mathematics in context 情境中的数学 34本mathematics in context - it_s all the same_W

1、Its All the SameGeometry 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 Instit

2、ute 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 exp

3、ressed are those of the authors and not necessarily those of the Foundation.Roodhardt, A.; Abels, M.; de Lange, J.; Dekker, T.; Clarke, B.; Clarke, D. M.; Spence,M. S.; Shew,J. A.; Brinker,L. J.; and Pligge, M. A. (2006). Itsallthesame. InWisconsin Centerfor Education Research & Freudenthal Institut

4、e(Eds.), Mathematics in Context. Chicago: 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 bytho

5、se laws. Any uses not in conformity with the U.S. copyright statute are prohibited without our express writtenpermission, including but not limited to duplication, adaptation, and transmission bytelevision or other devices or processes. Formore information regarding a license, write Encyclopdia Brit

6、annica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610.ISBN 0-03-038567-93 4 5 6 073 09 08 07 06The Mathematics in Context Development TeamDevelopment 19911997The initial version of Its All the Same was developed by Anton Roodhardt and Mieke Abels. It was adapted for use in American schools

7、 by Barbara Clarke, Doug M. Clarke, Mary C. Spence, Julia A. Shew, and Laura J. Brinker.Wisconsin Center for EducationResearch StaffFreudenthal Institute StaffThomas A. RombergDirectorGail BurrillCoordinatorJoan Daniels PedroAssistant to the DirectorMargaret R. MeyerCoordinatorJan de LangeDirectorEl

8、s FeijsCoordinatorMartin van ReeuwijkCoordinatorProject StaffJonathan Brendefur Laura Brinker James Browne Jack BurrillRose ByrdPeter Christiansen Barbara Clarke Doug Clarke Beth R. ColeFae Dremock Mary Ann FixSherian Foster James A, Middleton Jasmina Milinkovic Margaret A. Pligge Mary C. Shafer Jul

9、ia A. Shew Aaron N. Simon Marvin Smith Stephanie Z. Smith Mary S. SpenceMieke Abels Nina Boswinkel Frans van GalenKoeno Gravemeijer Marja van denHeuvel-Panhuizen Jan Auke de Jong Vincent Jonker Ronald Keijzer Martin KindtJansie Niehaus Nanda Querelle Anton Roodhardt Leen Streefland Adri Treffers Mon

10、ica Wijers Astrid de WildRevision 20032005The revised version of ItsAll the Same was developed by Jande Lange, Mieke Abels, andTruus Dekker. It was adapted for use in American schools by Margaret A. Pligge.Wisconsin Center for EducationResearch StaffFreudenthal Institute StaffThomas A. RombergDirect

11、orGail BurrillEditorial CoordinatorDavid C.WebbCoordinatorMargaret A. PliggeEditorial CoordinatorJan de LangeDirectorMieke AbelsContent CoordinatorTruus DekkerCoordinatorMonica WijersContent CoordinatorProject StaffSarah Ailts Beth R. Cole Erin Hazlett Teri Hedges Karen Hoiberg Carrie Johnson Jean K

12、rusi Elaine McGrathMargaret R. Meyer Anne ParkBryna Rappaport Kathleen A. Steele Ana C. Stephens Candace Ulmer Jill VettrusArthur Bakker Peter Boon Els Feijs Dd de Haan Martin KindtNathalie Kuijpers Huub Nilwik Sonia Palha Nanda QuerelleMartin van Reeuwijk(c) 2006 Encyclopdia Britannica, Inc. Mathem

13、atics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopdia Britannica, Inc.Cover photo credits: (left) Corbis; (middle, right) Getty ImagesIllustrations14 (top) Rich Stergulz; (middle) Christine McCabe/ Encyclopdia Britannica, Inc.; 15 Christine McCabe/ Encyclopdia

