Mentally fold and unfold a paper based on a sequence of folds - CogAT Grades K 2

SECTION 1

Why Do We Fold Paper?

People have been folding paper for a very long time! Folding is when you bend a flat piece of paper so one part goes on top of another part. You probably fold paper every day. You might fold a letter, fold a napkin, or fold paper to make a fun shape.

When we learn to fold and unfold paper in our minds, we are using our brain like a superpower. We can see the paper change shape without even touching it! This skill helps us on tests like the CogAT.

105 AD • Paper Is Invented

Long ago in China, people first made paper. They folded it in many ways.

1600s • Origami Begins

In Japan, people started folding paper into beautiful animals and shapes. This is called origami.

1950s • Paper Folding on Tests

Teachers started using paper folding puzzles to see how well kids can think and picture things in their minds.

Today • CogAT Paper Folding

The CogAT test asks you to imagine folding and unfolding paper. It shows how well your brain can picture shapes!

So the big question is: can you picture in your mind what a paper looks like after it is folded, poked with a hole, and then opened back up? Let's learn how!

SECTION 2

The Big Ideas About Paper Folding

Before we start folding paper in our minds, let's learn some important ideas. These will help you solve paper folding puzzles!

The Fold Line

A fold line is the line where the paper bends. Think of it like a hinge on a door.

Layers Stack Up

Each time you fold, the paper gets more layers. One fold makes 2 layers. Two folds make 4 layers.

Holes Go Through All Layers

When you punch a hole in folded paper, the hole goes through every layer. More layers means more holes!

Symmetry Is Your Friend

Symmetry means both sides match. When you unfold, the holes on one side will mirror the holes on the other side of the fold line.

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KEY TAKEAWAY

Think of folding paper like folding a sandwich in half. If you poke a toothpick through the folded sandwich, you get a hole on the top piece AND the bottom piece. When you open up the sandwich, you see two holes that are mirror images of each other!

SECTION 3

See How a Fold Works

Let's look at a picture that shows what happens step by step. First we start with a flat square paper. Then we fold it in half. Then we punch a hole. Then we unfold it to see what it looks like!

This picture shows all four steps. Notice how the one hole in Step 3 becomes two holes in Step 4. They are on opposite sides of the yellow fold line.

Look at the yellow dashed line. That is the fold line. The two holes sit the same distance away from the fold line, but on different sides. It's like looking in a mirror!

SECTION 4

How to Think Through Each Fold

When you see a paper folding question on the CogAT, here is how to think about it. Follow these steps every time!

The 3-Step Thinking Plan

Step 1 — Watch the folds. Look at each fold carefully. Which way does the paper move? Is it folding left, right, up, or down?
Step 2 — Count the layers. After one fold you have 2 layers. After two folds you have 4 layers. The hole goes through ALL layers.
Step 3 — Unfold backwards. In your mind, open the paper back up in the OPPOSITE order. The last fold you did is the first one you undo.

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Remember!

Each time you unfold, the number of holes doubles. One fold and one hole? You get 2 holes. Two folds and one hole? You get 4 holes!

Think about unfolding backwards like taking off your shoes. You put your left shoe on first, but you take it off last. Folds work the same way. The last fold comes undone first!

SECTION 5

Different Ways Paper Can Fold

Paper can fold in different directions. Let's look at the most common folds you will see on the CogAT test. Understanding each type will help you picture the right answer!

Three common fold types: left-to-right, top-to-bottom, and diagonal (corner). The yellow dashed line shows where the fold happens. The bottom row shows where holes appear when you unfold.

Each fold type creates holes in a different pattern. A left-to-right fold makes holes that are side by side. A top-to-bottom fold makes holes that are above and below each other. A diagonal fold makes holes that are in opposite corners!

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Helpful Hint

Always look at the fold line first. The holes will always be the same distance away from the fold line on both sides. Like a butterfly's wings — both sides match!

SECTION 6

Let's Solve One Together!

Let's practice with a problem. Imagine we have a square piece of paper. We fold it in half from left to right. Then we fold it in half from top to bottom. Then we punch one hole in the top-left corner. What does it look like when we unfold it?

Two Folds and One Hole

1
Step 1 — Start with a flat square

We begin with a flat square piece of paper. It has no folds and no holes yet.
2
Step 2 — First fold: left to right

We fold the left side onto the right side. Now the paper is a tall rectangle. It has 2 layers.

Paper is now a tall rectangle with 2 layers.
3
Step 3 — Second fold: top to bottom

We fold the top down onto the bottom. Now the paper is a small square. It has 4 layers (2 × 2 = 4).

Paper is now a small square with 4 layers.
4
Step 4 — Punch one hole in the top-left corner

We punch a hole through all 4 layers in the top-left corner of the small square.

The hole goes through all 4 layers!
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Step 5 — Unfold the second fold (undo top-to-bottom)

We undo the last fold first. We open the top-to-bottom fold. The hole that was in the top-left corner now appears in the top-left AND bottom-left. We now have 2 holes.

