Predict the location of holes or marks after folding/unfolding - CogAT Grades K 2

SECTION 1

Why Do We Learn About Paper Folding?

People have been folding paper for a very long time! When you fold a piece of paper and cut it or punch a hole, something cool happens. The hole goes through more than one layer. That means when you open the paper back up, you see more holes than you made! This is a fun puzzle that helps your brain think about shapes and space.

Long Ago • Paper Folding Begins

People in China and Japan started folding paper to make art. They noticed that folding creates matching patterns!

In School • Teachers Use Folding to Teach

Teachers started using paper folding to help kids learn about symmetry. Symmetry means both sides look the same!

On Tests • Paper Folding on the CogAT

The CogAT test uses paper folding puzzles to see how well you can picture things in your mind.

On the CogAT test, you will see a picture of paper being folded. Then a hole is punched. Your job is to figure out: Where will the holes be when you unfold the paper? Let's learn how to do this!

SECTION 2

The Big Ideas About Folding and Holes

Before we start solving puzzles, let's learn a few important ideas. These are the rules that help us figure out where holes go!

Layers Stack Up

When you fold paper, the layers sit on top of each other. A hole goes through all the layers at once!

The Fold Line Is a Mirror

The line where you fold is like a mirror. Holes on one side will match on the other side when you open the paper.

More Folds = More Holes

Each time you fold the paper, there are more layers. One fold makes 2 holes. Two folds can make 4 holes!

Unfold in Reverse

To find the holes, think about opening the paper backward — undo the last fold first, then the one before that.

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KEY TAKEAWAY

Think of it like a butterfly! When a butterfly closes its wings, both wings touch. If you put a dot of paint on one wing and press them together, the same dot appears on both wings in the same spot. That's just like folding paper and punching a hole!

SECTION 3

See How It Works: One Fold

Let's look at what happens step by step. We start with a flat piece of paper. We fold it in half. Then we punch one hole. When we unfold the paper, we see two holes! Look at the picture below to see how this works.

This diagram shows a square piece of paper folded in half from left to right. A hole is punched on the right side. When unfolded, two matching holes appear, one on each side of the fold line.

See how the two holes are in the same spot on each side of the fold line? They are like a mirror image. The fold line is the mirror! This is called symmetry. Both sides match.

SECTION 4

How Folding Works: Counting Holes

Let's think about how many holes you get. It all depends on how many times you fold the paper. Every fold doubles the layers.

Number of Folds	Number of Layers	Holes from 1 Punch
0 (no fold)	1	1
1 fold	2	2
2 folds	4	4

More folds mean more layers, and more layers mean more holes!

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Remember!

You don't need to do math on the test. But knowing this rule helps you check your answer. If you folded the paper once and only see one hole in your answer, something is wrong! You should see at least two holes after one fold.

The most important thing is not just counting holes. It is knowing where the holes will be. The holes are always in matching spots on each side of the fold line. Think about flipping a pancake — the same spot is in the same place, just on the other side!

SECTION 5

Different Ways to Fold Paper

Paper can be folded in different directions. Let's look at the main types of folds you will see on the test.

Three common fold types: left-to-right, top-to-bottom, and diagonal. Each fold direction changes where the matching holes appear. The bottom box shows what happens with two folds.

Look at the pictures above. The solid pink circles show where the holes will be when you open the paper. The lighter circles show the matching hole on the other side of the fold line. The fold line is always the line that acts like a mirror. No matter which way you fold, the hole on one side matches a hole on the other side.

SECTION 6

Let's Solve One Together!

Let's walk through a problem just like one you might see on a test. Follow along step by step!

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The Problem

A square piece of paper is folded in half from bottom to top. Then a hole is punched in the upper-right corner of the folded paper. What will the paper look like when you unfold it?

Step-by-Step Solution

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Step 1 — Look at the Fold

The paper is folded from bottom to top. That means the bottom half goes up and sits on top of the top half. The fold line is in the middle going left to right (a horizontal line).
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Step 2 — Find the Hole on the Folded Paper

The hole is punched in the upper-right corner. This is on the folded paper, so the hole goes through two layers.
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Step 3 — Think About Unfolding

Now unfold the paper by pulling the bottom back down. The hole in the upper-right corner stays where it is. The matching hole appears in the lower-right corner because the fold line is horizontal. The hole flips down, like a reflection in a pond!
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Step 4 — Check Your Answer

The unfolded paper has two holes. One is in the upper-right corner. One is in the lower-right corner. Both are the same distance from the fold line.

Answer: Two holes — upper-right and lower-right corners!

