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Inset Article

QUANTUM CARDS

A SIMPLE FALLING CARD IN PRINCIPLE LEADS TO A QUANTUM MYSTERY

According to quantum physics, an ideal card perfectly balanced on its edge will fall down in both directions at once, in what is known as a superposition. The card's quantum wave function (blue) changes smoothly and continuously from the balanced state (left) to the mysterious final state (right) that seems to have the card in two places at once. In practice, this experiment is impossible with a real card, but the analogous situation has been demonstrated innumerable times with electrons, atoms and larger objects. Understanding the meaning of such superpositions, and why we never see them in the everyday world around us, has been an enduring mystery at the very heart of quantum mechanics. Over the decades, physicists have developed several ideas to resolve the mystery, including the competing Copenhagen and many-worlds interpretations of the wave function and the theory of decoherence.

Inset Article

COPENHAGEN INTERPRETATION

IDEA: Observers see a random outcome; probability given by the wave function.

ADVANTAGE: A single outcome occurs, matching what we observe.

PROBLEM: Requires wave functions to "collapse,"but no equation specifies when.

When a quantum superposition is observed or measured, we see one or the other of the alternatives at random, with probabilities controlled by the wave function. If a person has bet that the card will fall face up, when she first looks at the card she has a 50 percent chance of happily seeing that she has won her bet. This interpretation has long been pragmatically accepted by physicists even though it requires the wave function to change abruptly, or collapse, in violation of the Schrodinger equation.

Inset Article

MANY-WORLDS INTERPRETATION

IDEA: Superpositions will seem like alternative parallel worlds to their inhabitants.

ADVANTAGE: The Schr6dinger equation always works: wave functions never collapse. ..BI.-PROBLEMS:

The bizarreness of the idea. Some technical puzzles remain.

If wave functions never collapse, the Schrbdinger equation predicts that the person looking at the card's superposition will herself enter a superposition of two possible outcomes: happily winning the bet or sadly losing. These two parts of the total wave function (of person plus card) carry on completely independently, like two parallel worlds. If the experiment is repeated many times, people in most of the parallel worlds will see the card falling face up about half the time. Stacked cards (right) show 16 worlds that result when a card is dropped four times.

Inset Article

DECOHERENCE: HOW THE QUANTUM GETS CLASSICAL

IDEA: Tiny interactions with the surrounding environment rapidly dissipate the peculiar quantumness of superpositions.

ADVANTAGES: Experimentally testable. Explains why the everyday world looks "classical" instead of quantum.

CAVEAT: Decoherence does not completely eliminate the need for an interpretation such as many-worlds or Copenhagen.

The uncertainty of a quantum' superposition (left) is different from the uncertainty of classical probability, as occurs after a coin toss (right).A mathematical object called a density matrix illustrates the distinction. The wave function of the quantum card corresponds to a density matrix with four peaks. Two of these peaks represent the 50 percent probability of each outcome, face up or face down. The other two indicate that these two outcomes can still, in principle, interfere with each other. The quantum state is still "coherent." The density matrix of a coin toss has only the first two peaks, which conventionally means that the coin is really either face up or face down but that we just haven't looked at it yet.

Decoherence theory reveals that the tiniest interaction with the environment, such as a single photon or gas molecule bouncing off the fallen card, transforms a coherent density matrix very rapidly into one that, for all practical purposes, represents classical probabilities such as those in a coin toss. The Schrbdinger equation controls the entire process.

SPLITTING REALITY

It is instructive to split the universe into three parts: the object under consideration, the environment, and the quantum state of the observer, or subject. The Schrodinger equation that governs the universe as a whole can be divided into terms that describe the internal dynamics of each of these three subsystems and terms that describe interactions among them. These terms have qualitatively very different effects.

The term giving the object's dynamics is typically the most important one, so to figure out what the object will do, theorists can usually begin by ignoring all the other terms. For our quantum card, its dynamics predict that it will fall both left and right in superposition. When our observer looks at the card, the subject-object interaction extends the superposition to her mental state, producing a superposition of joy and disappointment over winning and losing her bet. She can never perceive this superposition, however, because the interaction between the object and the environment (such as air molecules and photons bouncing off the card) causes rapid decoherence that makes this superposition unobservable.

