Origami Math

(1)

BY

K

AREN

B

AICKER

New York • Toronto • London • Auckland • Sydney

Mexico City • New Delhi • Hong Kong • Buenos Aires

(2)

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 to Scholastic Teaching Resources, 557 Broadway, New York, NY 10012-3999.

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

(3)

How to Use This Book

. . . 5

Tips for Teaching With Origami . . . 6

The Language and Symbols of Origami . . . 7

Basic Geometric Shapes Reference Sheet. . . 8

Activities

Turn a Rectangle into a Square

. . . . 9

Math Concepts: shapes, patterns, symmetry, spatial relations NCTM Standard 3

What a Card!

. . . 11

Math Concepts: shapes, patterns, size, symmetry, spatial relations NCTM Standards 2 and 3

Whale of Triangles

. . . 13

Math Concepts: size, symmetry, shapes, patterns, spatial relations NCTM Standards 2 and 3

Instant Cup

. . . 16

Math Concepts: spatial reasoning, shapes, volume NCTM Standards 3 and 4

Playful Pinwheel

. . . 19

Math Concepts: spatial relations, pattern, symmetry, motion NCTM Standards 2 and 3

Handy Hat

. . . 21

Math Concepts: spatial reasoning, sequence, symmetry, scale NCTM Standards 2 and 3 Math Concepts: multiplication, division, dimension NCTM Standard 3

Noise Popper

. . . 27

Math Concepts: measurement, shape, spatial reasoning, symmetry NCTM Standard 3

Itty-Bitty Book

. . . 30

Math Concepts: spatial reasoning, shapes, symme-try, fractions, multiplication, division NCTM Standards 1 and 3

Jumping Frog

. . . 33

Math Concepts: shape, measurement, distance, height NCTM Standards 3 and 4

Kitty Cat

. . . 36

Math Concepts: angles, symmetry NCTM Standard 3

Floating Boat

. . . 39

Math Concepts: shape, fractions, area NCTM Standards 2 and 3

Page-Hugger Bookmark

. . . 42

Math Concepts: shape, spatial reasoning, symmetry, congruence NCTM Standards 3

Paper Airplane Express

. . . 45

Math Concepts: symmetry, balance, spatial reasoning NCTM Standards 3 and 4

Glossary of Math Terms

. . . 48

1 2 3 4 5 6 10 9 8 11 13 12 14

(4)

Why fold? The learning behind the fun

The 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 experience

this with immediacy when a figure does not line up properly or does not match the diagram. Also, because

math skills are integrated with paper folding, a physical activity, students absorb the learning on a deeper

level. 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 you

have created a new shape or structure!

Tactile Learning Suppose you want to see if two shapes are the same size. You can measure the sides to get the information you need about area. But the easiest way to see if two objects are the same size is to place one on top of the other. That’s essentially what you are doing when you fold a piece of paper in half.

Spatial Reasoning Origami activi-ties challenge students to look at a diagram and anticipate what it will look like when folded. Often, two diagrams are shown and the reader must imagine the fold that was necessary to take the first image and produce the second. These are complex spatial relations problems—— but ever so rewarding when the end result is a cat or an airplane!

Symmetry Origami patterns often call for symmetrical folds, which create congruent shapes on either side of the fold, and clearly mark the line of symmetry.

Fractions Folding a piece of paper is a very concrete way to demonstrate fractions. Fold a piece of paper in half to show halves, and in half again to show quarters. For younger students, you can shade in sections to show parts of a whole. For older students, you can explore fractions. You can even show fraction equivalencies. Is of the same as of ? Through paper folding, you can see that it is!

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

Geometry Most of the basic principles of geometry——point, line, plane, shape——can be illustrated through paper folding. One example is Euclid’s first principle, that there is one straight line that connects any two points. This postulate becomes obvious when you make a fold that

connects two points on the paper. For another example, older students are told that the angles of a triangle add up to 180º. Folding a triangle can prove this geometric fact, as you see in the diagrams below. You can also demonstrate the concepts of hypothesis and proof. Predict what will happen, and then fold the paper to test the hypothesis.

