Grades 2–3
BY
K AREN B AICKER
New York • Toronto • London • Auckland • Sydney
Mexico City • New Delhi • Hong Kong • Buenos Aires
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
For Joseph Baicker with thanks for all the
help with math, even though he wouldn’t
let me do it the teacher’s way.
Scholastic Inc. grants teachers permission to photocopy the reproducibles from this book for classroom use. No other part of this publication 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.
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
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Contents
Introduction
......................4
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
7
Math Concepts: multiplication, division, dimension
NCTM Standard 3
8
Turn a Rectangle into a Square . . . . 9
9
Math Concepts: shapes, patterns, symmetry,
spatial relations
NCTM Standard 3
2
What a Card! . . . . . . . . . . . . . . . . . . . . . . . 11
Whale of Triangles . . . . . . . . . . . . . . . . . 13
10
Instant Cup . . . . . . . . . . . . . . . . . . . . . . . . . 16
11
Playful Pinwheel . . . . . . . . . . . . . . . . . . . 19
12
Handy Hat . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Math Concepts: spatial reasoning, sequence,
symmetry, scale
NCTM Standards 2 and 3
Floating Boat . . . . . . . . . . . . . . . . . . . . . . . 39
Math Concepts: shape, fractions, area
NCTM Standards 2 and 3
13
Page-Hugger Bookmark . . . . . . . . . . 42
Math Concepts: shape, spatial reasoning,
symmetry, congruence
NCTM Standards 3
Math Concepts: spatial relations, pattern,
symmetry, motion
NCTM Standards 2 and 3
6
Kitty Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Math Concepts: angles, symmetry
NCTM Standard 3
Math Concepts: spatial reasoning, shapes, volume
NCTM Standards 3 and 4
5
Jumping Frog . . . . . . . . . . . . . . . . . . . . . . . 33
Math Concepts: shape, measurement,
distance, height
NCTM Standards 3 and 4
Math Concepts: size, symmetry, shapes,
patterns, spatial relations
NCTM Standards 2 and 3
4
Itty-Bitty Book . . . . . . . . . . . . . . . . . . . . . 30
Math Concepts: spatial reasoning, shapes, symmetry, fractions, multiplication, division
NCTM Standards 1 and 3
Math Concepts: shapes, patterns, size,
symmetry, spatial relations
NCTM Standards 2 and 3
3
Noise Popper . . . . . . . . . . . . . . . . . . . . . . . 27
Math Concepts: measurement, shape,
spatial reasoning, symmetry
NCTM Standard 3
Activities
1
Box It Up! . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
14
Paper Airplane Express . . . . . . . . . . . 45
Math Concepts: symmetry, balance,
spatial reasoning
NCTM Standards 3 and 4
Glossary of Math Terms . . . . . . . . . . . . . . . . . . 48
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
3
Introduction
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 activities 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.
4
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.
1
2
2
1
Is __
of __
the same as __
of __
?
2
3
3
2
Through paper folding, you can see
that it is!
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.
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
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 2
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.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
5
Lesson 2
Tips for Teaching With Origami
possible. Moving through the steps
helps you anticipate any areas of
difficulty students may encounter
and your completed activity provides a model for them to consult.
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.
Review the Math Wise! section and
Demonstrate the folds with a larger
Prepare for the lesson
Try the activity ahead of time if
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 projects. 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
6
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.
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.
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.
Jumping frog unfolded
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 2
The Language and Symbols of Origami
colored side or front of paper
white side or back of paper
fold toward the front
fold toward the back
(valley fold)
(mountain fold)
fold, then unfold
turn over
fold
folded edge
crease
raw edge
cut
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
7
Lesson 2
Basic Geometric Shapes Reference Sheet
CONGRUENT
QUADRILATERALS
equal in measurement
A quadrilateral is any figure that has four sides
congruent line
segments
congruent angles
60°
60°
congruent figures
Parallelogram
a quadrilateral
that has two
pairs of parallel
sides and
two pairs of
congruent sides
Rectangle
a quadrilateral
that has four
right angles (90°).
All rectangles are
parallelograms
Square
a quadrilateral
that has four
right angles and
four congruent
sides. All squares
are rectangles
TRIANGLES
A triangle is any figure that has three sides
8
Right triangle
a triangle with
a right angle
Isosceles triangle
a triangle with
two congruent
sides and two
congruent angles
Isosceles right
triangle
a triangle with
two congruent
sides and one
right angle
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
OVAL
PENTAGON
HEXAGON
OCTAGON
a round shape
measuring 360º
an egg-shape
with a smooth
continuous edge
a shape with
five sides
a shape with
six sides
a shape with
eight sides
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 1
Turn a Rectangle
into a Square
Math Wise! Distribute copies of page 10. Use these
tips to highlight math concepts and vocabulary for each step.
