origami math

Grades 2–3BY KAREN BAICKER

New York • Toronto • London • Auckland • SydneyMexico City • New Delhi • Hong Kong • Buenos Aires

Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources

�

Scholastic Inc. grants teachers permission to photocopy the reproducibles from this book for classroom use. No other part of this publica-tion may be reproduced in whole or in part, or stored in a retrieval system, or transmitted in any form or by any means, electronic,mechanical, photocopying, recording, or otherwise, without permission of the publisher. For information regarding permission, write toScholastic Teaching Resources, 557 Broadway, New York, NY 10012-3999.

Cover and interior design by Maria Lilja

Interior illustrations by Jason Robinson

Copyright © 2004 by Karen Baicker. All rights reserved.

ISBN 0-439-53991-9

Printed in the U.S.A.

1 2 3 4 5 6 7 8 9 10 40 11 10 09 08 07 06 05 04

For Joseph Baicker with thanks for all the

help with math, even though he wouldn’t

let me do it the teacher’s way.


3�3

Introduction . . . . . . . . . . . . . . . . . . . . . . 4How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . 5Tips for Teaching With Origami . . . . . . . . . . . . . . . 6The Language and Symbols of Origami . . . . . . . . 7Basic Geometric Shapes Reference Sheet . . . . . . . 8

ActivitiesTurn a Rectangle into a Square . . . . 9Math Concepts: shapes, patterns, symmetry, spatial relationsNCTM Standard 3

What a Card! . . . . . . . . . . . . . . . . . . . . . . . 11Math Concepts: shapes, patterns, size, symmetry, spatial relationsNCTM Standards 2 and 3

Whale of Triangles . . . . . . . . . . . . . . . . . 13Math Concepts: size, symmetry, shapes, patterns, spatial relationsNCTM Standards 2 and 3

Instant Cup . . . . . . . . . . . . . . . . . . . . . . . . . 16Math Concepts: spatial reasoning, shapes, volumeNCTM Standards 3 and 4

Playful Pinwheel . . . . . . . . . . . . . . . . . . . 19Math Concepts: spatial relations, pattern, symmetry, motionNCTM Standards 2 and 3

Handy Hat . . . . . . . . . . . . . . . . . . . . . . . . . . 21Math Concepts: spatial reasoning, sequence, symmetry, scaleNCTM Standards 2 and 3

Box It Up! . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Math Concepts: multiplication, division, dimensionNCTM Standard 3

Noise Popper . . . . . . . . . . . . . . . . . . . . . . . 27Math Concepts: measurement, shape, spatial reasoning, symmetryNCTM Standard 3

Itty-Bitty Book . . . . . . . . . . . . . . . . . . . . . 30Math Concepts: spatial reasoning, shapes, symme-try, fractions, multiplication, divisionNCTM Standards 1 and 3

Jumping Frog . . . . . . . . . . . . . . . . . . . . . . . 33Math Concepts: shape, measurement, distance, heightNCTM Standards 3 and 4

Kitty Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Math Concepts: angles, symmetryNCTM Standard 3

Floating Boat . . . . . . . . . . . . . . . . . . . . . . . 39Math Concepts: shape, fractions, areaNCTM Standards 2 and 3

Page-Hugger Bookmark . . . . . . . . . . 42Math Concepts: shape, spatial reasoning, symmetry, congruenceNCTM Standards 3

Paper Airplane Express . . . . . . . . . . . 45Math Concepts: symmetry, balance, spatial reasoningNCTM Standards 3 and 4

Glossary of Math Terms . . . . . . . . . . . . . . . . . . 48

Contents

1

2

3

4

5

6

7

10

9

8

11

13

12

14


4�4

Introduction Why fold? The learning behind the funThe first and most obvious benefit of teaching with origami is that it’s fun and motivating for students. But the opportunities for learning through paper folding go much further. Many mathematical principles “unfold” and basic measurement and computation skills are reinforced as each model takes shape. The activities in this book are all correlated with NCTM (National Council of Teachers of Mathematics) standards, which are highlighted in each lesson.

In addition, origami teaches the value of working precisely and following directions. Students will experiencethis with immediacy when a figure does not line up properly or does not match the diagram. Also, becausemath skills are integrated with paper folding, a physical activity, students absorb the learning on a deeperlevel. Origami helps develop fine motor skills, which in turn enhances other areas of cognitive development.

Best of all, origami offers a sense of discovery and possibility. Make a fold, flip it over, open it up——and youhave created a new shape or structure!

Tactile Learning Suppose youwant to see if two shapes are thesame size. You can measure the sidesto get the information you needabout area. But the easiest way to seeif two objects are the same size is toplace one on top of the other. That’sessentially what you are doing whenyou fold a piece of paper in half.

Spatial Reasoning Origami activi-ties challenge students to look at a diagram and anticipate what it willlook like when folded. Often, twodiagrams are shown and the readermust imagine the fold that was necessary to take the first image and produce the second. These arecomplex spatial relations problems——but ever so rewarding when the endresult is a cat or an airplane!

Symmetry Origami patterns oftencall for symmetrical folds, which create congruent shapes on eitherside of the fold, and clearly mark theline of symmetry.

Fractions Folding a piece of paperis a very concrete way to demonstratefractions. Fold a piece of paper in halfto show halves, and in half again toshow quarters. For younger students,you can shade in sections to showparts of a whole. For older students,you can explore fractions. You caneven show fraction equivalencies. Is of the same as of ?Through paper folding, you can seethat it is!

Sequence With origami, it is critical to follow directions in a precise sequence. The consequencesof skipping a step are immediate and obvious.

Geometry Most of the basic principles of geometry——point, line,plane, shape——can be illustratedthrough paper folding. One exampleis Euclid’s first principle, that there isone straight line that connects anytwo points. This postulate becomesobvious when you make a fold that

connects two points on the paper.For another example, older studentsare told that the angles of a triangleadd up to 180º. Folding a triangle canprove this geometric fact, as you see in the diagrams below. You can alsodemonstrate the concepts of hypothesis and proof. Predict whatwill happen, and then fold the paperto test the hypothesis.2__

31__2

1__2

2__3


Lesson 2

5�5

How to Use This BookThe lessons in this book are organized from very simple to challenging; all are geared toward the interests, abilities, and math skills of second and third graders. Both the math concepts and the origamimodels are layered to reinforce and build on earlier lessons. Nonetheless, the lessons also stand independently and you may select them according to the interests and needs of your class, in any order.

In each lesson, you will find a list of Math Concepts, Math Vocabulary, and NCTM Standards that highlight the math skills addressed. At the heart of each lesson is Math Wise!, a script of teaching pointsand questions designed to help you incorporate math concepts with every step on the student activitypage. Look for related activities to help students further explore these concepts at the end of the lessonin Beyond the Folds!

