Jossey-Bass Teacher - Math Wise Phần 1 ppsx

includes activities that will help each student gain full comprehension of basic mathematical concepts, including numbers andcounting, computation, estimation, probability, data analysis

Jossey-Bass Teacher

Jossey-Bass Teacher provides educators with practical knowledge andtools to create a positive and lifelong impact on student learning We offerclassroom-tested and research-based teaching resources for a variety ofgrade levels and subject areas Whether you are an aspiring, new, orveteran teacher, we want to help you make every teaching day your best.From ready-to-use classroom activities to the latest teachingframework, our value-packed books provide insightful, practical, andcomprehensive materials on the topics that matter most to K–12teachers We hope to become your trusted source for the best ideasfrom the most experienced and respected experts in the field

Copyright © 2010 by James L Overholt and Laurie Kincheloe All rights reserved.

989 Market Street, San Francisco, CA 94103-1741—www.josseybass.com

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form

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ISBN: 978-0-470-471999

Printed in the United States of America

SECOND EDITION

PB Printing 10 9 8 7 6 5 4 3 2 1

About This Resource

Math Wise! includes activities that will help each student gain full

comprehension of basic mathematical concepts, including numbers andcounting, computation, estimation, probability, data analysis, measure-ment, geometry, algebra, problem solving, and logical thinking Students

in today’s math classrooms must be able to do more than achievecorrect answers through computation; they need to understand basic

concepts and experience a range of mathematical applications Math Wise! is designed to help the teacher accomplish these learning objec-

tives It contains a wide variety of learning experiences that have beenarranged according to difficulty level Whenever possible, the activitiesare presented in either hands-on or visual formats

Concrete/Manipulative Activities

Especially when exploring ‘‘new’’ concepts, each student should workwith hands-on materials A number of the activities therefore includeeasily obtained manipulatives, such as straws, paper clips, sugar cubes,

and beans For example, a problem in the activity Paper Clip Division

asks students to show 44 divided by 7 One-to-one correspondence isused when one paper clip corresponds to the numeral 1 The result mightappear as:

6 GROUPS

OF 7 CLIPS

REMAINDER shows 7 ) 44 6

42 2v

In Punchy Math, students use a paper hole punch, scrap paper, and

a pencil to show 3× 7 = The outcome, after folding, punching,looping, and labeling, shows 3 groups of 7 If turned sideways, it canalso show 7 groups of 3, or 7× 3 = Whereas the resulting punchedholes are concrete, the looped segments provide a visual component thatdirectly corresponds to the abstract number relationships involved

numbers For example, in Cross-Line Multiplication, three horizontal lines

represent the number 3 and five vertical lines represent the number 5.When the lines are crossed, the fifteen intersection points represent theanswer to the problem 3× 5 = The following figure illustrates thisvisual representation Of course, turning the drawing sideways shows

5× 3 = 15

In Decimal Squares, another visual activity, students are provided

with a sheet of Decimal Squares Each decimal square is a 10-unit by10-unit square divided into 100 square units Each small square unitrepresents one hundredth of the decimal square, or 01 Students arethen asked to show the relationship between 0.6 and 0.21 For example,

in the problem 0.6 0.21, students are required to fill in the blank

with >, <, or= to make the statement true To find the answer, studentsare asked to shade in the Decimal Squares, as shown below

0.6 > 0.21

Abstract Procedures

A major goal of mathematics education is to help students eventuallyperform abstract mathematical procedures and understand the under-lying concepts behind these procedures When possible, mathematicsteachers should not only instruct students in regard to mathematicalmechanics but also enable them to gain a true understanding of theconcepts involved

In the activity Post-it Mental Math, one student has Post-it numerals

placed on his or her back without being allowed to see them The othergroup members, after viewing the numerals, give the student clues aboutthe numerals Using these clues, the Post-it wearer must do mental math

to determine the numerals In the situation that follows, the Post-itplayer has made a first guess based on one player’s clues

