mathematics in context 情境中的数学 34本mathematics in context - figuring all the angles_W

1、Figuring All the AnglesGeometry 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

2、 Institute 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 FoundationOpini

3、ons expressed are those of the authors and not necessarily those of the Foundation.deLange, J.,van Reeuwijk, M., Feijs, E., Middleton, J. A., and Pligge, M. A. (2006). Figuring all the angles. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context. Chicago:

4、 Encyclopdia 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

5、 the U.S. 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,

6、Chicago, Illinois 60610.ISBN 0-03-039622-02 3 4 5 6 073 09 08 07 06The Mathematics in Context Development TeamDevelopment 19911997The initial version of Figuring All the Angles was developed by Jan deLange, Martin van Reeuwijk, and Els Feijs. It was adapted for use in American schools by James A. Mi

7、ddleton, and Margaret A. Pligge.Wisconsin Center for EducationFreudenthal Institute StaffResearch StaffThomas A. RombergJoan DanielsPedroJan de LangeDirectorAssistant to the DirectorDirectorGail BurrillMargaret R. MeyerEls FeijsMartin vanReeuwijkCoordinatorCoordinatorCoordinatorCoordinatorProject St

8、affJonathan 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

9、ChristiansenJulia 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 Figuring A

10、ll The Angles was developed by Els Feijsand 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 BurrillMarg

11、aret A.PliggeMieke 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.

12、 SteeleDd de HaanNanda 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 Brita

13、nnica, Inc.Cover photo credits: (left to right) Comstock Images; Corbis; Getty ImagesIllustrations1 James Alexander; 8, 9 Holly Cooper-Olds; 26 Jerry Kraus/ Encyclopdia Britannica, Inc.; 34 Encyclopdia Britannica, Inc.Photographs3 Corbis; 4 Charles E. Rotkin/Corbis; 14 Corbis; 17 Roger Ressmeyer/Cor

14、bis; 21 (top) TimBoyle/Newsmaker/Getty Images;(bottom) PhotoLink/PhotoDisc/ Getty Images; 23 ImageGap/Alamy; 26 Corbis; 28 PhotoDisc/Getty Images;30 Corbis; 42 (top) Historic Urban Plans, Inc.; (bottom) Design Pics; 43 PhotoDisc/ Getty ImagesContentsSection AA Sense of Direction0N WE SLetter to the

15、StudentviGetting Your Sense of Direction1Summary6Section BFinding Your Way90270Check Your Work6Sunray (1850)8Sunray (1900)9Sunray Today10Downtown Provo, Utah11Summary12Section CDirectionsCheck Your Work12San Francisco Bay Area14Sun Island18Summary20Section DNavigation and OrientationCheck Your Work2

16、0Traffic Control21Summary24Section EChanging Directions: TurnsCheck Your Work24Flight Instructions26Plane Landing Activity31Summary32Section FFrom Turns to AnglesCheck Your Work32Sled Tracks33Regular Polygons35Summary36NSection GAngles and Their MeasuresCheck Your Work36Angles38Measures of Angles409

17、0QPalmanova42Summary44Check Your Work4518000180Additional Practice46Answers to Check Your Work52Contents vDear Student,Welcome to the unit Figuring All the Angles.In this unit, you will learn how planes and boats navigate their way around the globe.You will build upon the cardinal directions of nort

18、h, east, south, and west to give a better indication of where you want to go.Along the way, you will pay close attention to the turns made along a route. Turns relate to angles, and you will use these to solve some geometry problems involving shapes.Sincerely,The Mathematics in Context Development T

19、eamNWESairport bridge0510 miles51016 kmMap from the Road Atlas 1994 by Rand McNally.vi Figuring All The AnglesA Sense of DirectionGetting Your Sense of Direction1. Point toward north. Is everyone in your class pointing in the samedirection?2. a. Sketch a top view of your classroom. Include the deskt

20、ops in your sketch. Draw an arrow pointing north on each of the desks.b. Do all the arrows point in the same direction?c. Will lines from the arrows ever meet?3. In what direction is south?4. The position of the sun in the sky is related to the direction south. Record the suns position in the sky be

21、fore noon, at noon, and after noon.5. In which direction from the classroom is your schools playground?6. Name a town about 50 miles away and point in the direction of that town. Describe this direction.7. If you traveled north from your school, which towns would you pass through?Section A: A Sense