Step-by-step illustrations showing exactly how to do each step in the origami activity appear on a reproducible activity page following the lesson. Encourage students to keep these diagrams and bringthem home so that the skills and sequence can be reinforced through practice. A reproducible patternfor creating the activity is also included for each lesson. The pattern is provided for your convenience,though you may use your own paper to create the activities. The pattern pages feature decorativedesigns that enhance the final product and provide students with visual support, including folding guidelines and ★s that help them position the paper correctly. To further support students’ work with origami and math, you may also want to distribute copies of reference pages 7 and 8, The Language and Symbols of Origami and Basic Geometric Shapes Reference Sheet.

Origami on the WebThere are many great websites for teaching origami to children. Here are some recommended resources:

www.origami.com This comprehensive site also sells an instructional video for origami in the classroom.

www.origami.net This site is a clearinghouse for information and resources related to origami.

www.paperfolding.com/math This excellent site explores the mathematics behind paper folding. Although geared for older students, it provides a useful overview for teachers.

www.mathsyear2000.org Click on “Explorer” to find some math-related origami projects with step-by-step illustrations, suitable for elementary students.


Lesson 2

6�6

Prepare for the lesson� Try the activity ahead of time if

possible. Moving through the stepshelps you anticipate any areas of difficulty students may encounterand your completed activity pro-vides a model for them to consult.

� Review the Math Wise! section andthink through the mathematical concepts you want to highlight. You can find helpful definitions forthe lesson’s vocabulary list in theglossary on page 48.

� Photocopy the step-by-step instructions found in the lessonand the activity patterns (or haveother paper ready for students to use).

Teach the lesson� Familiarize students with the basic

folding symbols on page 7.

� Introduce or review the origami terminology students will use in thelesson. Note that communication isenhanced when you can describe aspecific edge, corner, or fold with precision.

� When possible, use the language ofmath as well as the language of origami when creating these proj-ects. By saying, “Here I am dividingthe square with a valley fold,” you reinforce geometry concepts aswell as the folding sequence. Someof the ideas expressed in the MathWise! notes in the lesson plan maysound sophisticated. Yet, by using

the proper language as you makethe folds, you will begin to teach students concepts that become thefoundation for success with mathin later grades. You’ll be surprisedat how much they grasp in the context of creating the origami.

� Demonstrate the folds with a largerpiece of paper. Make sure thepaper faces the way the students’paper is facing them.

� Support students who need morehelp with following directions or with manipulating spatial relationships by marking landmarkson the paper with a pencil as yougo around the classroom. You canmake a dot at the point where twocorners should meet, for example.

� Arrange the class in clusters, and letstudents who have completed onefold assist other students. This willfoster cooperative learning andhelp you address all students’ questions.

Fold accurately � Make sure students fold on a

smooth, hard, clean surface.

� Encourage students to make a softfold and check that the edges lineup properly to avoid overlapping.They can also refer to the diagramand make sure that the folded shape looks correct. After theymake adjustments, they can make a sharper crease using their fingernails.

Choose your paper � You can reproduce the patterns

in this book onto copy paper.However, you can also use packs of origami paper, or cut your ownsquares. Keep in mind that thinnerpaper is easier to fold. Gift wrap,catalogues, magazines, menus, calendars, and other scrap papercan make wonderful paper for theseprojects. It is best to work withpaper where the two sides, frontand back, are easily distinguished.

Encourage students to explore geometry� Unfold an origami project just to

look at the interesting pattern andthe geometric figures you have created through your series ofcreases. Challenge students to create their own variations—andmake their own diagrams showinghow they did it.

Jumping frog unfolded

Tips for Teaching With Origami


Lesson 2

7�7

The Language and Symbols of Origami

white side or back of papercolored side or front of paper

fold toward the front (valley fold)

fold toward the back (mountain fold)

fold, then unfold turn over

fold

crease

cut

folded edge

raw edge


Lesson 2

8�8

Squarea quadrilateral that has four right angles andfour congruentsides. All squaresare rectangles

Rectanglea quadrilateral that has four right angles (90°).All rectangles are parallelograms

QUADRILATERALSA quadrilateral is any figure that has four sides

Isosceles trianglea triangle with two congruentsides and two congruent angles

Right trianglea triangle witha right angle

Isosceles right trianglea triangle withtwo congruentsides and oneright angle

TRIANGLESA triangle is any figure that has three sides

Scalene trianglea triangle with no congruentsides and no congruent angles

Equilateral triangle a triangle with three congruentsides and three congruent angles

CIRCLEa round shapemeasuring 360º

OVALan egg-shape with a smoothcontinuous edge

PENTAGONa shape with five sides

HEXAGONa shape with six sides

OCTAGONa shape with eight sides

Parallelograma quadrilateralthat has twopairs of parallelsides and two pairs of congruent sides

Basic Geometric Shapes Reference Sheet

CONGRUENTequal in measurement

congruent linesegments

congruent angles

congruent figures

60° 60°


9�9

Knowing how to turn a rectangleinto a square is an invaluableorigami technique! It’s also agreat way to teach some basicgeometry. Use rectangular paperof any size. Then invite studentsto use the reproducible tangramspuzzle and geometric shapes reference sheet to furtherexplore basic shapes.

Materials Needed page 10 (steps and pattern), rectangularsheet of paper, page 7 (The Language andSymbols of Origami), page 8 (BasicGeometric Shapes Reference Sheet)

Math Concepts shapes, patterns, symmetry, spatial relations

NCTM Standards� analyze characteristics and properties of

two- and three-dimensional geometricshapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1)

� use visualization, spatial reasoning, andgeometric modeling to solve problems(Geometry Standard 3.4)

Math Vocabulary rectangle

square

right angle

isosceles right triangle

triangle

congruent

Beyond the Folds!� Help students become familiar with the step-by-step directions they’ll read in

each lesson. Distribute copies of The Language and Symbols of Origami, page 7,and review the word and picture cues presented.

� Explore shapes further by distributing the Basic Geometric Shapes ReferenceSheet, page 8.

� Show how you can take a piece of paper with a square corner—any size—anduse it to test various corners in the classroom. If the edges line up with theirpaper, they have found a right angle. Note that if their right angle fits inside the other angle with extra room, the other angle is bigger. It has a wider mouth. If their right angle covers up the other angle, then the other angle issmaller. Find bigger angles and smaller angles around the classroom.

� Distribute How to Make a Square, page 10. After students have made a square from any rectangular sheet of paper, you can try the tangrams activityat the bottom of the page. (For best results, photocopy onto heavier paper or glue onto cardstock.) Give students a little background on the history of tangrams. The tangram is a seven-piece puzzle that has been played in China for over 200 years. Explain that squares, triangles, and rectangles can be usedtogether to make a wide variety of shapes. Challenge students to match the pictures shown and to create their own tangrams activities.

Lesson 1

Turn a Rectangleinto a Square

When we start, the corners of this rectangle are perfectsquare corners. Another word for a square corner is a rightangle. Let’s look around the room and find some right angles.(Point out the corners of the room, tables, books, and so on.) When wemake our fold, we’ve cut this angle exactly in half. And nowwe have two isosceles right triangles, which means that thetwo sides of the triangle that form the right angle are thesame length, or congruent.

