1、ReallotmentGeometry 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 Institute a

2、tthe 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 expresse

3、d are those of the authors and not necessarily those of the Foundation.Gravemeijer, K., Abels, M.,Wijers, M., Pligge, M. A., Clarke, B., and Burrill, G. (2006). Reallotment. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics incontext. Chicago: Encyclopdia Britann

4、ica.Copyright 2006 Encyclopdia Britannica, Inc. All rights reserved.Printed in the United States of America.This work is protected under current U.S. copyright laws, and the performance, display,and otherapplicable usesofitaregovernedbythose laws. Anyuses not in conformity with the U.S. copyright st

5、atute are prohibited without our express writtenpermission, including but not limited to duplication, adaptation, and transmissionbytelevisionorotherdevicesorprocesses.Formore information regarding a license, write Encyclopdia Britannica, Inc., 310 South Michigan Avenue, Chicago, Illinois 60604.ISBN

6、 0-03-039614-X1 2 3 4 5 6 073 09 08 07 06 05The Mathematics in Context Development TeamDevelopment 19911997The initialversion of Reallotment was developed by Koeno Gravemeijer. It was adapted for use in American schools by Margaret A. Pligge and Barbara Clarke.Wisconsin Center for EducationFreudenth

7、al Institute StaffResearch StaffThomas A. RombergJoan DanielsPedroJan de LangeDirectorAssistant to the DirectorDirectorGail BurrillMargaret R. MeyerEls FeijsMartin vanReeuwijkCoordinatorCoordinatorCoordinatorCoordinatorProject StaffJonathan BrendefurSherian FosterMieke AbelsJansie Niehaus Laura Brin

8、kerJames A, MiddletonNina BoswinkelNanda Querelle James BrowneJasmina MilinkovicFrans vanGalenAnton Roodhardt Jack BurrillMargaret A. PliggeKoeno GravemeijerLeen Streefland Rose ByrdMary C. ShaferMarja van denAdri Treffers Peter ChristiansenJulia A. ShewHeuvel-PanhuizenMonica WijersBarbara Clarke Do

9、ug 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 20032005The revised version of Reallotment was developed by Mieke Abels and MonicaWijers. It was adapted for us

10、e in American schools by Gail Burrill.Wisconsin Center for EducationFreudenthal Institute StaffResearch StaffThomas A. RombergDavid C.WebbJan de LangeTruus DekkerDirectorCoordinatorDirectorCoordinatorGail BurrillMargaret A.PliggeMieke AbelsMonica WijersEditorial CoordinatorEditorial CoordinatorConte

11、nt CoordinatorContent CoordinatorProject StaffSarah AiltsMargaret R.MeyerArthur BakkerNathalie Kuijpers Beth R. ColeAnne ParkPeter BoonHuub Nilwik Erin HazlettBryna RappaportEls FeijsSonia PalhaTeri HedgesKathleen A. SteeleDd de HaanNanda QuerelleKaren Hoiberg Carrie Johnson Jean Krusi Elaine McGrat

12、hAna 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 photo credits: (left to right) Comstock Images; Corbis; Getty Image

13、sIllustrations1 James Alexander; 39 Holly Cooper-Olds; 49 James AlexanderPhotographs5 M.C. Escher “Symmetry Drawing E21” and “Symmetry Drawing E69” 2005 The M.C. Escher Company-Holland. All rights reserved. ; 17 Age Fotostock/SuperStock; 25 (top) Sam Dudgeon/HRW Photo; (middle)Victor

14、ia Smith/HRW; (bottom) EyeWire/PhotoDisc/Getty Images; 30 PhotoDisc/ Getty Images; 32, 40 Victoria Smith/HRWContentsLetter to the StudentviSection AThe Size of ShapesLeaves and Trees1Tulip Fields2Reasonable Prices2Tessellations4Big States, Small States7Islands and Shapes8Summary10Check Your Work11Se

15、ction BArea PatternsRectangles13Quadrilateral Patterns15Looking for Patterns15Strategies and Formulas20Summary22Check Your Work24Section CMeasuring AreaGoing Metric25Area26Floor Covering30Hotel Lobby32Summary34Check Your Work35Section DPerimeter and AreaPerimeter37Area and Perimeter Enlarged38Circum