2 __ 3 1 __ 2 1 __ 2 2 __ 3

(5)

How to Use This Book

The 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 origami

models 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 points

and questions designed to help you incorporate math concepts with every step on the student activity

page. Look for related activities to help students further explore these concepts at the end of the lesson

in 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 bring

them home so that the skills and sequence can be reinforced through practice. A reproducible pattern

for 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 decorative

designs that enhance the final product and provide students with visual support, including folding guide

lines 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 Web

There 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.

(6)

Prepare for the lesson

Try the activity ahead of time if possible. Moving through the steps helps you anticipate any areas of difficulty students may encounter and your completed activity pro-vides a model for them to consult.

_{Review the Math Wise! section and}

think through the mathematical concepts you want to highlight. You can find helpful definitions for the lesson’s vocabulary list in the glossary on page 48.

_{Photocopy the step-by-step}

instructions found in the lesson and the activity patterns (or have other 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 the lesson. Note that communication is enhanced when you can describe a specific edge, corner, or fold with precision.

_{When possible, use the language of}

math as well as the language of origami when creating these proj-ects. By saying, “Here I am dividing the square with a valley fold,” you reinforce geometry concepts as well as the folding sequence. Some of the ideas expressed in the Math Wise! notes in the lesson plan may sound sophisticated. Yet, by using

the proper language as you make the folds, you will begin to teach students concepts that become the foundation for success with math in later grades. You’ll be surprised at how much they grasp in the context of creating the origami.

Demonstrate the folds with a larger piece of paper. Make sure the paper faces the way the students’ paper is facing them.

Support students who need more help with following directions or with manipulating spatial relationships by marking landmarks on the paper with a pencil as you go around the classroom. You can make a dot at the point where two corners should meet, for example.

Arrange the class in clusters, and let students who have completed one fold assist other students. This will foster cooperative learning and help you address all students’ questions.

Fold accurately

Make sure students fold on a smooth, hard, clean surface.

Encourage students to make a soft fold and check that the edges line up properly to avoid overlapping. They can also refer to the diagram and make sure that the folded shape looks correct. After they make 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 own squares. Keep in mind that thinner paper is easier to fold. Gift wrap, catalogues, magazines, menus, calendars, and other scrap paper can make wonderful paper for these projects. It is best to work with paper where the two sides, front and back, are easily distinguished.

Encourage students to explore geometry

_{Unfold an origami project just to}

look at the interesting pattern and the geometric figures you have created through your series of creases. Challenge students to create their own variations—and make their own diagrams showing how they did it.

Jumping frog unfolded

(7)

The Language and Symbols of Origami

white side or back of paper

colored 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

(8)

Square

a quadrilateral that has four right angles and four congruent sides. All squares are rectangles Rectangle

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

QUADRILATERALS

A quadrilateral is any figure that has four sides

Isosceles triangle a triangle with two congruent sides and two congruent angles Right triangle a triangle with a right angle Isosceles right triangle a triangle with two congruent sides and one right angle

TRIANGLES

A triangle is any figure that has three sides

Scalene triangle a triangle with no congruent sides and no congruent angles Equilateral triangle a triangle with three congruent sides and three congruent angles

CIRCLE

a round shape measuring 360º

OVAL

an egg-shape with a smooth continuous edge

PENTAGON

a shape with five sides

HEXAGON

a shape with six sides

OCTAGON

a shape with eight sides Parallelogram a quadrilateral that has two pairs of parallel sides and two pairs of congruent sides

Basic Geometric Shapes Reference Sheet

CONGRUENT

equal in measurement congruent line segments congruent angles congruent figures 60° 60°

(9)

Knowing how to turn a rectangle into a square is an invaluable origami technique! It’s also a great way to teach some basic geometry. Use rectangular paper of any size. Then invite students to use the reproducible tangrams puzzle and geometric shapes reference sheet to further explore basic shapes. Materials Needed

page 10 (steps and pattern), rectangular sheet of paper, page 7 (The Language and Symbols of Origami), page 8 (Basic Geometric Shapes Reference Sheet)

Math Concepts

shapes, patterns, symmetry, spatial relations

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 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 Reference}

Sheet, page 8.

