## Math with Paper: Fold Some Math into Your Day!

by Sarah Eason , Michelle Hurst , Susan Levine , Amy Claessens , Madeleine Oswald , Kassie Kerr & Abrea Greene

Many fun math games for families can be done with materials you probably have at home already, such as scrap paper. Learn how to create origami shapes, be a paper math wizard, and support children’s learning with these activities.

There are lots of fun math activities for families to do together that don’t require special materials. In fact, with just a few sheets of paper, families can find fun ways to explore math ideas and problem solving!

## Why Do Math with Paper?

Kids can learn a lot about math at home through hands-on, playful activities that inspire conversations about  numbers , shapes , and  spatial relations .

The activities outlined below introduce and reinforce math concepts while encouraging creativity. They are also easy to adapt for different ages and may be done with multiple kids at once. Because families can do them together, it’s a great opportunity to talk about math.

Many of these activities can be done with whatever paper is available—even scrap paper, newspapers, or magazine pages would work. Scissors, hole punchers, and pencils or markers will be helpful, too.

## Math with Paper Activities

Origami is great for thinking about shapes and ideas of space and place. The same sheet of paper can look completely different depending on where and how it gets folded.

A quick search online will lead you to lots of origami instructions. Here are some origami projects to get you started:

• Beginner:  Dog origami
• Intermediate:  Heart origami

Paper Wizard

Mental imagery—the ability to imagine what a shape looks like or to move objects around in your mind—is an important part of math. This activity helps children practice thinking about spatial information and imagining how space can be transformed without actually seeing it.

• Get started:  Paper wizard
• Try another one:  Advanced Paper wizard

Even More Ideas

There are lots of ways to explore math with paper! Check out our printable  Guide to Math with Paper for ways to keep going and to support children’s math thinking during all of these activities. You can also get creative and think of your own ways to do math with paper.

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## SMART Space

Origami geometry.

Use congruency, trigonometry, and proofs to calculate the surface area of your origami creations! For more ideas, see: Geometric Exercises in Paper Folding, T. Sundara Row Project Origami, Thomas Hull

Video of solution for calculating the surface area on an origami fold.

Teachers have been using Origami to explain geometric concepts for over a hundred years. However, it is usually used to teach concepts that are too simple (basic spatial reasoning) or too complex (advanced mathematical proofs) for a high school classroom.  The assignment is simple: students will make one of their favorite origami modules, and then they have to calculate the surface area of the 2D module made.

Our task was to successfully fold an origami figure while using  geometry  to calculate the surface area of the finished product.  We worked in teams of two, but it is possible to have students work individually. This assignment only requires pencil and paper.

Our first task was to choose a figure that we felt was cool yet simple enough to calculate the surface area of mathematically.  We went with an 8-pointed star. Composed of eight separate folded parts, it would be difficult to build, but we recognized that this actually made the calculations easier. As soon as we found the surface area of a single portion (only two distinct geometric shapes), we could simply multiply by eight to find the total surface area.

If the teachers want to choose an origami module to assign to students, it would be ideal to select a module in which many concepts of geometry can be practiced.    In our example project, we choose a module in which several units have to be assembled together. This is because then, the students would be able to practice geometric concepts and to study the geometric relations between different units.

We made an elephant and a 8-pointed star. These two examples can create great projects because they make uses of various geometric concepts.

In this part, we will focus on the process of calculating the surface area of this 8-pointed star. In class, the students are free to choose whichever origami module they want to make, and the surface area can still be easily calculated through the following steps. (these steps are explained in the video) Step 1:  Look up a tutorial on the Internet and make the parts. Assemble some parts together. Step 2:   Look at the overall shape of the origami module made. Shade the exposed parts, which make up the surface area of the main module. Step 3:  Unfold the folded unit back into the original piece of paper. Step 4:  Using the given dimension of this piece of paper (in the video the paper is 9 inches x 9 inches), determine the dimensions of the shaded regions. The dimensions can be calculated using various geometric concepts such as  congruency ,  similar triangles ,  trigonometric identities , and  Pythagorean theorem . In cases where the chosen origami module consists of an angle other than 90 or 180 degrees, students can also learn the  Law of Sines and Cosines  while calculating different lengths of the shaded regions.  Students can also practice calculating the  area  of many different geometric shapes in this origami project. Step 5:  This 8-pointed star is created from 8 units. Therefore, the calculated area is multiplied eightfold for the final area of the origami module.

Without folding the shape first, it is extremely difficult to determine which parts of the shape are a part of the surface area.

If you (carefully) pull on the points of the star, it transforms into this donut-shaped figure!

We initially wanted students to come up with sets of geometric rules that their classmates could use to fold certain origami shapes. On paper, it seemed great - make students use mathematical language to convey a highly visual task (paper-folding). However, it simply proved too difficult. Unless the rules were incredibly complicated, it was possible to fold multiple correct solutions.

Trang Ngo & Will Luna, Tufts University CEEO 2016

## Introduction: The Math and Art in Origami - How to Make Geometric Wireframes

I was never truly intrigued by origami until I ran into modular origami. I have folded for a few years now, and, while I am still no master of this art, I want to share at least a part of it with others. Origami is a lovely example of the too-often-ignored (although frequent) intersections between science and art. Modular origami leans more toward the mathematical side, but the potential for a practically infinite variety of shapes that are (both visually and theoretically) beautiful is also rather artistic. The majority of the models shown above are compounds of multiple intersecting shapes, which is where the variety really opens up.

In this instructable, I'll talk about only one variety of modular origami: wireframes made from Ow units. A wireframe is a 2-d or 3-d shape where only the edges are solidified, leaving open faces. The units (the individual modules in a model) that I will explain were more or less originally conceptualized by Francis Ow.

To equip you to come up with new compounds and modular constructions, I will cover a bit of theory on polyhedra and the polygons that make them up, explain how to fold Ow units and how to adjust the design to a variety of angles for making different polygons, and then leave you with some pictures, links, and ideas for going further.

That aside, let's get started! All you need is:

-Paper: origami paper is better, but printer paper usually works fine

-Papercutter or scissors (unless you want to crease and tear, which isn't quite as pretty)

-Time. More is better

## Step 1: Theory: Polyhedra

Polyhedron: from poly (many) and hedron (face). Polyhedra are 3D geometric shapes. They are built from polygons ( gon - angle/corner), which are 2D geometric shapes. Polyhedra are named and classified based on the number and type of faces, and their symmetry, as well as how they can be constructed.

Possibly the most fundamental group is the Platonic Solids (all consist of identical, regular polygons, shown in a render above). They are the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Another common group is the Archimedean Solids, which all consist of identical vertices but a variety of faces (two or more types of regular polygons). There are 13 of these (depending on the exact definition - for more, see the going further page), shown in the second render. There are several other groups, but those groupings are less commonly used.

The aforementioned groupings are only for polyhedra with regular-polygonal faces; of course, a theoretical infinity of irregular-polygonal polyhedra, as well as irregular-verticed, regular polygonal polyhedra are possible, but only the simpler ones are practical to construct with Ow's methods, due to their size and interlocking limitations imposed by the geometry of the given model.

## Step 2: Theory: Polygons

Many models are comprised of polyhedra made from regular polygons; these are probably the easiest ones to understand. This is because the inside angles of a regular polygon are, by definition, equal. Thus, these angles are easy to determine, using the formula

θ = (180(n - 2)) / n

when θ is one inside angle, and n is the number of side/angles.

Irregular polygons' angles can be calculated only if all sides, or a sufficient mixture of sides and angles, are known. Once the given polygon is decomposed into triangles, the individual angles can be calculated using a variety of trigonometric techniques. All the angles will still sum to 180(n-2), but this can only be used to discover, say, the one remaining angle if all others are known. To determine the length of paper needed to fold the units in compound models, the side lengths of the polygons making up each polyhedron must be known, and have some length added to them, as some is used up in the locking mechanism. The method for calculating this amount will be discussed later.

## Step 3: Folding: Ow Units

An Ow unit is mostly folded on the end of a strip of paper. Thus, the length of the unit (and, therefore, the length of the resultant side of the polygon) is independent of the size of the joining mechanism, which depends on the width of the strip (and, therefore, the "thickness" of the "wire", or edge, in the wireframe model).

Thus, I am only demonstrating on one end of a unit, as opposed to showing the whole thing:

• Fold unit in half so that the lengthwise open slit is on the inside. This will crease across the part of the pocket that sticks over the centre line

## Step 4: Folding: Assembly of Units

To lock two separate units together:

• Crimp the second unit along the centre fold, locking the units together.

The jump from angle to closed polygon to polyhedron is not difficult once the units and their attachment is understood. Keep in mind that, on a completed model, each vertex formed at the juncture of units should be surrounded by a complete ring of units, with no extra flaps or pockets (see image). Depending on the angles in the corners of the polygons that are meeting, there may be anywhere from two to perhaps five units meeting at a single vertex.

## Step 5: Theory: Generalization of Concept

How can these angles be varied to produce any polygon?

Analyzing how the internal 60° angle was produced allows the folding of any angle up to about 140° before the units stop locking together. The reverse fold made for the pocket used the reference points of the centre line and the left-most quarter line. These reference lines were used to fold a triangle with proportions of base 1 and hypotenuse 2.

θ = sine^-1(1/2) = 30°

This is shown in the first image.

Next, follow my blue thought-process arrows. The 30° angle can be geometrically carried over to the angle between the centre line and the edge of the flap. Since this is an angle in a right triangle, we can find the angle just clockwise from this because we know two of the three angles as well as the sum of the angles:

180° - 90° - 30° = 60°

This can be carried over to the angle between the centre line and the reverse fold, which forms the bottom of the pocket.

In the third picture, I have the two units we assemble flipped over from the previous step, such that the pocket is still on the right side of the unit. It can be seen that the right unit's edge lies along the reverse-fold in the pocket of the left unit. Thus, the red angle, which we already found, can be carried over to the angle between the units. Therefore, the units have an angle of 60°, or the inside angle of an equilateral triangle's corner, between them.

Similar reasoning, done backwards, can be used to find sufficiently accurate reference folds for other angles, as desired. Understanding that it is the pocket angle that determines the angle between the units is necessary for folding units which are the shared edge between two differently shaped polygons, as exist, for example, in the Archimedean solids. A bit of trigonometry, geometric manipulation, somewhere to write, and practice will allow you to easily produce a variety of polygons, regular and irregular, to make polyhedra from.

When making compounds of polyhedra, precise lengths of units are important to make the final model fit snugly. When calculating the dimensions of the rectangle of paper needed to produce a given length of edge, keep in mind the following artifact of the locking mechanism. Any paper past where the angled fold crosses the centre line (as shown in the last picture) will not contribute to the edge's length (this is intuitively obvious after you do some folding). Therefore, for precision edge lengths, it is necessary to find how much will be truncated on the units' ends, and to add this onto your original paper dimensions.

## Step 6: Going Further

This is only one type of modular origami, but, as you can see above, it is rather versatile.

If you have questions about any part of this art, please ask them in the comment section, and I'll try to help.

For references on geometric solids:

Platonic Solids

Archimedean Solids

Catalan Solids

Johnson Solids

For further ideas and inspiration, these people have fascinating resources that I've found very helpful:

Daniel Kwan (Flickr)

Byriah Loper (Flickr)

Aaron (Flickr)

Robert Lang

-and another page, a bit more theoretical

Philipp Legner - Mathigon

Participated in the Full Spectrum Laser Contest 2016

Participated in the Hand Tools Only Contest 2016

## Be the First to Share

Did you make this project? Share it with us!

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## Using Geometry to Celebrate National Origami Day

Lora mckillop.

• November 8, 2021

## What is National Origami Day?

National Origami Day is when we honor the Japanese art form of origami November 11th each year. Origami is the ancient Japanese culture of paper folding. It requires intricate folding and makes complex shapes out of square pieces of paper.

Paper was invented in China and taken to Japan by Buddhist monks during the sixth century. The Japanese developed origami into a very detailed art form. Most origami instructions were initially passed down in tradition orally instead of having handwritten instructions; those were later created so that the tradition could be preserved.

## How Does Geometry Relate to Origami?

Origami relates to geometry in many ways! The most basic way it can be related and taught to students is with two-dimensional and three-dimensional shapes. It can also be connected to lines, angles, the different types of triangles, symmetry, perimeter, area, and volume. You can teach and incorporate origami into geometry activities to make it fun and engaging.

## Fun Origami Activities to Try Using Geometry

Origami 3d “nets”.