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.
1 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.
Materials Needed
away. What shape is this? (rectangle)
page 10 (steps and pattern), rectangular
sheet of paper, page 7 (The Language and
Symbols of Origami), page 8 (Basic
Geometric Shapes Reference Sheet)
3 Let’s open up this triangle. Now we have a perfect
2 Now we have an extra shape we have to cut (or tear)
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)
Math Concepts
shapes, patterns, symmetry, spatial
relations
Beyond the Folds!
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)
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.
Math Vocabulary
rectangle
square
right angle
isosceles right triangle
triangle
congruent
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
9
Lesson 1
How to Make a Square
1 Take a rectangular sheet of
paper. Bring the bottom right
corner up, so that the bottom
edge and the left side line up.
2 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.
3 Now open up
your square. Find the
two triangles!
Tip: To tear, fold the crease
or crease
and tear
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.
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.
10
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 2
What a Card!
Math Wise! Distribute copies of page 12. Use these
tips to highlight math concepts and vocabulary for each step.
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.
1 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.
2 & 3 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.
Materials Needed
page 12 (steps and pattern), rectangular
paper or square paper (optional),
crayons or markers
Math Concepts
fractions, spatial relations, size, symmetry
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.
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
(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
(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.)
Beyond the Folds!
1
Ask students to find different ways to express __12 and __4 . 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.)
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
11
Lesson 2
How to a Make Card and Card Pattern
2 Fold in half again, bringing the
1 Cut out the card pattern below.
left side over to meet the right.
Place the paper facedown, with the ★
Crease well again.
in the bottom right corner. Fold in half,
bringing the top down to meet the bottom.
Make a e
n th
design o page
is
th
f
o
back
it
and fold r a
fo
s
rd
a
w
k
bac
le card!
reversib
3
Decorate and fill in your card!
RSVP:_______________________
time:_______________________
where:______________________
date:_______________________
c a r d wa s d e s i g n e d b
s
i
y
Th
____________________________
____________________________
____________________________
12
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 3
Whale of
Triangles
Math Wise! Distribute copies of pages 14 and 15. Use these
tips to highlight math concepts and vocabulary for each step.
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.
1 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
Materials Needed
page 14 (steps), page 15 (pattern)
or 6-inch square paper
Math Concepts
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
shapes, patterns, symmetry, spatial relations
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.
NCTM Standards
That’s called “the line of symmetry.” That means that the
line divides two halves that match.
analyze change in various contexts
(Algebra Standard 2.4)
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)
Math Vocabulary
square
diamond
triangle
right
left
point
center
line of symmetry
quadrilateral
isosceles triangle
scalene triangle
multiply
divide
2 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.)
3 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.)
4 Ah-hah! We’re using that line of symmetry again.
Now you see why we folded it in the first place!
5 & 6 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)
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 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.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
13
Lesson 3
How to Make the Whale
1 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.
2 Fold the two lower sides to meet
the center fold, or line of symmetry.
3 Fold the top point down to meet
the folded triangles.
4 Fold the right side over to meet
the left side.
5 Rotate the shapes so that the long
flat line is at the bottom.
6 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.
14
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 3
Whale Pattern
Follow the steps on page 14 to create a whale.
✂
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
15
Lesson 4
Instant Cup
Math Wise! Distribute copies of pages 17 and 18. Use these
tips to highlight math concepts and vocabulary for each step.
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 triangle 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.
1 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.
2 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.
3 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)
Materials Needed
4 & 5 In origami, you often do something to one side and
page 17 (steps), page 18 (pattern) or
6-inch square paper, water (optional)
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;
Math Concepts
spatial reasoning, shapes, volume
NCTM Standards
it wouldn’t balance on a table!)
analyze characteristics and properties
of two- and three-dimensional geometric
shapes and develop mathematical
arguments about geometric
relationships (Geometry Standard 3.1)
Beyond the Folds!
apply transformations and use
Ask students if they think that a square double the size of the original square
symmetry to analyze mathematical
will make a cup that holds double the volume. Then have them conduct an
situations (Geometry Standard 3.3)
experiment: Make two cups with different-sized paper and test the volume.
use visualization, spatial reasoning,
Fill the larger cup and then pour it into the smaller cup once. Then dump
and geometric modeling to solve
that water (or grains of rice) out, and refill from the larger cup. See how many
problems (Geometry Standard 3.4)
times the smaller cup fills to compare the capacity of the two cups.
understand measurable attributes of
objects and the units, systems, and
Make a large cup out of a big square of newsprint. Turn it upside down and it’s
processes of measurement
a hat! Use this activity to discuss the fact that form and function are often a
(Measurement Standard 4.1)
matter of perspective—just like turning the square into a diamond in the first
step. Let students make and decorate their own hats.
