connection Inc
(604) 387-5331 Price: $20.00 Canadian; ISBN/Order No.: TE0009
Before teaching latitude and altitude, you might want to teach an introductory lesson about time zones. Students could compute and configure time zones across Asia as an introduction to Nepal. You'd need classroom globe(s) with time zones and maps which displays time zones centered on Greenwich Mean Time (starting at 0 at GMT time zones extend about one hour each way west and east to + or - 12 GMT, giving 24 time zones for the world).
Students - working in pairs - add or subtract, in hours and/or minutes, the differences in various time zones. For example, if it is:
h 1:00 p.m. on London, England, then it is __________in Sydney, Australia
h 10:00 p.m. in Beijing, China, then it is ____________in London, England
h 5:00 p.m. in Tokyo, Japan, then it is _____________in Tashkent, Russia
h 2:00 p.m. in Moscow, Russia, then it is ___________ in Melbourne, Australia
h Midnight in Shanghai, China, then it is ____________in Cairo, Egypt
Students could also make up their own questions.
Extending the learning: After computing the time differences, students could make a list of the things a person would be doing at a certain time in a specific country and compare and contrast that with what the people in a different time zone might be doing.
You'll need a world map, newspaper and/or Internet weather report to teach Latitude and Altitude. Here are suggestions/answers to lead your class discussion:
A) Students brainstorm why latitude is so important to determining temperature. Have them record the answer: At 0 degrees latitude at the equator, the sun's rays are the strongest. The further away in latitude a region is from the equator, the cooler the temperature.
B & C) Namche Bazaar and Miami both have the same latitude.
E) VERY unlikely the temperatures would be the same! Discuss with students why it would be cooler in Nepal. Define and explain altitude.
F) Air molecules move and when they do they bump into each other, creating heat. You can relate this concept to a basketball. If a basketball is dribbled for a while, the surface of the ball will get warm because of the friction. As you go up in altitude, the density of air molecules decreases. This means there are fewer molecules to collide with each other and cause heat. The air becomes thinner and cooler.
G) Miami's elevation, or altitude, is 6 ft. above sea level. Namche Bazaar's elevation is 10,000 ft. above sea level.
H) Although scientists make calculations to predict the events of nature, the predictions may not be accurate. How often are weather reports right? Nature is complex system that is always changing. We can only make our best guesses…
Extending the learning: Use the calculation to predict the temperature at Pumori Base Camp (18,000 ft.) and the summit (24,000 ft.). Predict the temperature until the team reaches Base Camp. Record the actual temperature daily from the Internet weather report. Graph estimated temp. vs. actual temp.
o Namche Bazaar - 10,000 ft. o Tengbochey - 13,000 ft. o Pangbochey - 15,000 ft.
o Dugla - 15,500 ft. o Lobuchey - 16,500 ft. o Pumori base camp - 18,000 ft.
If 29,035 foot high Mount Everest were standing on the ocean floor in the Marianas Trench in the western Pacific Ocean, its peak would be 7170 feet below sea level. How deep is the Marianas Trench? Draw a picture to solve this problem. Add the height of Mount Everest to the distance from the top of Everest to the ocean surface (sea level) to get the answer. Your answer may be in any of the following forms:
It would also be interesting to teach Everest statistics lessons. It is possible to calculate pretty much anything on the mountain: temperature rating of sleeping bags, altitude, wind pressure, jet stream velocity, ad infinitum. How do these statistics help climbers? How accurate are the stats? How much faith should people put in stats? Can stats be skewed?
Why would people climb with such frightening stats? Is it the "couldn't happen to me/I won't be a statistic" mentality? How do stats help/hinder us as we make life decisions…
Think about the teens in your classroom. Where do they look for statistical information? Drug statistics are an excellent example of mixed messages. Exactly how reliable is the information from various sources such as news media and the Internet? Here are some news reports you could use in your stats lessons, including editorials about the 2001 drug movie Traffic.
An estimated 13.6 million Americans were current users of illicit drugs in 1998. Although this number is slightly less than the 13.9 million estimate for 1997, the difference is not statistically significant. By comparison, the number of current illicit drug users was at its highest level in 1979 when the estimate was 25.0 million.
9.9 percent of youths age 12-17 reported current use of illicit drugs in 1998. This estimate represents a statistically significant decrease from the estimate of 11.4 percent in 1997. The rate was highest in 1979 (16.3 percent), declined to 5.3 percent in 1992, then increased to 10.9 percent in 1995. The percent of youth reporting current use of illicit drugs has fluctuated since 1995 (9.0 percent in 1996 and 11.4 percent in 1997). 8.3 percent of youths age 12-17 were current users of marijuana in 1998. The prevalence of marijuana use among youth did not change significantly between 1997 when it was 9.4 percent and 1998 when it was 8.3 percent. Youth marijuana use reached a peak of 14.2 percent in 1979, declined to 3.4 percent in 1992, more than doubled from 1992 to 1995 (8.2 percent), and has fluctuated since then (7.1 in 1996 and 9.4 percent in 1997).