Now we have an extra shape we have to cut (or tear)away. What shape is this? (rectangle)

Let’s open up this triangle. Now we have a perfectsquare. All four sides are the same length. We also have tworight angles. What is the word that describes lines, angles,or shapes that are equal in measure? (congruent)

1

2

3

Math Wise! Distribute copies of page 10. Use these tips to highlight math concepts and vocabulary for each step.


10�10

Lesson 1

Tangram Puzzle Pieces You just made a rectangle into a square. Now you can take a square and make it into a whole lot more! These seven basictangrams shapes can be used to make hundreds of designs. Cut out the shapes below. Then try to make the shapes shownbelow. Or make up your own figures for others to solve. When you’re done, see if you can put them back into a square again.

Cut off the extra rectangle, as shown. Or fold the extra paper at the top, using the top of the triangle as a guide. Crease it well. Tearalong the crease.

How to Make a SquareTake a rectangular sheet of

paper. Bring the bottom rightcorner up, so that the bottomedge and the left side line up.

Now open up your square. Find thetwo triangles!

2 31

Tip: To tear, fold the creaseback and forth, scoring it each time with your finger or fingernail. Then hold the paperdown firmly with one hand,and use the other hand to tearthe rectangle away. Keep yourhands close to the fold for better control.

or creaseand tear


11�11

This activity is a simple butimportant introduction to basicmath and origami concepts—please don’t skip it! You mightteach how to make origamicards before a class party or holiday, and tailor the cardsaccordingly.

Materials Needed page 12 (steps and pattern), rectangularpaper or square paper (optional), crayons or markers

Math Concepts fractions, spatial relations, size, symmetry

NCTM Standards� understand numbers, ways of

representing numbers, relationshipsamong numbers, and number systems (Number and Operations, Standard 1.1)

� represent and analyze mathematical situations and structures using algebraic symbols(Algebra Standard 2.2)

� analyze characteristics and propertiesof two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1)

� apply transformations and use symmetry to analyze mathematical situations (Geometry Standard 3.3)

Math Vocabulary half

quarter

fraction

line of symmetry

Beyond the Folds!� Ask students to find different ways to express and . Folded paper is

one way. Coins and dollars are another. Bar graphs, pie charts, and measuringcups are some others.

� To further explore the “line of symmetry,” let students paint with watercolorson one half of a piece of paper. Then have them fold the page in half, press itfirmly, and open up again. They will have a symmetrical design.

� Make a chart that shows “Folds” in one column and “Sections” in a second column. If you fold a piece of paper once, you have two sections; fill in the firstrow with 1 (fold) and 2 (sections). Have students fold the paper again (while it is still folded) in the other direction. Then fill in the chart. Repeat again in theopposite direction again. Have students continue to fill in the chart. Have themtry to determine a pattern. (The number of sections doubles with each fold.)

What a Card!

1__2

1__4

Lesson 2

This fold is called a valley fold. That’s because we’re making a little valley here. We started with a square, andnow we have two rectangles. The middle line is called the“line of symmetry.” That’s because the line divides the rectangle into two halves that are exactly the same. We alsohave an inside and an outside now. Look what else has happened. The short side is now the long side.

& This fold is sometimes called a book fold. Why doyou think it has that name? Let’s unfold it for a minute justto see what’s happened. Look, we have four equal squares.Let’s say the whole card cost $1.00. How much would one of these squares be worth? (25¢) So that’s a hundred centsdivided into 4 parts.

Let’s fold it back up again, following the same steps. Noticehow one side now forms both the inside and the outside ofthe card? The other side is all folded up inside, you see. (Let students discuss and then decorate and fill in their cards. Encourage them to use a square sheet of colored paper and follow these directions tomake a card of their own.)

1

2 3



12�12

Lesson 2

Cut out the card pattern below. Place the paper facedown, with the ★in the bottom right corner. Fold in half, bringing the top down to meet the bottom.

Fold in half again, bringing theleft side over to meet the right.Crease well again.

How to a Make Card and Card Pattern

Make a

design on the

back of this page

and fold it

backwards for a

reversible card!

21

★

____________________________

____________________________ ____________________________

This card was designed by

date:_______________________

where:______________________

time:_______________________

RSVP:_______________________

Decorate and fill in your card!3


13�13

Lesson 3

Our Whale Pattern page hassome features drawn in. But youmay also want to distribute blueor gray paper and have studentsmake their own.

Materials Needed page 14 (steps), page 15 (pattern) or 6-inch square paper

Math Concepts shapes, patterns, symmetry, spatial relations

NCTM Standards� analyze change in various contexts

(Algebra Standard 2.4)� analyze characteristics and properties of


� apply transformations and use symmetryto analyze mathematical situations(Geometry Standard 3.3)


Math Vocabularysquare

diamond

triangle

right

left

point

center

line of symmetry

quadrilateral

isosceles triangle

scalene triangle

multiply

divide

Beyond the Folds!� Make a whole school of whales and put the school on a bulletin board.

Make them form an array, with rows and columns. You can teach basic multiplication from your array.

� Have students use bigger and smaller squares to see how the whales turn outdifferent sizes, depending on the initial square. Show how you can measurethe size by putting the squares, and the whales, on top of each other.

1

Math Wise! Distribute copies of pages 14 and 15. Use these tips to highlight math concepts and vocabulary for each step.

What shape are we starting with? (A square. You may wish topoint out that the shape is a quadrilateral—a shape that has four sides.)Notice how I turn or rotate the square like a diamond, so thepoint is at the top. Let’s make sure everyone’s paper is facingthe same way. (Encourage the class to look around the room to check forthe correct positioning. Looking from different angles will strengthen theirspatial relations skills.) Sometimes in origami we make folds, onlyto unfold them again! What’s the point? Well, as you’ll see,these folds always come in handy later on! Now we have twonew shapes. What are they called? (They are triangles; you may wishto point out that they are isosceles right triangles, each having a right angleand two sides of the same length.) Find the centerline we just folded.That’s called “the line of symmetry.” That means that the line divides two halves that match.

This time we don’t have to unfold it! But look at whatshapes we’ve made. That’s right, two new triangles. (You may wish to note that these are scalene triangles, triangles that have no sides that are the same length.)

Now look how many triangles we have! Let’s count them.(Show students that some triangles are within larger triangles. There are eight triangles showing, not counting the ones hidden underneath the folds.)

Ah-hah! We’re using that line of symmetry again. Now you see why we folded it in the first place!

& We just made our last triangle! When we slit the tail,we divided it in two. But it looks like we multiplied it, doesn’tit? That’s because we started with two layers. Go ahead and add some details to your whale—don’t forget to add a blowhole! What shape is that? (a circle)

Whale of Triangles

2

3

4

5 6


14�14

Lesson 3

Fold the two lower sides to meetthe center fold, or line of symmetry.

How to Make the WhaleCut out the whale pattern on page

15. Position the square so that it lookslike a diamond with the ★ at the topfacedown. Fold the left point over tomeet the right point. Open it up again.