Even if she could completely isolate the card from the environment (for example, by doing the experiment in a dark vacuum chamber at absolute zero), it would not make any difference. At least one neuron in her optical nerves would enter a superposition of firing and not firing when she looked at the card, and this superposition would decohere in about 10-20 second, according to recent calculations. If the complex patterns of neuron firing in our brains have anything to do with consciousness and how we form our thoughts and perceptions, then decoherence of our neurons ensures that we never perceive quantum superpositions of mental states. In essence, our brains inextricably interweave the subject and the environment, forcing decoherence on us.

M. T. and J.A.W.

The Authors

MAX TEGMARK and JOHN ARCHIBALD WHEELER discussed quantum mechanics extensively during Tegmark's three and a half years as a postdoc at the Institute for Advanced Studies in Princeton, N.J. Tegmark is now an assistant professor of physics at the University of Pennsylvania. Wheeler is professor emeritus of physics at Princeton, where his graduate students included Richard Feynman and Hugh Everett III (inventor of the many-worlds interpretation). He received the 1997 Wolf Prize in physics for his work on nuclear reactions, quantum mechanics and black holes. In 1934 and 1935 Wheeler had the privilege of working on nuclear physics in Niels Bohr's group in Copenhagen. On arrival at the institute he asked a workman who was trimming vines running up a wall where he could find Bohr. "I'm Niels Bohr," the man replied. The authors wish to thank Emily Bennett and Ken Ford for their help with an earlier manuscript on this topic and Jeff Klein, Dieter Zeh and Wojciech H. Zurek for their helpful comments.

REASSESSING AN ANCIENT ARTIFACT

Source: Science News, 01/27/2001, Vol. 159 Issue 4, p56, 1/3p

Author(s): Peterson, Ivars

From New Orleans at the Joint Mathematics Meetings

The famous Mesopotamian clay tablet known as Plimpton 322 has tantalized historians of mathematics ever since its discovery more than 60 years ago. Scholars have considered the tablet to be an anomalous mathematical exercise well in advance of its time. They have variously interpreted the cryptic columns of numbers, written in the wedge-shaped script called cuneiform, as a trigonometric table or a sophisticated scheme for generating Pythagorean triples. A Pythagorean triple is a set of three whole numbers, a, b, and c, such that a[2] + b[2] = c[2].

Now, Eleanor Robson of the Oriental Institute at the University of Oxford in England offers an alternative explanation of the tablet's purpose. The tablet served as a guide for a teacher preparing exercises involving squares and reciprocals, she suggests. Robson also pinpoints the tablet's date to within 40 years of 1800 B.C. and says that it probably came from Larsa, a Mesopotamian city about 100 miles southeast of Babylon.

Previous historians had typically failed to consider the tablet's cultural context and relied on later mathematical developments to infer its purpose. For example, the concept of angle measurement, which is essential for a trigonometric table, was not developed until nearly 2,000 years after the tablet was made. New scholarly approaches to Mesopotamian mathematics, however, combine historical, linguistic, and mathematical techniques to address questions such as, How did Mesopotamians approach mathematical problems, and what role did these problems play in their society? "We need to understand the document in its historical and cultural context," Robson says. "Neglecting these factors can hinder our interpretations."

By comparing Plimpton 322 with other ancient tablets, Robson established that its style is consistent with temple records and documents of about 1800 B.C. in Larsa. Scrutiny of various mathematical tablets revealed the importance of computational methods based on reciprocals (1/x) and squares (chi square) of numbers. Robson also found examples of student exercises that consisted of problem lists, each one registering essentially the same problem with slightly different numbers.

Such evidence enables modern mathematicians to view Plimpton 322 "not as a freakish anomaly in the history of early mathematics but as the epitome of Mesopotamian mathematical culture at its best," Robson says. "It's a well-organized, well-executed, beautiful piece of mathematics." Robson describes her findings in a report scheduled for publication in HISTORIA MATHEMATICA.

PRIME PROOF ZEROS IN ON CRUCIAL NUMBERS

Source: Science News, 12/02/2000, Vol. 158 Issue 23, p357, 1/2p

Author(s): Peterson, I.

Fermat's last theorem is just one of many examples of innocent-looking problems that can long stymie even the most astute mathematicians. It took about 350 years to prove Fermat's tantalizing conjecture.

Now, Preda Mihailescu of the Swiss Federal Institute of Technology in Zurich has proved a theorem that is likely to lead to a solution of Catalan's conjecture, another venerable problem involving relationships among whole numbers. He describes his result in a paper to be published in the Journal of Number Theory.

"This is a very important contribution," says mathematician Andrew Granville of the University of Georgia in Athens. Mihailescu's work probably puts the resolution of Catalan's problem into the foreseeable future, he notes.

