mathematics in context 情境中的数学 34本mathematics in context - made to measure_W

1、Made to MeasureGeometry and MeasurementMathematics in Context is a comprehensive curriculum for the middle grades. It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institu

2、te atthe University of Utrecht,The Netherlands, withthe support of the National Science Foundation Grant No. 9054928.The revision of the curriculum was carried out in 2003 through 2005, with the support of the National Science Foundation Grant No. ESI 0137414.National Science FoundationOpinions expr

3、essed are those of the authors and not necessarily those of the Foundation.de Lange, J., Wijers, M., Dekker, T., Simon, A. N., Shafer, M. C., and Pligge, M. A. (2006). Made to measure. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclo

4、pdia Britannica, Inc.Copyright 2006 Encyclopdia Britannica, Inc. All rights reserved.Printed in the United States of America.ThisworkisprotectedundercurrentU.S.copyrightlaws,andtheperformance, display, and other applicable uses of it are governed bythose laws. Any uses not in conformity with the U.S

5、. copyright statute are prohibited without our express written permission, including but not limited to duplication, adaptation, and transmission bytelevision or other devices or processes. Formoreinformation regarding a license, write Encyclopdia Britannica, Inc., 331 North LaSalle Street, Chicago,

6、 Illinois 60610.ISBN 0-03-042404-61 2 3 4 5 6 073 09 08 07 06 05The Mathematics in Context Development TeamDevelopment 19911997The initial version of Made to Measure was developed by Anton Roodhardt and Jan Auke de Jong.It was adapted for use in American schools by Laura J. Brinker, James A. Middlet

7、on, and Aaron N. Simon.Wisconsin Center for EducationFreudenthal Institute StaffResearch StaffThomas A. RombergJoan DanielsPedroJan de LangeDirectorAssistant to the DirectorDirectorGail BurrillMargaret R. MeyerEls FeijsMartin vanReeuwijkCoordinatorCoordinatorCoordinatorCoordinatorProject StaffJonath

8、an BrendefurSherian FosterMieke AbelsJansie Niehaus Laura BrinkerJames A, MiddletonNina BoswinkelNanda Querelle James BrowneJasmina MilinkovicFrans vanGalenAnton Roodhardt Jack BurrillMargaret A. PliggeKoeno GravemeijerLeen Streefland Rose ByrdMary C. ShaferMarja van denAdri Treffers Peter Christian

9、senJulia A. ShewHeuvel-PanhuizenMonica WijersBarbara Clarke Doug Clarke Beth R. Cole Fae Dremock Mary Ann FixAaron N. Simon Marvin Smith Stephanie Z. Smith Mary S. SpenceJan Auke de Jong Vincent Jonker Ronald Keijzer Martin KindtAstrid deWildRevision 20032005Therevised version of Made to Measure was

10、 developed Mieke Abels and Jande Lange. It was adapted for use in American Schools by Margaret A. Pligge.Wisconsin Center for EducationFreudenthal Institute StaffResearch StaffThomas A. RombergDavid C.WebbJan de LangeTruus DekkerDirectorCoordinatorDirectorCoordinatorGail BurrillMargaret A.PliggeMiek

11、e AbelsMonica WijersEditorial CoordinatorEditorial CoordinatorContent CoordinatorContent CoordinatorProject StaffSarah AiltsMargaret R.MeyerArthur BakkerNathalie Kuijpers Beth R. ColeAnne ParkPeter BoonHuub Nilwik Erin HazlettBryna RappaportEls FeijsSonia PalhaTeri HedgesKathleen A. SteeleDd de Haan

12、Nanda QuerelleKaren Hoiberg Carrie Johnson Jean Krusi Elaine McGrathAna C. Stephens Candace Ulmer Jill VettrusMartin KindtMartin vanReeuwijk(c) 2006 Encyclopdia Britannica, Inc. Mathematics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopdia Britannica, Inc.Cover

