Table of Contents
Jossey-Bass Teacher
Title Page
Copyright Page
About This Resource
Concrete/Manipulative Activities
Visual/Pictorial Activities
Abstract Procedures
A Final Note
About the Authors
Acknowledgements
Suggestions for Using Math Wise!
Silent Math
Section One - Making Sense of Numbers
Chapter 1 - Toothpick Storybooks
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 2 - Number Combination Noisy Boxes
Why Do It:
You Will Need:
How To Do It:
Example:
Extension:
Chapter 3 - Everyday Things Numberbooks
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 4 - Under the Bowl
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 5 - Cheerios™ and Fruit Loops™ Place Value
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 6 - Beans and Beansticks
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 7 - Incredible Expressions
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 8 - Number Cutouts
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 9 - Celebrate 100 Days
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 10 - Paper Plate Fractions
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 11 - Bean Cups to 1,000
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 12 - Dot Paper Fractions
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Dot Paper Fraction Problems
More Dot Paper Fraction Problems
Chapter 13 - Fraction Cover-Up or Un-Cover
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 14 - Post-it™ Mental Math
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 15 - Calendar Math
Why Do It:
You Will Need:
How To Do It:
Examples:
Extension:
Chapter 16 - Let’s Have Order
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 17 - Reject a Digit
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 18 - A Million or More
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 19 - Smallest and Largest
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 20 - Numbers to Words to Numbers
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 21 - Target a Number
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 22 - Fraction Codes
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
A Good Rule
Chapter 23 - Comparing Fractions, Decimals, and Percents
Why Do It:
You Will Need:
How To Do It:
Example:
Chapter 24 - Number Clues
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Number Clue Activity
Chapter 25 - Number Power Walks
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Section Two - Computation Connections
Chapter 26 - Paper Clip Addition Cards
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 27 - Arm-Lock Computation
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 28 - Punchy Math
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 29 - Multiplication Fact Fold-Outs
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 30 - Ziploc™ Division
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 31 - Dot Paper Diagrams
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 32 - File Folder Activities
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 33 - Beat the Calculator
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 34 - Floor Number Line Actions
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 35 - Egg Carton Math
Why Do It:
You Will Need:
How To Do It:
Extensions:
Egg Carton Probability
Chapter 36 - Cross-Line Multiplication
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 37 - Highlighting Multiplication
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Multiplication Charts
Chapter 38 - Chalkboard or Tabletop Spinner Games
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Problems:
Chapter 39 - Skunk
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 40 - Subtraction Squares
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Subtraction Square
Chapter 41 - Drawing Fraction Common Denominators
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 42 - Fraction × and ÷ Diagrams
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 43 - Decimal Squares
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Decimal Squares
Chapter 44 - Square Scores
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 45 - Math Concentration
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 46 - Scramble
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 47 - Palindromic Addition
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Guide to Palindromic Sums
Chapter 48 - Paper Clip Division
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 49 - I Have _____, Who Has ______?
Why Do It:
You Will Need:
How To Do It:
Example:
Extension:
Chapter 50 - Number Grids
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Number Grid
Chapter 51 - Here I Am
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Master Game Board
Player Game Boards
Chapter 52 - Equation Match-Up
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 53 - Block Four
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 54 - Silent Math
Why Do It:
You Will Need:
How To Do It:
Examples and Extensions:
Solutions:
Chapter 55 - Rapid Checking
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Section Three - Investigations and Problem Solving
Chapter 56 - Shoe Graphs
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 57 - Sticky Gooey Cereal Probability
Why Do It:
You Will Need:
How To Do It:
Extensions:
Cut-Out Spinner
Sticky Gooey Cereal Record Sheet
Chapter 58 - Sugar Cube Buildings
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
3-D Drawing Paper
Chapter 59 - A Chocolate Chip Hunt
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chocolate Chip Records
Chapter 60 - Flexagon Creations
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Can You Create a Flexagon?