Fold the top point down to meetthe folded triangles.

2 31

Rotate the shapes so that the longflat line is at the bottom.

Fold the right side over to meetthe left side.

Fold the left point up along thedotted line to form a tail. Slit the tailalong the cut line. Fold the triangles out to form the flukes.

5 64


15�15

Lesson 3

Whale PatternFollow the steps on page 14 to create a whale.

★

✂


16�16

Lesson 4

This cup really works. It can holdliquid—until the water soaksthrough! You can adapt this modelto make a hanging pocket holder.Start with a bigger sheet ofpaper. At the last step, don’t tuckthe final flap in. Keep the back tri-angle up, punch a hole in it, andhang it up to collect Valentines.You can also punch holes and addstraps to make a little carryall.

Materials Neededpage 17 (steps), page 18 (pattern) or 6-inch square paper, water (optional)

Math Conceptsspatial reasoning, shapes, volume

NCTM Standards� analyze characteristics and properties

of two- and three-dimensional geometricshapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1)


� use visualization, spatial reasoning, and geometric modeling to solve problems (Geometry Standard 3.4)

� understand measurable attributes ofobjects and the units, systems, andprocesses of measurement(Measurement Standard 4.1)

Math Vocabularyquadrilateral square

trapezoid isosceles right triangle

parallel diagonal

line of symmetry

Beyond the Folds!� Ask students if they think that a square double the size of the original square

will make a cup that holds double the volume. Then have them conduct anexperiment: Make two cups with different-sized paper and test the volume. Fill the larger cup and then pour it into the smaller cup once. Then dump that water (or grains of rice) out, and refill from the larger cup. See how manytimes the smaller cup fills to compare the capacity of the two cups.

� Make a large cup out of a big square of newsprint. Turn it upside down and it’sa hat! Use this activity to discuss the fact that form and function are often amatter of perspective—just like turning the square into a diamond in the firststep. Let students make and decorate their own hats.

Instant Cup

A diamond is just a different way of looking at a square!Let’s make two triangles. For each triangle, two sides are thesame length, and they have a right angle. That makes themisosceles right triangles.

We’re making this point (the tip of the angle) meet the dot(a small circle). This bottom part of our cup is called the baseof the triangle. The top point is called the apex.

Let’s fold this top triangle down and tuck it in. What kindof triangle is it? (Isosceles right triangle) What makes it so? (two sides the same length, one right angle)

& In origami, you often do something to one side andthen repeat the exact sequence on the other side. That isone kind of symmetry. Our final shape is a trapezoid. A trapezoid is a quadrilateral with one set of parallel edges.Which sides are parallel? What would happen with a cup ifthe bottom and top were not parallel? (The liquid would spill out; it wouldn’t balance on a table!)

1

2

3

4


5


17�17

Lesson 4

Fold up the bottom right corner to meet the opposite edge. Line up the corner so that it meets the dot.Crease the fold.

How to Make the CupCut out the cup pattern on page 18

and place it like a diamond, with the ★in the top point, facedown. Or use asquare sheet, positioned facedown like a diamond. Fold in half, bringing the bottom corner up to meet the top.

Fold down the top layer of the tri-angle above, tucking it into the pocketof the cup as far as it will go. Crease.

2 31

Turn over and repeat steps 2 and 3above tucking in the remaining triangle.4 Gently pinch the sides together to

open your cup.5


18�18

Lesson 4

Cup Pattern This cup actually holds water. Once you learn this simple model, you’ll alwaysbe able to whip up a cup anytime you’re thirsty and you’ve got paper to fold!

★


19�19

Lesson 5

This classic paper-folded pinwheel is fun to create andplay with, and it makes a great model for exploring symmetry and patterns.

Materials Needed page 20 (steps and pattern), square paper(optional), pencils with erasers, scissors,push pins, glue (optional), crayons ormarkers

Math Concepts spatial relations, pattern, symmetry,motion

NCTM Standards� understand patterns, relations, and

functions (Algebra Standard 2.1)� specify locations and describe spatial

relationships using coordinate geometry and other representationalsystems (Geometry Standard 3.2)


Math Vocabulary diagonal

intersect

isosceles right triangles

center

Beyond the Folds!

� Experiment with patterns and using the back and front of the paper: Have students decorate the backside of their pinwheel pattern before foldingit. (Or have them decorate both sides, if they are using different paper.) Ask them to make a design with a repeating pattern or image or to use complementary colors or patterns on the opposite sides. Then test what thedesigns look like in motion! Have them take their pinwheels apart and refoldthem from the opposite side. Encourage them to consider how the direction of the folds makes the backs and fronts interconnected and alternating.

� Discuss the fact that motion, in a way, can have a shape. What shape does the motion of the pinwheel form? (a circle or a spiral)

When we cut along these lines from the corner in, what

direction are the lines? (diagonal) The cut lines stop before

they reach the center of the square. Describe what would

happen if they continued. (They would come together or intersect inthe middle. If they were cut, you would have four equal triangles.)

What are the four different shapes we’re folding?

(triangles)

Why do our folds allow the pinwheel to spin?

(The pockets can catch the air, like a sail.)

Tip: If students’ pinwheels do not spin freely, adjust the

tension by pulling the thumbtack out slightly. Also adjust the

angle of the blades so that they do not hit the pencil.

1

2

3


Playful Pinwheel


20�20

Lesson 5

The magic of

pinwheels is the way

they look when they’re

spinning. Make some

more with different

designs and give

them a whirl!

Place the pattern facedown. Foldthe top right corner to just past the center of the square. Do not crease.Hold in place with your finger, or use a dab of glue stick. Repeat this fold withthe other three corners.

How to Make a Pinwheel and Pinwheel PatternCut out the pinwheel pattern

below or start with a 6-inch square.Cut along the diagonal lines, makingsure to stop well before the center, as marked. Color in the design.

Ask a grown-up to push a thumbtackthrough all layers into the side of the eras-er end of a pencil, so that the folded sidesface out and the flat side faces the eraser.Spin away! If your pinwheel gets stuck, ask a grown-up to help you adjust it.

2 31

✂


21�21

Lesson 6

This is the traditional newspaperhat. It may help to do it first on a smaller scale using ourreproducible Hat Pattern, and then try it with big sheets of newspaper. There is one additional step when using the newspaper sheets (see MathWise! step 3).

Materials Neededpage 22 (steps), page 23 (pattern) or newspaper sheets

Math Conceptsspatial reasoning, sequence, symmetry,

scale

NCTM Standards� analyze change in various contexts

(Algebra Standard 2.4)� specify locations and describe spatial

relationships using coordinate geometry and other representationalsystems (Geometry Standard 3.2)


Math Vocabularyhorizontal

vertical

line of symmetry

perpendicular


right angle

rectangle

Handy Hat

Beyond the Folds!

� Distribute copies of the tangrams puzzle on page 10, and ask students to use the shapes to make a design for their hats. They can color the shapes and make a design that reflects something about them.