13、photo credits: (left) Getty Images; (middle) Kaz Chiba/PhotoDisc/ Getty Images; (right) PhotoDisc/Getty ImagesIllustrations2, 14, 22 Holly Cooper-Olds; 23, 24 Christine McCabe/ Encyclopdia Britannica, Inc.; 28 Holly Cooper-Olds; 31 Christine McCabe/ Encyclopdia Britannica, Inc.; 32 Holly Cooper-Olds

14、; 35 Christine McCabe/ Encyclopdia Britannica, Inc.; 37, 38 Holly Cooper-OldsPhotographs1 (counter clockwise) PhotoDisc/Getty Images; PhotoDisc/Getty Images; Ingram Publishing; PhotoDisc/Getty Images; Sam Dudgeon/HRW Photo; Corbis; 10 Victoria Smith/HRW; 17 Victoria Smith/HRW; 34 Sam Dudgeon/HRWCont

15、entsLetter to the StudentviSectionALengthsIntroduction1Historical Measures2Feet and Shoes6Body Length and Fathom8Other Measures for Length9Summary10Check Your Work11SectionBAreasA Bodys Surface Area12Squares13Hands and Body15Surface Area by Formula17Height, Weight, and Area18Early Areas19Summary20Ch

16、eck Your Work21SectionCVolumesThe Volume of Your Heart22Solids22Liquids26The Volume ofYour Body27Other Measures for Volume28Summary30Check Your Work30SectionDAnglesFurniture32Summary38Check Your Work38Additional Practice40Answers to Check Your Work43Contents vDear Student,Welcome to the Mathematics

17、in Context unit Made to Measure. This unit is all about measuring: measuring your feet, your thumb, your hands, and the angle made by your arm and your wrist.You will investigate how measuring units evolved. You will further investigate measurements for length, area, and volume. You might beamazed b

18、y what you can measure!You will find that mathematics plays an important role in measurement. Every timeyou measure something, you might ask yourself: Will every person measuring this item get the same measurement that I did? Do all of these things have the same measurement? What other units of meas

19、ure can I use? Are there other ways to measure these things?Whenever you make a measurement in this unit, picture how bigor small or steep or short that measurement is. When you can do this with all of the measurements in this unit,you are well on your way to becoming a mathematician!Sincerely,The M

20、athematics in Context DevelopmentTeamvi Madeto MeasureLengthsIntroductionPeople who design the objects you use every day have thought a lot about how big or how small those objects should be. Knowing the sizes of peoplesarms, legs, and hands can be very useful when designing furniture, clothes, toys

21、, windows, doors, and many other items.1. Which body measures would be useful to know if you were designing the followingitems?a. doorsd. pantsb. school deskse. baby cribsc. shoesf. stairs2. For what other objects would you need to know body measures?SectionA: Lengths 1ALengthsHistorical MeasuresYou

22、 probably discovered that the lengths and heights of different body parts are important for designing many common objects. At one time, all measurements for length were related to the human body. Some of these units of measure include the thumb, hand span, foot, yard, pace, and fathom.3. Match each

23、of the units of measure listed in the paragraph above with its drawingbelow.a.b.c.d.e.f.2 Madeto MeasureLengthsA4. a. Which of the units of measure from problem 3 would you use to find the length of the nail shown here?b. How long is this nail in the units of measure you chose for your answer to a ?

24、5. a. Measure the length of your desk by using one or more of the units of measure from problem 3.b. List the class results in a table. Did everyone find the same length? Why do you think this happened?6. Reflect Name one advantage and one disadvantage of using your body to make measurements.In Scot

25、land during the Middle Ages, a unit of measure calledthe Scottishthumb wasused. AScottishthumb is the mean of the thumb widths of three men: a large man, an average-sized man, and a smallman.7. Why were three different-sized men used to determine the Scottishthumb?In 1616, the Germans decided to cre

26、ate a unit of measure called the mean foot. To do this, they cut a piece of rope that was as long as the feet of 16 men.8. a. How do you think the rope was used to find the length of the mean foot?b. Which measurement is closer to the average persons measurement: the German mean foot or the Scottish