Block Four, which requires two players or two opposing teams, a

numbered game board, and two paper clips, is another activity askingstudents to make abstract computations and draw upon their logical-thinking abilities The first player places the paper clips on two numbers,and then performs the multiplication The student then puts an X onthe square with the answer The next player can only move one paperclip, leaving the other one alone This player will then perform themultiplication and mark his or her square with an O The boards belowshow two partially played games

27

40

56

3918284263

41020304564

51221324872

61424354981

1

2 13 23 14 34 18 38 58 78 16 19

1 8

1 3

1

6 49

1 9

1 2

1 6

1 9 1 8

1 4

1 4

1 8

3 8

2

9 2164 361 325 725 54125

48 1

12 323 245 161 327 161

7 16 1 64 3 16 1 12

35 64

1 18

5 64

1 18

5 24

1 72

1 18

15 32

9 64 5 12 9 16

2 27 9 32 25 64

7 12 7 64

1 12 1 48

1 27 21 31 49 64

1 24

3 64

5

16 487 241 485 323 1564

1 32

3 16

1 36

7 24

7 72

About the Authors

James L Overholt has an Ed.D from the University of Wyoming, Laramie.

He has been exploring the use of manipulative and visual materialsfor mathematics instruction since the 1960s As an elementary andsecondary school teacher in Minnesota and Wyoming, and later as auniversity professor, his investigations have taken him into both K–12classrooms and adult mathematics learning workshops He is currently

a professor of education at California State University, Chico

Dr Overholt regularly conducts mathematics education courses andworkshops for pre-service and in-service teachers at the elementary

and secondary levels His earlier published books include Math Stories for Problem Solving Success, Second Edition, also published by Jossey- Bass/Wiley; Dr Jim’s Elementary Math Prescriptions; Math Problem Solving for Grades 4–8; Math Problem Solving for Beginners Through Grade 3; Outdoor Action Games for Elementary Children, and Indoor Action Games for Elementary Children.

Laurie Kincheloe has a B.A in mathematics and an M.A in

mathemat-ics education from California State University, Chico She taught highschool mathematics for twelve years and is presently teaching math-ematics at Butte College in northern California She has worked withK–12 students, parents, and teachers as a family math coordinator and

as a mentor for new teachers She teaches concepts in mathematics topre-service elementary teachers, and has coordinated service learningprojects connecting high school and college students with elementarystudents through mathematics She was co-coordinator of the Mathe-matics Project at California State University, Chico, and has conductedworkshops on the teaching of mathematics for elementary and secondaryteachers at numerous education conferences

In addition to teaching at Butte College, Laurie has served as thedevelopmental coordinator for the Mathematics Department, created amath-anxiety class designed to help apprehensive students be successful

at math, and organized the annual Math Awareness Week She hasreceived the Faculty Member of the Year Award and the Service LearningProject Faculty Award

ix

A Special Acknowledgement:

James F Lindsey, Ed.D (University of California, Berkeley) served as

an elementary teacher and principal for 25 years He co-authored

Math Stories for Problem Solving Success: Ready-to-Use Activities

for Grades 6–12, First and Second Editions (Jossey-Bass/Wiley).

When asked if he would help edit and proofread the new edition of

Math Wise!, he remarked ‘‘I would be honored!’’ From beginning to

end, James was always ahead of expectations He will be missed.

Suggestions for Using

Math Wise!

The activities in this book provide a varied collection of interesting andunderstandable tasks from which students in kindergarten through themiddle grades will benefit Although many of these activities can be used

in any order, it is advisable to designate tasks that are appropriate withregard to class size, students’ stages of learning, or other considerations.For this reason, several features in this book are designed to help selectappropriate activities

• The Contents categorizes each activity in five ways:

1 Section (‘‘Making Sense of Numbers,’’ ‘‘Computation

Connec-tions,’’ ‘‘Investigations and Problem Solving,’’ and ‘‘LogicalThinking’’)

2 Descriptive Title (such as Everyday Things Numberbooks, Paper

Clip Division, Peek Box Probability, and String Triangle Geometry)

3 Grade Level (K–2, 2–4, 4–6, and 6–8)

4 Activity Type (Concrete/Manipulative, Visual/Pictorial, and

Abstract)