22、of Direction 1AA Sense of DirectionNWESUse a compass to answer problems 8 and 9.8. a. Sketch the room where you sleep. Be sure to include windows in your sketch.b. Designate where north is in the sketch.9. Compare the sketches of all the students in your class. Count how many of the windows in the s

23、ketches face south.Here is a partial map of the United States to answer the next questions.NWE2 Figuring All The AnglesSMt. RushmoreMonument ValleyWest Virginia0400 miles0400 kilometersA Sense of DirectionAMonument Valley, in Arizona, has many spires and mesas that did not erode as fast as the land

24、around them. You may have seen the picture above in Western movies. Western movies often describe how the West was settled.10. Why is the West called the West?On one side of Mount Rushmore, the heads of four United States presidents have been carved out of the mountain. Mount Rushmore is located in

25、South Dakota.11. Is South Dakota in the South? Why is it called South Dakota?You have probably read about the North Pole.12. Explain why the word North in North Pole has a different meaning from West in West Virginia. (West Virginia is labeled on the partial map of the United States on page 2.)Secti

26、on A: A Sense of Direction 3AA Sense of DirectionThe state capitol of Wisconsin is located in Madison. The capitol building is unusual because of the four identical wings radiating from the huge central dome.The four wings of the building point in the four compass directions: north, south, east, and

27、 west.The south wing is indicated in the drawing below.13. Label the directions of the remaining three wings on Student Activity Sheet 1.4 Figuring All The AnglesA Sense of DirectionASection A: A Sense of Direction 5Hamilton StreetState StreetKing StreetThis map of the area around the capitol buildi

28、ng in Madison shows streets going from north to south and from east to west.The map of the state capitol looks very much like a compass used by sailors. As you can see, there are more than four directions.14. Fill in the two missing compass directions on Student Activity Sheet 1.Hamilton St.Hamilton

29、 St.Hamilton StreetNNW?(Northwest)WE?SES(Southeast)In Madison, many streets do not run northsouth or eastwest.Lake MononaSEWNKing St.State St.Lake Mendota15. Explain why some of the roads in Madison are going southwest northeast instead of northsouth or eastwest?AA Sense of DirectionNorth, south, ea

30、st, and west are directions you can use to find places on a map and in the real world. You can combine them to be more specific; for example, you can say northeast, northwest, southeast, and southwest.Directions are relative. For example, South Dakota is not in the South, but it is south of North Da

31、kota. West Virginia is in the eastern United States, but it is west of Virginia.The North Pole is a unique place on earth.1. Why is it unique? Name another unique place on earth.2. How would you describe the positions of Hawaii and Alaska relative to the United States mainland?A r c t i c O c e a nA

32、laska6 Figuring All The AnglesNORTH AMERICAEUROPEASIAP a c i f i c O c e a nHawaiiA t l a n t i c O c e a nAFRICAP a c i f i c O c e a nSOUTH AMERICANI n d i a n O c e a nAUSTRALIAWESANTARCTICA3. How would you describe the position of South America in relation to Europe?4. Describe the position of S

33、outh America relative to Australia.Name at least five more locationsinside or outside the USwith names that relate to compass directions.Section A: A Sense of Direction 7BFinding Your WaySunray (1850)Small towns often develop along a single road. In Sunray, the first road was called Main Street. Lik

34、e many towns, Sunray developed in the mid 1800s, when the population of the United States was shifting from the East to the West.Soon the city expanded and needed more streets. New streets were built either parallel or perpendicular to Main Street. The roads perpendicular to Main Street were called

35、avenues.1. Draw a map of Sunray with three streets north of Main Street, two streets south of Main Street, two avenues west of Lincoln Avenue, and three avenues east of Lincoln Avenue.2. a. Why is it convenient to have the avenues run northsouth when the streets run eastwest?b. How are the streets n

36、amed in the town where you go to school?8 Figuring All The AnglesFinding Your WayBSection B: Finding Your Way 9Sunray (1900)Crystashose7t.N.eggishos1sS.N.Ton HallM iS. 6th Ave. W.1st Ave. W. Lincoln Ave.Sunray has grown. It has many streets and avenues. The plan of the city looks like a grida combin

37、ation of horizontal and vertical lines.Crystal lives at the corner of 7th Street North and 6th Avenue West. Reggie lives at the corner of 3rd Street North and 3rd Avenue West.Use a copy of this grid on Student Activity Sheet 2 to help you answer the problems below.3. Crystal and Reggie are friends.