� Use strips of construction paper to make feathers to stick in the hat, Robin Hood-style. This presents another opportunity to explore symmetry.Students can fold a strip in half, length-wise, to make slits on both sides andopen to find a feather shape.

We’re starting with a big rectangle. The bigger the rectangle, the bigger the hat we’ll end up with. Open up the page after these two folds, just to take a look at thelines. We have two lines of symmetry—one horizontal andone vertical. These lines are perpendicular—they form rightangles where they cross. How many rectangles do we have?(five: the big one plus the four smaller ones inside)

What shapes are we folding down? (triangles) Notice that inorigami when we make a fold on one side, we often repeat iton the other and we get two halves that are exactly thesame. What is the word that means perfect balance or exactly the same on each side? (symmetry)

& What is the shape we are folding up? (rectangle)

Note: for a newspaper hat, fold the bottom up twice: Fold once to meet the base of the triangle. Then fold again, as described.

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22�22

Lesson 6

Fold in the top corners to meet thecenter line. Crease.

How to Make a HatCut out the hat pattern on page

23 and place the page facedown withthe ★ in the top left corner. Or use aplain rectangular sheet of paper, or asheet of newspaper opened up. Fold in half right to left. Crease and unfold.Then, fold in half top to bottom andcrease. This time, leave the paper folded.

Fold up the top layer of the bottom edge. Turn over and repeat on the other side.

2 31

Pull out on the sides to open it.Try it on.4


23�23

Lesson 6

Hat PatternThis page won’t make a hat big enough for you, but you might use it for a doll or anaction figure! Make these same folds on a sheet of newspaper and you’ll be covered!

★


24�24

Lesson 7

This is one of the simplest and mosttraditional origami box designs.With this box, the top and bottomare the same, so fitting themtogether may be a tight squeeze—unless the lid is madewith a slightly larger rectangle.This activity works best withsturdy paper, so photocopy thepattern onto card stock if possible. Using the cover of amagazine also works well!

Materials Needed page 25 (steps), page 26 (pattern) or 6-inch square of sturdy paper

Note: For paper that’s just the right weight and gives the box a glossy, colorful finish, use a magazine cover.

Math Concepts multiplication, division, dimension


two- and three-dimensional geometricshapes and develop mathematical argu-ments about geometric relationships(Geometry Standard 3.1)

� specify locations and describe spatialrelationships using coordinate geometryand other representational systems(Geometry Standard 3.2)


Math Vocabularyheight

volume

trapezoid

parallel

Box It Up!

Beyond the Folds!� Challenge students to figure out how to make the top slightly larger than the

bottom. They can start with a slightly larger rectangle. But they can also“cheat” on the folding with the cabinet folds. They can make the edges notquite meet the center crease. They can imagine a closet door that can’t quiteclose! Ask them to think about why this will yield a larger box.

� Ask students to explore different ways to make the box stronger. Then havethem list what properties add strength. Possible solutions include: using heavier paper, making smaller boxes, using double layers, inserting cardboardon the inside.

Let’s see if I can express this fold with an equation. WhenI fold it, it’s 1 ÷ 2 = . Now when I unfold it, I can say 1 x 2 = 2.

This fold is sometimes called a cabinet fold. Can you seewhy? How would you express one of the “cabinet doors” asa fraction? ( )

Now how many sections do we have? (8) So what is thefraction that represents each rectangle? ( ) We started outwith four sections for the “cabinet.” What equation could weuse to show what we now have? (4 x 2 = 8, or 4 + 4 = 8) Isn’tthat strange? When we fold it, we seem to be dividing it! But then when we open it, we’ve actually multiplied!

We don’t need to open it up this time! But if we did, howmany sections do you think we would find? (16) You can takea peek and refold it to check your answer!

When we fold our triangles, they don’t quite reach themiddle. You know what that tells me? These little sections arenot squares! If they were, the triangles would line up with thecrease, because all of the sides would be the same length.

See how we have two trapezoids now! Trapezoids haveone set of parallel lines. But watch when we open it! Theseother sides will become parallel.

& Voila! See how something that is two-dimensional,or flat, becomes three-dimensional! That’s why we had tomake all those creases. They formed the sides here, whichgive it the height, or third dimension.

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25�25

Lesson 7

Fold each side in half, folding inthe long edges to the center fold.Crease sharply. Unfold.

How to Make a BoxCut out the box pattern on page

26 and place it facedown with the ★in the upper left corner. Or use an 8 by 11-inch rectangular sheet ofheavy paper and position the paperfacedown. Fold in half, right to left.Crease and unfold.

Fold in half, top to bottom. Unfold.2 31

Fold up the bottom right cornerup along the fold line so that the cor-ner meets the dot. Note that the topedge does not quite reach the centerfold. Crease firmly. Repeat this stepwith the other three corners.

Fold in the short edges to meetthe center fold. Crease very sharply.This time, do not unfold.

Fold back the edges, away fromthe center, to cover the top part ofthe triangles. Crease these bandssharply.

5 64

Now form the box by pullingopen the bands. 7

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Help shape the corners of the boxby creasing the corners and bottomedges. Flex the sides inward.

8


26�26

Lesson 7

Box PatternMake two of these so your box will have a lid or attach a strip of paperto make a basket with a handle. Use it for paper clips, pennies, rubberbands, your favorite collector cards, hair clips, or any other favorite item!

Create a design on the back,which will show on the inside of yourfolded box. Color this side as well, if you wish.

★


27�27

Lesson 8

We know paper can fold, fly, spinand twirl. Add noise-making tothe list of paper’s possibilitieswith this snappy popper.

Materials Needed page 28 (steps), page 29 (pattern) or8 by 11-inch rectangle (loose-leaf notebook paper works best), crayons or markers

Note: The pattern on page 29 must be enlarged at least 125% to pop well.

Math Conceptsmeasurement, shape, spatial reasoning,symmetry

NCTM Standards� apply transformations and use symmetry

to analyze mathematical situations(Geometry Standard 3.3)


Vocabulary horizontal

parallel

quadrilateral

right angle

Noise Popper

Beyond the Folds!� Have students all snap their poppers in unison. Note how loud it sounds.

Next divide the class into two groups. Have one half pop their poppers. Then have the other half snap. Does each group sound about half as loud? Then divide the group into quarters and let each group create the soundagain. Is it a quarter of the original volume? Point out that noise level can be measured. What are some other things that can be measured? (Answers include: size, weight, volume, distance, speed, time.)

Notice that we’re placing our paper horizontally. The longest side is going across. When people name the measurements for something they usually give the widthfirst. A page of copy paper held this way (show vertically) is considered 8 by 11 inches. Turn it this way, and we’dsay 11 by 8 inches. But it’s the same piece of paper!

The shape we end up with here looks a bit like a football.The shape of a football is made to spiral through the air. But this shape will be used to make sound.

When we fold the shape in half, we have a shape withfour sides again—a quadrilateral. But there’s something special about this quadrilateral: two of its sides are parallel,equal distance apart at all points—like train tracks. Whichtwo sides are parallel? (The top and bottom. If students are ready foranother term, you may want to point out that a quadrilateral with two parallelsides is called a trapezoid.)