An estimated 1.8 million (0.8 percent) Americans age 12 and older were current users of cocaine in 1998. The estimate was 1.5 million (0.7 percent) in 1997; but the difference is not statistically significant. Cocaine use reached a peak of 5.7 million or 3.0 percent of the population in 1985.
The percent of youths reporting current use of inhalants decreased significantly from 2.0 percent in 1997 to 1.1 percent in 1998.
An estimated 4.1 million people met diagnostic criteria for dependence on illicit drugs in 1997 and 1998, including 1.1 million youths age 12-17.
While it is true that casual drug use has declined 50% since its peak in 1979-80, drug addiction has not declined. At least 6 million people in the U.S. are drug-dependent--and drug-related deaths have doubled since 1989. Heroin and MDMA(methylenedioxymethamphetamine) use by teens more than doubled in the last 10 years. We have doubled our incarceration rate. Drug purity is up, price is down and availability is unchanged.
Ecstasy use has doubled in the past two years. This year's "Monitoring the Future" report (a national survey of legal and illegal drug use among nearly 50,000 secondary students) found heroin use among high school seniors is at record levels. According to the Centers for Disease Control and Prevention, injection drug use has directly and indirectly accounted for 58% of all AIDS cases among women in the United States.
This country clearly has a drug problem. How has the drug war contributed or not contributed to a solution?
Ben and Nancy are about to start trekking to Everest Base Camp. BUT, before they do they have to complete last minute errands in Kathmandu! From their hostel in the Thamel district they go straight to New Baneshwar to pick up their Nepalese visa extensions. Then, to the hospital along the Vishnumati River to fetch medical supplies. And finally, to Ratna Park to grab a lunch to eat en route to their expedition departure point: Durbar Square.
Rickshaws change American dollars (not rupees!) by the mile in Kathmandu. Ben and Nancy are working on a budget, so they want to take the shortest route possible. You will need to find out all the possible ways they can get from their hostel to Durbar Square to determine which route is the shortest!
List all the possible routes using a tree diagram. You might want to calculate the total number of possible combinations and then list them.
Compute the distances for each route and determine which is shortest.
Calculate Ben and Nancy's fare for the shortest route. Assume they spend five minutes at each stop along the way and the rickshaw fares are $1.75 for the first mile, $.75 rupees for each mile thereafter and $.50 for each minute the meter runs while the driver is waiting.
How much do Ben and Nancy save by traveling this route vs. the longest route?
Mount Everest once went by the pedestrian name of Peak XV among westerners. That was before surveyors established that it was the highest mountain on earth, a fact that came as something of a surprise-Peak XV had seemed lost in the crowd of other formidable Himalayan peaks, many of which gave the illusion of greater height.
In 1852 the Great Trigonometrical Survey of India measured Everest's elevation as 29,002 feet above sea level. This figure remained the officially accepted height for more than one hundred years. In 1955 it was adjusted by a mere 26 feet to 29,028 (8,848 m).
The mountain received its official name in 1865 in honor of Sir George Everest, the British Surveyor General from 1830-1843 who had mapped the Indian subcontinent. He had some reservations about having his name bestowed on the peak, arguing that the mountain should retain its local appellation , the standard policy of geographical societies.
Before the Survey of India, a number of other mountains ranked supreme in the eyes of the world. In the 17th and 18th centuries, the Andean peak Chimborazo was considered the highest. At a relatively unremarkable 20,561 feet (6,310 m), it is in fact nowhere near the highest, surpassed by about 30 other Andean peaks and several dozen in the Himalayas. In 1809, the Himalayan peak Dhaulagiri (26,810 ft.; 8,172 m) was declared the ultimate, only to be shunted aside in 1840 by Kanchenjunga (28,208 ft.; 8,598 m), which today ranks third. Everest's status has been unrivaled for the last century-and-a-half, but not without a few threats.
The most recent challenge came from a 1986 American expedition climbing (28,250 ft., 8,611 m) in the Karakoram range in Pakistan. According to their measurements, K2 was actually 29,284 feet, beating Everest by a cool 256 feet. Had this figure been accepted, mountaineering history would have required drastic revision: Everest would have taken a back seat to K2 and no longer been the ultimate of geographical extremes.
K2 was left even further back in the clouds when Everest's official height was revised in 1999. On May 5, 1999, a team of nine climbers summitted Everest, armed with state-of-the-art satellite measuring devices. Six months later the results of their survey were announced: Everest is in fact 29,035 feet (8,835 meters)-seven feet or two meters higher than the last official (1955) measurement.