_{Show how you can take a piece of paper with a square corner—any size—and}

use it to test various corners in the classroom. If the edges line up with their paper, 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 is smaller. 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 activity at 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 used together to make a wide variety of shapes. Challenge students to match the pictures shown and to create their own tangrams activities.

Turn a Rectangle

into a Square

When we start, the corners of this rectangle are perfect square corners. Another word for a square corner is a right angle. Let’s look around the room and find some right angles.

(Point out the corners of the room, tables, books, and so on.)When we make our fold, we’ve cut this angle exactly in half. And now we have two isosceles right triangles, which means that the two sides of the triangle that form the right angle are the same 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 perfect square. All four sides are the same length. We also have two right 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)

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 basic tangrams shapes can be used to make hundreds of designs. Cut out the shapes below. Then try to make the shapes shown below. 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. Tear along the crease.

How to Make a Square

Take a rectangular sheet of paper. Bring the bottom right corner up, so that the bottom edge and the left side line up.

Now open up your square. Find the two triangles!

2

3

1

Tip:To tear, fold the crease back and forth, scoring it each time with your finger or fingernail. Then hold the paper down firmly with one hand, and use the other hand to tear the rectangle away. Keep your hands close to the fold for better control.

or crease and tear

(11)

This activity is a simple but important introduction to basic math and origami concepts— please don’t skip it! You might teach how to make origami cards before a class party or holiday, and tailor the cards accordingly.

Materials Needed

page 12 (steps and pattern), rectangular paper or square paper (optional), crayons or markers

Math Concepts

fractions, spatial relations, size, symmetry

NCTM Standards

_{understand numbers, ways of} representing numbers, relationships among 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 properties} of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

_{apply transformations and use} symmetry to analyze mathematical situations (Geometry Standard 3.3)

Math Vocabulary

half quarter fraction

line of symmetry

_{Ask students to find different ways to express and . Folded paper is}

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

_{To further explore the “line of symmetry,” let students paint with watercolors}

on one half of a piece of paper. Then have them fold the page in half, press it firmly, 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 first row 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 the opposite direction again. Have students continue to fill in the chart. Have them try to determine a pattern. (The number of sections doubles with each fold.)

What a Card!

1 __ 2 __14

This fold is called a valley fold. That’s because we’re making a little valley here. We started with a square, and now 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 also have 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 do

you think it has that name? Let’s unfold it for a minute just to 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 cents

divided into 4 parts.

Let’s fold it back up again, following the same steps. Notice how one side now forms both the inside and the outside of the 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 to make a card of their own.)

1

2

3

(12)

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 the left 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!

2

1 ____________________________

____________________________

T

h

i

s

c

ard w

as

desi

gn

_{ed b}

y

date:_______________________

wher

e:______________________

time:_______________________

RSVP:______________________

_

Decorate and fill in your card!

(13)

Our Whale Pattern page has some features drawn in. But you may also want to distribute blue or gray paper and have students make their own.

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}

_{analyze characteristics and properties of} 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

_{use visualization, spatial reasoning, and} geometric modeling to solve problems

(Geometry Standard 3.4) Math Vocabulary square diamond triangle right left point center line of symmetry quadrilateral isosceles triangle scalene triangle multiply divide

_{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 out}

different sizes, depending on the initial square. Show how you can measure the 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 to point 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 the point is at the top. Let’s make sure everyone’s paper is facing the same way. (Encourage the class to look around the room to check for the correct positioning. Looking from different angles will strengthen their spatial relations skills.) Sometimes in origami we make folds, only to unfold them again! What’s the point? Well, as you’ll see, these folds always come in handy later on! Now we have two new shapes. What are they called? (They are triangles; you may wish to point out that they are isosceles right triangles, each having a right angle and 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 what shapes 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’t it? 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)

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

How to Make the Whale

Cut out the whale pattern on page 15. Position the square so that it looks like a diamond with the

★

at the top facedown. Fold the left point over to meet the right point. Open it up again.