5 Learning Format (Total Group, Cooperative, and Independent)

• A Key for each activity notes the most appropriate grade levels,

the preferred working arrangement, and the kinds of experiences

in which learners will take part For example, the following key to

Silent Math indicates that

1 The activity is best suited for students in grades 4 through 8.

2 The activity can be worked on by the whole class or by

cooper-ative groups

3 The students will work with visual diagrams and will perform

abstract computations

xi

• Each activity begins with a Why Do It statement that details the

specific mathematical concepts the students will be learning andpracticing

• The You Will Need statement specifies any supplies or equipment

necessary for the activity These items, such as paper clips, indexcards, and straws, are easily obtained and free or inexpensive

• The How To Do It section details what the teacher or other

education professional must do to set up and carry out the activity.Suggestions are made as to the steps that should be taken for theactivity to be successful It also describes how the investigationworks best as an independent activity, a cooperative project wherestudents work in pairs or small groups, or a total group venture.This section will provide the general premise and content of theactivity before the example are presented

• The Examples illustrate how the activity might progress, and

display typical outcomes

• An Extensions section at the end of each activity contains more

investigations that can be done using the same or similar proceduresdescribed in the activity It often contains more sample questions orsuggestions as to how to expand the mathematical concepts beingstudied Teachers and students are encouraged to propose similartasks of their own

• Where appropriate, reproducible pages immediately follow therelevant activity These pages include game boards, workmats, dotpaper, playing cards, graph paper, and more

• Students should be encouraged to record their methods and tions in a math journal or to keep a special file containing samples

solu-of their work

• Solutions are also provided when appropriate

xii

Grade Level K–2 2–4 4–6 6–8 Activity TypeConcrete/Manipulative Visual/PictorialAbstract Learning FormatTotal Group Cooperative Independent

I Making Sense of Numbers

1 Toothpick Storybooks X X X X X X X X

2 Number Combination Noisy Boxes X X X X X X X

3 Everyday Things Numberbooks X X X X X X X X

4 Under the Bowl X X X X X X X

5 Cheerios ™ and Fruit Loops ™ Place Value X X X X X X X X X

6 Beans and Beansticks X X X X X X X X X

12 Dot Paper Fractions X X X X X X X X

13 Fraction Cover-Up or Un-Cover X X X X X X X

14 Post-it ™ Mental Math X X X X X X

15 Calendar Math X X X X X X X

16 Let’s Have Order X X X X X X

17 Reject a Digit X X X X X X

18 A Million or More X X X X X X

19 Smallest and Largest X X X X X X

20 Numbers to Words to Numbers X X X X X X

21 Target a Number X X X X X X

1 3 5 7 9 11 13 19 22 27 30 33 36 43 47 50 54 57 62 66 71 74

xiii

Grade Level K–2 2–4 4–6 6–8 Activity TypeConcrete/Manipulative Visual/PictorialAbstract Learning FormatTotal Group Cooperative Independent

22 Fraction Codes X X X X X X X

23 Comparing Fractions, Decimals, and Percents X X X X X X X

24 Number Clues X X X X X X

25 Number Power Walks X X X X X X X

II Computation Connections

26 Paper Clip Addition Cards X X X X X X

27 Arm-Lock Computation X X X X X X X

28 Punchy Math X X X X X X X X X

29 Multiplication Fact Fold-Outs X X X X X X X

30 Ziploc ™ Division X X X X X X X X

31 Dot Paper Diagrams X X X X X X X X

32 File Folder Activities X X X X X X

33 Beat the Calculator X X X X X X X

34 Floor Number Line Actions X X X X X X X

35 Egg Carton Math X X X X X X X X X

41 Drawing Fraction Common Denominators X X X X X X X X

42 Fraction × and ÷ Diagrams X X X X X X X

48 Paper Clip Division X X X X X X X X X

49 I Have , Who Has ? X X X X X X

77 80 83 91 95 97 100 104 106 109 112 119 122 125 128 133 136 139 143 147 151 154 158 163 168 170 174 179 182

Grade Level K–2 2–4 4–6 6–8 Activity TypeConcrete/Manipulative Visual/PictorialAbstract Learning FormatTotal Group Cooperative Independent