Math Vocabulary
quadrilateral
trapezoid
parallel
line of symmetry
16
square
isosceles right triangle
diagonal
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 4
How to Make the Cup
1 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.
2 Fold up the bottom right corner
to meet the opposite edge. Line up
the corner so that it meets the dot.
Crease the fold.
4 Turn over and repeat steps 2 and 3
above tucking in the remaining triangle.
5 Gently pinch the sides together to
open your cup.
3 Fold down the top layer of the triangle above, tucking it into the pocket
of the cup as far as it will go. Crease.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
17
Lesson 4
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!
★
18
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 5
Playful
Pinwheel
Math Wise! Distribute copies of page 20. Use these
tips to highlight math concepts and vocabulary for each step.
This classic paper-folded
pinwheel is fun to create and
play with, and it makes a
great model for exploring
symmetry and patterns.
1 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.)
Materials Needed
2 What are the four different shapes we’re folding?
page 20 (steps and pattern), square paper
(optional), pencils with erasers, scissors,
push pins, glue (optional), crayons or
markers
(triangles)
3 Why do our folds allow the pinwheel to spin?
(The pockets can catch the air, like a sail.)
Math Concepts
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.
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)
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 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)
Math Vocabulary
diagonal
intersect
isosceles right triangles
center
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
19
Lesson 5
How to Make a Pinwheel and Pinwheel Pattern
1 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.
2 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.
3 Ask a grown-up to push a thumbtack
through all layers into the side of the eraser 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.
✂
f
agic o
The m is the way
ls
ee
ey’re
pinwh k when th me
o
so
they lo ing. Make ent
n
er
f
in
if
sp
with d
e
more ns and giv
desig a whirl!
them
20
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 6
Handy Hat
Math Wise! Distribute copies of pages 22 and 23. Use these
tips to highlight math concepts and vocabulary for each step.
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).
Materials Needed
page 22 (steps), page 23 (pattern)
or newspaper sheets
Math Concepts
spatial 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 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
1 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)
2 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)
3 & 4 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.
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 and
open to find a feather shape.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
21
Lesson 6
How to Make a Hat
1 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.
2 Fold in the top corners to meet the
center line. Crease.
3 Fold up the top layer of the
bottom edge. Turn over and repeat
on the other side.
4 Pull out on the sides to open it.
Try it on.
22
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 6
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!
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
23
Lesson 7
Box It Up!
Math Wise! Distribute copies of pages 25 and 26. Use these
tips to highlight math concepts and vocabulary for each step.
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!
3 Now how many sections do we have? (8) So what is the
1
) We started out
fraction that represents each rectangle? ( __
8
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!
Materials Needed
4 We don’t need to open it up this time! But if we did, how
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.
many sections do you think we would find? (16) You can take
a peek and refold it to check your answer!
Math Concepts
multiplication, division, dimension
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
(Geometry Standard 3.2)
use visualization, spatial reasoning, and
geometric modeling to solve problems
(Geometry Standard 3.4)
1 Let’s see if I can express this fold with an equation. When
1
I fold it, it’s 1 ÷ 2 = __
2 . Now when I unfold it, I can say 1 x 2 = 2.
2 This fold is sometimes called a cabinet fold. Can you see
why? How would you express one of the “cabinet doors” as
1
a fraction? ( __
4 )
5 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.
6 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.
7 & 8 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.
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 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.
Math Vocabulary
height
volume
trapezoid
parallel
24
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 7
How to Make a Box
1
Cut out the box pattern on page
26 and place it facedown with the ★
in the upper left corner. Or use an
1
8 __2 by 11-inch rectangular sheet of
heavy paper and position the paper
facedown. Fold in half, right to left.
Crease and unfold.
2
Fold each side in half, folding in
the long edges to the center fold.
Crease sharply. Unfold.
3
4 Fold in the short edges to meet
the center fold. Crease very sharply.
This time, do not unfold.
5 Fold up the bottom right corner
up along the fold line so that the corner 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.