Chapter 61 - Watermelon Math
Why Do It:
You Will Need:
How To Do It:
Example:
Extension:
Chapter 62 - Restaurant Menu Math
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 63 - Peek Box Probability
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Peek Box Records
Peek Box Records
Chapter 64 - A Problem-Solving Plan
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
A Problem-Solving Plan
Chapter 65 - Fraction Quilt Designs
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 66 - What I Do in a Day
Why Do It:
You Will Need:
How To Do It:
Extensions:
Circle Graph Activity Sheet
Chapter 67 - Shaping Up
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Attribute Pieces
Chapter 68 - Verbal Problems
Why Do It:
You Will Need:
Directions and Problems for Young Students (Grades 1 - 3)
Directions and Problems for Middle-Grade Learners (Grades 4 - 5)
Directions and Problems for Older Learners (Grades 6 - 8)
Chapter 69 - Scheduling
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
My Weekly Schedule
Chapter 70 - Student-Devised Word Problems
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 71 - Tired Hands
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 72 - Paper Airplane Mathematics
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Airplane Contest Records
Chapter 73 - A Dog Pen Problem
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 74 - Building the Largest Container
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 75 - The Three M’s (Mean, Median, and Mode)
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 76 - Post-it Statistics
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 77 - A Postal Problem
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 78 - Build the “Best” Doghouse
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 79 - Dog Races
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Dog Race Chart
Chapter 80 - Four-Coin Statistics
Why Do It:
You Will Need:
How To Do It:
Extensions:
Four-Coin Chart
Chapter 81 - Tube Taping
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 82 - Height with a Hypsometer
Why Do It:
You Will Need:
How To Do It:
Example:
Extension:
Chapter 83 - Fairness at the County Fair
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Fairness at the County Fair
Chapter 84 - Winning a Prize Spelling “NUT”
Why Do It:
You Will Need:
How To Do It:
Extensions:
Winning a Prize Spelling “NUT” Activity Sheet
Chapter 85 - Building Toothpick Bridges
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 86 - A Bridge with a Bulge
Why Do It:
You Will Need:
How To Do It:
Extensions:
Section Four - Logical Thinking
Chapter 87 - Stacking Oranges
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Solutions:
Chapter 88 - Tell Everything You Can
Why Do It:
You Will Need:
How To Do It:
Examples:
Possible Solutions:
Chapter 89 - Handshake Logic
Why Do It:
You Will Need:
How To Do It:
Extension:
Chapter 90 - 2- and 3-D Arrangements
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Chapter 91 - Overhead Tic-Tac-Toe
Why Do It:
You Will Need:
How To Do It:
Extensions:
Chapter 92 - Magic Triangle Logic
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Magic Triangle Worksheet
Chapter 93 - Paper Clip Spinners
Why Do It:
You Will Need:
How To Do It:
Example:
Extension:
Chapter 94 - Triangle Toothpick Logic
Why Do It:
You Will Need:
How To Do It:
Questions and Extensions:
Solutions:
Chapter 95 - Rectangle Toothpick Logic
Why Do It:
You Will Need:
How To Do It:
Problem Solving with Toothpicks
Chapter 96 - What Graph Is This?
Why Do It:
You Will Need:
How To Do It:
Examples:
More Examples To Try:
Extensions:
Possible Solutions:
Chapter 97 - Fold-and-Punch Patterns
Why Do It:
You Will Need:
How To Do It:
Extension:
Solutions:
Chapter 98 - Coordinate Clues
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 99 - Puzzlers with Paper
Why Do It:
You Will Need:
How To Do It:
Extensions:
Solutions:
Chapter 100 - Create a Tessellation
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Chapter 101 - Problem Puzzlers
Why Do It:
You Will Need:
How To Do It:
Solutions:
Chapter 102 - Dartboard Logic
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Dartboards
Chapter 103 - Angelica’s Bean Logic
Why Do It:
You Will Need:
How To Do It:
Example:
Extensions:
Solutions:
Chapter 104 - Line It Out
Why Do It:
You Will Need:
How To Do It:
Examples:
Extensions:
Line It Out
Some Solutions
Chapter 105 - Duplicate Digit Logic
Why Do It:
You Will Need:
How To Do It:
Extensions:
Solutions:
Chapter 106 - String Triangle Geometry
Why Do It:
You Will Need:
How To Do It:
Solutions:
Chapter 107 - A Potpourri of Logical-Thinking Problems, Puzzles, and Activities
Why Do It:
You Will Need:
How To Do It:
Plan a Circuit Board
22 Wheels and 7 Kids
Candy Box Logic
Brownie Cutting
Making Sums with 0-9
Upside-Down Displays
Coin Walk
Dice Plotting
Coin Divide
Animal Pens
12 Days of Christmas
Rubber Sheet Geometry
How Long Is a Groove?