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TRICK TO REMEMBER

Pretend the fold line is a mirror sitting on a table. Look at where the hole is. Now imagine where it would be in the mirror. That's where the matching hole goes! The mirror trick works every time.

SECTION 7

Helpful Tips and Common Mistakes

Some things can trip you up on these problems. Let's look at what helps and what can go wrong.

Good Tips ✅	Common Mistakes ❌
Look at the fold direction carefully — is it left-right or top-bottom?	Forgetting which way the paper was folded and putting holes in the wrong spot.
Use the fold line as a mirror to find matching holes.	Moving the hole to the wrong side — like putting it on the left when it should be on the bottom.
Count the folds to know how many holes to expect.	Forgetting that 2 folds means 4 holes, not 2.
Undo folds one at a time, starting with the last fold.	Trying to undo all folds at once and getting confused.

Do these good things and avoid the mistakes!

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KEY TAKEAWAY

When there are two folds, go slow! Undo one fold at a time. First undo the last fold to find the new holes. Then undo the first fold to find even more holes. It's like unwrapping a present one layer at a time!

SECTION 8

From Paper Folding to Bigger Ideas

Paper folding is not just a test question. It helps you build a skill called spatial thinking. This means you can picture things in your mind and move them around. People use this skill for many things!

Paper Folding Skills	Where Else You Use Them
Picturing matching holes across a fold	Making art, cutting snowflakes, building with blocks
Thinking about mirror images	Learning about symmetry in math and nature (butterflies, leaves)
Keeping track of layers	Solving puzzles, playing strategy games

As you get older, you might see harder versions of this puzzle. There might be more folds, or the holes might be in trickier spots. But the same rules always work. The fold line is always a mirror, and holes always match on each side. Keep practicing, and you will get faster and faster!

SECTION 9

Practice Problems

Now it's your turn! Try these problems. Think about the fold, the hole, and then unfold the paper in your mind. Check your answer after you try each one.

Question 1: CONCEPTUAL

A square piece of paper is folded in half from left to right. One hole is punched in the middle of the folded paper. How many holes will you see when you unfold the paper?

Show answer

You will see 2 holes. When you fold paper in half, it has two layers. The hole goes through both layers. So when you unfold it, you see one hole on the left side and one hole on the right side. The fold line down the middle acts like a mirror.

Question 2: BASIC CALCULATION

A piece of paper is folded from top to bottom. A hole is punched in the bottom-left corner of the folded paper. Where will the holes be when you unfold it?

Show answer

There will be two holes. One is in the bottom-left corner (where you punched it). The other is in the top-left corner. Since the fold went top to bottom, the fold line goes across the middle. The hole flips from bottom to top, like a reflection, but stays on the left side.

Question 3: INTERMEDIATE

A piece of paper is folded in half from left to right. Then it is folded again from top to bottom. One hole is punched in the top-right corner. How many holes will there be when the paper is fully unfolded?

Show answer

There will be 4 holes, one in each corner of the paper. Two folds made four layers. First, undo the top-to-bottom fold: the top-right hole gets a match at the bottom-right. Now undo the left-to-right fold: the two right-side holes get matches on the left side. So you end up with holes in all four corners!

Question 4: APPLIED

A piece of paper is folded from right to left. A hole is punched near the top of the folded paper, close to the fold line but not on the edge. Where will the two holes be when you unfold the paper? Will they be far apart or close together?

Show answer

The two holes will be close together, near the top of the paper. Since the hole was punched close to the fold line, its mirror image will also be close to the fold line. Both holes will be near the middle of the paper, close to the top. The closer a hole is to the fold line, the closer the two holes will be to each other when unfolded.

Question 5: CRITICAL THINKING

A piece of paper is folded in half from left to right. Then it is folded again from bottom to top. A hole is punched right on the corner where both fold lines meet (the center of the original paper). How many holes will you see when you unfold? Why?

Show answer

You will see only 1 hole right in the center of the paper! This is a tricky one. When the hole is punched exactly on the fold lines, the matching holes all land on the same spot. The mirror image of the center is still the center. Even though the hole went through 4 layers, all 4 holes overlap in the same place. So you just see one hole in the middle!

SUMMARY

What Did We Learn?

Summary

When you fold paper, you create layers. When you punch a hole, it goes through all the layers. When you unfold the paper, you see matching holes on each side of the fold line. The fold line works like a mirror. One fold makes 2 holes. Two folds can make up to 4 holes.

To solve these puzzles, always look at the fold direction first. Then find the hole. Use the mirror trick to find the matching hole across the fold line. If there are two folds, undo one fold at a time, starting with the last fold. You've got this!