6 Fold back the edges, away from
the center, to cover the top part of
the triangles. Crease these bands
sharply.
7 Now form the box by pulling
open the bands.
8 Help shape the corners of the box
by creasing the corners and bottom
edges. Flex the sides inward.
Fold in half, top to bottom. Unfold.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
25
Lesson 7
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!
Crea
design o te a
n th
which w e back,
on the in ill show
sid
folded b e of your
ox
this side . Color
as w
if you w ell,
ish.
★
26
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 8
Noise Popper
Math Wise! Distribute copies of pages 28 and 29. Use these
tips to highlight math concepts and vocabulary for each step.
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 __12 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 3.3)
use visualization, spatial reasoning, and
geometric modeling to solve problems
(Geometry Standard 3.4)
Vocabulary
horizontal
parallel
quadrilateral
right angle
1 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 width
first. A page of copy paper held this way (show vertically)
1
is considered 8 __
2 by 11 inches. Turn it this way, and we’d
1
say 11 by 8 __
inches.
But it’s the same piece of paper!
2
2 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.
3 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.)
4 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)
5 & 6 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 triangular 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.)
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 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.)
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
27
Lesson 8
How to Make a Noise Popper
1 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.
2 Fold the bottom right corner up so
that the right side lines up with the
center crease. Repeat with the other
corners.
3 Fold in half, top to bottom
and crease.
4 Fold the top left corner over to
meet the top right corner.
5 Fold the front flap down, so that the
top edge lines up with the left edge.
Turn over and repeat.
6 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.
snap down
and pop
pinch
here
28
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 8
Noise Popper Pattern
Bang! Pop! See how loud you can snap this noise popper.
Who knew paper could make such a racket?
Enlarge th
pattern to is
or larger to 125%
popper wit create a
ha
sound. Or great
sheet of lo use a
ose-leaf
paper.
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
29
Lesson 9
Itty-Bitty Book
Math Wise! Distribute copies of pages 31 and 32. Use these
tips to highlight math concepts and vocabulary for each step.
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.
Math Concepts
spatial reasoning, shapes, symmetry,
fractions, multiplication, division
NCTM Standards
understand numbers, ways of representing 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
(Geometry Standard 3.2)
Math Vocabulary
rectangle
line of symmetry
eighths
30
How many rectangles do we have? (five—the four smaller rectangles
plus the whole sheet as the fifth rectangle)
1
page 31 (steps), page 32 (pattern) or
rectangular sheet of paper, scissors,
crayons or markers
2 How would we show one of these sections as a fraction?
1
( __
) Now suppose we cut this section out. What fraction
4
3)
would be left? ( __
4
3 Now we have one half of the page showing. The fold in
Materials Needed
1 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?
1
__
the middle divides that in half. Half of the half or __
2 x 2
1
equals one quarter ( __
4 ). So each of these narrow rectangles is
one quarter of our original sheet. Let’s fold it in half again.
4 How many sections are there now? (8) So each section is
1
one eighth ( __
) of the whole page.
8
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.)
5 Before we fold it in half, let’s count how many pages our
book has. (8)
6
&
7 Watch what happens to the flat paper as we push
the ends together. We’ve created a three-dimensional form.
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 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.
fractions
quarters
right angle
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 9
How to Make an Itty-Bitty Book
1 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.
2
Fold in half, left to right. Crease it
sharply, and leave it folded.
3 Fold again in the same direction.
Unfold this last step.
4 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.
5 Fold in half top to bottom, so that
the two long edges meet.
6 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.
7
Design your book!
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
31
Lesson 9
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.
★
32
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 10
Jumping Frog
Math Wise! Distribute copies of pages 34 and 35. Use these
tips to highlight math concepts and vocabulary for each step.
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
shape, measurement, distance, height
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
(Geometry Standard 3.2)
understand measurable attributes of
objects and the units, systems, and
processes of measurement
(Measurement Standard 4.1)
apply appropriate techniques, tools, and
formulas to determine measurements
(Measurement Standard 4.2)
Math Vocabulary
rectangle
intersection
pentagon
2 & 3 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.
4 When we’re starting this fold, we have three triangles,
plus the bottom triangle that’s part of the house-shape
half. That’s five triangles lining up over the bottom triangle.
5 Now if we folded these triangles so that they line up
NCTM Standards
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.
(a pentagon). But we’re collapsing the two side triangles in
Math Concepts
1 We’re folding a square to make two rectangles. We could
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.
6 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.
7 & 8 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?