Solutions to Selected Potpourri Activities:
Selected Bibliography
Index
Jossey-Bass Teacher
Jossey-Bass Teacher provides educators with practical knowledge and tools to create a positive and lifelong impact on student learning. We offer classroom-tested and research-based teaching resources for a variety of grade levels and subject areas. Whether you are an aspiring, new, or veteran teacher, we want to help you make every teaching day your best.
From ready-to-use classroom activities to the latest teaching framework, our value-packed books provide insightful, practical, and comprehensive materials on the topics that matter most to K-12 teachers. We hope to become your trusted source for the best ideas from the most experienced and respected experts in the field.
About This Resource
Math Wise! includes activities that will help each student gain full comprehension of basic mathematical concepts, including numbers and counting, computation, estimation, probability, data analysis, measurement, geometry, algebra, problem solving, and logical thinking. Students in today’s math classrooms must be able to do more than achieve correct 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 objectives. It contains a wide variety of learning experiences that have been arranged according to difficulty level. Whenever possible, the activities are presented in either hands-on or visual formats.
Concrete/Manipulative Activities
Especially when exploring “new” concepts, each student should work with hands-on materials. A number of the activities therefore include easily 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 is used when one paper clip corresponds to the numeral 1. The result might appear as:
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 can also show 7 groups of 3, or 7 × 3 = ____ . Whereas the resulting punched holes are concrete, the looped segments provide a visual component that directly corresponds to the abstract number relationships involved.
Such manipulative activities provide a basis for true understanding of mathematical concepts. For this reason, each section contains a number of similar exercises.
Visual/Pictorial Activities
For many learners, visual representations of mathematical problems are keys to the comprehension of these problems. Often visual representations involve 1-to-1 correspondence in connecting pictures with 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 the answer to the problem 3 × 5 = _____ . The following figure illustrates this visual 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 by 10-unit square divided into 100 square units. Each small square unit represents one hundredth of the decimal square, or .01. Students are then 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, students are asked to shade in the Decimal Squares, as shown below.
Abstract Procedures
A major goal of mathematics education is to help students eventually perform abstract mathematical procedures and understand the underlying concepts behind these procedures. When possible, mathematics teachers should not only instruct students in regard to mathematical mechanics but also enable them to gain a true understanding of the concepts 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 other group members, after viewing the numerals, give the student clues about the numerals. Using these clues, the Post-it wearer must do mental math to determine the numerals. In the situation that follows, the Post-it player 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 asking students 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 on the square with the answer. The next player can only move one paper clip, leaving the other one alone. This player will then perform the multiplication and mark his or her square with an O. The boards below show two partially played games.
BLOCK FOUR Multiplication Facts
BLOCK FOUR Multiplication of Fractions
A Final Note
Students will find the activities and investigations from this book informative, interesting, and fun. Most important, students will gain a better understanding of the mathematics they are expected to master. Math Wise! will prove to be a most valuable supplement to any mathematics program.
Jim Overholt
Laurie Kincheloe
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 materials for mathematics instruction since the 1960s. As an elementary and secondary school teacher in Minnesota and Wyoming, and later as a university professor, his investigations have taken him into both K- 12 classrooms and adult mathematics learning workshops. He is currently a professor of education at California State University, Chico.
Dr. Overholt regularly conducts mathematics education courses and workshops 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 mathematics education from California State University, Chico. She taught high school mathematics for twelve years and is presently teaching mathematics at Butte College in northern California. She has worked with K - 12 students, parents, and teachers as a family math coordinator and as a mentor for new teachers. She teaches concepts in mathematics to pre-service elementary teachers, and has coordinated service learning projects connecting high school and college students with elementary students through mathematics. She was co-coordinator of the Mathematics Project at California State University, Chico, and has conducted workshops on the teaching of mathematics for elementary and secondary teachers at numerous education conferences.