14、 Britannica, Inc.; 33, 39, 43 Rich Stergulz; 44, 48 Christine McCabe/ Encyclopdia Britannica, Inc.Photographs11 Comstock, Inc.; 27 Sam Dudgeon/HRW Photo; 29 Andy Christiansen/ HRW; 31 HRW Art; 36 Andy Christiansen/HRW; 40Victoria Smith/HRW; 44 (top left, right, bottom left) PhotoDisc/Getty Images; (

15、bottom right) Corbis; 54 PhotoDisc/Getty ImagesContentsLetter to the StudentviTessellationsTriangles Forming Triangles TessellationsIts All in the Family SummaryCheck Your WorkSectionA12467Enlargement and ReductionMore Triangles Enlargement and Reduction Overlapping TrianglesThe Bridge Problem Josep

16、hs Bedroom SummaryCheck Your WorkSectionB9111214161820ASimilaritySimilar Shapes Point to Point Shadows TakeoffAngles and Parallel LinesYou Dont Haveto Get Your Feet Wet SummaryCheck Your WorkSectionC222327282931323310 paces20 pacesDBC24 pacesE2.45C88 cmSimilar ProblemsPatternsUsing Similar Triangles

17、 More TrianglesThe PorchEarly Motion Pictures SummaryCheck Your WorkSectionDCDE?35363839414242216 cmAB2.45180 cmCoordinate GeometryParallel and Perpendicular Roads to Be Crossed Length and Distance SummaryCheck Your WorkSectionE4548495051Additional Practice52Answersto Check YourWork57Contentsvvi Its

18、 All The SameDear Student,Did you ever want to know the height of a tree that you could not climb? Do you ever wonder how people estimate the width of a river?Have you ever investigated designs made with triangles?In this Mathematics in Context unit,Its All The Same, you will exploregeometric design

19、s called tessellations. You will arrange triangles in different patterns, and you will measure lengths and compare angles in your patterns. You will also explore similar triangles and use them to find distances that you cannot measure directly.As you work through the problems in this unit, look for

20、tessellations in your home and in your school. Look for situations where you can use tessellations and similar triangles to find lengths, heights, or other distances. Describe these situations in a notebook and share them with your class. Have fun exploring triangles, similarity, and tessellations!S

21、incerely,The Mathematics in Context Development TeamATessellationsTriangles Forming TrianglesCut out the nine triangles on Student Activity Sheet 1.Use all nine triangles to form one large triangle.Rearrange the nine triangles to form one large triangle so you form a black triangle whenever two tria

22、ngles meet.Rearrange the nine triangles to form a symmetric pattern. How can you tell your arrangement is symmetric?Section A: Tessellations1TessellationsATessellationsA tessellation is a repeating pattern that completely covers a larger figure using smaller shapes. Here are two tessellations coveri

23、ng a triangle and a rhombus.TriangleRhombusHowdoesthearea of thelargetrianglecompare to theareaof the rhombus?The triangle consists of nine congruent triangles. What does the word congruent mean?The rhombus consists of a number of congruent rhombuses. How many?You can use the blue and white triangle

24、s to cover or tessellate the rhombus. How many of these triangles do you need to tessellate the large rhombus?Can you tessellate a triangle with 16 congruent triangles? If so, make a sketch. Ifnot, explain why not.1.a.b.c.d.e.2 Its All The SameTessellationsAHereis alargetriangletessellation.You can

25、break it down by cutting rows along parallel lines.Row 1 Row 2 Row 3Row 4These lines form one family of parallel lines. There are other families of parallel lines.2. How many different families of parallel lines are in this large triangle?Here is the triangle cut along a different family of parallel

26、 lines.13573. a. Explain the numbers below each row.b. Explain what the sequence of numbers 1, 4, 9, 16 has to do with the numbers below eachrow.c. Lily copied this tessellation but decided to add more rows. She used 49 small triangles. How many triangles are in Lilys last row?Section A: Tessellatio

27、ns3TessellationsAIts All in the FamilyHere is a drawing, made with two families of parallel lines. It is the beginning of a tessellation of parallelograms.4. a. On Student Activity Sheet 2, draw in a third family of parallel lines to form a triangle tessellation.b. Are the resulting trianglescongrue