1Pedestrian: ordinary, commonplace, undistinguished 2Appellation: name or title
It is remarkable how accurate all the official measurements of Everest have been. Conducted 147 years earlier, the Great Trigonometrical Survey of India in 1852 recorded Everest's height at 29,002 feet--a mere 33 feet off the mark.
If the number 29,028 is seared into your mind along with 1066, 1492, and other seemingly immutable figures, get ready for a big change in the holy canon of statistics. That number, of course, was the elevation-in feet-of the world's highest mountain. As of Nov. 11, 1999, the new official height of Mt. Everest was announced as 29,035 feet.
The new height was determined by using satellite-based technology: the Trimble Global Positioning System (GPS). A team of seven climbers measured the mountain from the summit on May 5, 1999. The data was collected from various GPS satellite receivers-one of which had to be placed in bedrock-at the very top of Everest. It took the climbers a number of attempts over several years until they were able to successfully set up the equipment at the summit.
The number 29,028 was always a peculiarly American figure anyway, and will not be missed by the rest of the world, who use the metric system. The number 8,848 meters is the figure indelibly etched on the minds of the 5.7 billion non-Americans of the world (or at least those among them who, through coercion or passion, have a set of common statistics committed to memory).
Now, that number has been replaced by one with a more mnemonic ring: 8,850 meters. Future generations of school children from Canada to the Czech Republic ought to have an easier time remembering the height of Everest now that two extra meters have rounded out the number.
As for Americans, there may be even more statistical surprises awaiting in their lifetimes. Someday they may be asked to change their mindsets yet again-just when 29,035 feet begins to sink in and feel natural they may be asked to switch to 8,850 meters.
TASK - ROUND 'EM UP!
Rounding mountains: Mt. Everest is roughly thirty thousand feet high. Actually, that is the height rounded to the nearest ten thousand feet. Mount Cook (New Zealand) measures in at roughly ten thousand feet. Again, this is the height rounded to the nearest ten thousand feet. Given these estimates, find the difference between the greatest possible height and the least possible height of the two mountains. Explain in mathematical terms why this difference is larger than the difference of twenty thousand between the two estimated figures.
Bonus: Name the tallest mountain on the continent where you live and give the height of the mountain. Be sure to include WHERE you found your information.
3Immutable: not susceptible to change; ageless 4Mnemonic: a device, such as a formula or rhyme, used to aid in remembering
To solve a multiplication equation, use the inverse operation of division. Divide both sides by the same non-zero number.
Got it? Try this: Mt. Everest in Nepal is the world's tallest mountain, about 29,000 ft. high. It is twice as high as Mount Whitney in California. How high is Mount Whitney?
We can write a multiplication equation to find the answer to problems like this. Our unknown number is the height of Mount Whitney. Let x represent this height. We know that 2x is the height of Mount Everest. We can write our equation like this:
2x = 29,000 ft
To solve this equation, we can use the inverse of multiplying by 2, which is dividing by 2. If we divide the left side of the equation by 2, we will get x alone on the left. Remember, any operation done to one side must also be done to the other side, so we must also divide the right side by 2.
We divide, and find that x is equal to 14,500 ft. This is very close to the actual height of Mount Whitney, which is 14,494 ft.
Now you try some mental math (without using a calculator!):
During this unit you will learn about the relation between latitude and altitude with temperature, specifically in the country of Nepal. Where is Nepal? In Asia! Look it up right now on a world map.
You are going to be able to predict and understand why expedition teams climbing mountains go through different temperature zones during the trek.
LET'S START:
A) Brainstorm: what determines a region's temperature? Hint: latitude is important in determining temperature!
B) Using a world map, determine the latitude of Namche Bazaar.
C) What city in the United States is located at exactly the same latitude?
D) Find out the temperature at Namche Bazaar. And in Miami? Are they the same? If not, determine the difference in degrees of F.
Read the Internet weather report to find out the temperature at Namche Bazaar. Using a newspaper, find out what the temperature is in Miami. Record the actual temperature of the two places. Are they the same?
E) Chances are, it is cooler in Namche Bazaar. Determining a region's temperature is not as simple as just going by its latitude! Altitude also plays a crucial role. The further up a region is in altitude, the cooler the temperature. Why do you think that is?
F) Air is made of molecules. When the molecules move, what happens?
Now research and record the altitude at Miami and at Namche Bazaar! Air cools at a rate of 3.3 degrees F per 1000 ft. Using the temperature in Miami and assuming that Miami is at 0 ft. elevation for calculating purposes, determine the estimated temperature at Namche Bazaar. Compare it to the actual temperature. Is the estimate close? There is probably a difference between the estimated and the actual temperature. Why?