In addition to teaching at Butte College, Laurie has served as the developmental coordinator for the Mathematics Department, created a math-anxiety class designed to help apprehensive students be successful at math, and organized the annual Math Awareness Week. She has received the Faculty Member of the Year Award and the Service Learning Project Faculty Award.
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 and understandable tasks from which students in kindergarten through the middle grades will benefit. Although many of these activities can be used in any order, it is advisable to designate tasks that are appropriate with regard to class size, students’ stages of learning, or other considerations. For this reason, several features in this book are designed to help select appropriate activities.
• The
Contents categorizes each activity in five ways:
1. Section (“Making Sense of Numbers,” “Computation Connections,” “Investigations and Problem Solving,” and “Logical Thinking”)
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 cooperative groups.
3. The students will work with visual diagrams and will perform abstract computations.
Silent Math
Grades 4-8 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
• Each activity begins with a Why Do It statement that details the specific mathematical concepts the students will be learning and practicing.
• The You Will Need statement specifies any supplies or equipment necessary for the activity. These items, such as paper clips, index cards, 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 the activity to be successful. It also describes how the investigation works best as an independent activity, a cooperative project where students work in pairs or small groups, or a total group venture. This section will provide the general premise and content of the activity 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 procedures described in the activity. It often contains more sample questions or suggestions as to how to expand the mathematical concepts being studied. Teachers and students are encouraged to propose similar tasks of their own.
• Where appropriate, reproducible pages immediately follow the relevant activity. These pages include game boards, workmats, dot paper, playing cards, graph paper, and more.
• Students should be encouraged to record their methods and solutions in a math journal or to keep a special file containing samples of their work.
• Solutions are also provided when appropriate.
Section One
Making Sense of Numbers
The activities in this section introduce students to many number concepts and relationships, including 1-to-1 correspondence, 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.
Chapter 1
Toothpick Storybooks
Grades K - 3 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
Students will discover the concepts of 1-to-1 counting and number conservation, and will study basic computation relationships.
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 configurations, all of which would still yield 4 toothpicks.
2. After exploring for a while, students should begin making Toothpick Storybooks, starting by creating number pages. Students can write, for instance, the number 6 on a sheet of white paper and glue 6 toothpicks onto a piece of colored paper. (To avoid a sticky mess, students should dip only the ends of the toothpicks in the glue.) When they are ready, the learners follow the same procedure for equations and the corresponding toothpick pictures. (Note: Students sometimes portray subtraction by pasting a small flap on the colored page that covers the number of toothpicks to be “taken away.” Furthermore, they enjoy lifting the flap to rediscover the missing portion.)
3. When a number of toothpick diagrams have been finished, the pages can be stapled together into either individual or group Toothpick Storybooks. Ask each student to tell a number story about one of the diagrams in which he or she makes reference to both the toothpick figure and the written equation or number.
Example:
Extensions:
1. Simple multiplication facts, and even longer problems, can be portrayed with toothpick diagrams. For 6 × 3 =_____ , the player might show ||| ||| ||| ||| ||| ||| = 18. Similarly, for 4 × 23 = _____, it is necessary to show 4 groups of 23 toothpicks to yield 92.
2. Division can also be shown with toothpick diagrams. If the problem calls for the division of 110 into sets of 12, the player would need to form as many groups of 12 as possible, also taking into account any remainder. (Note: The student might also complete such a problem using partitive division. See Paper Clip Division, p. 179.)
Chapter 2
Number Combination Noisy Boxes
Grades K - 3 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
This activity provides students with a visual and concrete aid that will help them understand basic number combinations and practice addition and subtraction.
You Will Need:
Ten (or more) stationery or greeting card boxes with clear plastic lids, approximately 50 marbles, and pieces of Styrofoam or sponge that can be trimmed to fit inside the boxes are required.