An easy place to start with kids would be three-dimensional nets to fold them into the actual shapes that they need to learn about through these classroom activities. Then have students fill in a chart to tell how many faces, edges, and sides each three-dimensional shape has.

Students can fold a three-dimensional cube that makes a large die and then use it to play a math game ! Here is the template for the die.

Games that can be played with the dice are as simple as asking two students to work together, roll the cube, and add the numbers: play for five rounds, then add all the numbers together. Whoever has the highest number wins. You can do the same thing for subtraction and multiplication. If you are working on place value, you could have students roll up to six of them, make a number, and complete a place value chart.

## Animal Origami

You can have students make fun, animal origami shapes and then have them do mathematical activities with them, such as measure the angles that make up the animal, tell how many triangles it has, and tell what types of triangles it has. Here are three great tutorials for easy origami projects:

• Easy Origami Fish
• Easy Origami Butterfly
• Easy Origami Hummingbird

## Holiday Origami

Students can make cute Halloween and Thanksgiving origami and tell how many different shapes make up the figure:

• Bat  (this one is even a bookmark!)
• Fall Leaves

## Geometric Prisms

By using three-dimensional nets of cubes and rectangular prisms, students can learn more about the concept of volume. Once students make the shapes, have them fill them with  centimeter cubes .

Then teach a lesson on what volume is and have students count the cubes to determine the volume. This will serve as a foundation for their understanding when they learn the formula and have to find it without using manipulatives.

## Student How-To Videos

You could have students create their origami how-to video with accompanying directions and drawings. This might be easier for students with a partner, or group , to provide someone to bounce ideas off of and record each other. Be sure to give the students a rubric so they know what is expected from the assignment.

## Infographics or Comic Strips

Have students think of ways that origami is related to geometry or math in general and share their ideas creatively, such creating an infographic or comic strip. There are some great apps for creating these, such as Canva and Toontastic 3D .

## Find the Area

To have students practice finding areas, you can have them trace a three-dimensional net onto graph paper. Then have them use this to break the net down into manageable parts, calculate each part’s area, and add the parts together to figure the total area. At the end, they can then fold the net into a three-dimensional shape.

The most important thing is to use origami to have fun teaching whatever skills you want to incorporate with it. Students will love it and learn a lot at the same time!

• #ClassroomActivities , #NationalOrigamiDay

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## You are here

Origami in the geometry classroom, by ophir feldman, wentworth institute of technology.

Let’s face it, teaching an elementary college geometry course for design students can easily become a snooze fest. I was looking for a way to make the class a bit more engaging by appealing to these students’ creative/artistic side while showing them how to ’do mathematicsâ? with their bare hands. I always liked origami and I could see the potential for adopting it to geometry but I started to think: how can a flapping bird be used to demonstrate geometrical concepts? Sure, it’s fun to fold and create projects but it must have some meaningful point that relates to material in the course. In addition, I must keep the folding to some basic creases, nothing too fancy like an open double sink fold.

Well, I looked around and found a great resource to help me get started. Thomas Hull’s Project Origami answered my questions and provided ideas for wonderful classroom activities. For example, did you know that a few basic folds can subdivide a square piece of paper into exact thirds, fourths, fifths, n th s? And not only that, but you can easily prove this using only similar triangles! This is a proof they can see right on their piece of folded paper. Let’s face it, teaching an elementary college geometry course for design students can easily become a snooze fest. I was looking for a way to make the class a bit more engaging by appealing to these students’ creative/artistic side while showing them how to ’do mathematicsâ? with their bare hands. I always liked origami and I could see the potential for adopting it to geometry but I started to think: how can a flapping bird be used to demonstrate geometrical concepts? Sure, it’s fun to fold and create projects but it must have some meaningful point that relates to material in the course. In addition, I must keep the folding to some basic creases, nothing too fancy like an open double sink fold.

Here’s another easy one: With only a couple of moves you can trisect any acute angle on your square paper ’ a feat one cannot accomplish even with a straightedge and compass. This can be easily shown with some congruent triangles that students can trace after performing the folds (for a quick guide to trisecting an angle using origami click here ).

And yes, even the good old flapping bird has something to teach these students. But to me, the nicest activity involves constructing modular Bucky Balls using Tom’s PHiZZ units. The units are very easy to fold and putting them together to form, say, a dodecahedron (see picture below) is like doing a fun puzzle. Especially challenging is to try and put it together with only three colors such that no two colors touch. The students learn the real geometric restrictions that these structures must obey. It is useful to point out that Bucky Balls (also known as geodesic domes) are used in nanotechnology, architecture and design. Mathematically, students use Euler’s formula, do some counting of vertices, edges and faces and then solve a system of linear equations to arrive at the required relationships between the number of pentagon faces, hexagon faces, edges and vertices.

For the three-colorability, students learn about Hamilton circuits. This project definitely gets a lot of math bang-for-the-buck.

On exams, I feel comfortable asking students deeper questions relating to concepts that we covered using origami. I also dare to ask them to prove or justify their answers. The results are much better than when I ask these kinds of more abstract questions about concepts that were learned without the aid and motivation of origami.

The feedback from students is mostly positive. They never expected their ’boringâ? math class to have an origami twist. What is interesting is that some students have trouble following the most basic of folding instructions. I found this surprising since working with their hands and good visualization skills are supposed to be their forte. So be warned: some students do get frustrated, but with repeated explanations and one-on-one help from the instructor or fellow classmates, they get it.

So, go ahead give it a try. You really don’t have to be an origami master to bring origami to your math class.

1. Hull, Thomas, 2006, Project Origami , A.K. Peters.

Ophir Feldman ( [email protected] ) received his B.S. in Mathematics from Hofstra University in Hempstead, New York. He received his M.A. and Ph.D. from Brandeis University in Waltham, Massachusetts and is currently an Assistant Professor at Wentworth Institute of Technology in Boston, Massachusetts. His research area is geometric group theory and he has a strong interest in the mathematics and art of origami and its applications to the classroom.

The Innovative Teaching Exchange is edited by Bonnie Gold .

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## Origami and Geometry Lesson Plan

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Help students practice origami to develop a knowledge of geometric properties. This craft project is meant for second-graders for the duration of one class period, 45 to 60 minutes.

## Key Vocabulary

• origami paper or wrapping paper, cut into 8-inch squares
• a class set of 8.5-by-11-inch paper

Use origami to develop an understanding of geometric properties.

## Standards Met

2.G.1 . Recognize and draw shapes having specified attributes , such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

## Lesson Introduction

Show students how to make a paper airplane using their squares of paper. Give them a few minutes to fly these around the classroom (or better yet, a multipurpose room or outside) and get the sillies out.

## Step-By-Step Procedure

• Once the airplanes are gone (or confiscated), tell students that math and art are combined in the traditional Japanese art of origami. Paper folding has been around for hundreds of years, and there is much geometry to be found in this beautiful art.
• Read The Paper Crane to them before starting the lesson. If this book can't be found in your school or local library, find another picture book that features origami. The goal here is to give students a visual image of origami so that they know what they'll be creating in the lesson.
• Visit ​a website, or use the book you selected for the class to find an easy origami design. You can project these steps for students, or just refer to the instructions as you go, but this boat is a very easy first step.
• Rather than square paper, which you usually need for origami designs, the boat referenced above begins with rectangles. Pass one sheet of paper out to each student.
• As students begin to fold, using this method for the origami boat, stop them at each step to talk about the geometry involved. First of all, they are starting with a rectangle. Then they are folding their rectangle in half. Have them open it up so that they can see the line of symmetry, then fold it again.
• When they reach the step where they are folding down the two triangles, tell them that those triangles are congruent, which means they are the same size and shape.
• When they are bringing the sides of the hat together to make a square, review this with students. It is fascinating to see shapes change with a little folding here and there, and they have just changed a hat shape into a square. You can also highlight the line of symmetry down the center of the square.
• Create another figure with your students. If they have reached the point where you think they can make their own, you can allow them to choose from a variety of designs.

## Homework/Assessment

Since this lesson is designed for a review or introduction to some geometry concepts, no homework is required. For fun, you can send the instructions for another shape home with a student and see if they can complete an origami figure with their families.

This lesson should be part of a larger unit on geometry, and other discussions lend themselves to better assessments of geometry knowledge. However, in a future lesson, students may be able to teach an origami shape to a small group of theirs, and you can observe and record the geometry language that they are using to teach the “lesson.”

• A Sample Lesson Plan for Teaching Geometry Using 'The Greedy Triangle'
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## Final Project

The finished star, a cool trick, refining the challenge.

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## Origami Geometry

Use congruency, trigonometry, and proofs to calculate the surface area of your origami creations!

For more ideas, see: Geometric Exercises in Paper Folding, T. Sundara Row Project Origami, Thomas Hull

Authored by Trang Ngo & Will Luna, Tufts University CEEO 2016

Video of solution for calculating the surface area on an origami fold.

Teachers have been using Origami to explain geometric concepts for over a hundred years. However, it is usually used to teach concepts that are too simple (basic spatial reasoning) or too complex (advanced mathematical proofs) for a high school classroom.

The assignment is simple: students will make one of their favorite origami modules, and then they have to calculate the surface area of the 2D module made.

Our task was to successfully fold an origami figure while using geometry to calculate the surface area of the finished product.

We worked in teams of two, but it is possible to have students work individually. This assignment only requires pencil and paper.

Our first task was to choose a figure that we felt was cool yet simple enough to calculate the surface area of mathematically.

We went with an 8-pointed star. Composed of eight separate folded parts, it would be difficult to build, but we recognized that this actually made the calculations easier. As soon as we found the surface area of a single portion (only two distinct geometric shapes), we could simply multiply by eight to find the total surface area.

If the teachers want to choose an origami module to assign to students, it would be ideal to select a module in which many concepts of geometry can be practiced.

In our example project, we choose a module in which several units have to be assembled together. This is because then, the students would be able to practice geometric concepts and to study the geometric relations between different units.

We made an elephant and a 8-pointed star. These two examples can create great projects because they make uses of various geometric concepts.

In this part, we will focus on the process of calculating the surface area of this 8-pointed star. In class, the students are free to choose whichever origami module they want to make, and the surface area can still be easily calculated through the following steps.

(these steps are explained in the video)

• Look up a tutorial on the Internet and make the parts. Assemble some parts together.
• Look at the overall shape of the origami module made. Shade the exposed parts, which make up the surface area of the main module.
• Unfold the folded unit back into the original piece of paper.
• Using the given dimension of this piece of paper (in the video the paper is 9 inches x 9 inches), determine the dimensions of the shaded regions. The dimensions can be calculated using various geometric concepts such as congruency, similar triangles, trigonometric identities , and Pythagorean theorem . In cases where the chosen origami module consists of an angle other than 90 or 180 degrees, students can also learn the Law of Sines and Cosines while calculating different lengths of the shaded regions. Students can also practice calculating the area of many different geometric shapes in this origami project.
• This 8-pointed star is created from 8 units. Therefore, the calculated area is multiplied eightfold for the final area of the origami module.

Without folding the shape first, it is extremely difficult to determine which parts of the shape are a part of the surface area.

If you (carefully) pull on the points of the star, it transforms into this donut-shaped figure!

If you think this assignment will be too difficult for your students, limit their options to simpler shapes instead of letting them choose from anything online. This will prevent them from picking cool but overly ambitious projects.

The origami project can also be used to teach 3D geometry using available 3D origami modules. Using 3D modules would allow students to practice calculating volume of different 3D shapes.

If some students finish early, have them trade shapes with another student, and see if they come up with the same answer.

We initially wanted students to come up with sets of geometric rules that their classmates could use to fold certain origami shapes. On paper, it seemed great - make students use mathematical language to convey a highly visual task (paper-folding). However, it simply proved too difficult. Unless the rules were incredibly complicated, it was possible to fold multiple correct solutions.