Named for Belgian mathematician Eugone Charles Catalan, the conjecture concerns powers of whole numbers. For example, the sequence of all squares and cubes of whole numbers greater than 1 begins with the integers 4, 8, 9, 16, 25, 27, and 36. In this sequence, 8 (the cube of 2) and 9 (the square of 3) are not only powers but also consecutive whole numbers.

In 1844, Catalan asserted that among powers of whole numbers, the only pair of consecutive numbers that arises is 8 and 9. Since then, Catalan's conjecture has posed a challenge to number theorists akin to that provided by Fermat's last theorem (SN: 11/5/94, p. 295).

Solving Catalan's problem amounts to a search for whole number solutions to the equation x[superscript]p - y[superscript]q = 1, where x, y, p, and q are all greater than 1. The conjecture suggests that there is only one such solution: 3[superscript]2 - 2[superscript]3 = 1.

In a major step toward resolving Catalan's conjecture, Robert Tijdeman of the University of Leiden in the Netherlands showed in 1976 that even if it is not true, there is a finite rather than an infinite number of solutions to the equation. In effect, each of the exponents p and q must be less than a certain value.

Last year, Maurice Mignotte of the Universite Louis Pasteur in Strasbourg, France, demonstrated that p had to be less than 7.15 5 1011 and q less than 7.78 5 10[superscript]16. Meanwhile, computations showed that no consecutive powers other than 8 and 9 occur below 10[superscript]7.

In the latest advance, Mihailescu proved that, if additional solutions to the equation exist, the exponents p and q are a pair of what are known as double Wieferich primes. These pairs obey the following relationship: p[superscript](q - 1) must leave a remainder of 1 when divided by q[superscript]2, and q[superscript](p - 1) must leave a remainder of 1 when divided by p[superscript]2. The pair of prime numbers 2 and 1,093 fits this relationship.

Only six examples of double Wieferich primes have been identified so far. All of these pairs are below the range specified by the computations addressing Catalan's conjecture. A major collaborative computational effort (http://www.ensor.org) has now been mounted to find additional double Wieferich primes, but mathematicians are betting that a theoretical approach to proving Catalan's conjecture will beat out the computers.

MATH & MUSIC: THE MAGICAL CONNECTION

Source: Scholastic Parent & Child, Dec2000/Jan2001, Vol. 8 Issue 3, p50, 5p, 3c

Author(s): Church, Ellen Booth

Math and music unite the two hemispheres of the brain--a powerful force for learning.

did you ever consider the skills your child uses when she sings a song such as "This Old Man"? She is matching and comparing (through pitch, volume, and rhythm), patterning and sequencing (through melody, rhythm, and lyrics), and counting numbers and adding. Add dramatic hand movements or clapping to the beat, and you have created an entire package of learning rolled into one song!

In recent years, there has been a considerable amount of research on the effect of music on brain development and thinking. Neurological research has found that the higher brain functions of abstract reasoning as well as spatial and temporal conceptualization are enhanced by music activities. Activities with music can generate the neural connections necessary for using important math skills.

Music and math seem to create a connection between the two hemispheres of the brain. Music is considered a rightbrain activity, while math is a left-brain activity. When combined, the whole child is engaged not only in the realm of thinking but in all the other domains of social-emotional, creative, language, and physical development. Music and math: Together they make a complete developmental package.

The melody of math

The next time your child is singing a song she learned at school, join in. Clap or tap a beat to go with it--even if you don't know the words. Rhythm is made up of patterns--just like math. By focusing on the beat, you will be making the structure of math audible. Tap the beat to a favorite song and see if your child can guess it. Clap the rhythm of her name.

Make up a rhythm for your child to echo back to you.

Take time to sing counting songs too. Remember "One Potato, Two Potato" or "10 Little Monkeys"? Songs like these help children learn about numbers by giving them a "hands-on" experience. Instead of counting by rote, children can count to a beat, a tune, a motion, or an object--or all of the above.

Math around the house

Besides making music, other "homegrown" activities can help your child understand matching, comparing, sorting, and making patterns and sequences and build a foundation for future math learning.

Match them up. Matching and comparing are essential skill activities in math development. Before your child can understand that 3 is more than 2, she needs to be able to recognize more than (bigger than), less than (smaller than), and same as (equal to) in the world around her.