16、ference40Drawing a Circle40Circles41Circles and Area44Summary46Check Your Work47Section ESurface Area and VolumePackages49Measuring Inside51Reshaping54Summary60Check Your Work62Additional Practice64Answers to Check Your Work70Contents vDear Student,Welcome to the unit Reallotment.In this unit, you w

17、ill study different shapes and how to measure certain characteristics of each. You will also study both two- and three-dimensional shapes.You will figure out things such as how many people can stand in your classroom. How could you find out without packing people in the entire classroom?You will als

18、o investigate the border or perimeter of a shape, the amount of surface or area a shape covers, and the amount of space or volume inside athree-dimensional figure.How can you make a shape like the one here that will cover a floor, leaving no open spaces?In the end, you will have learned some importa

19、nt ideas about algebra, geometry, and arithmetic. We hope you enjoy the unit.Sincerely,The Mathematics in Context DevelopmentTeamSectionA: TheSizeof Shapes 1AThe Size ofShapesLeaves and TreesElmHere is an outline of an elm leaf and an oak leaf. A baker uses these shapes to create cake decorations.Su

20、ppose that one side of each leaf will be frosted with a thin layer of chocolate.Oak1. Whichleafwillhavemorechocolate? Explain your reasoning.This map shows two forests separated by a river and a swamp.Swamp Meadow Forest River2. Which forest is larger? Use the figures below and describe the method y

21、ou used.Figure AFigure BThe Size of ShapesASectionA: TheSizeof Shapes 7Tulip FieldsField AField BField CReasonable Prices80D.A.C.B.F.G.Here are three fields of tulips.3. Which field has the most tulip plants? UsethetulipfieldsonStudentActivity Sheet 1 to justify your answer.Mary Ann works at a craft

22、 store. One of her duties is to price different pieces of cork. She decides that $0.80 is a reasonable price for the big square piece (figure A). She has to decide on the prices of the other pieces.4. Use Student Activity Sheet 2 to find the prices of the other pieces. Note: All of the pieces have t

23、he same thickness.E.I.H.J.Here are drawings of tiles with different shapes. Mary Ann decides a reasonable price for the small tile is $5.A.B.C.D.$5E.F.G.K.H.I.J.5. a. UseStudentActivitySheet3 tofindthepricesoftheother tiles. b. Reflect Discuss your strategies with some of your classmates. Which tile

24、 was most difficult to price? Why?To figure out prices, you compared the size of the shapes to the$5 square tile. The square was the measuring unit. It is helpful to use a measuring unit when comparing sizes.The number of measuring units needed to cover a shape is called thearea of the shape.Tessell

25、ationsWhen you tile a floor, wall, or counter, you want the tiles to fit together without space between them. Patterns without open spaces between the shapes are called tessellations.Sometimes you have to cut tiles to fit together without any gaps. The tiles in the pattern here fit together without

26、any gaps. They form a tessellation.A.B.6. Use the $5 square to estimate the price of each tile.$5Each of the two tiles in figures A and B can be used to make a tessellation.7. a. Which of the tiles in problem 5 on page 3 can be used in tessellations? Use Student Activity Sheet 4 to help you decide.b

27、. Choose two of the tiles (from part a) and make a tessellation.Tessellations often produce beautiful patterns. Artists from many cultures have used tessellations in their work. The pictures below are creations from the Dutch artist M. C. Escher.Here is one way to make a tessellation. Start with a r

28、ectangular tile and change the shape according to the following rule.What is changed in one place must be made up for elsewhere.For example, if you add a shape onto the tile like this,youhaveto takeawaythesameshapesomeplaceelse.Herearea few possibilities.Tessell tion A.B.ShapeA.B.C.D.8. How many com

29、plete squares make up each of the shapes Athrough D?Shape D can be changedintoafish by takingawayandaddingsome more parts. Here is the fish.9. a. Draw the shape of the fish in your notebook.b. Show in your drawing how you can change the fish back into a shape that uses only wholesquares.c. How many