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## Accessible hotspots for single-protein SERS in DNA-origami assembled gold nanorod dimers with tip-to-tip alignment

• Francis Schuknecht   ORCID: orcid.org/0000-0002-2120-9407 1   na1 ,
• Karol Kołątaj 2   na1   nAff3 ,
• Michael Steinberger 1 ,
• Tim Liedl   ORCID: orcid.org/0000-0002-0040-0173 2 &
• Theobald Lohmueller   ORCID: orcid.org/0000-0003-2699-7067 1

Nature Communications volume  14 , Article number:  7192 ( 2023 ) Cite this article

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• Nanoparticles
• Organizing materials with DNA
• Raman spectroscopy

The label-free identification of individual proteins from liquid samples by surface-enhanced Raman scattering (SERS) spectroscopy is a highly desirable goal in biomedical diagnostics. However, the small Raman scattering cross-section of most (bio-)molecules requires a means to strongly amplify their Raman signal for successful measurement, especially for single molecules. This amplification can be achieved in a plasmonic hotspot that forms between two adjacent gold nanospheres. However, the small (≈1−2 nm) gaps typically required for single-molecule measurements are not accessible for most proteins. A useful strategy would thus involve dimer structures with gaps large enough to accommodate single proteins, whilst providing sufficient field enhancement for single-molecule SERS. Here, we report on using a DNA origami scaffold for tip-to-tip alignment of gold nanorods with an average gap size of 8 nm. The gaps are accessible to streptavidin and thrombin, which are captured at the plasmonic hotspot by specific anchoring sites on the origami template. The field enhancement achieved for the nanorod dimers is sufficient for single-protein SERS spectroscopy with sub-second integration times. This design for SERS probes composed of DNA origami with accessible hotspots promotes future use for single-molecule biodiagnostics in the near-infrared range.

## Introduction

The label-free detection of single proteins or other small biomolecules from liquid samples is of great significance in biomedical diagnostics and pharmacology. Several experimental approaches have been developed towards this goal, such as nanopore conductance measurements 1 , interferometric detection 2 and localized surface plasmon based sensing 3 . However, most methods do not provide detailed chemical information about the analyte, making an unambiguous identification challenging. Raman and infrared (IR) spectra on the other hand, provide a unique chemical fingerprint of the measured sample. While this seems ideal for label-free biomolecule detection, both methods also display limitations. IR spectroscopy is not compatible with measurements in aqueous solution due to the high absorption of water molecules in this wavelength range. Raman spectroscopy, on the other hand, is limited by the weak Raman scattering cross-sections of most molecules, which are typically between ≈10 −27 to 10 −30 cm 2   4 . Additionally, large background noise and autofluorescence are often observed for biological samples, which renders single-molecule detection particularly challenging. Therefore, an enhancement of the Raman scattering intensity on the order of 10 7 to 10 10 is required for single-molecule (SM) Raman measurements 5 , 6 .

Surface-enhanced Raman scattering (SERS) exploits the product of the squares of the incident electromagnetic (EM) field enhancement and the polarizability enhancement at emission 7 . This is achieved by exposing molecules to EM hotspots generated by rough metal surfaces or plasmonic nanoantennas 8 . For example, tip-enhanced Raman spectroscopy has been applied to identify single proteins 9 . This technique exploits the high EM field enhancement at a sharp tip of a plasmonic probe to boost the Raman scattering intensity of analytes adsorbed on a substrate. Measuring spectra then requires scanning the sample with the probe. On a single-particle level, gold or silver nanostars feature sharp spikes, which provide a strong field enhancement sufficient for single-molecule detection, albeit with limited enhancement volume 10 . However, positioning an analyte precisely in the tip region can be challenging and sharp tips can display a limited stability 11 . A strong and highly confined EM field enhancement, by over two orders of magnitude, is further obtained in so-called plasmonic “hotspots” that occur due to plasmonic coupling between two nanoparticles forming a plasmonic dimer nanoantenna. In recent years, many examples of plasmonic dimers have been demonstrated as excellent probes for SM-SERS on dried samples 12 .

Optimizing the performance and applicability of plasmonic dimer nanoantennas towards SM-SERS involves several factors. Most importantly, the EM-field enhancement strength acts inversely to the interparticle distance. This limits the hotspot sizes to a few nm and requires accurate particle positioning. Furthermore, the analyte must be located precisely in the nanoparticle gap to benefit from the highest field enhancement. This second point is not an easy feat, particularly if one wants to add the analyte subsequently to preassembled dimers.

A highly successful approach for addressing both particle alignment and analyte positioning, is DNA self-assembly - the nanoscale folding of DNA into complex three-dimensional geometries 13 , 14 . The DNA origami method has been used to synthesize plasmonic dimer nanoantennas with controlled interparticle distances as highly reproducible and reliable SERS probes 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 . In particular, for example, SM-SERS was achieved with “nanofork” 21 and “funnel” 18 DNA origami designs, where the origami template hosting the analyte also spanned the interparticle gap. As a result, the plasmonic hotspots of these dimers were not freely accessible from the outside. Researchers have therefore devised DNA origami templates that displayed “open gaps” 15 , 17 . However, gaps in the range of ≈1–2 nm are typically required with nanosphere dimers to obtain sufficient signal enhancement for SM-SERS 12 . Such small hotspots cannot accommodate most proteins, which are a few nm in size 23 .

Dimer gaps large enough to accommodate single proteins while displaying a sufficient field enhancement for single-molecule detection have been obtained by switching from nanospheres to other particle shapes such as nanostars 24 , 25 , 26 , bipyramids 27 , and bowtie antennas 19 . For example, single-protein SERS was demonstrated by Tanwar et al., where thrombin was bound to a DNA template and then sandwiched between two bimetallic nanostars 28 . Heck et al. 22 demonstrated SM-SERS on biotin/streptavidin with self-assembled “nanolenses” made of silver particles. Further, Zhan et al. reported non-resonant SERS of Cy5 in 5 nm wide gaps formed by DNA origami-assembled gold nanotriangles 19 .

However, most DNA origami-assembled dimer structures for SERS measurements are synthesized by following a similar protocol: A single dye or biomolecule is first attached to a DNA template, followed by the plasmonic nanoparticle assembly around the analyte. The resulting dimers are then typically purified by gel electrophoresis before conducting a SERS measurement. This synthetic procedure carries the advantage that most dimers are indeed labelled with the analyte, which is also positioned exactly in the plasmonic hotspot between the particles.

Arguably, a strategy of building the plasmonic dimer around the analyte is of limited use if one aims at identifying individual biomolecules such as proteins from a liquid sample. A reverse scheme based on capturing freely diffusing proteins from solution with a specific binding antagonist would be more desirable and applicable, as also pointed out by Tanwar et al. 28 . However, this requires accessible - i.e. large and open - hotspots for single proteins to enter via diffusion, while providing sufficient EM field enhancement over the protein volume at the same time. The design of plasmonic dimer nanoantennas with addressable binding sites thus requires finding a right balance between hotspot volume and Raman enhancement.

Here, we report on a DNA-origami design that enables the tip-to-tip alignment of gold nanorods (GNRs) with accessible interparticle gaps. GNRs display a larger plasmonic field enhancement for their longitudinal mode compared to gold nanospheres, due to an increase in tip curvature to volume ratio, and reduced surface plasmon damping 29 . The DNA origami-assembled gold nanorod dimers feature plasmonic hotspots, which are ≈8 nm wide and thus accessible for proteins of a smaller size from solution. This accessibility is demonstrated by SM-SERS detection of streptavidin and thrombin, where the analyte molecules enter the nanoantenna gaps during the SERS measurement procedure. We support the use of DNA origami-assembled nanorod dimers, as highly effective SERS probes, by comparing their calculated enhancement factors to those of well-established gold nanosphere dimers and other particle geometries.

## DNA origami design and assembly of GNR dimer nanoantennas

The assembly of GNR dimer nanoantennas by DNA origami is illustrated in Fig.  1a . The DNA template was designed as a ≈215 nm long scaffold beam made of 14 DNA helices arranged in a honeycomb cross-section (Supplementary Fig.  1 ). The geometry of the origami was confirmed by transmission electron microscopy (TEM, Supplementary Fig.  2 ).

a Gold nanorods are functionalised with different sequences of thiolated ss (single-strand) DNA (either T or R). Due to the differing labels, an individual nanorod can only bind to a designated binding site on the DNA origami support beam (either A 8 or R’ 8 ) and nanorod dimers are formed. A docking site for the SERS analyte is precisely located between the nanorod tips, which form the plasmonic hotspot. b TEM image of GNR dimers assembled on the DNA origami support beam after sample purification. c Schematic of the SERS dark-field microscopy setup. Individual dimer nanoantennas are localized on a glass substrate by DFM. A 671 nm laser is coupled through the objective to perform SERS measurements on single nanoantennas, with analyte diffusing and binding into their hotspot gaps. A longpass (LP) filter is used to block the laser from entering the spectrometer.

Each DNA origami beam exhibits two binding sites for attaching the GNRs. Each binding site is composed of a group of 14 DNA capture strands, which are spaced ~4 nm apart from one another. Both capture strands feature orthogonal sequences to prevent single nanorods from bridging the binding sites when binding to the template: an 8 nt long poly-A (A 8 ), and a random (R’ 8 ) (ATGTAGGT) sequence.

The GNRs were synthesized with an aspect ratio (length:width) of ≈3, according to previously published protocols 30 . The average length of the nanorods was ≈63 nm (Supplementary Fig.  3 ), which provides a longitudinal plasmon mode within the bio-optical window for tissue, at around 690 nm (Supplementary Fig.  4 ). The synthesized nanorods were divided in two batches and functionalized with two different types of thiolated, single-stranded DNA (ssDNA), each complementary to one of the two binding sites on the DNA origami support beam. A small red-shift of the bulk GNR extinction spectrum by ≈2 nm was observed after ssDNA coating, indicative of a small increase of the effective external permittivity (Supplementary Fig.  4 ).

The GNRs were mixed with the DNA origami template in solution to form dimers. Prior to use, gel electrophoresis was performed to separate nanorod clusters and single nanorods from the target structures (Supplementary Fig.  5 ). The final GNRs dimers were analysed by TEM (Fig.  1b ). From two synthesized nanorod batches, a nanogap size of 8.2 ± 2.6 nm and 8.6 ± 2.4 nm (tip-to-tip) was determined (Supplementary Fig.  6 ), with a nanorod dimer yield (with tip-to-tip orientation) of ≈55%. Subsequent SERS measurements on single dimers were conducted under a dark-field microscope (DFM, Fig.  1c ).

## SM-SERS of Cy3.5 in water

To determine the performance of the GNR dimer-origami structures for SM-SERS, samples hosting a single Cy3.5 molecule in the plasmonic hotspot between the nanorod tips were prepared (Fig.  2a ). The measurement was conducted by drop-casting a solution of the purified nanoantennas onto a clean glass substrate. After a short incubation time (5-10 min), individual nanoantennas were settled on the glass surface and visible as bright, red spots in the DFM via their scattered light (Fig.  2b ). Following this procedure, an average density of ca. 1000-1500 dimers per mm² was obtained (Supplementary Fig.  7 ). Furthermore, scattering spectra of individual structures were acquired to analyse the longitudinal plasmon resonance peak with respect to the Raman laser wavelength and to select suitable dimer nanoantennas for following SERS measurements (Fig.  2c ). In principle, one can estimate the dimer gap size by reproducing the measured single dimer scattering spectrum with numerical finite difference time domain (FDTD) simulations (Supplementary Fig.  8 ). However, small deviations of the nanorod geometry or the dimer alignment can influence the result, which renders an unambiguous characterization without additional scanning electron microscopy (SEM) measurements challenging.

a Sketch of the tip-to-tip alignment of GNRs with Cy3.5 at the nanoantenna gap. b Dark-field microscopy image of individual dimers on a glass substrate. Nanorod dimers are visible as red spots. c Scattering spectra of the GNR dimer circled in ( b ) before (black) and after (red) the SERS measurement. d Non-background subtracted SERS heatmap (intensity in counts: cts, integration time of 0.5 s) obtained from the dimer nanoantenna circled in b. e Single Cy3.5 Raman spectra (integration time of 0.5 s) from the measurement in d (at different points in time). Distinct Cy3.5 Raman peaks are marked with purple dashed lines (peak assignment provided in Table  1 ). The bulk spectrum was acquired from a 10 µM solution of Cy3.5 that was dried on a gold film that was sputtered onto a glass substrate. The scale bar marked by (*) of the bulk spectra corresponds to 4000 cts. Source data are provided as a Source Data file.

After having identified nanoantennas with DFM, SERS measurements were carried out in water with a focused 671 nm continuous wave (cw) laser at a power of 10 mW. The measurements were conducted using circularly polarized laser light to account for the random orientation of the nanorod dimers on the substrate. As shown in the heatmap displayed in Fig.  2d , the time-dependent SERS signal of the Cy3.5 displayed fluctuations, which is a common feature observed in SM measurements 31 , 32 . Two exemplary SERS spectra from the heatmap (obtained with integration times of 0.5 s) are shown in Fig.  2e . Characteristic Cy3.5 vibrational modes at ≈1270-1280 cm −1 (motions of aromatic groups) 33 , 34 , ≈1350 cm −1 (central methine chain motion) 33 , 34 , 35 , ≈1463 cm −1 (asymmetric CH 3 deformation) 33 , 34 , 35 , 1560 and 1590 cm –1 (N +  = C stretching motion) 33 , 34 , 36 , and ≈1610-1620 cm −1 (C=C stretching mode) 34 , 36 were identifiable (see also Table  1 ). The Cy3.5 SERS spectra also displayed good agreement with a reference spectrum of Cy3.5 obtained from a gold-coated glass substrate.