27、 thumb? Explain your answer.9. With the help of 16 classmates, find the length of the mean foot in your class by using the method describedabove.SectionA: Lengths 3ALengths10. Measure the following in centimeters (cm). List the results in a table.a. your thumb widthc. your footlengthb. your hand spa

28、nd. your pace11. Use the table you made in problem 10 to answer the questions about relationships betweenmeasures.a. How many thumb widths are in one hand span?b. How many thumb widths are in one foot?c. How many feet are in onepace?You may add other units of measure and the relationships between th

29、em to yourlist.12. What is the size of the “typical” thumb for your classmates? Explain your answer.Think about what it must have been like when everyone used his or her own thumbs for measuring. Today, of course, everyone has standard systems of measurement.A few countries, including the United Sta

30、tes, still use the foot as a unit of measure, but the length of a foot no longer refers to the length of each persons foot. A standard has been officially established for the length of a foot. Most countries use the metric system, which was adopted in France in 1795.13. a. The foot is a part of the

31、Imperial, or English, system of measurement. In the United States, we call this the customarysystem.Listsomeotherunits of measure for length that are part of the customary system.b. In your notebook, write as many relationships between the units of measure in the customary system as you can.14. a. L

32、ist some units of measure for length that are part of the metric system.b. Write as many relationships between these units of measure as you can.Since the United States officially uses the metric system, it is important for you to have a sense of how the metric and the customary systems relate. The

33、next activity will help you find some simple relationships between the metric and customary systems.4 Madeto MeasureComparing SystemsMeters and Yards Use a meter stick to measure your classroom. Predict how the room dimensions would change if you measured with a yardstick. Use a yardstick to measure

34、 your classroom. Compare your prediction to your actual measurement. Find a conversion rule for meters and yards you can use when doing mental calculations.Centimeters and Inches Use a centimeter ruler to measure a paper clip. Predict how the measurements would change if you measured using a ruler w

35、ith inches. Use an inch ruler to measure the paper clip. Compare your prediction to your actualmeasurement. Find a conversion rule for centimeters and inches you can use when doing mental calculations.Kilometers and MilesSince athletes compete internationally, all distances are in meters(m) or kilom

36、eters (km). Today, many U.S. high school cross-country teams run 5-km races. Did you know that five kilometers is about three miles? Investigate how your school measures the running events. How long is the running track that your schooluses? Name a location that is about one mile away from your scho

37、ol. Would a location that is about one kilometer away from your school be closer orfarther?15. Write as many relationships between units in the customary measurement system and the metric system as you can.SectionA: Lengths 5ALengthsFeet and ShoesIn problem 10 c, you measured the length of your foot

38、 in centimeters. The length of your foot is different from your shoe size.16. a. Do you think that there is a relationship between shoe size and foot length? Explain youranswer.b. Make a table that lists the foot length (in cm) and corresponding shoe size for each student in your class.c. Graph the

39、results. Put foot length in centimeters on the horizontal axis and shoe size on the vertical axis. What does your graph tell you about the relationship between foot length and shoe size?Just as countries use different systems of measurement, they also have different systems for determining shoe size

40、s. For some shoes, you can find at least three different sizes: European sizeusually a number between 33 and 47 U.K. (United Kingdom) sizeusually a number between 1and 15 U.S. sizeusually a number between 1 and 15 (slightly larger than U.K. sizes)The U.K. system of shoe sizes began in the seventh ce

41、ntury.Shoe sizes were measured with a standard thumb(now called an inch).17. How many standard thumbs (or inches) long is your foot?To get a more accurate measurement, the U.K. introduced a smaller unit of measure, the stitch. Three stitches are in one standard thumb.18. How many stitches long is yo

42、ur foot?3In the U.K. system, shoe sizes are based on the number of stitches. The first 25 stitches are not counted in adult shoe sizes. Size 1 is, therefore, really 26 stitches, or 8 2 inches (in.).6 Madeto MeasureLengthsA19. a. Copy the table into your notebook and continue it to shoe size 8.b. Use