Now we changed the shape again, but it’s still a quadrilateral. Are there any two angles that are the same?(Yes, there are two right angles.) Are there any two sides that arethe same length? (no)

& Let’s take a look at it carefully before we use ournoise popper. How do you think we’ll make the noise? Whatkind of sound will be created? (Students may speculate that the trian-gular flap will pop out and make a popping or snapping sound. Show themhow to hold the popper straight down and snap with a flick of their wrists.)

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28�28

Lesson 8

Fold the bottom right corner up sothat the right side lines up with thecenter crease. Repeat with the othercorners.

How to Make a Noise PopperCut out the noise popper pattern

on page 29 and place it horizontally, facedown, with the ★ in the upper left corner. Or use a sheet of loose-leafpaper. Fold in half, bottom to top, andcrease sharply. Unfold.

Fold in half, top to bottom and crease.

2 31

Fold the front flap down, so that thetop edge lines up with the left edge.Turn over and repeat.

Fold the top left corner over tomeet the top right corner.

54 To snap the popper, pinch flapstogether firmly at the point, keepingyour fingers toward the bottom so theydo not block the action of the innerfolds. Snap your wrist forward and the inner flap will pop out, making asnapping noise. If it doesn’t come out,loosen it a few times and try again.

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pinch here

snap downand pop


29�29

Lesson 8

Noise Popper PatternBang! Pop! See how loud you can snap this noise popper. Who knew paper could make such a racket?

★

Enlarge this pattern to 125% or larger to create apopper with a greatsound. Or use asheet of loose-leafpaper.


30�30

Lesson 9

This is an elegant way to make an eight-page book out of a singlesheet of paper and one snip! The secret, of course, is in thefolds. There are dozens of cross-curricular uses for this project—journals, poetry books,invitations, autobiographies, and more.

Materials Needed page 31 (steps), page 32 (pattern) or rectangular sheet of paper, scissors, crayons or markers

Math Concepts spatial reasoning, shapes, symmetry, fractions, multiplication, division

NCTM Standards� understand numbers, ways of represent-

ing numbers, relationships among numbers, and number systems (Number and Operations Standard 1.1)

� understand meanings of operations and how they relate to one another(Number and Operations Standard 1.2)

� analyze characteristics and properties oftwo- and three-dimensional geometricshapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1)


Math Vocabulary rectangle fractions

line of symmetry quarters

eighths right angle

Itty-Bitty Book

Beyond the Folds!� Students may use the Itty-Bitty Book pattern to create a “My Important

Numbers Book.” Encourage students to list some of the many numbers thatare important in their lives. Numbers may include phone numbers, addresses,birthdates, grade level, number of classmates, and so on.

� Have students make a blank book. Ask students to use their books to writeand illustrate a number story. For example, Jake had five books out from thelibrary (page 1). He went to the library and returned three books (page 2).But he checked out four more (page 3). How many did he have altogether?(page 4) Answer: Jake had six books (back of page 4).

� Have students start with a blank page and plan a book. Ask them to figureout how the pages will fall when the book is folded. They can use the patternas an example, or open up a folded book and mark the page numbers.

We’re going to make some folds and one snip, and turnthis piece of paper into a book. How many pages do youthink we can make?

How would we show one of these sections as a fraction? ( ) Now suppose we cut this section out. What fractionwould be left? ( )

How many rectangles do we have? (five—the four smaller rectanglesplus the whole sheet as the fifth rectangle)

Now we have one half of the page showing. The fold inthe middle divides that in half. Half of the half or xequals one quarter ( ). So each of these narrow rectangles isone quarter of our original sheet. Let’s fold it in half again.

How many sections are there now? (8) So each section isone eighth ( ) of the whole page.

We made our slit on an outer edge. But look now—it’s on theinside. And it’s twice as long! How did that happen? (The cutwas on a fold. When we unfolded, that fold became the middle. The slitwent through two layers. That’s why it’s double length.)

Before we fold it in half, let’s count how many pages ourbook has. (8)

& Watch what happens to the flat paper as we pushthe ends together. We’ve created a three-dimensional form.

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31�31

Lesson 9

Fold in half, left to right. Crease itsharply, and leave it folded.

How to Make an Itty-Bitty BookCut out the book pattern on

page 32 and place it facedown withthe ★ in the upper left corner. Oruse a rectangular sheet, and place it horizontally, facedown. Fold thepaper in half, top to bottom. Crease and unfold.

Fold again in the same direction.Unfold this last step.

2 31

Fold in half top to bottom, so thatthe two long edges meet.

Cut in from the left side to thecenter, following the cut line. Makesure to stop at the middle crease.Open the whole sheet.

Push the two outer edges in, sothat the slit opens and the inner pagesare formed. Crease the edges of allpages to make the book.

5 64

Design your book!7


32�32

Lesson 9

Book PatternHow can you turn one page into an eight-page book with one little snip?Try this brilliant paper-folding project. Use it to make a mini-journal, or a long card for your best friend.

★


33�33

Lesson 10

This is a popular origami modelthat jumps! Introduce or reinforce measurement skills such as distance and height byholding an origami frog-jumpingcontest. You can measure distance, height, and accuracy.

Materials Needed page 34 (steps), page 35 (pattern) or a 6-inch square of paper, crayons or markers

Math Concepts shape, measurement, distance, height




� understand measurable attributes ofobjects and the units, systems, andprocesses of measurement(Measurement Standard 4.1)

� apply appropriate techniques, tools, andformulas to determine measurements(Measurement Standard 4.2)

Math Vocabulary rectangle triangle

intersection perpendicular lines

pentagon right angle

Beyond the Folds!� In step 2, you can also introduce the concepts of line segments

using the top square with the intersecting folds. Label the corners A (top left), C (top right), D (bottom left), and B (bottom right). Ask students to show you the fold that makes line segment AB. Point out thatthere are two ways that points A and B can be related through folding. One isto make a fold that forms the line segment. The other way is to make a foldin which A and B end up on top of each other. To make that fold, you end upmaking line segment CD! Line segments AB and CD are perpendicular.

� Have a frog-jumping contest and measure the distances in both inches and centimeters. You may also ask students to guess the distance beforemeasuring and record the accuracy of their estimates.

We’re folding a square to make two rectangles. We couldjust start with a rectangle this size, but we make the paperthicker this way. That will help the frog keep its shape andjump better when we’re done.

& Look, we’ve made an “X” here. The point where thetwo lines cross is called the intersection. See how we havefour perfect corners at the intersection? That means that thetwo lines are perpendicular to each other.

When we’re starting this fold, we have three triangles,plus the bottom triangle that’s part of the house-shape (a pentagon). But we’re collapsing the two side triangles inhalf. That’s five triangles lining up over the bottom triangle.