After the SERS measurement, a second dark-field scattering spectrum was taken to confirm that the nanorod dimer remained stable during the process. The post-SERS scattering spectrum looked almost identical in shape, although a small red-shift of the longitudinal plasmon peak by ≈8 nm was observed in this case (Fig.  2c ). One potential source is attractive forces between the nanorods due to plasmon coupling induced by the focussed laser beam, which could pull the nanorods closer together, or a small contraction of the DNA origami template during the measurement. This in turn would increase the EM field enhancement, and could explain why Raman signals appeared only a few seconds after the start of the measurement. Strong plasmonic heating, which could destroy the dimer nanoantenna by particle melting or degradation of the DNA origami was avoided by performing the measurements in water, which has a significantly higher thermal conductivity than air. This argument is supported by the fact that dark-field scattering pre- and post-SERS spectra did not display a strong change. Additionally, a temperature increase would have led to dissociation of the analyte molecule connected to the DNA origami template by a single anchor strand 37 . Using lower laser powers for acquiring spectra is generally desirable, even in buffer, to minimize any temperature effects on the DNA origami, but can necessitate longer integration times. We therefore conducted protein measurements with different laser powers (0.5-5 mW) to balance both effects on the SERS signal.

## Single-protein SERS of streptavidin and thrombin

The applicability of the DNA origami GNR dimers for detecting proteins was determined by SM-SERS measurements of streptavidin (≈60 kg mol −1 ) and thrombin (≈36 kg mol −1 ). These proteins were chosen, as their molecular weight is close to the mean weight of proteins in eukaryotic cells (49 ± 48 kg mol −1 ) 38 . The tetrameric protein streptavidin has a diameter of ≈5 nm 39 , whilst thrombin has a hydrodynamic diameter of 4.1 nm 40 . The gaps of dimer nanoantennas were therefore large enough to accommodate a single protein.

To capture streptavidin from solution, the dimer hotspots were functionalized with a single-stranded biotin-labelled DNA anchor. The ability to define specific anchoring sites for single molecules on an origami scaffold is a major strength of DNA origami technology and has been demonstrated previously for many DNA origami designs to conduct single-molecule SERS 18 , 20 , 22 , 28 and single-molecule localization microscopy 41 . In our case, the presence of a single anchoring site at the centre of the nanorod dimer, along with limited accessible space in the nanogaps ensured single-protein measurements.

The nanoantennas were drop-cast on a glass cover slip, and individual dimers were identified by DFM. The complete measurement comprised two steps. Firstly, SERS measurements of biotin-functionalized dimers were conducted in TE buffer with a laser power of 5 mW to confirm the presence of biotin in the hotspots and to determine any background Raman signal stemming from the DNA origami (Fig.  3a ). The obtained spectra were dominated by a central double peak at ≈1375 cm −1 , which most likely stems from nucleobases of the docking strand or origami template (ring breathing modes of T, A, and G bases) 42 . However, weaker Raman peaks indicative of biotin were also visible, at 1270 cm −1 (methylene group wagging) 43 , 44 , 1470 cm −1 (stretching of CH 2 ) 43 , 44 , and 1565 cm −1 (C-N stretch) 43 , 45 .

a DFM image of the corresponding dimer nanoantenna and sketch illustrating the nanogap filled with one biotin molecule, with example SERS spectra of only biotin (intensity in counts: cts). b Sketch illustrating the process of streptavidin entering the hotspot and binding to biotin, with SERS spectra obtained after adding streptavidin. The first spectrum in the stack was obtained shortly before streptavidin was captured by the antenna. Additional peaks corresponding to streptavidin are observed after the protein entered the nanogap (three bottom spectra). Raman peaks indicative of biotin and streptavidin are highlighted with green and blue bars respectively (see Table  1 for peak assignment). c DFM image of the corresponding dimer nanoantenna and sketch illustrating the nanogap filled with one aptamer, with SERS spectrum obtained from a dimer nanoantenna with HD22 before thrombin attachment. d Sketch illustrating the thrombin capturing process, with combined HD22/thrombin SERS spectra. All SERS spectra were obtained with 0.5 s integration times. Raman peaks indicative of HD22 or thrombin, are highlighted with grey and red bars respectively (see Table  1 for peak assignment). Source data are provided as a Source Data file.

Next, streptavidin in TE buffer was added to the sample. After this, SERS spectra changed and additional peaks indicative for streptavidin binding in the nanoantenna gap appeared (Fig.  3b ). The additional spectra displayed characteristic streptavidin Raman modes at 1239 cm −1 (amide III/β-sheet) 43 , 44 , 46 , 47 , 1336 cm −1 (tryptophan W7) 43 , 44 , 46 , 1560-1580 cm −1 (tryptophan W2) 43 , 46 as well as 1670 cm −1 (amide I/ β-sheet) 43 , 46 , 47 (Fig.  3b ), which also appeared in bulk measurements of streptavidin (further examples of single dimer and bulk spectra are shown in Supplementary Fig.  9 ). Furthermore, a strong peak at ≈1503 cm −1 was featured in the spectrum, which is not specific to streptavidin and assignable to aromatic ring vibrations or N  = H stretching 44 , 47 .

To demonstrate the binding specificity of streptavidin capturing by the dimer antennas, control measurements were conducted where the DNA origami dimers were incubated with myoglobin instead of streptavidin. Myoglobin was chosen to confirm specificity, as it is smaller than streptavidin (17 kg mol −1 ; hydrodynamic radius ≈ 1.75 nm) 48 and does not bind to the biotin anchor. Even after prolonged measurements, we were not able to obtain a myoglobin SERS spectrum. In comparison, measurements where myoglobin and streptavidin were added at the same time, yielded a SERS spectrum matching the bulk Raman spectrum of streptavidin (Supplementary Fig.  10 ).

To further test the viability of our approach, we conducted a second experiment with thrombin, a globularly shaped enzyme involved in blood coagulation, which is slightly smaller than streptavidin 49 . As a binding antagonist, the anti-thrombin aptamer HD22 was used to label the anchor point of the DNA template. HD22 consists of 29 nucleotides that form a duplex/G-quadruplex mixed structure. From the SERS spectrum, the presence of the HD22 cannot be unambiguously confirmed, since any Raman peaks could also originate from the DNA origami itself. SERS measurements of the HD22 binding aptamer commenced in PBS buffer. Typical Raman modes for DNA were observed at 1160-1170 cm −1 (G 42 , 47 ), 1375 cm −1 (T, A, and G ring breathing modes 42 , 47 ), and 1574 cm −1 (G, A 42 , 47 ). Additionally, a peak at 1230 cm −1 for antisymmetric phosphate stretching 47 is assignable to origami, linker DNA or/and the HD22 aptamer (Fig.  3c ).

Next, thrombin was added. The following measurement on single dimers was conducted with a laser power of 2 mW. Again, thrombin binding in hotspots was observable by SERS. Characteristic protein peaks for thrombin at 1230-1250 cm −1 (amide III/β-sheet 47 , 50 ) as well as at 1360 cm −1 (tryptophan 47 , 50 ), and ≈1550-1560 cm −1 (amide II range or tryptophan 47 , 50 ) appeared in the SERS spectrum obtained from individual nanorod dimers. Notably, a characteristic amide I peak between 1639-1670 cm −1 47 , 50 , 51 was not reliably observed for all spectra. This corresponds to findings, where the amide I vibrational mode can be suppressed in protein SERS and TERS measurements, by Kurouski et al. 52 (Fig.  3d , additional dimer spectra and the thrombin bulk spectrum are shown in the Supplementary Fig.  9 ).

## Theoretical Raman enhancement of GNR dimers

Based on the obtained SERS results, the question arises what field enhancement can be expected in the plasmonic hotspots of the nanorod dimers and whether single-molecule detection is feasible for this nanoantenna geometry. We performed FDTD calculations for the GNR dimer nanoantennas in water to determine the EM-field enhancement for different gap sizes, and to benchmark the performance of gold nanorod against nanosphere dimers. For these calculations, spherically end-capped nanorods with a width of 21 nm and a length of 64 nm were compared to gold nanospheres with diameters of 40, 60 and 80 nm. As shown in Fig.  4a , the maximum field enhancement for nanorod dimers exceeds the field enhancement of two nanospheres in the centre of the nanoantenna gap for all interparticle distances between 2 and 10 nm. Even for a 6-7 nm gap, the central | E | 4 enhancement for nanorod dimers exceeds 10 8 . As an additional comparison, the field enhancement obtained in the centre of 5 nm gaps between nanorods is similar to that of nanospheres with a gap length of 3 nm. For the latter, non-resonant SM-SERS detection was reported in literature 21 . Further illustrating this point, the hotspot volume in which E / E 0 (or enhancement factor EF) exceeds 100 (corresponding to an | E | 4 enhancement of 10 8 ) is on the order of 100 nm³ for nanorod dimers with a 5 nm large gap (approximately the size of a streptavidin or thrombin molecule), whilst similarly arranged 60 nm spheres feature an E / E 0 of ≈ 50 (Fig.  4b, c ). The latter is approximately the same maximum field enhancement one would already obtain at the tip of a single gold rod. We performed control measurements for streptavidin attached to biotin with a DNA origami structure hosting a single rod instead of a dimer (Supplementary Fig.  11 ). In this case, a protein SERS spectrum could not be obtained. This finding is supported by simulations, which show that the EF 4 is ≈3 orders of magnitudes lower at monomer tips compared to dimer gaps.

a Comparison of the calculated maximum E -field enhancement (at corresponding resonance wavelengths) at the nanogap centre for different gold nanosphere- and gold nanorod dimers. b Calculated maximum E -field enhancement distribution between the tips of two nanorods (excited at 808 nm), assuming a tip-to-tip distance of 5 nm. c Calculated maximum E -field enhancement distribution between two 60 nm gold spheres separated by 5 nm (excited at 607 nm). All calculations were performed for nanostructures in water on a glass substrate at their hotspot centre resonance.

Compared to gold spheres, nanorods are also advantageous for geometrical reasons. For the rods, even a short anchor strand that protrudes from the DNA origami template into the nanogap, was sufficient to localize the analyte molecule in the area of highest field enhancement. For 60 nm nanospheres, assembled in the same configuration, such a linker strand would have to be around as long as the sphere radius (30 nm). With a typical ≈50 nm persistence length of double stranded DNA 53 , the exact positioning of analyte in the nanoantenna hotspot 54 would thus be less probable between spheres.

The SM-SERS detection of freely diffusing streptavidin (4.2 nm × 4.2 nm × 5.6 nm) 39 and thrombin (4.5 nm × 4.5 nm × 5.0 nm) 49 requires a minimum gap size of ≈5 nm to allow a single protein to enter the hotspot. Increasing the gap size is generally detrimental to the SERS enhancement, which makes the results presented here particularly compelling. In theory, the GNR dimer system exceeds a reported minimum EF 4 requirements for SM-SERS of 10 7 , even for ≈8 nm wide gaps at their centre 6 . Also, there are other factors, including the general nature of proteins, which are beneficial for SERS measurements. For one, the proteins and their molecular subgroups examined here are relatively large. Bigger molecules tend to be more polarizable, and thus display larger Raman cross-sections 8 . Furthermore, the protein subgroups often feature repeatedly in the protein’s secondary structure, such as the amide β-sheet in streptavidin. The observed peaks in the SERS spectra are therefore not limited by individual molecular vibrations. Instead, the observed peaks are a superposition of the signal from all functional groups in the molecule that display the same Raman active modes. This argument is further supported by the fact that the hotspot volume between the nanorod tips is similar to the size of a whole protein (Fig.  4b ).

As stated previously, spectra were obtained with reduced laser powers to avoid sample degradation. Calculations with the finite element method (FEM) for nanorods separated by an 8 nm gap indicate that dimer temperatures in water do not rise above ≈49 °C for a laser power of 5 mW (Supplementary Fig.  12 ). At these temperatures, the onset of protein denaturation cannot be entirely excluded, which along with a small degree of protein movement in the nanogap, can explain the observed SERS fluctuations. Raman spectral fluctuations could, however, also be indicative for carbonization of the analyte, particularly in SM-SERS 55 , due to heating (up to several 100 °C), or photochemical analyte transformations via hot-electrons 56 .