How To Do It:
1. Construct Noisy Boxes for the numerals 0 through 9 (or beyond). For each box, cut the foam to make a divider that will lie perpendicular to the bottom of the box. Glue the divider to the bottom of the box, ensuring that it is trimmed down such that the marbles will pass over it when the top is on (see figure). Use a marking pen to write the numeral, such as 3, on the divider and to inscribe the appropriate number of dots on one outside edge of the box ( • • • ) and the corresponding number word on another outside edge (three). Insert that same number of marbles into the box and tape on the clear plastic lid.
2. Allow the students to work with different Noisy Boxes. Instruct students to tip or shake a Noisy Box so that some or all of the marbles roll past the divider. Once this is done, the player is to record the outcome as an addition problem. The student should shake the same Noisy Box again and record a new outcome. For example, three marbles will yield outcomes such as 1 + 2, 3 + 0, 2 + 1, or 0 + 3. The activity continues in this manner until no further combinations are possible (see Example below).
Example:
The recorded number combinations for the 7s Noisy Box should include the following:
Extension:
If any player has difficulty on a visual level in utilizing a Noisy Box, have that student temporarily remove the plastic lid. Then he or she can touch and physically move the marbles from one side of the box to the other. Nearly all students will experience success as a result of such a tangible experience with number combinations.
Chapter 3
Everyday Things Numberbooks
Grades K - 4 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
Students will discover that in their daily lives there are many things that come in numbered amounts, such as wheels on a bicycle.
You Will Need:
Each student will require paper that can be stapled into booklets, pencils, scissors, and glue sticks or paste (if desired).
How To Do It:
1. As a group, discuss things in everyday life that are generally found as singles or 1s: 1 nose for each person, 1 trunk per tree, 1 beak on a bird, 1 tail per cat, 1-a-day multiple vitamins, and so on. Then provide each student with a sheet of paper and have everyone write the number 2 at the top. Each participant should list as many things that come in 2s as he or she can think of, such as 2 eyes, ears, hands, and legs for each person; 2 wings per bird, and so on. Do the same for 3s: 3 wheels on a tricycle; 3 sides for any triangle; a 3-leaf clover, and so on. Students might also paste pictures representing numbered amounts on their pages. Have them complete a page (or more) for each number up to 10 or larger, and then discuss their ideas. You may want to construct large class lists for each number. This activity can continue for several days, and may be assigned as homework.
2. At first some numbers seem unusable, but wait and you will be delighted with students’ suggestions. For instance, 7 can be illustrated by 7-UP®, and 8 depicted by 8 sides on a stop sign. Students will often continue to make suggestions, even after the activity has ended!
Example:
The following is a partial Numberbook listing for the number 4.
Extensions:
Ask more advanced students to consider the following problems:
1. What items can commonly be found in 25s, 50s, 100s, or any other number you or students might come up with? Is there any number for which an example cannot be found?
2. Find examples for fractional numbers. If there are 12 sections in an orange, 1 of those sections is 1/12 of the orange; 3 of those sections are 3/12 or 1/4 of the orange.
Chapter 4
Under the Bowl
Grades K - 3 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
Under the Bowl provides students with a visual and concrete aid that will help them understand basic number combinations and practice addition and subtraction.
You Will Need:
A bowl or small box lid and small objects (such as beans, blocks, or bread tags) are required for each player.
How To Do It:
Allow students a brief period to explore their bowls and objects. Have students begin the activity with small numbers of items: students with 3 beans, for example, might be told to put 2 beans under the bowl and place the other on top of it. Then they should say aloud to a partner or together as a class, “One bean on top and two beans underneath make three beans altogether.” Once students understand the activity, ask them to keep a written record of their work; for 3 beans, as noted above, they should record 1 + 2 = 3 (after they have had instruction on four fact families, they should also record 2 + 1 = 3, 3 - 2 = 1, and 3 - 1 = 2). Although initially students should use only a few objects, they might go on to use as many as 20, 30, or even 100 items.