28、nt? Why or why not?c. Did everyone in your class draw the same family of parallel lines?Youcan use one small triangle to make a triangle tessellation. All you need to do is draw the three families of parallel lines that match the direction of each side of thetriangle.This large triangle shows one fa

29、mily of parallel lines.Heres how to finish this triangle tessellation. On Student Activity Sheet 2, use a straightedge to draw the other two families of parallel lines.How many small triangles are along each edge?How many small triangles tessellate the large triangle?5.a.b.c.If the triangle in probl

30、em 5 had ten rows, how many triangles would be along eachedge?How many small triangles would tessellate a triangle with ten rows?6.a.b.Think about a large triangle that has n rows in each direction. How many small triangles would be along each edge of the large triangle?Write a formula for the total

31、 number of triangles to tessellate a triangle with n rows.7.a.b.4 Its All The SameTessellationsALaura used one triangle to make rows of congruent triangles.She noticed very interesting things happen.Rows form parallel lines in three different directions.There is the same number of small triangles al

32、ong each edge.a. Make up your own large triangle tessellation using one small triangle.b. Verify that the formula you found in problem 7b works for this tessellation.8.Tessellations can make beautiful designs. Here is a tessellation design based on squares. This tessellation consists of eight pieces

33、 using only two different shapes.9. a. How many total pieces do you need to make each of these tessellationdesigns? Howmanydifferentshapes do you need?i.ii.iii.b. Design your own tessellation, based on squares, which consists of 16 pieces using exactly four different shapes.Section A: Tessellations

34、5ATessellations6 Its All The SameA tessellation is a repeating pattern that completely covers a large shape using identical smaller shapes.Congruent figures are exact “copies” of each other. Two figures are congruent if they have the same size and the same shape.When you use a small triangle to make

35、 a large triangle tessellation, interesting patterns occur.2 rows, 4 triangles3 rows, 9triangles The number of triangles making up each row is the odd number sequence, 1, 3, 5 Thetotalnumber of trianglesmaking up thetriangleis alwaysa perfect square number, 1, 4, 9, 16, 25 The number of rows will te

36、ll you how many small triangles tessellate a large triangle; for example, a triangle with six rows needs 36 small triangles to make atessellation.You can make a tessellation using small shapes.Kira completely covered this trapezoid using two shapes, a triangle and a hexagon.Her tessellation consists

37、 of 21 pieces using 2 different shapes. She used 14 congruent triangles and 7 congruent hexagons.TrapezoidTrapezoidal Tessellation1. a. Describe another way to identify congruent figures.b. Make two congruent shapes. Describe all the parts of the shape that are exactly thesame.2. Design a large tria

38、ngle using four rows of congruent triangles.Section A: Tessellations 7You can make a tessellation using families of parallel lines.Logan used two families of parallel lines to create a tessellation for a large parallelogram. His tessellation consists of 12 small congruent parallelograms.ATessellatio

39、ns8 Its All The SameRobert has 50 banners of his favorite sports team. The bannersareallcongruent, andeachbanner is thesame on the front and back. Robert wants to use his banners to make one giant display in the shape of a triangle.3. a. Is it possible for Robert to arrange all 50 banners into a lar

40、ge triangle? If so, sketch the large triangle. If not, sketchalargetriangulardisplaythatusesas close to 50 banners as possible.b. Howmanybannersarealongeachedge?(Use your sketch from a.)Consider a large rectangle with dimensions 10 centimeters (cm) by 20 cm. Find different ways to tessellate this re

41、ctangle with smaller rectangles. For each tessellation, record the dimensions of the smaller rectangles. (Remember: A tessellation must completely cover the shape.)BEnlargement and ReductionMore TrianglesThis large triangle is partially tessellated. The dimensions of the large triangle are given.180

42、 cm210 cm240 cm1. What are the lengths of the sides of the small triangle used in the tessellation?2. a. Make a table like this one to record your answers to problem 1.b. Explain why this table is also a ratio table.c. Compare the small triangle to the large triangle. What do you notice?d. Thelarget