Carbonization is characterized by the emergence of broad carbon D-band at ≈1350 cm −1 and a stronger G-band at ≈1580 cm −1 that dominate the time-averaged spectrum. We did not observe carbon formation in our single-protein measurements (Supplementary Fig.  13 ). For one, Heck et al. have shown that gold is less prone to induce carbon formation compared to silver, possibly due to a favourable surface chemistry 57 . Additionally, Bjerneld et al. reported that carbonization is suppressed when SERS measurements are conducted in water 58 .

An additional important point to be considered is the probability for a single protein to actually diffuse into a nanogap. The mean diffusion time for single proteins (Supplementary Fig.  14 ) was approximated with a 3D random walk model. In general, the mean diffusion time is concentration dependent. For physiologically relevant concentrations of thrombin 59 the estimated time for a protein to enter an available nanogap ranges from ≈2 h to ≈10 s respectively. For our measurements, a concentration range between 330 nM–4.2 µM (streptavidin) and 6.9–61 µM (thrombin) was investigated. Individual SERS spectra were acquired from dimer nanoantennas after incubation times of at least a few min, which excludes the mean diffusion time as a limiting factor. In fact, we did not observe any clear influence of analyte concentration in our experiments. For the single-dimer measurements reported here, other factors, such as the individual dimer geometry and the general orientation of the nanoantennas on the substrate play a more important role.

Averaged over all of our measurements, approximately ≈10-15% of DNA origami dimers delivered SERS. At first glance, this yield appears low. However, only dimers within a certain gap size range can be expected to deliver a single-protein spectrum, when they are both large enough to fit a single protein and small enough to provide sufficient field enhancement. Additionally, not all nanorod dimers on the substrate were aligned perfectly. An average alignment angle of 167° was determined by TEM measurements (Supplementary Fig.  6 ). The nanorod alignment itself does not alter the field enhancement in the nanoparticle gap significantly up to an angle of 150°, as shown by FDTD calculations (Supplementary Fig.  15 ). It could, however, lead to misalignment of the linker strand, and thus lower the accessibility of the binding site in the hotspot. Potential strategies to improve the SERS yield could include a further stabilization of the origami template, which could be achieved by DNA silanization 60 .

To verify that the gap size is indeed critical, we performed measurements with biotin-labelled dimer nanorod antennas on biotin-binding immunoglobulin G (IgG, 0.13 µM). IgG is larger than streptavidin and thrombin (≈150 kg mol −1 ; 14.5 nm × 8.5 nm × 4 nm 61 ; hydrodynamic diameter ≈10.6 nm 62 ) and might not fit those nanogaps small enough to deliver sufficient SERS. Indeed, we did not obtain any protein SERS spectrum of IgG, even after extended incubation times beyond 100 min. This also confirms that unspecific protein binding to the dimer antennas does not yield a SERS signal.

However, the IgG control measurement raises the question of how the antenna design could be optimized to obtain reliable SM-SERS of even larger molecules. In principle, beyond optimising the existing DNA-origami scaffold, two strategies can be envisioned: (i) one could use nanorods with sharper tips that provide a larger field enhancement, or, (ii) additional nanorods could be aligned in a trimer or tetramer structure to obtain larger openings for molecules to enter. We have performed a theoretical study comparing these scenarios (Supplementary Fig.  16 ). For sharper tips (at the gap), one finds that the EF is indeed higher close to the particle surface (similar to the tips of gold nanostars or -triangles) but decreases around the nanogap centre. Furthermore, sharp tips are generally more prone to melting and deformation upon laser excitation, even at low intensities. Blunter tips, on the other hand, provide a larger volume of more homogenous field enhancement, which is beneficial for proteins occupying almost the entire hotspot between the nanorods. As mentioned earlier, SERS of proteins benefits from a superposition of the SERS signal obtained from all Raman active vibrations of the molecule. Therefore, nanorods with spherical ends appear well suited to obtain a signal of the whole protein.

Adding more nanorods to form trimer and tetramer structure (Supplementary Fig.  16 ) can provide larger hotspots for analytes bound in the centre of the structure, and an additional axis of SERS enhancement (as dimers only enhance Raman along one spatial direction). However, for circularly polarised light, the strongest plasmonic coupling in this case is obtained between neighbouring nanorods. Central gaps formed by the trimer and tetramer antennas do - even when accounting for twice the field enhancement axes - not provide a stronger maximum Raman enhancement than similar dimers. However, for future studies of larger proteins (such as IgG), with more molecular subgroups, such designs might present a viable strategy.

Finally, practical factors in favour of nanorod dimers are considered. For one, the nanoantennas display plasmon resonances at wavelengths above 670 nm. This is an advantage, as the autofluorescence background of biological samples lies at lower wavelengths. Additionally, for SERS, the plasmonic nanoantennas operate fully within the near-infrared (NIR) or bio-optical window of tissue, where light has a high penetration depth. Since the nanoantennas are assembled and stabilized by the DNA origami in solution, they are freely deployable and could be injected as SERS probes into tissue and potentially operate in living cells. Combined with their addressability for specific biologically relevant proteins, future GNR dimer based in situ or in vivo measurements appear tangible. To realize such measurements, the nanosensors could be further protected by a layer of silica to preserve their integrity in cells. Importantly, site-specific silanization of DNA origami has already been demonstrated 63 , 64 , showing that it is indeed possible to selectively protect nanoantennas without hindering the binding of analyte molecules to be detected.

Ascorbic acid (99%), gold (III) chloride trihydrate (HAuCl 4 , 99%), hexadecyltrimethylammonium bromide (CTAB, ≥99%), magnesium chloride (MgCl 2 , ≥98%), sodium borohydride (NaBH 4 , ≥98%), hydrochloric acid (HCl, 37%), sodium dodecyl sulfate (SDS, 10% in H 2 O), as well as streptavidin (SKU 189730 and S4762), thrombin (SKU 1.12374), IgG (anti-biotin antibody produced in goat, SKU B3640) and myoglobin (from horse skeletal muscle; SKU M0630) were acquired from Sigma-Aldrich. Silver nitrate was purchased from TCI America (99%). All DNA strands were acquired from Eurofins Genomics and Biomers GmbH. Ultrapure water (resistivity: 18.2 MΩ cm at 25 °C) was obtained from a Milli-Q water purification system. All chemicals were used as received without further purification or treatment.

## DNA origami scaffold design and synthesis

The DNA origami structure was designed using caDNAno software 65 , 66 A honeycomb lattice of 14 DNA strands (14 HB) with a total length of 215 nm was used. Two binding sites for nanorods were introduced along the surface of DNA origami. The gap between the nanorod binding sites featured the anchor strand for the analyte. Each binding site consisted of 14 staple strands extended by 8 nucleotides protruding out of the structure. The binding sites were extended with two different sequences, i.e. A 8 (poly-A8 - AAAAAAAA) and R’ 8 (Random’ - ATGTAGGT). In the hotspot, two complementary DNA strands were introduced to elevate the Cy3.5 dye, the biotin molecule, and the HD22 aptamer above the DNA origami surface. The origami strand diagram and details on the hotspot design can be found in Supplementary Fig.  1 . DNA sequences are given in Supplementary Table  1 .

The folding of 14 HB DNA origami was carried out in TE buffer (Sigma-Aldrich) containing 10 nM of an 8634 nt scaffold, 100 nM DNA staples, 10 mM Tris (Sigma-Aldrich), 1 mM EDTA pH 8, and 24 mM MgCl 2 (Sigma-Aldrich). Firstly, the mixture was heated up to 65 °C to ensure denaturation of all DNA strands and then cooled down to 4 °C in 19 h using a thermo cycler. The DNA origami structures were purified from an excess of DNA staples using filtration (100 kg mol −1 Amicon filters, 5 min, 10,000 ×  g ). Filtration was repeated at least 5 times until no DNA signal was observed in the filtrate. To ensure the stability of the structures TE 6 mM MgCl 2 buffer was added after each centrifugation step.

## Synthesis and functionalization of Au nanorods

Gold rods were synthesized using the seed-mediated method developed by Ming et al. 30 . The amount of silver ions during synthesis was varied to tune the nanorod aspect ratio. In the first stage of the synthesis, small gold nanoparticles (i.e., seeds) were formed by injecting a freshly-prepared, 4 °C NaBH 4 solution (0.6 mL, 10 mM) into a rapidly stirred solution of HAuCl 4 (0.25 mL, 10 mM) and CTAB (9.75 mL, 100 mM). Stirring was continued for 10 min. Subsequently, the solution was transferred to a thermostat and kept at 30 °C for 2 h. In the following step, obtained seeds were diluted 10x in a CTAB solution (0.1 mL seeds and 0.9 mL, 100 mM CTAB), and then added to a mixture of CTAB (100 mL, 0.1 M), HAuCl 4 (5 mL, 10 mM), AgNO 3 - 1 mL (for batch R1) or 0.75 mL (for batch R2) with 10 mM, HCl (2 mL, 1 M), and ascorbic acid (0.8 mL, 100 mM). After introducing the seed solution, the mixture was gently stirred for 10 s and then kept at 30 °C for 2 h. The solution turned red during this time, indicating the growth of gold rods. To remove any excess of reagents, the nanorod solutions were centrifuged two times for 10 min at 5000 ×  g and re-dispersed in a 0.2 mM CTAB solution. Before functionalization with DNA, the nanorods were once again centrifuged for 10 min at 5000 × g and re-suspended in 0.1% SDS solution. The dimensions (length and width) of the obtained gold nanorods were 63.4 nm x 20.5 nm (batch R1) and 62.7 nm x 24.0 nm (batch R2) (Supplementary Fig.  3 ).

The nanorods were labelled with two different 5’ thiolated DNA sequences to enable their binding to the DNA origami structures: poly-T19 and Random (TTCTCTACCACCTACAT). Both of them were complementary to DNA sequences of the binding sites on the DNA origami beam. Nanoparticle functionalization was carried out via a freeze and thaw method 67 . In this method, the growth of ice crystals during a freezing process increases the local concentration of both nanoparticles and DNA molecules, which enables a high labelling yield. Here, 300 µL, 0.1% SDS solution of 3 nM rods were mixed with 200 µL of 100 µM ssDNA to get a final ratio of GNR/DNA = 1/22,222. Afterwards, the solution was placed in −80 °C for 30 min, and then melted in an ultrasonic bath. Functionalized nanorods were purified from an excess of DNA strands using electrophoresis (1% agarose, 70 V, 2 h in 1 x TAE, 11 mM MgCl 2 buffer).

## Binding of nanoparticles to DNA origami

The synthesis of nanorod dimers on DNA origami was realized through hybridization between DNA-functionalized nanorods and complementary strands on a DNA origami surface. Firstly, 20 µL of 20 nM rods solution was added to 20 µL of 2 nM 14 HB under vigorous stirring. Then, the solution was annealed from 45 °C to 20 °C for 20 h. Finally, the obtained dimers were purified from an excess of unbound nanorods by gel electrophoresis (1% agarose, 70 V, 2 h in 1 x TAE, 11 mM MgCl 2 buffer).

## TEM measurements

For TEM imaging, 3 µL of sample solution were dropped onto formvar-coated grids (300 mesh Cu, Ted Pella Inc). After 1 min the solution was removed with a paper filter, and the sample was negatively stained for 10 s using 2% Uranyl formate solution. All TEM measurements were carried out using a JEM-1101 electron microscope (JEOL) with an accelerating voltage of 80 kV.

## Deposition of GNR dimers on glass slides

For SERS, dark-field, and SEM measurements obtained dimers were deposited on the surface of 170 µm thick glass slides (Carl Roth GmbH, Germany). Firstly, glass slides were thoroughly cleaned by bath sonication in Hellmanex III (Sigma-Aldrich), acetone, isopropanol, and in Milli-Q water for 15 min each. Afterwards, glass slides were plasma cleaned in oxygen atmosphere for 5 min. Finally, 50 µL of diluted dimers solution was deposited on the surface for 1 min, followed by rinsing in distilled water and blow-dried with nitrogen.

## Optical microscopy and spectroscopy

Dark-field microscopy and spectroscopy as well as SERS measurements were carried out with a Zeiss Axio Scope A1. For dark-field illumination a Zeiss oil condenser (NA 1.2–1.4), and an unpolarised halogen white light source were used. A Canon EOS 6D was used for image acquisition. Fluid levels were kept at ≈100 μL, by injecting deionised water, to compensate for evaporation during the measurements. A Zeiss Achroplan 100x NA 1.0 objective was used for SERS measurements in water. Dark-field scattering images in air were taken with a Zeiss Epiplan 50x/NA 0.7 objective. Spectra were taken with a Princeton Instruments Acton SP2500 grating, coupled to a Princeton eXcelon CCD detector. A 671 nm Laser (500 mW model, Laser Quantum) was used for SERS measurements. Background subtraction was employed for the Raman spectra in Fig.  2e (Cy3.5), as well as those in Supplementary Figs.  8 and 10b (except for: Myoglobin only), for improved visual clarity.