2 holes: top-left and bottom-left of the tall rectangle.
6
Step 6 — Unfold the first fold (undo left-to-right)

Now we undo the first fold. We open the left-to-right fold. Each of the 2 holes makes a mirror copy on the other side. So we now have 4 holes — one in each corner of the paper!

4 holes, one in each corner! ✨

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KEY TAKEAWAY

With 2 folds and 1 hole, you get 4 holes! Think of it like this: if you stack 4 cookies on top of each other and push a straw through all of them, every cookie gets a hole.

SECTION 7

Helpful Tips and Common Mistakes

Paper folding questions can be tricky! Here are some tips to help you do your best, and some mistakes to watch out for.

Good Habits ✅	Common Mistakes ❌
Always unfold in reverse order — last fold opens first.	Unfolding in the wrong order and getting holes in the wrong spots.
Count the layers to know how many holes you will get.	Forgetting that the hole goes through ALL layers, not just the top one.
Use the fold line to find where the mirror holes will be.	Putting the mirror hole too close or too far from the fold line.
Take your time and go one fold at a time.	Trying to do everything at once and getting confused.

Tips vs. Common Mistakes

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REMEMBER THIS

Go slow and be careful. Think about one fold at a time. It is like building with blocks — you stack them one at a time, and you take them off one at a time. Don't try to do all the folds at once!

SECTION 8

Building Bigger Skills

Paper folding is one part of a bigger set of thinking skills. As you get better at this, you will also get better at other things! Let's see how paper folding connects to harder puzzles you might see later.

What You Know Now	What Comes Next
Folding paper 1 or 2 times	Folding paper 3 or more times with many holes
Punching one round hole	Cutting shapes like triangles or squares out of the paper
Folding in straight lines (left, right, up, down)	Folding on diagonal lines (corner to corner)
Flat paper (2D shapes)	Folding boxes and 3D shapes in your mind

From Today's Skill to Tomorrow's Challenges

The skill you are learning now — picturing shapes in your mind — is called spatial thinking. It helps artists, builders, scientists, and even astronauts! Every time you practice paper folding, your brain gets stronger at this important skill.

SECTION 9

Practice Problems

Now it's your turn! Try these problems. Think carefully about each fold. Remember to unfold in reverse order!

Question 1: CONCEPTUAL

You fold a square piece of paper in half from left to right. You punch one hole in the middle of the folded paper. When you unfold the paper, how many holes do you see?

Show answer

You see 2 holes. When you folded the paper in half, you made 2 layers. The hole went through both layers. When you unfold, each layer has its own hole. The two holes are on opposite sides of the fold line, like a mirror.

Question 2: BASIC CALCULATION

You fold a square piece of paper in half from top to bottom. Then you fold it in half again from left to right. How many layers does the paper have now?

Show answer

The paper has 4 layers. The first fold made 2 layers. The second fold doubled that to 4 layers. Each fold doubles the number of layers. So if you punched a hole now, you would get 4 holes when you unfold!

Question 3: INTERMEDIATE

You fold a square piece of paper in half from left to right. You punch a hole in the top-right corner of the folded paper. When you unfold, where are the holes?

Show answer

There are 2 holes. One is in the top-right corner (where you punched it). The other is in the top-left corner. When you folded left to right, the left side went on top of the right side. So the mirror hole appears on the left side, in the same spot but flipped across the fold line.

Question 4: APPLIED

You fold a square paper in half from top to bottom. Then you fold it again from left to right. You punch a hole in the bottom-right corner of the small square. Describe where all the holes are when you completely unfold the paper.

Show answer

There are 4 holes, one in each corner of the paper. First, undo the left-to-right fold: the bottom-right hole gets a mirror in the bottom-left. Now you have 2 holes on the bottom. Then undo the top-to-bottom fold: each bottom hole gets a mirror on top. So you end up with holes in all four corners: top-left, top-right, bottom-left, and bottom-right.

Question 5: CRITICAL THINKING

You fold a square paper in half from left to right. Then you fold it from top to bottom. You punch a hole in the CENTER of the small folded square (not in a corner or edge). When you unfold completely, how many holes do you see, and why?

Show answer

You see 4 holes. Even though the hole is in the center of the small square, remember there are 4 layers. When you undo the top-to-bottom fold, the center hole makes a mirror hole, giving you 2 holes side by side vertically. When you undo the left-to-right fold, each of those 2 holes makes a mirror copy, giving you 4 holes total. The 4 holes form a small square pattern near the center of the paper.

SUMMARY

Let's Review What We Learned!

Summary

Paper folding questions ask you to picture in your mind what happens when paper is folded, a hole is punched, and then the paper is unfolded. The fold line is where the paper bends. Each fold creates more layers. A hole goes through all layers, so more layers means more holes when you unfold. The holes always appear as mirror images on each side of the fold line.

Remember the 3-step plan: watch the folds, count the layers, and unfold backwards. Go one fold at a time, and always undo the last fold first. With practice, your brain will get faster and faster at spatial thinking!