Fold the top point down to meet the folded triangles.

2

3

1

Rotate the shapes so that the long flat line is at the bottom.

Fold the right side over to meet the left side.

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

5

6

(15)

Whale Pattern

Follow the steps on page 14 to create a whale.

★

(16)

This cup really works. It can hold liquid—until the water soaks through! You can adapt this model to make a hanging pocket holder. Start with a bigger sheet of paper. At the last step, don’t tuck the final flap in. Keep the back tri-angle up, punch a hole in it, and hang it up to collect Valentines. You can also punch holes and add straps to make a little carryall. Materials Needed

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

Math Concepts

spatial reasoning, shapes, volume

NCTM Standards

_{analyze characteristics and properties} of 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) _{use visualization, spatial reasoning,}

and geometric modeling to solve problems (Geometry Standard 3.4) _{understand measurable attributes of}

objects and the units, systems, and processes of measurement

(Measurement Standard 4.1)

Math Vocabulary

quadrilateral square

trapezoid isosceles right triangle parallel diagonal

line of symmetry

_{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 an experiment: 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 many times 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’s}

a hat! Use this activity to discuss the fact that form and function are often a matter of perspective—just like turning the square into a diamond in the first step. 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 the same length, and they have a right angle. That makes them

isosceles 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 base of the triangle. The top point is called the apex.

Let’s fold this top triangle down and tuck it in. What kind of 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 and then repeat the exact sequence on the other side. That is one 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 if the bottom and top were not parallel? (The liquid would spill out; it wouldn’t balance on a table!)

1

2

3

4

(17)

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 Cup

Cut out the cup pattern on page 18 and place it like a diamond, with the

★

in the top point, facedown. Or use a square 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 pocket of the cup as far as it will go. Crease.

2

3

1

Turn over and repeat steps 2 and 3 above tucking in the remaining triangle.

4

Gently pinch the sides together to

open your cup.

(18)

Cup Pattern

This cup actually holds water. Once you learn this simple model, you’ll always

be able to whip up a cup anytime you’re thirsty and you’ve got paper to fold!

(19)

This classic paper-folded pinwheel is fun to create and play 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 or markers

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 representational systems (Geometry Standard 3.2) _{apply transformations and use}

symmetry to analyze mathematical situations (Geometry Standard 3.3)

Math Vocabulary

diagonal intersect

isosceles right triangles center

_{Experiment with patterns and using the back and front of the paper:}

Have students decorate the backside of their pinwheel pattern before folding it. (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 the designs look like in motion! Have them take their pinwheels apart and refold them 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 in the 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)

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. Fold the 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 with the other three corners.

How to Make a Pinwheel and Pinwheel Pattern

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

Ask a grown-up to push a thumbtack through all layers into the side of the eras-er end of a pencil, so that the folded sides face 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

3

1

(21)

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

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

Math Concepts

spatial reasoning, sequence, symmetry, scale

NCTM Standards

_{analyze change in various contexts}

_{specify locations and describe spatial} relationships using coordinate geometry and other representational systems (Geometry Standard 3.2) _{apply transformations and use}

symmetry to analyze mathematical situations (Geometry Standard 3.3)

Math Vocabulary

horizontal vertical

line of symmetry perpendicular isosceles right triangle right angle

rectangle

Handy Hat

_{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 and open 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 the lines. We have two lines of symmetry—one horizontal and one vertical. These lines are perpendicular—they form right angles 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 in origami when we make a fold on one side, we often repeat it on the other and we get two halves that are exactly the same. 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.

1

2

3

(22)

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

How to Make a Hat

Cut out the hat pattern on page 23 and place the page facedown with the

★

in the top left corner. Or use a plain rectangular sheet of paper, or a sheet of newspaper opened up. Fold in half right to left. Crease and unfold. Then, fold in half top to bottom and crease. This time, leave the paper folded.