## Numerical simulations

3D random walk modelling was used to estimate the concentration dependent mean diffusion time (MDT) of a protein for entering a nanogap (Supplementary Fig.  14 ). For the model, a fixed step-size of 4 nm, in a 400 nm x 400 nm x 400 nm large simulation box with reflective surfaces, and a freestanding 4 nm × 4 nm × 4 nm target “hitbox” was assumed. Different numbers of proteins were placed randomly in different locations of the simulation box. The simulations were run 100 times each. Step-times were approximated with the diffusion time of the 4 nm step-size. For this, diffusion coefficients \(D\) of streptavidin (7.72 · 10 −11  m² s −1 ) and thrombin (1.06 · 10 −10  m² s −1 ) were calculated using the diffusion equation

Here, hydrodynamic radii \(r\) of 2.82 nm for streptavidin 62 and 2.05 nm for thrombin 40 were used. To determine the MDTs, step-times were multiplied with average step-counts until a particle landed in the hitbox. Mean diffusion times are inversely related to protein concentration [protein], with MDT S.  = 8.4 s µM [streptavidin] −1 for streptavidin, and with MDT T.  = 6.1 s µM [thrombin] −1 for thrombin.

FDTD calculations were done with Lumerical (2020a FDTD Solver Version 8.23.2194, and newer). Meshing for 21 nm × 64 nm GNR, as well as 40 and 60 nm sphere-based dimers was set to 0.2 nm. For 80 nm sphere dimers, as well as the trimer and tetramer meshing was set to 0.4 nm. The mesh-volume embedding the particle was sized with an extra margin of ≈10% per spatial dimension. The overall simulation region was 4 µm x 4 µm × 4 µm in size. For dimers, min boundary conditions were set to asymmetric along, and symmetric perpendicularly to the polarization axis and the poynting vector of the linearly illuminating TFSF plane wave source. The light source was polarised along the dimers’ long axes, directed perpendicularly to the substrate plane (Supplementary Fig.  8 ). For the trimer and tetramer structures, two 90° phase and polarization shifted TFSF sources were used to account for additional geometric symmetries (here no symmetric boundary conditions were employed). Field enhancement factors were gained from hotspot centres, and renormalized with field strengths at an empty substrate under similar illumination conditions. The simulations were run until 10 −5 as a fraction of the initial energy remained in the system. Material parameters for glass 68 , water 68 , and single-crystalline gold 69 were taken from literature.

Heating of GNR dimers was modelled using COMSOL Multiphysics (version 5.2a). Meshing was set to be physics-controlled and extra fine. The simulation volume was 1 µm x 1 µm x 1 µm large, half water, half glass, with a boundary surface temperature of 20 °C. Material parameters were derived from the database of the software.

## Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

## Data availability

The data that support the findings of this study are available from the corresponding authors upon request.  Source data are provided with this paper.

## Code availability

Python code used to implement the random walk model for estimating hotspot diffusion times is available from the authors upon request.

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## Acknowledgements

We greatly appreciate funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 201269156—SFB 1032 (projects A6 and A8). K.K. received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 765703. T.Li. further acknowledges support from the DFG through the cluster of excellence e-conversion EXC 2089/1-390776260.

Open Access funding enabled and organized by Projekt DEAL.

## Author information

Karol Kołątaj

Present address: Département de Physique, Université de Fribourg, Chemin du Musée 3, 1700, Fribourg, Switzerland

These authors contributed equally: Francis Schuknecht, Karol Kołątaj.

## Authors and Affiliations

Chair for Photonics and Optoelectronics, Nano-Institute Munich, Department of Physics, Ludwig-Maximilians-Universität (LMU), Königinstraße 10, 80539, Munich, Germany

Francis Schuknecht, Michael Steinberger & Theobald Lohmueller

Physics Department and CeNS, Ludwig-Maximilians-University Munich, Geschwister-Scholl-Platz 1, 80539, Munich, Germany

Karol Kołątaj & Tim Liedl

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## Contributions

T.Li. and T.Lo. conceived the study. K.K. synthesized the GNR dimer nanoantennas. F.S. and M.S. performed SERS measurements and diffusion calculations. F.S. conducted numerical field enhancement and heating simulations. All authors contributed in analysing the data and writing of the manuscript.

## Corresponding authors

Correspondence to Tim Liedl or Theobald Lohmueller .

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Schuknecht, F., Kołątaj, K., Steinberger, M. et al. Accessible hotspots for single-protein SERS in DNA-origami assembled gold nanorod dimers with tip-to-tip alignment. Nat Commun 14 , 7192 (2023). https://doi.org/10.1038/s41467-023-42943-7

Accepted : 27 October 2023

Published : 08 November 2023

DOI : https://doi.org/10.1038/s41467-023-42943-7

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## Engaging Elementary Students in Geometry through Origami

Author: Terry Anne Wildman

Overbrook Elementary School

Seminar: Origami Engineering

Keywords: geometry , Math , origami

School Subject(s): Geometry , Math

As we sometimes struggle in America to design mathematics curriculum that will engage students and reach all learning styles, teaching students through manipulatives and play has always been the foundation for engaging young students. This unit seeks to provide lessons for students in fourth grade who are learning more abstract concepts and struggle to fully comprehend these new concepts. The unit can be modified for fifth and sixth graders using more complex designs and deeper discussions around geometric concepts. Included in this unit is background knowledge on the history of paper making and the history of origami. Students will learn about the evolution of origami and the role it can play in teaching students about geometric shapes, lines, and symmetry. Embedded in the lessons is the opportunity for students to practice precision, perseverance, and following step- by-step instructions. These skills will be useful for students as they persevere in solving future complex mathematical problems.

## Full Unit Text

Introduction.

Mathematics for elementary students has been changing dramatically for the last 20 years.  Gone are the days where students in first through fifth grades learn addition, subtraction, multiplication and division with fractions and decimals thrown in by fifth grade.  Students were asked to memorize their multiplication tables, quickly add and subtract double-digit numbers without using their fingers, and complete long division problems using prescribed steps.

For better or for worse, curriculum gurus developed a spiral type of learning that includes addition, subtraction, multiplication, division, fractions, decimals, algebra, geometry, complex and multi-step word problems and higher order thinking skills.  The idea is that students will be introduced to each type of mathematics from Kindergarten through Sixth grade.  Each grade year, students will learn and be able to apply deeper and more complex strategies to solve problems.  The goal is by Sixth grade for students to be exposed to and understand the complex nature of mathematics.  Some educators feel that this is great because the United States has fallen behind in terms of math test scores throughout the world, while other educators worry that students in this new curriculum never get a chance to master anything.

Whichever school of thought you may lean towards, I believe this new curriculum is here to stay, at least in the foreseeable future.  The School District of Philadelphia recently purchased new math curriculum from Pearson called enVisionmath.  It uses the same “spiral” learning as Chicago math introduced years ago.   Our students struggle with this curriculum because 1) it assumes fourth graders have been using envision math for their elementary years and 2) it includes very abstract concepts and skills for students who are not developmentally equipped to solve complex, multi-step problems when they have not mastered their addition, subtraction, and multiplication skills.  Let me be clear – students understand for the most part how to add and subtract up to double and triple digits.   They understand that multiplying means to add groups of numbers.  What they do not have is the hours of focus and practice it takes to 1) have number sense in which, for instance, students see immediately that 345 + 135 cannot equal 4, 080 and 2) memorize their multiplication facts so that dividing and multiplying fractions become skills they can succeed in.

I currently teach fourth graders in North Philadelphia.  My student population is made up of 27% African American, 63% Hispanic/Latino, 3% White, 1% Asian, and 6% Multi Racial/Other.  We are a Title I school with 100% recorded as economically disadvantaged.  My students have a difficult time focusing on their lessons, completing home and school work, and self-monitoring their behavior.  I find when teaching math that my students get upset when presented with new concepts that assume they have mastered basic math skills.  Our curriculum also assumes students have prior experience with the curriculum (it is our second year using this curriculum) and have been successful.  I find that many times, my students do not know what to ask when they do not understand a concept.  It seems that most do not have the basic terminology to articulate questions they have about the math concepts are learning.

Have you ever seen students making fortune-tellers in your classroom?  Students in my class this year make them all the time.  I introduced an “origami book,” which is a fun way to organize information about non-fiction texts.  Now I find my students making 3-D shapes in the classroom such as boxes and shapes that involve several pieces of paper locked together like a pinwheel.  Geometry can be fun and engaging for elementary students.  Identifying shapes, understanding angles and degrees of a circle, identifying triangles, etc. give students a break from the usual adding, subtracting, multiplying and dividing.  For our students in the Philadelphia School District, providing additional practice and hands on activities will give them a chance to play with these concepts and hopefully gain a deeper understanding.

Norma Boakes wrote that spatial visualization is needed if students are going to understand shapes and structures; they need to spend time exploring and developing their spatial skills.  One way to do this is through origami.  Origami involves a student “following a construction process moving a two-dimensional square into a variety of three-dimensional shapes and figures.” (Boakes 2)

This unit uses the idea of engaging students by uses a simple manipulative – paper.  Students will be able to create shapes, both two and three-dimensional to develop an understanding of symmetry, angles, fractions, and measurement.

Using paper, a manipulative, that students can work and practice with, will help them to better grasp concepts such as measuring the degrees of a circle.  Designing problems and projects around geometric concepts will help students to better understand these abstract concepts.

Brief History of Paper

There are a few books out that look at history through objects or food.  It is a fascinating way to chart history through time without necessarily charting wars and battles.  Mark Kurlansky has written a few of these.  Salt: A World History, Cod: A Biography of the Fish that Changed the World, The Big Oyster: History on the Half Shell, and Milk! and A 10,000-Year Food Fracas are just a few of his books that look at the history of these items and how the need, distribution, sale, use, and quest for them have changed the world.

Paper: Paging Through History was also written by Kurlansky in 2016.  Paper as we know and use it today is a relatively new technology.  Paper satisfied a need in society – a need to record information.  As governments grew and changed and as businesses and trade grew and changed, the need to record and keep records of transactions and laws became important.

Kurlansky wrote that the Chinese invented papermaking.  That is not to say that a type or form of paper was not being used in other parts of the world around the same time.  Evidence shows that people were writing on materials such as clay, stone, papyrus and parchment.

At one point, the mark of an advanced civilization was one that made paper.  As paper became the cheaper option (much cheaper that papyrus, a water plant, which could only be grown along the Nile and parchment, which is made from the skins of sheep, goats, etc.), its use spread throughout the world.  As merchants from the East traveled throughout Asia, paper was traded and eventually paper mills grew around the world changing the economy of those towns that settled near a good source of running water.  Because papyrus was unique to Egypt, it became a valuable commercial product that was shipped throughout the world.

The papyrus reed was peeled and once the outer layer was removed, there were about twenty inner layers that would be unrolled and laid out flat.  The layers were “woven” together, the second set laid at a 90-degree angle from the first set underneath.  Water was used to moisten the sheets and then they were pressed together with weights for a few hours.  The reeds, when cut, had a sticky sap that served as the glue that kept the layers together and if needed, a flour paste was used.  The sheets were then rubbed with a stone, piece of ivory, or shell until they were smooth and the layers did not create grooves that a stylus could move across the sheet.

Parchment was made from the skins of sheep, goats, and cows.  Vellum, a finer quality of parchment, was made from the skins of calves.  The process was tedious: after being flayed, the skin is soaked in water for a day.  The skin was soaked in a dehairing liquid, which eventually included lime, for eight days.  It had to be stirred a couple of times a day and you had to be careful not to soak the skin too long because it weakened the skin.  Next it was stretched out on a stretching frame by wrapping small, smooth rocks in the skins with rope or leather strips. The skin would be scraped to remove the last of the hair and get the skin the right thickness.

Paper is made by “breaking wood or fabric down into its cellulose fibers, diluting them with water, and passing the resulting liquid over a screen so that it randomly weaves and forms a sheet.” (Kurlansky xv)  Different trees and plants were used to make paper so through the last two to three centuries, paper has evolved from a thicker, coarser paper to the thin, smooth paper that we use today.  Paper was slow to become popular in Europe, it was felt that important and religious books should be written on parchment because they would last longer than books made of paper.