• Invite your child to build a tower that is as tall as the coffee table (or the couch or the dining table). How many blocks did she use? Is her tower bigger or smaller than the table? By how many blocks? Is her "couch tower" bigger than, the same as, or shorter than her "table tower"? By how many blocks?

• Have a pizza party! Invite your child to match one piece to each person (one-to-one correspondence). You can extend the learning by asking, "How many slices will we need for everyone to have two?"

Patterns all around. One of the important skills in math is the ability to see (or "read") and verbalize a pattern.

• Look for the patterns in your environment. Is there a pattern on the wallpaper, the parking lot, even the stripes on your child's shirt? Point these out and invite your child to say or clap the pattern with you: "red, blue, red, blue, red, blue." What color comes next?

• Make patterns with the shells you collected at the beach this year, with the coins in her piggy bank, with socks--anything you have multiples of!

How big? How long? Measurement is a natural extension of matching and estimating. To measure, your child has to match a series of objects to the length or width of something. The first step in measuring is to create a standard of measure--but the item you use to measure with doesn't have to be standard at all.

• How many (clean!) socks long is the kitchen counter? Count them and see. Then measure the counter with soup spoons. Suggest another item--toy cars, cereal boxes, magazines--and ask your child to guess (estimate) how many of these will be equal to the length of the counter. (Just remember that all your "measuring items" should be around the same size.)

• Your child can use nonstandard measuring items for practical purposes too. How much room will the new picture take up on the wall? Why not measure around the frame with erasers? (And for comparison's sake, show your child the equivalent measurement in inches on a ruler.)

Exactly the opposite. The concept of opposites is important to math: For instance, in order for your child to understand low, she needs to experience high. When you use the comparative language of opposites, you are helping your child learn about proportion and number relationships.

• Ask your child: "Can you put this can on a low shelf and reach up to put the cereal box on a high shelf?."

• While you're taking a walk, say, "Can you take BIG steps? Now do the opposite."

Tally the score. Tally marks are up-and-down lines that are made in sets of four with the fifth mark made as a slash through the set of four. Children learn tally marks quickly because they are connected with an action or event.

• Show your child how to keep score in family games such as go fish or tic-tac-toe.

• For children over 3 who no longer put objects in their mouth, instead of tally marks, use objects such as buttons, beads, or coins and count five of each into egg carton sections.

• As your child becomes comfortable with tally marks or objects, you can introduce a "shopping cart tally." Use a child-size calculator to add up how much each item costs as it goes into the shopping cart. Your child may not really understand what the numbers mean, but she will see that they get larger and larger as the cart gets fuller and fuller.

• When you get home, play with the change that is left over. Your child may be becoming interested in money, and although she may not understand the value of each coin, she can begin to sort the pennies and other coins she is collecting. (This activity is for children age 4 years and older.) She can match them by size or color. Eventually your child will be able to match pennies to nickels and dimes just like she matched the tally marks to objects!

Numbers are everywhere in your child's world. You might find them on speed limit signs, route number signs, posters, and houses. Point out the numbers on household electronic devices. Set the timer together on the coffee machine, microwave, alarm clock, or VCR. It's through simple day-to-day activities--from singing songs to slicing pizza--that your child will have her first important math experiences.

PHOTO (COLOR): Rat-a-tat-tat, toot, toot. When playing musical instruments, children experiment with rhythmic patterns.

PHOTO (COLOR): "This tower is eight blocks higher than the coffee table." Parents can help children practice the language of math

PHOTO (COLOR): "The picture is five erasers wide." Your child can estimate and measure without using a ruler.

Great Books about numbers, size, and counting

Beep Beep, Vroom Vroom!

by Stuart J. Murphy, illustrated by Chris L. Demarest

HarperCollins, 2000; $4.95, paper. Ages 4-8.

Benny's Pennies

by Pat Brisson, illustrated by Bob Barner

Yearling, 1995; $5.99. Ages 4-8.

The Cheerios Counting Book: 1, 2, 3

by Barbara Barbieri McGrath, illustrated by Rob Bolster and

Frank Mazzola, Jr.

Scholastic Inc., 2000; $6.99. Ages 2-4.

Eating Fractions

by Bruce McMillan

Scholastic Inc., 1991; $15.95. Ages 4-8.

Learn to Count, Funny Bunnies

by Cyndy Szekeres

Scholastic Inc., 2000; $6.99. Ages 2-4.

More, Fewer, Less

by Tana Hoban

Greenwillow, 1998; $15. Ages 4-8.

