Select the Resulting Pattern That Matches the Folding Process - CogAT Grades K 2

SECTION 1

Why Do We Fold Paper?

People have been folding paper for a very long time! When you fold a piece of paper in half, something cool happens. Both sides look the same. This is called symmetry. Artists and builders use symmetry to make beautiful things.

On tests like the CogAT, you will see pictures of paper being folded. Then a hole is punched through the folded paper. You need to figure out what the paper looks like when you open it back up. Let's learn how!

Step 1 • Start with flat paper

You begin with a flat piece of paper. It is usually a square or rectangle.

Step 2 • Fold the paper

The paper gets folded in half. Sometimes it folds more than once!

Step 3 • Punch a hole

A hole is punched through all the layers of the folded paper.

Step 4 • Unfold and look

When you unfold the paper, you see a pattern of holes! You pick the answer that matches.

The big question is: Where will the holes appear when the paper is opened up? That is what we will learn together!

SECTION 2

Important Ideas to Know

Before we practice, let's learn some important ideas. These ideas will help you solve every folding problem you see!

The Fold Line

The fold line is where the paper bends. It acts like a mirror. What is on one side also shows on the other side.

Layers of Paper

When paper is folded, it has layers. A hole goes through all the layers at the same time.

Mirror Match

Holes appear on both sides of a fold line. They are in the same spot, like a mirror image.

Count the Holes

One fold makes 2 holes from 1 punch. Two folds make 4 holes from 1 punch. More folds mean more holes!

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KEY TAKEAWAY

Think of the fold line like a mirror in your room. If you stand in front of a mirror, your reflection is on the other side. A hole on one side of the fold will have a matching hole on the other side, the same distance away. It is like the hole is looking in a mirror!

SECTION 3

See How It Works

Let's look at a picture that shows what happens when we fold paper and punch a hole. Watch how the hole shows up on both sides of the fold!

This diagram shows how a flat piece of paper gets folded in half. A hole is punched through the folded paper. When the paper is unfolded, two holes appear — one on each side of the fold line, in matching spots.

Look at the picture above. See how the two pink holes are in the same spot on each side of the purple fold line? That's the big secret! The holes are always the same distance from the fold line.

SECTION 4

How Folding and Holes Work

What Happens with One Fold?

When paper is folded once, there are 2 layers. If you punch 1 hole, you get 2 holes when you open the paper.

What Happens with Two Folds?

When paper is folded twice, there are 4 layers. If you punch 1 hole, you get 4 holes when you open the paper. The holes make a nice pattern!

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Easy Rule to Remember

1 fold = 2 holes. 2 folds = 4 holes. Each fold doubles the number of holes!

Where Do the Holes Go?

✓ If the paper folds to the left, the matching hole is on the right.
✓ If the paper folds up, the matching hole is down.
✓ If the paper folds at a corner, the matching hole is in the opposite corner.

SECTION 5

Different Ways Paper Can Fold

Paper can fold in many ways. Let's look at the most common folds you will see on the test. Each one makes a different hole pattern!

This diagram shows the three most common fold types: left-to-right, top-to-bottom, and diagonal (corner). Notice how the pink holes are always the same distance from the dotted fold line.

Fold Direction	Fold Line Goes	Holes Appear
Left to Right	Up and Down (vertical)	Left side & Right side
Top to Bottom	Side to Side (horizontal)	Top & Bottom
Corner (Diagonal)	Corner to Corner	Opposite corners

Common fold types and where the holes appear

SECTION 6

Worked Example: Solving a Fold Problem

Let's solve a folding problem together, step by step! Imagine a square piece of paper. It is folded from right to left. Then a hole is punched in the top-left corner of the folded paper. What does the paper look like when it is unfolded?

Fold Right to Left, Hole in Top-Left Corner

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Step 1 — Look at the Fold

The paper folds from right to left. This means the fold line is on the left side, going up and down.

Fold line is vertical (up-down) on the left.
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Step 2 — Find the Hole

The hole is punched in the top-left corner of the folded paper. This is near the fold line.

Hole is at the top, near the fold.
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Step 3 — Unfold in Your Mind

Now imagine opening the paper back up. The fold line is like a mirror. The hole near the fold on the left will also appear on the right side, in the same spot.