Example:
The players shown above are working with 7 beans. Thus far they have recorded the four fact family for 1 bean on top of the bowl and 6 beans under it. They are now beginning to record their findings for 2 beans on top. Next they might put 3 beans on top and record. (Note: Should a student become confused about a number combination, he or she may count the objects on top and then lift the bowl to either visually or physically count the objects underneath. This usually helps clarify the problem.)
Extensions:
1. When older students are working as partners, an interesting variation has one student making a combination and the other trying to figure out what it is. For example, the first student might put 3 beans on top of the bowl and some others under it. He or she then states, “I have 11 beans altogether. How many beans are under the bowl, and what equations can you write to represent this problem?” The second student should respond that there are 8 beans under the bowl, yielding the equations 3 + 8 = 11, 8 + 3 = 11, 11 - 8 = 3, and 11 - 3 = 8.
2. You can also extend this activity to introduce algebra concepts to students. For example, after instruction a student presents the problem shown in Extension 1, with the equation n + 3 = 11. Explain to students that using a letter to represent a missing number is a basic concept in algebra.
Chapter 5
Cheerios™ and Fruit Loops™ Place Value
Grades K - 5 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
Students will begin to understand place value concepts through a visual and concrete experience.
You Will Need:
This activity requires several boxes of Cheerios and one box of Fruit Loops breakfast cereals, string or strong thread, needles, and two paper clips (to temporarily hold the cereal in place) for each group or student. (Note: If you do not wish to use needles, you can use waxed or other stiff cord.)
How To Do It:
After a place value discussion about 1s, 10s, and 100s, challenge the students to make their own place value necklaces (or other decorations). Ask them to determine the “place value” of their own necks and, when they look puzzled, ask, “How many Cheerios on a string will it take to go all the way around your neck?” Then explain that they will be stringing Cheerios and Fruit Loops on their necklaces in a way that shows place value: for each 10 pieces of cereal to be strung, the first 9 will be Cheerios and every 10th piece must be a colored Fruit Loop. They will then be able to count the place value on their necks as 10, 20, 30, and so on. As students finish their necklaces, be certain to have students share the numbers their necklaces represent and how the necklace displays the number of 10s and the number of 1s in their number.
Example:
The partially completed Cheerios and Fruit Loops necklace shown above has the place value of two 10s and three 1s, or 23.
Extensions:
1. As a class, try making and discussing other personal place value decorations, such as wrist or ankle bracelets or belts.
2. An engaging group project is to have students estimate the length of their classroom (or even a hallway) and make very long Cheerios and Fruit Loops chains. Be sure to initiate place value discussions about 100s; 1,000s; and even 10,000s or more. (Hint: When making such long chains, it is helpful to have individuals make strings of 100 and then tie these 100s strings together.)
Chapter 6
Beans and Beansticks
Grades K - 6 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
This activity will give students 1-to-1 concrete and visual understandings of place value and computation concepts.
You Will Need:
Required for this project are dried beans, Popsicle sticks (or tongue depressors), and clear-drying carpenter’s glue (most forms of permanent glue work well).
How To Do It:
1. Beansticks, a stick with 10 beans, will help students visualize the 10s place of a number. A flat, a raft made of 10 beansticks, will allow students to view the 100s place of a number. Single beans will serve for the numbers 1 through 9, but 10s beansticks and 100s flats (or rafts) will be needed after that. The 10s beansticks are constructed by gluing ten beans to a stick, and the 100s flats are made by gluing ten beansticks together with cross supports (see the illustrations below). The beansticks and rafts will be more durable if a second bead of glue is applied several hours after the first layer of glue dries. Finally, to enhance the lesson, students should construct the beansticks themselves; or, if necessary, have older students help younger students with the construction and then use the beansticks to lead a lesson in place value.
2. The students should first use single beans to represent single-digit numbers. Next they should incorporate the 10s beansticks to display numbers with two-digit place value. Three-digit numbers require the 100s flats. Example 1, below, depicts such place value representations.
3. Examples 2, 3, 4, and 5 cite methods for using the beans and beansticks to add, subtract, multiply, and divide. Please pay particular attention to the situations in which trading (sometimes called renaming, regrouping, borrowing, or carrying) is necessary. For these examples, students could work in pairs. If each student initially made ten beansticks and ten rafts, a pair of students will have enough to work these examples. Finally, students should keep a written record of the problems, processes, and outcomes.