Now if we folded these triangles so that they line up perfectly with the top point, then we couldn’t see the feet.We’ll fold them so they stick out here. Did I just make theangle of the folded triangle bigger or smaller? (smaller/narrower)Take a look at how that changes the length of this edge.

This looks a little like a house now. What is this five-sidedshape called? (a pentagon) When we fold these sides in, theshape looks more like a rocket.

& First we make a valley fold and then a mountain fold.What letter are we making along the side edge? (Z) Let’smake a big “Z” on the board. Can you see why this shape isspringy? How will this Z-shape design help the frogs to hop?

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Jumping FrogMath Wise! Distribute copies of pages 34 and 35. Use these tips to highlight math concepts and vocabulary for each step.


34�34

Lesson 10

Fold the top right corner over tomeet the dot on the left edge. Creaseand unfold. Repeat on other edge, andunfold so that you have an X at the top.

How to Make a FrogCut out the frog pattern on

page 35 and place it facedown with the★ in the upper right corner. Or startwith a 6-inch square, facedown. Foldthe page in half, right to left.

Fold the top points of the X downto meet the bottom points of the X.Crease and unfold.

2 31

Take the bottom two points of thetriangle and fold them up to create thefront legs.

As you fold the top part downagain, collapse the side triangles inward.Use your fingers to poke the triangles inas you fold. The top becomes a triangle.Crease the sides of the triangle well.

Fold the side edges in toward thecenter. Use the fold lines as a guide.

5 64

Fold in half, top to bottom. Do notcrease this fold sharply; simply bend it. 7 Flip over. Fold the top layer in half,

bottom to top, away from the legs andhead. Again, do not crease sharply.

8 Your frog is ready to hop! Push down on the spot on the frog’sback and release to make him go.

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35�35

Lesson 10

Frog PatternHop to it with this little jumping frog. Hold a frog-jumping contest withyour own frogs or challenge a friend to a hop-a-thon.

★


36�36

Lesson 11

Make these cat faces with blackpaper and use them as Halloweendecorations.

Materials Needed page 37 (steps), page 38 (pattern) or a 6-inch square of paper, crayons or markers

Math Concepts angles, symmetry

NCTM Standards� analyze characteristics and properties

of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1)

� use visualization, spatial reasoning, and geometric modeling to solve problems (Geometry Standard 3.4)

Math Vocabularycongruent


midpoint

base

angle

scalene triangle

hexagon

Beyond the Folds!

� You can display a litter of these cats by hanging a string across the room andthreading it through the top triangle fold of each origami cat. Tape the foldsdown over the string to make them stay in place. For a multiplication connection, ask students if cats have nine lives, how many lives are representedby this string of cats? Help them multiply by 9s. You can make this easier bycreating a story in which the cats have two lives each, and have them count by 2s. (Have students who need more support count two ears for each cat.)

� Invite students to create an origami cat with a new piece of paper and drawtheir own cat faces that are exactly symmetrical. They might start by folding theshape gently in half to find the center line (line of symmetry). Have them workoff of this line, drawing eyes, whiskers, eyelashes, and so on in the same place,proportion, and number on each side.

Kitty Cat

Make a valley fold to create two layers of congruent orequal-sized triangles. What special type of triangle is this?(isosceles right triangle)

Before we make this fold, let’s find the midpoint, or middle,of the base of the triangle. How can we do that? (fold it in half)Let’s not crease it firmly here; just make enough of a fold tomark the midpoint. Now when we fold these corners up,notice that we are making three congruent or equal angles at the base.

Now that we’ve folded the top down, we can see wherethe ears will be, can’t we? What shape are the ears? (triangles)This kind of triangle has no congruent or equal sides orangles. It’s called a scalene triangle.

& If we cover up the ears, what shape does the facebecome? (a hexagon)

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37�37

Lesson 11

Fold up the bottom right corneralong fold line. Repeat on the left side,so that the triangles cross.

How to Make a CatCut out the cat pattern on page 38

and place it like a diamond, with the ★at the top, facedown. Or start with a 6-inch square, facedown. Fold the bottom corner up to meet the top.

Fold the top triangle down(both layers) along the fold line.

2 31

Fold the bottom point up to meetthe top point. 4 Turn over and decorate the face of

your cat.5


38�38

Lesson 11

Cat PatternMake a bunch of black cats to hang up on Halloween. Or make one bigcat with a large square and a litter of kittens with smaller squares.

★


39�39

Lesson 12

This simple boat will actuallyfloat in water. Use a coin to helpbalance if the boat is too tippy.

Materials Needed page 40 (steps), page 41 (pattern) or 6-inch square, crayons or markers, basin of water (optional)

Math Concepts shapes, fractions, area

NCTM Standards� understand patterns, relations, and

functions (Algebra Standard 2.1)� analyze characteristics and properties

of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1)


Math Vocabulary rectangle

triangle

hexagon

area

Beyond the Folds!� Set up a basin of water and let students have a boat race. Measure speed,

distance, and accuracy. Discuss and adjust balance as necessary. Experimentmaking boats from different weights and sizes of paper. Which floats better?Which moves faster?

� This activity, like many, starts with a square. Show students how to determinethe area of a square by counting squares in a grid. Take a fresh 8-inch square of paper and fold it in half, and then in half again two more times to create8 columns. Open it up and repeat these folds in the other direction to create 8rows. Open it again, and you should have 64 small squares. Tell students thateach square is 1 inch x 1 inch. Count the squares to determine that the areais 64 square inches. Show how you can multiply the width by the height(8 inches x 8 inches) to get the same answer. Explain that they can use thisformula to find the area of any rectangle.

Let’s make a book fold. What two shapes do we have now? (rectangles)

How much of the page do we have with this strip here? ( )

How much of the page do we have showing now? ( )

What shape are these corners? (triangles) And what shape do we have here when the corners are folded in? (hexagon)

When we make this fold, all of the back side of the paperdisappears. It’s all tucked inside here.

How do you think this shape helps keep the water out?Many boats have this shape, like a canoe. How does this shape help it float through the water? (The pointed ends help theboat glide through the water smoothly. A flat front would slow down the speed and make the boat hard to control.)

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40�40

Lesson 12

Fold down the top edge, frontlayer only, to meet the bottom foldededge. Crease.

How to Make a BoatCut out the boat pattern on page

41 and place the ★ in the upper rightcorner, face up. Or start with a 6-inchsquare, face up. Fold in half, bottom totop. Crease and leave folded.

Turn over and repeat step 2. This time, unfold that last fold.

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Fold in half, top to bottom. Fold the top corners down to thecenter line and crease. Fold the bottomcorners up to the center line andcrease. Make sure to fold all the layers.

Separate the top edges to openthe boat. Press down along the bottom and pull out the sides tocreate a flat bottom.

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41�41

Lesson 12

Boat PatternThis simple boat actually floats. Try blowing it across a small tub of water with astraw. Have a boat race with a larger and smaller boat to see which moves faster.

★


42�42

Lesson 13

Students can use this friendlyfolded corner-hugger as a bookmark.