Making paper was not necessarily easier or faster than making papyrus or parchment.  In fourteenth century Europe, papermaking was common and wherever there was a river with clean water, a downhill run or swift moving current, and a town of people who could provide rags, there was a paper mill.  As the need for paper grew, workers, who did not have fixed hours, might work all night.  Apprentice paper workers, or children, “who were small enough to crawl into vats, scrubbed the hammers and the equipment clean” during the night when the mill was closed.  (Kurlansky 96)

With our new digital technologies, one might think that paper is seeing its last days.  Kurlansky would agree that paper might not be here forever.  But he does feel it is

more secure than electronic messages.  “Electronic messages can be hacked, accessed and reconstructed.” (Kurlansky 334)  If origami continues to be popular and used in education, health care and science, then paper will continue being manufactured in the future.

To give a perspective of how paper has evolved to today, here is a timeline of major events known throughout history taken from Paper: Paging Through History :

3000 BCE      Oldest papyrus found – a blank scroll in a tomb at Saqquara, near Cairo

500 BCE         Chinese begin writing on silk

252 BCE         Dating of the oldest piece of paper ever found in Lu Lan, China

105 BCE         Cai Lun of the Chinese Han court is credited with inventing paper

256 CE            First known book on paper produced in China

500-600 CE   Mayans develop bark paper

610 CE            Korean monk takes papermaking to Japan

751 CE            Papermaking in Samarkand begins – they are credited for producing

high quality paper exclusively from (linen) rags

1264 CE          First record of papermaking in Fabriano, Italy – they are credited

with first using watermarks to identify the papermaker.

1309 CE          Paper is first used in England.

1495 CE          John Tate establishes the first paper mill in England in Hertfordshire.

1502-20 CE   Aztec tribute book lists forty-two papermaking centers.  Some villages

produce half a million sheets of paper annually.

1575 CE         Spanish build the first paper mill in Mexico.

1729 CE         Papermaking in Massachusetts begins.

1833 CE         An English patent is granted for making paper from wood.

1863 CE         American papermakers start using wood pulp.  (Kurlansky 337 – 3446)

History of Origami

The word origami comes from the ori- meaning, folded and –kami meaning, paper.  Although the Chinese developed papermaking, the Japanese developed the art of origami.  The first Japanese folds date from the 6 th Century A.D.  Since paper was scare and precious at that time, the use of origami was limited to ceremonial occasions.  The designs were limited to representations of animals, people, and ceremonial designs.  The designs were passed down from generation to generation, usually from mother to daughter.

Some of the oldest existing directions for paper folding were printed in Japan in 1797, entitled Sembazuru Orikata or Folding of 1000 Cranes.  You may be familiar with the story of the young Japanese girl, who contracted leukemia after World War II from the effects of the Hiroshima atomic blast.  The crane is a symbol of good luck in Japan and the tradition was that if you fold 1,000 cranes, you would be granted one wish.  Young Sadako Sasaki decided to fold 1,000 cranes so that her wish to get better would be granted.  She died before achieving her goal, 365 short.  After she died, her classmates folded the rest for her and placed them in her coffin.

Akira Yoshizawa is credited with making origami popular again.  He was born to dairy farmers on March 14, 1911 in Japan.  When he was 13, he had to take a job in a factory in Tokyo.  In his early 20s, he was promoted to “technical draftsman,” responsible for teaching new employees basic geometry.  He had learned origami as a child so decided to use it as a tool to help these employees understand geometry.

Yoshizawa quit his job in 1937 to practice origami full time.  He lived in poverty for close to twenty years and during World War II, he served in the army medical corps.  To cheer up the sick patients, he made origami models but eventually became sick himself and was sent home.  Finally in 1951, a Japanese magazine asked him to fold the twelve signs of the Japanese zodiac.  This exposure basically led to his fame.  In 1954, he founded the International Origami Centre in Tokyo and through his travels became a goodwill ambassador for Japan.  He died in 2005 at the age of 94.

In the 1960s, two origami societies were established:  The Friends of the Origami Center of America and the British Origami Society.  With the resurgence of interest in origami, it has evolved into different forms including modular folds, three-dimensional folds, folds that combine several subjects into a single fold, action figures, and figures that move when tugged.  I believe that one reason that origami has become so popular and so many new ways to fold paper have become popular is because of the variety of paper that is manufactured today.  I cannot imagine making action figures or modular folds with the coarse, thicker paper that was made years ago.  It is with new technologies and materials that our paper today can be as think or thick as we desire.  Origami paper can be purchased in square shapes with different colors or patterns on either side to make folding paper much easier and more precise than in times past.  Today, the concept of paper folding has also been used in health care, i.e. cardiac stents, and science, folding lenses to fit into spacecrafts that can be remoting unfolded once in space.  Science and technology has taken the concept of paper folding and used it to fold different materials such as plastic and metals to advance our ability to save a life or see farther into the universe.

Basics of Origami

Origami is the art of folding an uncut sheet of paper into an object and animal.  Yoshizawa  invented a systematic code of dots, dashes, and arrows that was adopted by western authors Harbin and Randlett in the early 1960s, which is still used today.

This standardized the techniques and terminology of folds that people around the world use – if you know the system, you can recreate the design even if the book is written in a different language.

The following general rules are given when creating an origami design:  students should work on a hard, smooth, and flat surface so that their folds can be accurate, it is important that each fold and crease be precise and that a pencil or thumbnail is used to move over the fold for exactness, study the diagram/instructions before folding the paper, and if students use colored paper, start with the colored side facing down at the beginning of their folding.

This system includes instructions using lines, arrows, and terms used to describe these series of lines and arrows.  There are five different types of lines: paper edges, either raw or folded, are drawn with a solid line. Creases are drawn as a thinner line and will often end before the edge of the paper.  Valley folds are drawn with a dashed line and mountain folds by a chain of dot-dot-dash line.  The X-ray line or dotted line when shown on a drawing indicates anything hidden behind other layers or represents a hidden edge, fold, or arrow. (Lang 2003, 15).

( Math in Motion , Pearl 41)

Geometry and Origami

Using origami to introduce more complex abstract concepts to elementary students is a way to allow students to comprehend shapes and angles.  We may think that all students understand that a square has four sides of the same length and four right angles, but I have seen the Ah-ha! Moments some fourth graders have when they make one from an 8 ½ by 11” sheet of paper.  Using folds or creases; students can create triangles, such as equilateral triangles.  Here again, do students really understand what an equilateral triangle is?  They will after completing an activity where they are asked to create one from a square.  Thomas Hull wrote in Project Origami: Activities for Exploring Mathematics that “…when choosing to use origami as a vehicle for more organized mathematics instruction, an easy choice is to let the students discover things for themselves .” (Hull xi).  Origami is a great strategy to use to help students discover properties of two-dimensional shapes.  Students can create hexagons, octagons, and nonagons by using origami.

Students in fourth grade have a difficult time understanding angles.  They see two rays coming together at a point.  They are shown a protractor and shown how to measure the angle created by the two rays.  Students spend time practicing these measurements with their protractors and terms such as right, acute, obtuse angles.  One way to help students to grasp these concepts and discuss where we use geometry in the world would be use a square sheet of paper is to point that one corner is a right angle and by making creases (folding) the paper, they can create different acute angles.

Students can also fold a square making two creases that intersect at the middle of the square.  A circle can be drawn around the intersecting point so that students can see when we divide a square into four sections, we are creating four right angles each representing 90 degrees or a right angle.  If we multiply 90 times 4, we get a product of 360 degrees, which represents the total degrees of a circle.

Geometry terms such as lines, points, angles, triangles, rectangles, etc. can be modeled and discussed as part of an origami lesson.  Asking students to identify these terms and create them will support students who are visual learners.  Another concept that can be modeled is patterning.  Students will be able to create patterns through paper folding, especially when creating three-dimensional shapes.

• Students will be able to construct two-dimensional shapes in order to analyze the properties of two-dimensional shapes.
• Students will be able to construct multi-step shapes in order to develop perseverance in solving problems.
• Students will be able to construct three-dimensional shapes in order to analyze the properties of three-dimensional shapes.
• Students will be able to analyze characteristics of two and three-dimensional shapes in order to develop mathematical arguments about geometric relationships.
• Students will be able to develop basic geometric principles in order to construct and deconstruct models.

Developing authentic mathematical experiences for students is the best way to engage students and provide practice with abstract concepts.  Students will have an opportunity to answer questions and problems using origami.  For example, students will begin by folding two-dimensional four sided shapes into three to eight sided shapes.  They will discuss what they notice about constructing these shapes.  They will discuss what patterns they notice in the line creases they create when constructing each shape and what this tells them about the shape itself.

Teaching origami can seem daunting when you have a classroom of thirty plus students.  Folding paper into smaller and smaller shapes will be difficult for students to see.  Some will need one-on-one modeling while others will need to see the folds up close.  One strategy I will use to accommodate students is to use a document camera while identifying math concepts and terms and while constructing two and three-dimensional shapes.  Document cameras are a great way to model step-by-step instructions that will allow students to follow along.

Another strategy that I would like to try is to choose four to five students the meet with before beginning the unit.  I was thinking I would create an “origami club” where these students and I would meet once a week during lunch and construct origami shapes and become experts in creating the two and three dimensional shapes that we will be doing in class during the unit.  After presenting the lesson with the whole class, discussing concepts and modeling the origami shape, the experts along with the teacher will go around and assist the students who are struggling to construct the shape.

Another strategy that I would incorporate is for students to write reflections about their process.  After finishing an origami shape, students will be given prompts to reflect on in their “origami journal.”  Using mathematical terminology, students will be asked questions such as, what shapes did you notice when you unfolded your paper (creases in the unfolded paper will have different shapes).  What shapes do you notice in your finished three-dimensional shape?  What angles did you find when you unfolded your shape?  How many times did you have to unfold your paper and start again?  What part of making this shape was the most difficult?  Why?   Integrating writing with our lessons is important because students are asked to explain their answers in math.  Students can use the practice of writing about their thinking and learning in this unit as well.

Finally, to assess students’ understanding of the geometric terms and origami skills, I would use Exit Slips for each lesson.  Think of what you would like students to learn, for example, in the first lesson, using half sheets of paper, ask students to make two isosceles triangles and label the right angle.  For lesson two, ask students to define and draw horizontal and vertical lines.

## Lesson Plans

These lesson plans are designed for fourth graders and can be modified to use for other grades.  By choosing other shapes and designs in the books under Teacher Resources, teachers in third, fifth, and even sixth grade can use these lesson plans.  I designed these lessons to be used for a period of five to ten days using days eight through ten as “enrichment” lessons where students can learn more complex designs found in the aforementioned books.  I found once I taught the first lesson on the history of origami and the basic terms, students couldn’t wait to create shapes and designs.

These lessons could be spread throughout a quarter as well.  You could designate Fridays for seven weeks, for instance, as “Origami and Math Day” to complete this unit.  The benefit of doing this is that students are given a week to practice their origami skills.  By introducing and reviewing geometric concepts each lesson, students gain a deeper understanding of abstract concepts.  You can extend the enrichment lessons for as long as you like throughout the year.  There are books in the resources that give many complex and fun designs to make.

I recommend that you choose three to four students who you know would love to have the role as “experts” for your lessons.  Meet with them for approximately fifteen minutes before each lesson, which is easier to do when teaching one lesson per week, and show them your next lesson.  Allow them time to practice by giving them paper to take home.  They will be great assets in assuring the success of your lessons as they walk around the room helping those who are struggling to complete the shape or design.

Finally, you will have to practice completing each one of these designs before teaching the lesson.  It will take some time to master the more complex shapes and designs.  Since each lesson includes modeling with a document camera, the more expert you are, the smoother the lesson will go.  Using origami paper while modeling is also helpful as the two sides, colored side and white, are easily seen on the Smart board.

## Lesson One:  History of Origami and basic terms

Objective:   Students will be able to identify squares, rectangles, and isosceles triangles in order to create an origami shape.