Examples:
1. The numbers 3, 25, and 137 are displayed using beans and beansticks.
2. The problem 16 + 12 = _____ is solved below (in equation format) by simply combining the 1s beans and the 10s beansticks. In this case it is not necessary to trade.
3. The problem 21 - 6 = _____ requires trading. Because 6 cannot be subtracted from 1, one of the 10s beansticks is traded for 10 single beans, allowing 6 to be taken away from 11. The player ends with one 10s beanstick and 5 single beans, or 10 + 5 = 15.
4. Trading is necessary to solve the problem 3 × 45 = _____ . Students should start by setting up this problem as an addition problem. Three 45s are set up using four beansticks and five loose beans. Notice that 10 of the 15 loose beans need to be traded for a 10s beanstick, and also that 10 of the beansticks are traded for a 100s flat.
5. For the division problem 123 ÷ 27, the student must figure out how many 27s are in 123. First 123 is represented using one raft, two beansticks, and three loose beans. Then the student represents the number 27 with two beansticks and seven loose beans. After careful counting and carrying, the student will determine that after displaying four 27s and adding them together, the result is not 123, but close. What is needed to make 123 is one beanstick and five loose beans, or 15. Because what is needed is less than 27, then this must be the remainder of the division problem. Therefore, the answer is 4 with a remainder of 15.
Extensions:
Beans and beansticks may be utilized with both larger numbers and decimals.
1. If students are working with numbers into the 1,000s, they can build these by stacking the 100s flats: to form each 1,000 requires that 10 of the 100s flats be piled together.
2. It can also be helpful to utilize visual representations in the beanstick problems. For example, the number 253 might be quickly illustrated as shown below.
3. Beansticks might also be used (in reverse order) to portray decimals. For instance, if the 100s flat represents 1, then each 10s beanstick would equal .1 and each single bean would equal .01. Decimal computations can therefore be displayed. For example, adding .52 to .71 would entail adding five beansticks to seven beansticks to obtain one raft and two beansticks, then adding two single beans and one single bean to obtain three single beans. The answer would then be represented with one raft, two beansticks, and three single beans, which equals 1.23.
Chapter 7
Incredible Expressions
Grades K - 8 • Total group activity
• Cooperative activity
• Independent activity
• Concrete/manipulative activity
• Visual/pictorial activity
• Abstract procedure
Why Do It:
Students will develop their number sense and make new mathematical connections.
You Will Need:
This activity can be done on the chalkboard or on a large piece of paper, with chalk or marking pens.
How To Do It:
In this activity, you or a student will specify a number for the whole class to represent in different ways. For example, 10 can be represented as 5 + 5, 19 - 9, √100, and so on. Keep a permanent record of the various ways of naming the same number by writing students’ Incredible Expressions on a large piece of newsprint or butcher paper; or use the chalkboard to keep a short-term record.
Example:
The illustration below shows the Incredible Expressions for the number 21 that a group of students has devised.
Extensions:
Incredible Expressions may be simple addition or subtraction problems; or they may be more complex, involving multiple operations and exponents.
1. Each day, develop and list expressions to correspond with the calendar date. Students should also build these numbers with manipulatives (for example, 21 might be built with two bundles of 10 straws rubber banded together plus a single straw).
2. Restrict players to certain operations. For example, players might be directed to use only addition and subtraction.
3. Students who have had sufficient experience with the mathematical operations of addition, subtraction, multiplication, and division can be required to use all four of these operations. Advanced players might be instructed to find expressions that use parentheses, exponents, square roots, fractions, decimals, percents, and so on. Be certain that they use the proper order of operations when computing an expression. The order is parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right). For example, the value of the expression 2(5 + 4) is different from the value of 2 × 5 + 4. The first expression simplifies to 2 × 9 = 18, and the second expression computes as 10 + 4 = 14.
4. Older students can use calculators to create truly Incredible Expressions. If so, each player might be required to use at least five different numbers together with a minimum of three different operations.
Chapter 8
Number Cutouts