The 1, 2, 3's of Math Learning

Your child's acquisition of math skills follows a developmental sequence just as children (most, anyway) crawl before they walk, they learn math relationships before they learn the names of numbers and how to use them. Children need to learn the structure of math before they can use (and, most important, understand) the vocabulary and symbols of math. Too often we present children with numbers before they have had the opportunity to understand what those number symbols or words mean. For instance, sometimes a young child can count to 10, but he doesn't really understand what he is doing. He is just saying a series of memorized words.

RUNNING BY THE NUMBERS

Source: Mathematics Teaching in the Middle School, Dec2000, Vol. 6 Issue 4, p262, 4p, 2 charts

Author(s): Milliken, Paul; Little, Catherine

In 490 B.C., the messenger Pheidippedes ran twenty-six miles to Athens carrying the news of Greek victory at the battle of Marathon. He delivered the news and dropped dead from the effort. Today, we celebrate that famous run with one of the most demanding events in human athletics, the marathon. Like Pheidippedes, the modern runner strives to complete the distance in as little time as possible. Unlike that early messenger, today's competitors undergo extensive training to ensure that they remain alive when they have finished the run. Kevin Smith uses mathematics to help runners prepare for marathons.

Marathon Dynamics, Inc., in Mississauga, Ontario, is Kevin Smith's company. He provides a variety of services for the running community, including hosting running clinics, conducting fitness and health presentations and seminars, managing and promoting local running events, coaching runners, and creating customized training plans for individual runners using software that he developed himself.

"Basically my days are filled with crunching numbers," Kevin says, "involving calculations of distances, times, paces, and heart rates." He works with runners at all levels, from "recreational joggers to competitive athletes." Each training plan is customized "in order for an individual to improve running performance." Kevin brings an essential "appreciation and understanding of the relationships" among all the variables in his number-crunching.

Kevin says that he has "had to, at one time or another, apply the skills, formulae, and thought processes learned in" a whole range of mathematics disciplines, including "calculus, probability, algebra, and tons of more basic percentage and exponential calculation, and unit conversions (miles to kilometers, miles per hour to kilometers per hour, or meters per second, and so on)." All this work, on top of the accounting, record keeping, and financial management of operating his own company, means that Kevin is always doing mathematics on the job.

Does that mean that Kevin studied mathematics in college to prepare for his career? No. "I had no idea," he says, "I would be using math in this way or to this extent. I created the job I do when we founded our company." The mathematics-related courses that have been most useful to him are the accounting and economics courses that he took as part of his business administration degree at the University of Western Ontario. "As co-manager of the business, my responsibilities include most of the financial management duties of any small business, but the entrepreneurial spirit and desire will only get one so far." He still has to do the bookkeeping.

"I suppose my attitude toward mathematics would have been a little different," he admits, "had I known how it would all end up." He always did well but looked on mathematics "as a chore. I was not a natural, so I had to work at it. If I had known how vitally essential a comfort level with numbers was going to be in my career, I might have had a little more pure motivation to excel in math."

Instead of a specific interest in mathematics, a technology connection spurred Kevin's career and got his company started. "I started to toy around with a simple Lotus spreadsheet idea I had six or seven years ago," he explains, "about a way to help runners of vastly different experience and ability" plan their training. The resulting software program helped runners calculate "how frequently, how much, and how fast to train." Kevin claims that he "had no idea it would turn into what our customized training software has become--a matrix of over sixty interrelated Microsoft Excel spreadsheets, each of which has hundreds of lines of code and formulae embedded in it. I created the job I do after university, so I had no preconceived notion of how math would be involved in my current career."

Mathematics can save your life only if you know how to apply it. The first marathoner, Pheidippedes, did not know how to pace himself and died as a result. With help from Kevin Smith and some number-crunching through his customized-training-plan software, the famous messenger might have lived to deliver more news.

Teacher Notes

Begin work on the activity sheet on page 265 by having students measure their heart rates. In pairs, have them take each other's pulses by having one person count and the other time the beats per minute. Record the beats per minute for each student. Graph the results, and look for trends. Have the students exert themselves by running in place, for one minute, and measure the rates again. Compare the results. This exercise prepares the students for question 1.

For the other questions, make sure to review conversion strategies. For example, when converting from kilometers per hour to meters per second, students must change units of both distance and time.

When calculating the amount of running done in one year for question 2, remember that every fifth day is a rest day.

See figure 1 for one student's solution to the activity sheet.

