Origami and Topology Script

Warm-Up: Engaging Minds (5 minutes)

"Good morning/afternoon, everyone! Today we're going to dive into a really cool connection between art and math. To start, let's do a quick warm-up. Take a moment to look at a simple sheet of paper in front of you.

Think about this: How many distinct sides does a flat piece of paper have? And how many corners? Jot down your thoughts briefly.


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"Okay, who would like to share their observations? (Allow a few student responses). Great! Keep those ideas in mind as we explore how this simple piece of paper can transform into something truly amazing and mathematically fascinating."

Introduction to Topology & Origami (10 minutes)

"(Display Slide Deck: Fold Space, Not Just Paper! - Slide 1 and 2)

"First, let's talk about the art part: Origami. Many of you might already know about it. What is origami? What are some of its basic rules?" (Allow student responses, guide towards the idea of folding without cutting/gluing).

"Exactly! Origami is the art of paper folding, and a super important rule is that we usually don't cut or glue the paper. We just fold it to create incredible shapes and sculptures from a flat sheet."

"(Display Slide Deck: Fold Space, Not Just Paper! - Slide 3)

"Now for the math part: Topology. It sounds like a big word, but it's really about studying shapes and spaces in a special way. Imagine you have a piece of play-doh. You can squish it, stretch it, bend it, even twist it, right? But it's still the same piece of play-doh. It hasn't suddenly grown an extra hole or split into two pieces unless you actively tear it."

"Topology is interested in those properties that don't change even when you stretch or bend something. Like, if a shape has one hole, it will always have one hole no matter how much you squish it, as long as you don't tear it or poke a new hole.

"A classic example is a donut and a coffee mug. Believe it or not, to a topologist, they're the same! Why do you think that might be?" (Wait for responses, guide them to the idea of a single hole). "Yes, they both have one hole! You could theoretically transform one into the other by stretching and bending without tearing. Pretty wild, huh?"

Origami + Topology = ? (10 minutes)

"(Display Slide Deck: Fold Space, Not Just Paper! - Slide 4 and 5)

"So, what does this have to do with origami? Well, when we do origami, we take a flat, 2D piece of paper, and we transform it into a 3D object. But remember that key rule of origami? No cutting, no gluing!"

"This means that even though we're changing the geometric shape of the paper (like from a square to a crane), we are preserving its topological properties. The number of holes doesn't change unless we specifically design a fold that creates one, and it remains one continuous surface. We're essentially 'folding space' without breaking it."

"Now, let's get hands-on! We're going to do a simple fold together. As we fold, I want you to really think about the paper. Follow along with the instructions on the Activity: Folding a Topological Wonder."

(Guide students through a simple origami fold, e.g., a paper cup or simple box, referring to the Activity: Folding a Topological Wonder. Circulate and provide assistance. Prompt students with questions during the folding process).

"As you're folding, consider:
- How does the flat paper start to become a 3D object?
- Are you adding or removing any holes in the paper?
- Is the paper still one continuous piece, even with all the folds?"

Reflection and Discussion (5 minutes)

"Great job with your folds, everyone! Now, let's bring it all together. (Display Cool Down: Topological Takeaways)

"Based on what we just did, how do you think origami helps us understand 3D shapes and these ideas of topology?




"What was one thing that surprised you about the connection between origami and topology?




" "The next time you see an amazing origami creation, I hope you'll not only appreciate it as a piece of art but also as a fascinating example of topological transformation! You've just folded space!"

Origami Warm-Up!

Take a moment to look at a simple sheet of paper in front of you.

Think about this:

1. How many distinct sides does a flat piece of paper have?
   
   
   
   

2. How many corners does it have?
   
   
   
   

3. Can you think of a way to change its shape without tearing it or adding anything to it?
   
   
   
   
   
   
   

Folding a Topological Wonder: The Paper Boat!

Today, we're going to fold a simple paper boat. As you follow the steps, pay close attention to how the flat paper transforms into a 3D object, and think about the questions we discussed regarding topology.

Materials Needed:

• One square sheet of origami paper (or any square paper)

Instructions:

1. Start with your paper white-side up (if it has one). Fold the paper in half to form a rectangle. Crease well and unfold.
2. Fold the paper in half again, perpendicular to the first fold, to make a smaller rectangle. Crease well.
3. From the top open end, take the two top corners of the folded edge and fold them down to meet the center crease, forming two triangles (like a paper hat). The open edges should be facing you.
4. Fold up the bottom flap of the front layer (the rectangle below the triangles) over the triangles. Crease well.
5. Flip the paper over and fold up the remaining bottom flap in the same way. You now have a triangle shape (like a paper hat).
6. Open the bottom of the hat by pulling the center outwards, and flatten it into a square. Crease well. (You now have a smaller square, with the original triangles forming points at the top and bottom).
7. Take the bottom corner of the top layer of the square and fold it up to meet the top corner. Crease well.
8. Flip it over and do the same with the other side. You should have another triangle.
9. Again, open the bottom of this triangle by pulling the center outwards, and flatten it into a square. Crease well.
10. Gently pull the two opposite points of the top layer of this square outwards. The boat will start to form! Flatten the bottom to make it sturdy.

Observe and Reflect:

• How did the flat paper become a 3D object through these folds?
• Did you ever cut or glue the paper? How does this relate to topology?
• Is the paper still one continuous piece? How many distinct surfaces can you count on your boat?

Topological Takeaways

Take a few minutes to reflect on today's lesson about origami and topology.

1. In your own words, briefly explain the connection between origami and 3D topology.
   
   
   
   
   
   
   

2. What was one new idea or concept you learned today?
   
   
   
   
   
   
   

3. Can you think of another everyday object that might be topologically similar to your paper boat (meaning you could stretch/bend one into the other without tearing)?