Materials Needed page 43 (steps), page 44 (pattern) or 6-inch square, crayons or markers, scissors, card stock, index cards, or heavy paper, glue

Math Concepts shape, spatial reasoning, symmetry, congruence




Math Vocabulary diamond

square

triangle

perpendicular

pentagon

line of symmetry

congruent

Page-Hugger Bookmark

Beyond the Folds!

� Give students book-reading word problems or let them generate their own.Example: Joe read five pages at bedtime. The next morning he got up and read three more. On what page would you find his bookmark?

� Note that this bookmark actually marks two pages——the front and the back of the folio corner it covers. If the bookmark is set on page 11, a right-hand page, what page does it also mark? (12) Suppose you use a regular bookmark——a rectangular strip. If the bookmark is set on page 11,what page does it also mark? (10)

As you might know, the diamond is not really a shape.It’s just a square that we’ve rotated so that we see it differently. These two lines that we’ve folded are perpendicular to each other. See how the corners wherethey intersect, or cross, are perfect square corners?

What fraction of the corners have we folded? ( )

Before we make our fold, let’s look at this shape here.How many sides does it have? (5) What do we call a shapewith five sides? (a pentagon)

How many sides does our shape have now? (4) We’re taking this big triangle and bisecting it, or cutting it in half, to form two equal, or congruent triangles. Actually, it’s threecongruent triangles, with this other flap here!

& We’ve formed a pocket here that can fit on the corner of our page. There’s another pocket in our bookmarktoo. Can you find it? Could we use this pocket as a bookmarkas well? (no) Why not? (because it’s not shaped like a corner, or right angle)

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43�43

Lesson 13

Fold up the bottom point to meetthe center point. Crease. Repeat withthe left and top corners. Leave theright corner unfolded.

How to Make a BookmarkCut out the bookmark pattern on

page 44 and place the square like a diamond with the ★ at the top, face-down. Or start with a 6-inch square,facedown. Fold in half left to right, sothe corners meet. Crease and unfold.Fold in half top to bottom so that thecorners meet. Crease and unfold.

Fold down the top left corner tomeet the bottom right corner. Leavethat part folded.

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Tuck the right hand point insidethe pocket formed by the left-hand triangle.

Fold up the bottom left corner tomeet the top point. Crease well.

54 Fill in your name on the back.Decorate your page-hugger bookmark.6


44�44

Lesson 13

Bookmark PatternKeep this bookmark buddy handy and you’ll never lose your place.

This book

belon

gs to

____________________________

★


45�45

Lesson 14

There are hundreds of ways tomake paper airplanes. This modelhas good gliding action. Use theairplane pattern on page 47, orany 8 by 11-inch sheet of copy paper!

Materials Needed page 46 (steps), page 47 (pattern) or 8 x 11-inch rectangle, paper clip

Math Concepts symmetry, balance, spatial reasoning

NCTM Standards� apply transformations and use symmetry

to analyze mathematical situations(Geometry Standard 3.3)

� understand measurable attributes ofobjects and the units, systems, andprocesses of measurement (Measurement Standard 4.1)

� apply appropriate techniques, tools, andformulas to determine measurements(Measurement Standard 4.2)

Math Vocabulary half

center

weight

corner

bisect

angle

surface area

Paper Airplane Express

Beyond the Folds!

� Stage an airplane gliding contest and measure distance and accuracy. For distance, measure the distance between the launch and landing points.For accuracy, measure the distance between a target point and the actuallanding point. Calculate the average distance by compiling the flight recordsof all students.

� Have students decorate their own planes, each wing with a different design.

If we take a plain piece of paper and try to throw it, itdoesn’t travel very far! But if we crumple it up, look! We canthrow it much better. Why do different forms of the samepage fly better than others? (The crumpled-up ball has a smaller surface area—most of it is tucked into the center of the ball. Less of the paperhits the air, so it doesn’t slow down the way the open sheet does.) Whenwe fold an airplane, we want to make a design that has notmuch surface area and little wind resistance.

Many paper airplanes have a sharp point at the front.Later, we’ll fold this point and make a blunt nose. How mightthe shape at the front affect the way a plane flies? (The shapeof the plane’s nose affects the flight pattern.)

We’re folding these angles in half, or bisecting them.

In this model you might notice that whatever we do onone side, we do on the other. That’s because planes dependon symmetry for smooth flying. What would happen if the two sides were different? (It would fly crooked.)

& What happens to the plane when we fold the noseback? (The nose gets heavier.)

& What does the paper clip do to the nose of theplane? (It adds more weight and keeps the sides together tightly, so there’sless surface area and wind resistance.)

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46�46

Lesson 14

Fold down the top corners to the center line.

How to Make a Paper AirplaneStart with the paper airplane

pattern, page 47, and place the patternfacedown with the ★ in the upperright-hand corner. Fold in half left toright. Make a light crease. Unfold.

Fold in each side from the point, to meet the center line. Crease alongthe fold lines.

2 31

Fold in half, left to right. Crease well. Fold down the tip of the plane’snose toward the middle, along the foldline.

Fold down the top layer of the rearwing, along the short fold line. Turnover and repeat on other side.

5 64

Fold down the wings to the base,starting from the nose. Use the longfold line as a guide. Repeat on theother side.

7 Spread the wings. Add a paper clipto the nose for balance and weight.8


47�47

Lesson 14

Paper Airplane PatternWhat’s the longest distance your plane can fly?

★


48�48

angle the space formed by two lines stemming from a common point

area the measurement of a space formed by edges

base the bottom edge of a shape

bisect to divide in half

circle a round shape measuring 360º

congruent having the same measurement

diagonal a slanted line that joins two opposite corners

diamond a square positioned so that one corner is at the top (A diamond is not a distinct geometric shape.)

equilateral triangle a triangle with three congruent sides and three congruent angles

fraction a number that is not a whole number, such as 1/2, formed by dividing one quantity into multiple parts

hexagon a shape with six sides

horizontal running across, or left to right

intersection the point where two lines cross

isosceles triangle a triangle with two congruent sides and two congruent angles

line of symmetry a line that divides two alikehalves

midpoint a point that lies halfway along a line

octagon a shape with eight sides

oval an egg-shaped figure with a smooth,continuous edge

parallel two lines that are always the samedistance apart, and therefore never intersect

parallelogram a quadrilateral that has twopairs of parallel sides and two pairs of congruent sides

pentagon a shape with five sides

perpendicular at right angles to a line

quadrilateral any four-sided figure

rectangle a quadrilateral that has four rightangles (All rectangles are parallelograms.)

right angle an angle of 90º, which forms asquare corner

right triangle a triangle with a right angle (90º)

scalene triangle a triangle with no congruentsides and no congruent angles

square a quadrilateral that has four rightangles and four congruent sides (All squaresare rectangles.)

symmetry a balanced arrangement of partson either side of a central dividing line oraround a central point

trapezoid a quadrilateral with one pair ofparallel sides

triangle a figure with three sides

vertical running from top to bottom

volume the space contained within a three-dimensional object

Glossary of Math Terms


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