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

2

3

1

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

(23)

Hat Pattern

This page won’t make a hat big enough for you, but you might use it for a doll or an

action figure! Make these same folds on a sheet of newspaper and you’ll be covered!

(24)

This is one of the simplest and most traditional origami box designs. With this box, the top and bottom are the same, so fitting them together may be a tight squeeze—unless the lid is made with a slightly larger rectangle. This activity works best with sturdy paper, so photocopy the pattern onto card stock if possible. Using the cover of a magazine 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

NCTM Standards

_{analyze characteristics and properties of} two- and three-dimensional geometric shapes and develop mathematical argu-ments about geometric relationships

_{specify locations and describe spatial} relationships using coordinate geometry and other representational systems

(Geometry Standard 3.4) Math Vocabulary height volume trapezoid parallel

Box It Up!

_{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 not quite meet the center crease. They can imagine a closet door that can’t quite close! Ask them to think about why this will yield a larger box.

_{Ask students to explore different ways to make the box stronger. Then have}

them list what properties add strength. Possible solutions include: using heavier paper, making smaller boxes, using double layers, inserting cardboard on the inside.

Let’s see if I can express this fold with an equation. When I 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 see why? How would you express one of the “cabinet doors” as a fraction? ( )

Now how many sections do we have? (8)So what is the

fraction that represents each rectangle? ( )We started out with four sections for the “cabinet.” What equation could we use to show what we now have? (4 x 2 = 8, or 4 + 4 = 8) Isn’t that 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, how

many sections do you think we would find?(16) You can take

a peek and refold it to check your answer!

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

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

& Voila! See how something that is two-dimensional,

or flat, becomes three-dimensional! That’s why we had to make all those creases. They formed the sides here, which give it the height, or third dimension.

1

2

3

4

5

6

7

1 __ 2 1 __ 4 1 __ 8

(25)

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

How to Make a Box

Cut 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 of heavy paper and position the paper facedown. Fold in half, right to left. Crease and unfold.

Fold in half, top to bottom. Unfold.

2

3

1

Fold up the bottom right corner up along the fold line so that the cor-ner meets the dot. Note that the top edge does not quite reach the center fold. Crease firmly. Repeat this step with the other three corners. Fold in the short edges to meet

the center fold. Crease very sharply. This time, do not unfold.

Fold back the edges, away from the center, to cover the top part of the triangles. Crease these bands sharply.

5

6

4

Now form the box by pulling open the bands.

7

1 __ 2

Help shape the corners of the box by creasing the corners and bottom edges. Flex the sides inward.

(26)

Box Pattern

Make two of these so your box will have a lid or attach a strip of paper

to make a basket with a handle. Use it for paper clips, pennies, rubber

bands, your favorite collector cards, hair clips, or any other favorite item!

Create a design on the back,

which will sho w on the inside of your

folded box. Color this side as well,

if you wish.

(27)

We know paper can fold, fly, spin and twirl. Add noise-making to the list of paper’s possibilities with this snappy popper. Materials Needed

page 28 (steps), page 29 (pattern) or 8 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 Concepts

measurement, shape, spatial reasoning, symmetry

NCTM Standards

_{apply transformations and use symmetry} to analyze mathematical situations

(Geometry Standard 3.4) Vocabulary horizontal parallel quadrilateral right angle

Noise Popper

_{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 sound again. 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

measurementsfor something they usually give the width first. A page of copy paper held this way (show vertically) is considered 8 by 11 inches. Turn it this way, and we’d say 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 with four 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. Which two sides are parallel? (The top and bottom. If students are ready for another term, you may want to point out that a quadrilateral with two parallel sides 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 are the same length?(no)

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

1

2

3

4

5

6

1 __ 2 1 __ 2 1 __ 2

(28)

Fold the bottom right corner up so that the right side lines up with the center crease. Repeat with the other corners.