• document camera
• short history of Origami taken from the Background
• Origami Math – see resources
• Smart board

Vocabulary:

• mountain fold
• valley fold
• isosceles triangle

Procedures:

• Short History of Origami – origins given in the History of Origami above
• Why origami? Ask students to turn and talk to a partner about why origami would be beneficial to use in math lessons. Show images of origami used in science today.
• Basic terms – review and model symbols page given above: mountain, valley, crease, and fold, which is pictured under Basics of Origami. Or you can view this YouTube video on making a square: youtube.com/watch?v=gLrKrHgAI40 .   Note to the students that the presenter is carefully folding and creasing his paper.
• Share Tips for success –patience, sharp and precise folds, perseverance, following step by step procedures
• Model and practice – have students make a square using a sheet of copy paper. Point out that the two triangles that are created when folding the top portion of the paper are isosceles right triangles. Define an isosceles triangle and start an anchor chart.  You can tape an example of an isosceles triangle to the chart.
• The strip of paper left over can be used to create a simple heart shape using rectangular strips of paper.  Use the procedure used on page 9 of Origami Math.  Or you can use this website to create a heart using origami paper.  youtube.com/watch?v=nnV262Egucw .  Students can design one side of the paper to make it their own. Students can also write a message to a friend, family member to write inside the heart and share it with them.
• Exit Slip-given a square sheet of paper, students will fold the paper to create two isosceles triangles and label the two right angles.

## Lesson Two:  Geometric Shapes

Objective:   Students will be able to use a square sheet of paper to create a 3-dimensional pinwheel.

• Square paper – 6” x 6”
• Instructions for a Basic Form, Net I, and Net II in the appendix
• Pins or wire
• Document Camera
• Diagonal cross
• Horizontal lines
• Vertical lines
• Review properties of a square, including four right angles and four equal sides. Although this may sound basic, asking students to turn and talk to discuss the difference between a rectangle and a square is a good way to check for understanding.  Introduce vocabulary words and add to the anchor chart in Lesson one.
• Model with students how to fold Basic I, a straight and diagonal cross using the document camera. This Basic form can be found on page 16 of Easy & Fun Paper Folding .  As students fold, refer to geometric shapes that the creases create.  Ask students what they notice about the shapes they are creating from the Basic I folds.
• Model with students how to fold Net I and Net II, which is a basic form for many shapes. These forms can be found on pages 17 and 18 of Easy & Fun Paper Folding .  Use the document camera again so students can see the folds and creases.  As students fold, ask them what they notice about the shapes the creases are creating.
• Use Basic form, Net I, and Net II to create a pinwheel in the appendix using the pinwheel design, straws, and pins/wires. Remind students to be careful with the pins.
• Exit Slip-given a square sheet of paper, students will fold to create horizontal and vertical lines and label each line.

## Lesson Three:  Jumping Frog

Objective:   Students will be able to use origami procedures to create a moving 3-dimensional shape.

• 3” x 5” index cards
• Math in Motion instructions on pages 52, 53, and 54.
• Document camera
• Line segment
• Perpendicular lines
• Introduce vocabulary words and add to the anchor chart from Lesson one. Students could draw these terms on the chart using a straight edge.
• Introduce motion/action origami – shapes that can be folded in a way that when “prodded” can move
• Use Pearl’s instructions on pages 53 and 54 using a document camera so students can watch each step. As you model the first time (students watch you make the entire frog and then you make it step-by-step together), use vocabulary term while modeling.  Or you can show this YouTube video which is about 10 minutes long:
• Unfold your frog and model step-by-step instructions while students follow along. Allow students time to practice moving their frog.
• Ask students what other movable origami shapes they could make that would move. Allow students to work in pairs to design a shape that would move.

Exit Slip-students will draw perpendicular lines, intersecting lines, and line segments.

The following represent Pennsylvania standards for fourth graders:

CC.2.3.4.A.2  Classify two-dimensional figures by properties of their lines and angles.

CC.2.3.4.A.3  Recognize symmetric shapes and draw lines of symmetry.

CC.1.5.4.A  Engage effectively in a range of collaborative discussions on grade-level topics and texts, building on others’ ideas and expressing their own clearly.

Boakes, Norma. “The Impact of Origami-Mathematics Lessons on Achievement and Spatial Ability of Middle-School Students.” Origami 4 , May 2009, pp. 471–481., doi:10.1201/b10653-46.

This article discusses ways to use origami in the mathematics classroom.  Ideas are given to educators and results of studies completed on the spatial ability of middle school students.

Hull, T. (2006). Project Origami: Ideas for Exploring Mathematics . Wellesley, PA: A.K. Peters.

This is a technical book for teachers who are looking for more complex origami designs along with ideas for math lessons for older students.  There are many suggestions for lessons plans and essential questions for students to write their reflections.

Kurlansky, M. (2016). Paper: Paging through history . New York: W.W. Norton & Company.

This book presents the history of paper tracing the origin of paper from papyrus, silk, and animal skins to paper making in early America.  The author presents background information on how the idea of writing on different types of paper evolved throughout time.

Lang, R. J., & Macey, R. (1988). The complete book of origami: Step-by-step instructions in over 1000 diagrams: 37 original models . New York: Dover.

For teachers who would like more complex designs to make with their students, this book offers 37 complex designs with step-by-step instructions including elephants, scorpions, tarantulas, etc.

Lang, R. J. (2003). Origami Design Secrets: Mathematical Methods for an Ancient Art . Natick, MA: A.K. Peters.

This book is a comprehensive guide to making complex origami designs.  It includes basic instructions, crease patterns, and step-by-step instructions for many animals.

#### IMAGES

1. Double Tetrahedron OrigamiInspired from the origami videos and 3d art

2. Origami Geometric Shapes

3. Color It Like you MEAN it!: construction paper

4. Geometry

5. Origami ideas: Step By Step Geometric Origami

6. Kinga Kubowicz Has Created Moduuli, A Collection Of Geometric Origami

#### VIDEO

1. part 1of origami project continue with me in other

2. How To Make 3D Origami Pieces

3. "The Math of Origami: Folding Paper into Geometry"

4. 3D Origami Piece Tutorial

5. 3d origami collection 12

6. HiCE007: Mathematics (Algebra and Geometry)

1. Origametry!

Introduction Origametry? Yes, the art of origami involves several geometric components! In this lesson, students will create a decahedron through a paper folding activity and determine how geometric concepts, such as parallel lines, perpendicular lines, and bisecting lines, connect to the ancient art form. Learning Objectives

2. Origami

Step 1: Basics: Creating an Origami Fortune Teller for Kids A paper fortune teller can be constructed with these steps using a plain A4 paper: The corners of a sheet of paper are folded up to meet the opposite sides and (if the paper is not already square) the top is cut off, making a square sheet with diagonal creases.

3. PDF Origametry!

Origametry! - Exploring Geometry through Origami By: Amanda Martin Elementary school music teacher; M.A.Ed. In Curriculum and Instruction Math Grades 9-12 | Introduction Origametry? Yes, the art of origami involves several geometric components!

4. Math with Paper: Fold Some Math into Your Day!

Here are some origami projects to get you started: Beginner: Dog origami Intermediate: Heart origami Advanced: Swan origami Paper Wizard Mental imagery—the ability to imagine what a shape looks like or to move objects around in your mind—is an important part of math.

5. Origami Geometry

Step 1: Look up a tutorial on the Internet and make the parts. Assemble some parts together. Step 2: Look at the overall shape of the origami module made. Shade the exposed parts, which make up the surface area of the main module. Step 3: Unfold the folded unit back into the original piece of paper.

6. The Math and Art in Origami

Step 1: Theory: Polyhedra Polyhedron: from poly (many) and hedron (face). Polyhedra are 3D geometric shapes. They are built from polygons ( gon - angle/corner), which are 2D geometric shapes. Polyhedra are named and classified based on the number and type of faces, and their symmetry, as well as how they can be constructed.

7. Using Geometry to Celebrate National Origami Day

National Origami Day is when we honor the Japanese art form of origami November 11th each year. Origami is the ancient Japanese culture of paper folding. It requires intricate folding and makes complex shapes out of square pieces of paper. Paper was invented in China and taken to Japan by Buddhist monks during the sixth century.

8. Origami in the Geometry Classroom

So, go ahead give it a try. You really don't have to be an origami master to bring origami to your math class. Reference. 1. Hull, Thomas, 2006, Project Origami, A.K. Peters. About the Author. Ophir Feldman ([email protected]) received his B.S. in Mathematics from Hofstra University in Hempstead, New York. He received his M.A. and Ph.D. from ...

9. Origami and Geometry Lesson Plan for Grade 2

Objectives Use origami to develop an understanding of geometric properties. Standards Met 2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Lesson Introduction

10. Origami Geometry

The origami project can also be used to teach 3D geometry using available 3D origami modules. Using 3D modules would allow students to practice calculating volume of different 3D shapes. If some students finish early, have them trade shapes with another student, and see if they come up with the same answer. ...

11. Using Origami to teach Geometry

The Daffodil Origami Craft Instructions are a great way to brighten up the classroom while using origami to teach your children the Geometry of nature. Attach the daffodil to some straws and create a paper bouquet. Mix up the colour of the card, or create a varied range with the Origami Lotus Flower Craft Instructions and the Origami Paper ...

12. The satisfying math of folding origami

Dig into the mathematical rules and patterns of folding origami, the ancient Japanese art of paper folding. --Origami, which literally translates to "folding...

13. Origami and Paper Folding

To start, fold the sheet along its two diagonals. Geometry is one of the oldest parts of mathematics - and one of the most useful. Its logical, systematic approach has been copied in many other areas.

14. Mathematical Origami

Platonic Solids are the most regular polyhedra: all faces are the same regular polygon, and they look the same at every vertex. The Greek philosopher Plato discovered that there are only five solids with these properties. He believed that the they correspond to the four ancient Elements, Earth, Water, Air and Fire, as well as the Universe.

15. Project Template

Through this project, students will work with a variety of origami designs while learning about the underlying math concepts used to create them. The goal of this project is not only to find a new way to present the same old material, but also to expose students to the history and art of the cultures where origami originated.

16. 12 Origami Projects for Kids

Simple origami projects can teach math concepts like fractions, geometry, and measurements. Creating origami requires following instructions and understanding sequencing, valuable life skills. Practicing origami can build self-esteem and help children learn to deal with failure. Table of Contents Easy Origami Projects for Kids What Is Origami?

17. The Geometry Junkyard: Origami

Folding geometry. Wheaton college course project on modular origami. Geometric paper folding. David Huffman. Rona Gurkewitz' Modular Origami Polyhedra Systems Page . With many nice images from two modular origami books by Gurkewitz, Simon, and Arnstein. How to fold a piece of paper in half twelve times.

18. Geometric Origami

Geometric Origami is a book on the mathematics of paper folding, focusing on the ability to simulate and extend classical straightedge and compass constructions using origami. It was written by Austrian mathematician Robert Geretschläger [ de] and published by Arbelos Publishing (Shipley, UK) in 2008.

19. Geometry Origami Teaching Resources

Geometry Origami Polygons Project. by. Algebra Queen. 5.0. (1) \$2.50. Google Slides™. My students actually gave me the idea and helped me create this project and I am so excited to share it! In this project/activity, students will need origami paper (I got a pack of 200 on Amazon for \$6) to create 4 easy level shapes.

20. Experiment in Geometry Science Projects

Experiment in Geometry Science Projects. (7 results) Explore questions in geometry and topology, the study of shapes and spatial relationships that bending and stretching do not affect. Do a proof, take measurements, analyze geometric relationships, or make a pattern.

21. Project Template

The goal of this project is to use origami to bring together the artistic, kinesthetic, and mathematic skills of students through a series of hands-on activities. The projects will begin with simple origami designs to build both student skill and confidence in their abilities. Rather than working problems out of a book, students will use these ...

22. Beyond the Fold: The Math, History, and Technology behind Origami

origami applets, teachers can introduce this ancient craft to their students from a technology-oriented perspective. Keywords. geometry, history, origami 1 Introduction In the following paper, we discuss several aspects of origami. First we explain the history of origami, where it originated, and how it has evolved into what it is today.

23. 6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra

Students are also required to do a project presentation and a small number of problem sets. Topics This is an advanced class on computational geometry focusing on folding and unfolding of geometric structures including linkages, proteins, paper, and polyhedra. Examples of problems considered in this field:

24. Accessible hotspots for single-protein SERS in DNA-origami ...

The geometry of the origami was confirmed by transmission electron microscopy (TEM, Supplementary Fig. ... —Project-ID 201269156—SFB 1032 (projects A6 and A8). K.K. received funding from the ...

25. Engaging Elementary Students in Geometry through Origami

Engaging Elementary Students in Geometry through Origami - Teachers Institute of Philadelphia Engaging Elementary Students in Geometry through Origami Author: Terry Anne Wildman School/Organization: Overbrook Elementary School Year: 2018 Seminar: Origami Engineering Grade Level: 4 Keywords: geometry, Math, origami School Subject (s): Geometry, Math