43、 your answer to problem 18 to find your U.K. shoe size in the table you made in part a. How does your U.K. shoe size compare with your U.S. shoesize?c. Formulate a rule that helps you find someones foot length in inches if you know his or her U.K. shoe size. Write the rule in arrow language.SectionA

44、: Lengths 7U.K. shoe size? ? foot length(in stitches)(in inches)20. Reflect After you finish problem 19, look back at your answers to problem 16. Would you change your answers now? Whyor why not?In problem 10 you measured your pace and your foot length.21. Make a table of the pace and foot-length me

45、asurements of all of the students in your class. Use the results to determine whether the two measures are related. For example, you can find out if the person with the biggest foot also has the longest pace or if the person with the shortest foot has the shortest pace. Drawing a graph may be helpfu

46、l.ALengthsBody Length and FathomThe fathom is another unit of measure associated with the human body. You can measure a persons fathom by having him or her stand tall and extend his or her arms out from both sides, horizontal to the ground. The fathom originated as the distance from the middle finge

47、rtip of one hand, to the middle fingertip of the other hand.The picture is based on a famous drawing by Leonardo da Vinci. The girl more or less fits in a square.22. Based on the picture on this page, what is the relationship between a persons height and his or her fathom?23. Measure your height and

48、 your fathom to decide how precisely you would fit into a square.8 Madeto MeasureLengthsAOther Measures for LengthThere are many other ways to measure length. In Papua, New Guinea, for example, a local unit of distance is “a days travel.”24. Why might a days travel make sense as a unit of distance?I

49、n mountainous regions of New Guinea, walking distances are expressed in hours, not in kilometers or miles. Puli, a citizen of New Guinea, says, “It will take us two hours to cover the distance from the village to the lake in the mountains, but we save time on the return trip. The distance back will

50、only take five quarters of an hour.”25. a. Why do you think there are two travel times?b. Is the distance in kilometers different for the two directions? Explain.Mali (1200 A.D. 1500 A.D.)Mali2,000 km1,0000SCALEIn the 14th century, the biggest trading empire in Africa was the Empire of Mali. Mansa M

51、usa was one of its emperors. Sheik Uthman ed-Dukkali, a learned Egyptian who lived in Mali for 35 years, declared that Mali was “four months of travel long and three months wide.”Source: Data from Basil Davidson,African Kingdoms(New York: Time Incorporated, 1966).26. a. Use the map to estimate the l

52、ength and width of the Empire of Mali in both customary and metric units of measure. (Note: 500 miles (mi) equals 800 km.)b. What is the distance in miles or kilometers of “one month of travel”?c. Based on your answer to part b, how do you think people in Mali traveled in the 14th century?SectionA:

53、Lengths 9A LengthsIn this section, you learned many different ways to estimate and measure length. In the past, people used thumbs, feet, and arms to measure length, but each person often measured the same distance differently. Some people use units of time to measure distances. For example, Cedric

54、takes one hour to hike around the nature trail.Today, two standard systems of measurement exist. Most countries of the world use the metric system, in which length is measured in centimeters (cm), meters (m), and kilometers (km).1 kilometer = 1000 meters1 meter = 100 centimeters1 centimeter = 10 mil

55、limetersA few countries use the customary, or Imperial, system, in which length is measured in inches (in.), feet (ft), yards (yd), and miles (mi).1 mile = 5,280 feet1 yard = 3 feet1 foot = 12 inchesHere are some relationships between both measuring systems. You may need to convert from one system to another.1 mile is about 1.5 km (to be exact: 1 mi = 1.6 km)1 yard is a little less than 1 m (to be exact: 1 yd = 0.9144 m) 1 foot is a little more than 30 cm (to be exact: 1 ft = 30.48 cm) 1 inch is about 2.5 cm (1 in. = 2.54 cm)10 Made to Measure1.