A matching hole appears at the top-right.
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Step 4 — Pick the Answer

Look at the answer choices. Find the one that shows two holes at the top — one on the left and one on the right, both the same distance from the center.

Answer: Two holes at the top, one on each side!

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THINK ABOUT IT THIS WAY

Imagine you put paint on your finger and pressed a dot on one side of a folded paper. When you open the paper, the paint dot shows up on both sides! The fold line is like a mirror — the dot on one side has a twin on the other side, in the same spot.

SECTION 7

Helpful Tips and Common Mistakes

Here are some tips to help you do your best. We'll also look at mistakes kids sometimes make, so you can avoid them!

Helpful Tips ✅	Common Mistakes ❌
Always look at which way the paper folds first.	Forgetting to check the fold direction.
Undo folds one at a time if there are two folds.	Trying to undo all the folds at once.
Use the fold line as a mirror to find matching holes.	Putting holes in random spots.
Count the total holes. One fold = 2 holes per punch.	Choosing an answer with the wrong number of holes.
Check that holes are the same distance from the fold.	Making holes closer or farther from the fold line.

Tips to get it right vs. mistakes to avoid

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TOP TIP

If you're stuck, try using your hands! Pretend you are holding a piece of paper. Fold it the same way as in the picture. Point to where the hole would be. Then open your hands. Where are your fingers pointing? That's where the holes will be!

SECTION 8

Getting Ready for Harder Questions

You have learned how to solve folding problems with one fold. As you get better, you will see harder problems. Let's see what comes next!

What You Know Now	What Comes Next
One fold	Two or three folds
One hole punch	Two or more hole punches
Square paper	Rectangle or other shapes
Round holes	Different shaped cuts (triangle, square)

How folding problems get harder over time

Don't worry! The same rules always work. The fold line is always a mirror. Just take it one fold at a time, and you will do great!

SECTION 9

Practice Problems

Now it's your turn! Try these problems. Think about the fold line, and picture the paper opening up in your mind.

Question 1: CONCEPTUAL

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

Show answer

You will see 2 holes. When you fold a paper once and punch one hole, the hole goes through both layers. When you unfold the paper, each layer has a hole. So one fold with one punch gives you 2 holes.

Question 2: BASIC

A square piece of paper is folded from left to right. A hole is punched in the bottom-right corner of the folded paper. Where will the holes be when the paper is unfolded?

Show answer

There will be two holes at the bottom of the paper — one in the bottom-left corner and one in the bottom-right corner. The fold line goes up and down on the left side. The hole in the bottom-right flips to the bottom-left because the fold line acts like a mirror.

Question 3: INTERMEDIATE

A square piece of paper is folded from top to bottom. Then a hole is punched near the fold line on the left side. Where will the holes be when the paper is unfolded?

Show answer

There will be two holes on the left side of the paper — one near the top and one near the bottom. Since the paper folded from top to bottom, the fold line goes side to side in the middle. The hole near the fold on the bottom half is mirrored to the top half, both on the left side.

Question 4: APPLIED

A square piece of paper is folded from right to left. Then it is folded again from top to bottom. One hole is punched in the center of the folded paper. How many holes will you see when the paper is completely unfolded?

Show answer

You will see 4 holes. The first fold doubles the layers to 2. The second fold doubles them again to 4. One punch goes through all 4 layers. When you unfold, undo the last fold first — the center hole becomes 2 holes. Then undo the first fold — the 2 holes become 4 holes, one in each section of the paper.

Question 5: CRITICAL THINKING

A square piece of paper is folded from left to right. A hole is punched right on the fold line itself. When the paper is unfolded, how many holes will you see? Why is this different from punching a hole away from the fold?

Show answer

You will see only 1 hole! When a hole is punched right on the fold line, both layers are cut in the same place. When you unfold the paper, the two half-holes combine into one bigger hole right along the fold. This is different from punching away from the fold, where each layer gets its own separate hole.

SUMMARY

Let's Review!

Summary

When you see a paper folding problem, remember these steps. First, look at which way the paper folds. The fold line acts like a mirror. Next, find where the hole is punched. Then picture the paper opening up. The hole will have a matching twin on the other side of the fold line, in the same spot.

Remember: one fold makes 2 holes and two folds make 4 holes. If there are two folds, undo them one at a time, starting with the last fold. You can also use your hands to pretend you are folding paper. Practice makes this easier every time!