"Math at Work" explores how mathematics is used in the workplace. Each article will highlight a particular career and the mathematics specific to that discipline. Readers are encouraged to submit manuscripts for this department by sending them to "Math at Work," MTMS, NCTM, 1906 Association Drive, Reston, VA 20191-9988.

Calling All Teacher-Educators

To find out more about writing for the journal, contact Kathleen Lay at klay@nctm.org and ask for the "MTMS Writer's Packet." If you have a manuscript ready to go, send it directly to Mathematics Teaching in the Middle School, NCTM, 1906 Association Drive, Reston, VA 20191-9988. All submissions must include five double-spaced copied of the manuscript.

You have probably spent much time and effort encouraging your student teachers to write about their thinking. The Editorial Panel of Mathematics Teaching in the Middle School invites you, a teacher-educator who specializes in middle-grades mathematics, to do the same and share your ideas with your colleagues by writing for the journal. Teacher-educators have a special role to play in helping to translate the theory of good practice into pedagogical ideas that teachers can employ in their classrooms. Furthermore, many teacher-educators read the journal to get new ideas for their teaching.

Fig. 1 One student's solution to the activity sheet

Running by the Numbers Activity Sheet

NAME -----

1. An athlete's maximum exertion heart rate is calculated by subtracting his or her age from a fixed number: 220 for males and 226 for females. For example, a 24-year-old female runner has a maximum exertion heart rate of 202 beats per minute, or 226 - 24 = 202. The target performance heart rate is 80% of maximum. Calculate the target performance heart rate for the following runners:

a. 14-year-old female

b. 23-year-old male

c. 35-year-old female

d. 49-year-old male

e. yourself

2. A runner follows the training schedule in the table below.

DAY 1 DAY 2 DAY 3 DAY 4 DAY 5

5 km 7.5 km 10 km 5 km Rest

If the runner maintains an average rate of 8 km/hr., how much time does he spend training in one year?

3. A runner trained for three days in a row. On day 1, she ran 7.5 km in 37 minutes. On day 2, she ran 8.3 km in 41 minutes. On day 3, she ran 6.8 km in 32 minutes. What was her average pace expressed in km/hr. and in m/sec.?

4. The record for the Math-at-Work 5-km Mini-Marathon is 38 minutes and 12 seconds. What is the likely finishing time for the runner in question 3?

5. One mile is approximately 1.6 km. How many miles does the runner in question 3 run in training in one year?

Fig. 1 One student's solution to the activity sheet

1. a) Max.: 226 - 14 = 212

Target: 0.8 x 212 = 169.6 congruent to 170

b) Max.: 220 - 23 = 197

Target: 0.8 x 197 = 157.6 congruent to 158

c) Max.: 226 - 35 = 191

Target: 0.8 x 191 = 152.8 congruent to 153

d) Max.: 220 - 49 = 171

Target: 0.8 x 171 = 136.8 congruent 137

e) Max.: 226 - 13 = 213

Target: 0.8 x 213 = 170.4 congruent 170

2. Total distance ran: 5 km + 7.5 km + 10 km + 5 km = 27.5 km

Therefore 27.5 km/8 km/hr. = 3.44 hrs.

Number of 5-day cycles within a year: 365 days/5 days = 73 5-day

cycles.

(This character cannot be represented in ASCII text) The total

amount of time the runner spends on training in one year = 3.44

hrs. x 73 5-day cycles = 251.12 hrs.

3. Total distance: 7.5 km + 8.3 km + 6.8 km = 22.6 km

Total time: 37 min. + 41 min. + 32 min. = 110 min.

110 min./60 min. = 1.83 hrs.

Therefore Average pace (km/hr.) = 22.6 km/1.83 hrs.

= 12.35 km/hr.

Average pace (m/sec): 12.35 km x 1 000 = 12 350 m

1 hour = 3 600 sec.

then 12 350 m/3 600 sec. = 3.43 m/s

Therefore Average pace (m/sec.): 3.43 m/sec.

4. Time = distance/speed = 5 km/12.35 km/hr. =0.405 hrs. = 24.30

min.

= 24 min. 18 sec.

5. Total distance: 7.5 km + 8.3 km + 6.8 km = 22.6 km

Therefore Total distance (miles) = 22.6 km/1.6 km = 14.125 miles

Amount of 3-day cycles in a year: 365/3 = 121.67

Therefore Total amount of miles ran in training in one year:

14.125 miles x 121.67 = 1718.59 miles

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