How to Make a Noise Popper

Cut 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-leaf paper. Fold in half, bottom to top, and crease sharply. Unfold.

Fold in half, top to bottom and crease.

2

3

1

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

Fold the top left corner over to meet the top right corner.

5

4

To snap the popper, pinch flaps

together firmly at the point, keeping your fingers toward the bottom so they do not block the action of the inner folds. Snap your wrist forward and the inner flap will pop out, making a snapping noise. If it doesn’t come out, loosen it a few times and try again.

6

pinch here

snap down and pop

(29)

Noise Popper Pattern

Bang! Pop! See how loud you can snap this noise popper.

Who knew paper could make such a racket?

★

Enlarge this pattern to 1

25% or larger to create a popper with a great

sound. Or use a sheet of loose-leaf

(30)

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

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 of} two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1) _{specify locations and describe spatial}

relationships using coordinate geometry and other representational systems

Math Vocabulary

rectangle fractions line of symmetry quarters eighths right angle

Itty-Bitty Book

_{Students may use the Itty-Bitty Book pattern to create a “My Important}

Numbers Book.” Encourage students to list some of the many numbers that are 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 write}

and illustrate a number story. For example, Jake had five books out from the library (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 figure}

out how the pages will fall when the book is folded. They can use the pattern as an example, or open up a folded book and mark the page numbers.

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

How would we show one of these sections as a fraction?

( )Now suppose we cut this section out. What fraction would be left? ( )

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

Now we have one half of the page showing. The fold in the middle divides that in half. Half of the half or x

equals one quarter ( ). So each of these narrow rectangles is one quarter of our original sheet. Let’s fold it in half again.

How many sections are there now?(8) So each section is

one eighth ( ) of the whole page.

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

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

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

1

2

3

4

5

1 __ 2 1 __ 2 1 __ 4 1 __ 8 1 __ 4 3 __ 4

(31)

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

How to Make an Itty-Bitty Book

Cut out the book pattern on page 32 and place it facedown with the

★

in the upper left corner. Or use a rectangular sheet, and place it horizontally, facedown. Fold the paper in half, top to bottom. Crease and unfold.

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

2

3

1

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

Cut in from the left side to the center, following the cut line. Make sure to stop at the middle crease. Open the whole sheet.

Push the two outer edges in, so that the slit opens and the inner pages are formed. Crease the edges of all pages to make the book.

5

6

4

Design your book!

(32)

Book Pattern

How 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)

This is a popular origami model that jumps! Introduce or reinforce measurement skills such as distance and height by holding an origami frog-jumping contest. 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

NCTM Standards

_{analyze characteristics and properties of} two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (Geometry Standard 3.1) _{specify locations and describe spatial}

relationships using coordinate geometry and other representational systems

_{understand measurable attributes of} objects and the units, systems, and processes of measurement

_{apply appropriate techniques, tools, and} formulas to determine measurements

Math Vocabulary

rectangle triangle

intersection perpendicular lines pentagon right angle

_{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 that there are two ways that points A and B can be related through folding. One is to make a fold that forms the line segment. The other way is to make a fold in which A and B end up on top of each other. To make that fold, you end up making 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 before measuring and record the accuracy of their estimates.

We’re folding a square to make two rectangles. We could just start with a rectangle this size, but we make the paper thicker this way. That will help the frog keep its shape and jump better when we’re done.

& Look, we’ve made an “X” here. The point where the

two lines cross is called the intersection. See how we have four perfect corners at the intersection? That means that the two 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 in half. 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 the angle 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-sided

shape called?(a pentagon)When we fold these sides in, the

shape 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’s make a big “Z” on the board. Can you see why this shape is springy? How will this Z-shape design help the frogs to hop?

1

2

4

5

3

6

7

8 Jumping Frog

(34)

Fold the top right corner over to meet the dot on the left edge. Crease and unfold. Repeat on other edge, and unfold so that you have an X at the top.

How to Make a Frog

Cut out the frog pattern on page 35 and place it facedown with the

★

in the upper right corner. Or start with a 6-inch square, facedown. Fold the page in half, right to left.