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 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.
triangle
perpendicular lines
right angle
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
33
Lesson 10
How to Make a Frog
1 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.
2 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.
3 Fold the top points of the X down
to meet the bottom points of the X.
Crease and unfold.
4 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.
5 Take the bottom two points of the
triangle and fold them up to create the
front legs.
6 Fold the side edges in toward the
center. Use the fold lines as a guide.
7 Fold in half, top to bottom. Do not
crease this fold sharply; simply bend it.
8 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.
34
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 10
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.
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
35
Lesson 11
Kitty Cat
Math Wise! Distribute copies of pages 37 and 38. Use these
tips to highlight math concepts and vocabulary for each step.
1 Make a valley fold to create two layers of congruent or
equal-sized triangles. What special type of triangle is this?
Make these cat faces with black
paper and use them as Halloween
decorations.
(isosceles right triangle)
2 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.
Materials Needed
page 37 (steps), page 38 (pattern) or
a 6-inch square of paper, crayons or
markers
Math Concepts
angles, symmetry
3 Now that we’ve folded the top down, we can see where
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
congruent
isosceles right triangle
midpoint
base
angle
scalene triangle
hexagon
36
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.
4 & 5 If we cover up the ears, what shape does the face
become? (a hexagon)
Beyond the Folds!
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.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 11
How to Make a Cat
1 Cut out the cat pattern on page 38
and place it like a diamond, with the ★
at the top, facedown. Or start with a 6inch square, facedown. Fold the
bottom corner up to meet the top.
2 Fold up the bottom right corner
along fold line. Repeat on the left side,
so that the triangles cross.
4 Fold the bottom point up to meet
the top point.
5 Turn over and decorate the face of
your cat.
3 Fold the top triangle down
(both layers) along the fold line.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
37
Lesson 11
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.
★
38
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 12
Floating Boat
Math Wise! Distribute copies of pages 40 and 41. Use these
tips to highlight math concepts and vocabulary for each step.
This simple boat will actually
float in water. Use a coin to help
balance if the boat is too tippy.
1 Let’s make a book fold. What two shapes do we have
now? (rectangles)
Materials Needed
3
page 40 (steps), page 41 (pattern) or
6-inch square, crayons or markers,
basin of water (optional)
4 What shape are these corners? (triangles) And what shape
do we have here when the corners are folded in? (hexagon)
Math Concepts
5 When we make this fold, all of the back side of the paper
shapes, fractions, area
disappears. It’s all tucked inside here.
NCTM Standards
6 How do you think this shape helps keep the water out?
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)
apply transformations and use
symmetry to analyze mathematical
situations (Geometry Standard 3.3)
1
2 How much of the page do we have with this strip here? ( __4 )
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.)
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. 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.
Math Vocabulary
rectangle
triangle
hexagon
area
1
How much of the page do we have showing now? ( __
2)
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
39
Lesson 12
How to Make a Boat
1 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.
2 Fold down the top edge, front
layer only, to meet the bottom folded
edge. Crease.
3 Turn over and repeat step 2.
This time, unfold that last fold.
4
5
6
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.
40
Fold in half, top to bottom.
Separate the top edges to open
the boat. Press down along the
bottom and pull out the sides to
create a flat bottom.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 12
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.
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
41
Lesson 13
Page-Hugger
Bookmark
Math Wise! Distribute copies of pages 43 and 44. Use these
tips to highlight math concepts and vocabulary for each step.
Students can use this friendly
folded 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
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)
1 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 where
they intersect, or cross, are perfect square corners?
2 What fraction of the corners have we folded? ( __34 )
3 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)
4 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!
5 & 6 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)
Math Vocabulary
diamond
square
triangle
perpendicular
pentagon
line of symmetry
congruent
42
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)
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 13
How to Make a Bookmark
1 Cut out the bookmark pattern on
page 44 and place the square like a
diamond with the ★ at the top, facedown. 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.
2 Fold up the bottom point to meet
the center point. Crease. Repeat with
the left and top corners. Leave the
right corner unfolded.
3 Fold down the top left corner to
meet the bottom right corner. Leave
that part folded.
4
5
6 Fill in your name on the back.
Decorate your page-hugger bookmark.
Fold up the bottom left corner to
meet the top point. Crease well.
Tuck the right hand point inside
the pocket formed by the left-hand
triangle.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
43
Lesson 13
Bookmark Pattern
Keep this bookmark buddy handy and you’ll never lose your place.