38、If Crystal wants to visit Reggie at his house, how many blocks will she have to walk?Crystal and Reggie plan to meet each other for lunch. They like both Tonys Tortellini onthe corner of Main Street and 7th Avenue West and Ellas Deli on 5th Street North and Lincoln Avenue.4. Which restaurant is clos

39、er to their homes?The city will build a theater three blocks south of Main Street. The contractor has a choice of different building sites, all between 3rd Avenue East and 5th Avenue West.5. a. How far (in number of blocks) from Crystals house could the theater be built?b. Which location would be cl

40、osest toCrystals house?BFinding Your WaySunray Today10 Figuring All TheE40th St. North 25th St. North 10th St. North20th St. South45th St. South 60th St. South100th St. SouthANNWNEWED60th Ave. WestLincoln Ave.40th Ave. East65th Ave. East80th St. NorthTown HallMain St.CAIRPORT30th Ave. West10th Ave.

41、WestB10 milesHere is a map of Sunray as it looks now.You will now use a scale line on a map to find a distance. Many maps have scale lines. The scale line on the map of Sun Ray shows 10 miles.6. On the maps scale, what distance is indicated by each hatch mark?Use the map on Student Activity Sheet 3

42、to solve the following problems. Save this for later use.The baseball stadium lies exactly 7 miles south of Town Hall.7. Draw the stadium on the map. Use the scale line and a ruler.SWSESThe coliseum is 10 miles northwest of Town Hall. The park is 12 miles southeast of Town Hall. The mall is 3 miles

43、southwest of intersection E. All of these distances are as the crow flies.8. a. What does as the crow flies mean?b. Mark the positions of the buildings on the map.When you travel around the city, you follow the roads and do not usually travel as the crow flies. Distances that follow the lines of a m

44、ap grid or roads are often called taxicab distances.9. Explain why distances on a grid are called taxicab distances.10. How far do you have to drive to go from the mall to Town Hall?11. List at least three buildings that you might want to build in Sunray. Locate these buildings on Student Activity S

45、heet 3. Write directions for each of your buildings. Use any informationavailable except street names. Give your directions to a classmate to see if he or she can locate the positions of your buildings on the map.Finding Your WayB12: Peoples and Culture Museum1km01 mi0500 S.4: Pioneer Park5: Post Of

46、fice6: Provo Theater7: Public Library8: Forest House9: Greyhound Bus10: Doll Museum11: Visitor Bureau200 S.300 S.400 S.65100 S.4Center St.7To I-15100 N.89200 N.10300 N.11500 N.400 N.S12EW800 N.N1100 W.900 W.800 W.700 W.200 W.700 E.900 E.Downtown Provo, Utah600 W.500 W.400 W.300 W.200 W.100 W.Univers

47、ity Ave.100 E.200 E.300 E.400 E.500 E.600 E.12. Study the map above. How are the systems of naming streets alike for Sunray and Provo? How are they different?The Greyhound Bus Station is number 9 on the map. A passenger has just arrived at the bus station. He needs directions to the Visitor Bureau.1

48、3. Give him directions.At the Peoples and Culture Museum (number 12), someone tells a tourist that Pioneer Park (number 4) is about a half hours walk directly southwest of the museum.14. Do you agree? Why or why not?Section B: Finding Your Way 11B Finding Your WaySometimes streets and avenues in a c

49、ity are named so that residentsand visitors can find places easily. Some cities have eastwest streets and northsouth avenues. For example, 5th Avenue West and 3rd Street North identify the exact location on a map or in a city.Distances can be measured as taxicab distances or as the crow flies.Many m

50、aps have scale lines. A scale line tells you how distances are represented on the map.3rd Street North2nd Street North 1st Street NorthMain StreetPublic LibraryScale line01234miles5th Avenue West3rd Avenue West1st Avenue WestLibrary Avenue1. a. Draw a map of a town that has Town Hall in the center a

51、nd eight main roads leading from the center in the directions N, S, E, W, NW, NE, SW, and SE.b. How is the map of Sunray different from the town you just drew?12 Figuring All The Angles2. Would taxicab distances on a streetavenue system be the same as distances measured as the crow flies? Explain.3.

52、 Use Student Activity Sheet 3 to solve the following problems.a. How far is the “as the crow flies” distance from A to E on the map of Sunray?b. How far is the “taxicab” distance from A to E on the map of Sunray?c. How far is the “as the crow flies” distance from A to D on the map of Sunray?d. How far is