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

2

3

1

Take the bottom two points of the triangle and fold them up to create the front legs.

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

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

5

6

4

Fold in half, top to bottom. Do not crease this fold sharply; simply bend it.

7

Flip over. Fold the top layer in half,

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

8

Your frog is ready to hop!

Push down on the spot on the frog’s back and release to make him go.

(35)

Frog Pattern

Hop to it with this little jumping frog. Hold a frog-jumping contest with your own frogs or challenge a friend to a hop-a-thon.

(36)

Make these cat faces with black paper and use them as Halloween decorations.

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

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

Math Vocabulary

congruent

isosceles right triangle midpoint

base angle

scalene triangle hexagon

_{You can display a litter of these cats by hanging a string across the room and}

threading it through the top triangle fold of each origami cat. Tape the folds down over the string to make them stay in place. For a multiplication

connection, ask students if cats have nine lives, how many lives are represented by this string of cats? Help them multiply by 9s. You can make this easier by creating 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 draw}

their own cat faces that are exactly symmetrical. They might start by folding the shape gently in half to find the center line (line of symmetry). Have them work off 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 or equal-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 to mark 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 where the ears will be, can’t we? What shape are the ears? (triangles)

This kind of triangle has no congruent or equal sides or angles. It’s called a scalene triangle.

& If we cover up the ears, what shape does the face

become?(a hexagon)

1

2

3

4

5

(37)

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

How to Make a Cat

Cut 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

3

1

Fold the bottom point up to meet the top point.

4

Turn over and decorate the face of

your cat.

(38)

Cat Pattern

Make a bunch of black cats to hang up on Halloween. Or make one big

cat with a large square and a litter of kittens with smaller squares.

(39)

This simple boat will actually float in water. Use a coin to help balance 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

_{apply transformations and use} symmetry to analyze mathematical situations (Geometry Standard 3.3)

Math Vocabulary

rectangle triangle hexagon area

_{Set up a basin of water and let students have a boat race. Measure speed,}

distance, and accuracy. Discuss and adjust balance as necessary. Experiment making 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 determine}

the 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 create 8 columns. Open it up and repeat these folds in the other direction to create 8 rows. Open it again, and you should have 64 small squares. Tell students that each square is 1 inch x 1 inch. Count the squares to determine that the area is 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 this formula 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 paper disappears. 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 the boat glide through the water smoothly. A flat front would slow down the speed and make the boat hard to control.)

1

2

3 Floating Boat

4

5

6

1 __ 4 1 __ 2

(40)

Fold down the top edge, front layer only, to meet the bottom folded edge. Crease.

How to Make a Boat

Cut out the boat pattern on page 41 and place the

★

in the upper right corner, face up. Or start with a 6-inch square, face up. Fold in half, bottom to top. Crease and leave folded.

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

2

3

1

Fold in half, top to bottom. Fold the top corners down to the

center line and crease. Fold the bottom corners up to the center line and crease. Make sure to fold all the layers.

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

5

6

(41)

Boat Pattern

This simple boat actually floats. Try blowing it across a small tub of water with a

straw. Have a boat race with a larger and smaller boat to see which moves faster.

(42)

Students can use this friendly folded corner-hugger as a bookmark.

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

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 Vocabulary diamond square triangle perpendicular pentagon line of symmetry congruent

Page-Hugger

Bookmark

_{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

perpendicularto each other. See how the corners where they 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 shape

with 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 three congruent 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 bookmark too. Can you find it? Could we use this pocket as a bookmark as well? (no)Why not? (because it’s not shaped like a corner, or right angle)

1

2

3

4

5

3 __ 4

(43)

Fold up the bottom point to meet the center point. Crease. Repeat with the left and top corners. Leave the right corner unfolded.

How to Make a Bookmark

Cut 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, so the corners meet. Crease and unfold. Fold in half top to bottom so that the corners meet. Crease and unfold.

Fold down the top left corner to meet the bottom right corner. Leave that part folded.