57 Sticky Gooey Cereal Probability X X X X X X X

58 Sugar Cube Buildings X X X X X X X X X X

59 A Chocolate Chip Hunt X X X X X X X X X X

60 Flexagon Creations X X X X X X X X X X

61 Watermelon Math X X X X X X X X X

62 Restaurant Menu Math X X X X X X X X X

63 Peek Box Probability X X X X X X X X X X

72 Paper Airplane Mathematics X X X X X X X X X

73 A Dog Pen Problem X X X X X X X X X

74 Building the Largest Container X X X X X X X X X

75 The Three M’s (Mean, Median, and Mode) X X X X X X X

76 Post-it Statistics X X X X X X X X

77 A Postal Problem X X X X X X X X

186 189 194 196 203 206 209 211 214 219 223 228 232 235 238 242 247 250 254 260 271 274 278 281 285 288 290 294 297

Grade Level K–2 2–4 4–6 6–8 Activity TypeConcrete/Manipulative Visual/PictorialAbstract Learning FormatTotal Group Cooperative Independent

78 Build the ‘‘Best’’ Doghouse X X X X X X X X

79 Dog Races X X X X X X X

80 Four-Coin Statistics X X X X X X X

81 Tube Taping X X X X X X X X

82 Height with a Hypsometer X X X X X X X X

83 Fairness at the County Fair X X X X X X X

84 Winning a Prize Spelling ‘‘NUT’’ X X X X X X X

85 Building Toothpick Bridges X X X X X X X X

86 A Bridge with a Bulge X X X X X X

IV Logical Thinking

92 Magic Triangle Logic X X X X X X X

93 Paper Clip Spinners X X X X X X X X

94 Triangle Toothpick Logic X X X X X X X X X

95 Rectangle Toothpick Logic X X X X X X X X X

96 What Graph Is This? X X X X X X X X

Grade Level K–2 2–4 4–6 6–8 Activity TypeConcrete/Manipulative Visual/PictorialAbstract Learning FormatTotal Group Cooperative Independent

105 Duplicate Digit Logic X X X X X X

106 String Triangle Geometry X X X X X X X X

107 A Potpourri of Logical-Thinking

Problems, Puzzles, and Activities

Plan a Circuit Board X X X X X X X X X X

22 Wheels and 7 Kids X X X X X X X X X

Candy Box Logic X X X X X X X X X

Rubber Sheet Geometry X X X X X X

How Long Is a Groove? X X X X X X X

Solutions to Selected Potpourri Activities

408 411 414 414 415 416 416 417 418 418 419 420 420 421 422 423 424

Section One

Making Sense

of Numbers

The activitiesin this section introduce students to many

number concepts and relationships, including 1-to-1

corre-spondence, basic number combinations, place value, mental

math, fractions, large numbers, and decimals Students will

practice essential mathematical skills and develop conceptual

understanding through hands-on investigations and games

that make use of manipulative experiences, visual portrayals,

or relevant abstract procedures

A number of activities from other portions of this book

can be used to extend and enhance students’ comprehension

of the concepts introduced in this section, such as Punchy

Math (p 104) and Beat the Calculator (p 122) from Section

Two; Peek Box Probability (p 238) and Restaurant Menu Math

(p 235) from Section Three; and Duplicate Digit Logic (p 408)

from Section Four

Students will discover the concepts of 1-to-1 counting and

number conservation, and will study basic computation

rela-tionships

You Will Need:

This activity requires several boxes of flat toothpicks, white

and colored paper (pages approximately 6 by 9 inches work

well), glue, and marking pens or crayons

How To Do It:

1 Have younger students explore and share the different

arrangements they can make with a given number

of toothpicks For example, students could arrange

4 toothpicks in a wide variety of different rations, all of which would still yield 4 toothpicks

configu-2 After exploring for a while, students should begin

mak-ing Toothpick Storybooks, startmak-ing by creatmak-ing numberpages Students can write, for instance, the number 6

on a sheet of white paper and glue 6 toothpicks onto apiece of colored paper (To avoid a sticky mess, studentsshould dip only the ends of the toothpicks in the glue.)

3