43、riangleisanenlargementofthesmalltriangle. The enlargement factor is 6. Explain what this means.Section B: Enlargement and Reduction 9Lengths of SidesSmall TriangleLarge Triangle180 cm210 cm240 cmEnlargement and ReductionBYou can tessellate this triangle with small congruent triangles.R3. a. Find thr

44、ee different triangles that can tessellate QRS. For each triangle, give the lengths of the sides and explain why it tessellates the large triangle.30 cm36 cmQS42 cmb. For each tessellation, compare the large and small triangles to find the enlargement factor.This small triangle can tessellate a larg

45、e triangle with dimensions30 cm 40 cm 50 cm.4. a. How many small triangles fit along each side of the large triangle?8 cm6 cm10 cmb. Copy and complete this ratio table.1212.12c. Which number shows the enlargement factor?Before continuing, it is important to clarify some essential vocabulary of this

46、unit.Some of you probably have enlarged a special photograph to fit an 8 in. 10 in. portrait frame.You may have reduced a special photograph to fit into a wallet or small frame.The enlargement factor or reduction factor is the number you need to multiply the dimensions of the original object.The mul

47、tiplication factor encompasses either an enlargement or a reduction.10 Its All The SameSmall Triangle8.610Large Triangle40.Enlargement and ReductionBEnlargement and ReductionHere is a photograph shown in different sizes.1 2 2Aoriginal photoBThe original photo was both enlarged and reduced.5. a. What

48、 is themultiplicationfactorfromtheoriginalphoto to B?b. What is the multiplication factor from A to the original photo?c. What is the multiplication factor from A to B?d. A multiplicationfactor of two produces an enlargementof 200%. Explain why.If the multiplication factor is a number from 0 to 1, t

49、he original figure is reduced in size.Ifthemultiplicationfactorisanumbergreaterthan1,theoriginal figure is enlarged.In the drawing, DEC can tessellate ABC.In the small triangle, DE 40 cm, EC 35 cm, and CD 30 cm. In the large triangle, AC 270 cm.30 cmCDE270 cmAB6. Use a ratio table to find the length

50、s of sides AB and BC.Section B: Enlargement and Reduction11Enlargement and ReductionBIn the triangle, the markings indicate that sides NP and KL are parallel. As a matter of notation: NP | KL.M2 cmNPCan you use NPM to tessellateKLM ? If so, show the tessellation. If not, explain why you cannot.7. a.

51、3 cm5 cmb. By which factor do you need to multiply NPM in order to getKLM ?c. What is the difference between tessellating a triangle and enlarging a triangle?KLOverlapping TrianglesHere are two new triangles. These triangles are overlapping triangles. Sometimes, it makes it easy to see the correspon

52、ding sides if you redraw the trianglesseparately.C15 cmDE20 cmThe second drawing shows how, for example, side DC and side AC are corresponding sides.70 cma. What is the enlargement factor for these triangles?b. Use your answer from part a to find the length of side AC.c. What is the length of segmen

53、t AD?8.AB80 cmd. What does CB equal?CCCE15 cmD20 cm E70 cmAB80 cm12 Its All The SameEnlargement and ReductionB9. Find DE, if AC 3.CCDD ?EAB1210. a. The length of side KJ in small KLJ is 9. What is the length of the corresponding side HJ in the large triangle?J9 9K3HLI?b. What is the multiplication f

54、actor from KLJ to HIJ?c. Find the length of side HI, the side with the question mark.11. For the two triangles below, find the length of the side with the question mark. (Hint: For the second figure, you may want to use a ratio table.)OQTU21?12.5MNR2 S4P810VSection B: Enlargement and Reduction 13Enl

55、argement and ReductionBThe Bridge ProblemHere is a side view of a bridge that Diedra drives across as she travels to and from work. (Note: The drawing is not to scale.)Asshowninthediagrambelow,when Diedrascaris 50 meters(m) up theramp, sheestimatesshe is about 3 m abovegroundlevel. She drives another 400 m and reaches the b