30、squares make up one fish?Another way to ask this last question in part c is, “What is the area of onefishmeasuredinsquares?”Thesquareisthemeasuringunit.Big States, Small StatesThe shape of a state can often be found on tourism brochures, government stationery, and signs at stateborders.10. a. Withou

31、t looking at a map, draw the shape of the state in which you live.Texasb. If you were to list the 50 states from the largest to the smallest in land size, about where would you rank your state?UtahCaliforniaThree U.S. states, drawn to the same scale, are above.11. Estimate the answer to the followin

32、g questions. Explain how you found each estimate.a. How many Utahs fit into California?b. How many Utahs fit intoTexas?c. How many Californias fit into Texas?d. Comparetheareas of thesethreestates.Forty-eight of the United States are contiguous, or physically connected. You will find the drawing of

33、the contiguous states on Student Activity Sheet 5.12. Choosethree of the 48 contiguousstatesandcomparethearea of your state to the area of each of these three states.Islands and ShapesIf a shape is drawn on a grid, you can use the squares of the grid to find the area of the shape. Here are two islan

34、ds: Space Island and Fish Island.Space IslandFish Island13. a. Which island is bigger? How do you know? Use Student Activity Sheet 6 to justify your answers.b. Estimate the area of each island in square units.Since the islands have an irregular form, you can only estimate the area for these islands.

35、Youcan find the exact area for the number of whole squares, but you have to estimate for the remaining parts. Finding the exact area of a shape is possible if the shape has a more regular form.A.B.C.D.E.F.G.H.14. What is thearea of each of theshaded pieces? Use Student Activity Sheet 7 to help you.

36、Give your answers in square units. Be prepared to explain your reasoning.The Size of ShapesAWhen you know the area of one shape, you can sometimes use that information to help you find the area of another shape. This only works if you can use some relationship between the two shapes.Here are some sh

37、apes that are shaded.A.B.C.D.E.F.G.H.I.J.K.15. a. Choose four blue shapes and describe how you can find the area of each. If possible, use relationships between shapes.b. Now find the area (in square units) of each of the blue pieces.c. Describe the relationship between the blue area in shapes Cand

38、D.SectionA: TheSizeof Shapes 9AThe Size of ShapesThissectionisaboutareas(sizes)ofshapes.Youuseddifferentmethods to compare the areas of two forests, tulip fields, pieces of cork, tiles, and various states and islands. You: may have counted tulips; compared different-shaped pieces of cork to a larger

39、 square piece of cork; and divided shapes and put shapes together to make new shapes.You also actually found the area of shapes by measuring. Area is described by using squareunits.You explored several strategies for measuring the areas of various shapes. Youcounted the number of complete squaresins

40、ide a shape, then reallotted the remaining pieces to make new squares.Inside this shape thereare four complete squares.The pieces that remain can be combined into four new squares.1 23 4 You may have used relationships between shapes.You can see that the shaded piece is half of the rectangle.Or you

41、can see that two shapes together make a third one. Youmay have divided a shape into smaller parts whose area you can find more easily. Youmayalsohaveenclosedashapewitharectangleand subtracted the empty areas.1. Sue paid $3.60 for a 9 inch (in.)-by-13 in. rectangular piece of board. She cuts the boar

42、d into three pieces as shown. What is afair price for each piece?Section A: The Size of Shapes 11$3.6013 in.9 in.13 in.6 in.3 in.A The Size of Shapes2. Below you see the shapes of twolakes.a. Which lake is bigger? How do youknow?b. Estimate the area of eachlake.3. Find the area in square units of ea

43、ch of these orange pieces.A.B.C.D.E.4. Choosetwo of these shapesandfind the area of the green triangles. Explain how you found eacharea.A.B.C.D.Why do you think this unit is called Reallotment?BArea PatternsRectangles1. Findtheareaenclosed by each of therectanglesoutlined in the figures below. Explain your methods.A. B.C.D.E.4 cm5 cm2. a. Describe at least two different methods you can use to find the area enclosed by a rectangle. b. Reflect Which method do you prefer? Why?Section B: Are