★
k
oo o
b t
__
__
is gs
__
__
Th on
__
l
__
__
be
__
__
__
__
__
__
__
44
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 14
Paper Airplane
Express
Math Wise! Distribute copies of pages 46 and 47. Use these
tips to highlight math concepts and vocabulary for each step.
There are hundreds of ways to
make paper airplanes. This model
has good gliding action. Use the
airplane pattern on page 47, or
1
any 8 __
2 by 11-inch sheet of
copy paper!
1 If we take a plain piece of paper and try to throw it, it
doesn’t travel very far! But if we crumple it up, look! We can
throw it much better. Why do different forms of the same
page fly better than others? (The crumpled-up ball has a smaller
Materials Needed
we fold an airplane, we want to make a design that has not
much surface area and little wind resistance.
page 46 (steps), page 47 (pattern)
or 8__12 x 11-inch rectangle, paper clip
Math Concepts
symmetry, balance, spatial reasoning
apply transformations and use symmetry
to analyze mathematical situations
(Geometry Standard 3.3)
understand measurable attributes of
objects and the units, systems, and
processes of measurement
(Measurement Standard 4.1)
apply appropriate techniques, tools, and
formulas to determine measurements
(Measurement Standard 4.2)
Math Vocabulary
half
center
weight
corner
bisect
angle
surface area
2 Many paper airplanes have a sharp point at the front.
Later, we’ll fold this point and make a blunt nose. How might
the shape at the front affect the way a plane flies? (The shape
of the plane’s nose affects the flight pattern.)
NCTM Standards
surface area—most of it is tucked into the center of the ball. Less of the paper
hits the air, so it doesn’t slow down the way the open sheet does.) When
3 We’re folding these angles in half, or bisecting them.
4 In this model you might notice that whatever we do on
one side, we do on the other. That’s because planes depend
on symmetry for smooth flying. What would happen
if the two sides were different? (It would fly crooked.)
5 & 6 What happens to the plane when we fold the nose
back? (The nose gets heavier.)
7 & 8 What does the paper clip do to the nose of the
plane? (It adds more weight and keeps the sides together tightly, so there’s
less surface area and wind resistance.)
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 actual
landing point. Calculate the average distance by compiling the flight records
of all students.
Have students decorate their own planes, each wing with a different design.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
45
Lesson 14
How to Make a Paper Airplane
1 Start with the paper airplane
pattern, page 47, and place the pattern
facedown with the ★ in the upper
right-hand corner. Fold in half left to
right. Make a light crease. Unfold.
2 Fold down the top corners to the
center line.
3 Fold in each side from the point,
to meet the center line. Crease along
the fold lines.
4 Fold down the tip of the plane’s
nose toward the middle, along the fold
line.
5
6 Fold down the top layer of the rear
wing, along the short fold line. Turn
over and repeat on other side.
7 Fold down the wings to the base,
starting from the nose. Use the long
fold line as a guide. Repeat on the
other side.
8 Spread the wings. Add a paper clip
to the nose for balance and weight.
46
Fold in half, left to right. Crease well.
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
Lesson 14
Paper Airplane Pattern
What’s the longest distance your plane can fly?
★
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources
47
Glossary of Math Terms
angle the space formed by two lines
stemming from a common point
parallel two lines that are always the same
distance apart, and therefore never intersect
area the measurement of a space formed
by edges
parallelogram a quadrilateral that has two
pairs of parallel sides and two pairs of
congruent sides
base the bottom edge of a shape
pentagon a shape with five sides
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
perpendicular at right angles to a line
quadrilateral any four-sided figure
rectangle a quadrilateral that has four right
angles (All rectangles are parallelograms.)
right angle an angle of 90º, which forms a
square corner
right triangle a triangle with a right angle
(90º)
scalene triangle a triangle with no congruent
sides and no congruent angles
square a quadrilateral that has four right
angles and four congruent sides (All squares
are rectangles.)
intersection the point where two lines cross
symmetry a balanced arrangement of parts
on either side of a central dividing line or
around a central point
isosceles triangle a triangle with two
congruent sides and two congruent angles
trapezoid a quadrilateral with one pair of
parallel sides
line of symmetry a line that divides two alike
halves
triangle a figure with three sides
horizontal
running across, or left to right
midpoint a point that lies halfway along
a line
vertical
running from top to bottom
volume the space contained within a
three-dimensional object
octagon a shape with eight sides
oval an egg-shaped figure with a smooth,
continuous edge
48
Origami Math: Grades 2-3 © Karen Baicker, Scholastic Teaching Resources