Select the Resulting Figure That Matches the Folding Process - CogAT Grades 3 Plus

SECTION 1

Why Paper Folding Questions Exist

Have you ever folded a piece of paper in half, punched a hole through it with a hole-punch, and then unfolded it to see more holes than you expected? If so, you already know something about spatial reasoning (the ability to picture how objects look when they move or change shape). Scientists and teachers have studied this skill for over a hundred years because it helps us understand how people think without using words.

1905 • Early Intelligence Tests

Alfred Binet created some of the first tests to measure thinking skills in children. These included puzzles that asked kids to imagine shapes.

1954 • Spatial Ability Research

Researchers began to show that spatial reasoning — picturing objects in your mind — is a separate skill from reading or math.

1974 • Mental Rotation Studies

Roger Shepard proved that people actually "rotate" images inside their heads, just like spinning a real object.

2000s • CogAT Paper Folding

The Cognitive Abilities Test (CogAT) added paper-folding items to its Nonverbal Battery. Each item shows a square paper folded one or more times, with a hole punched through the folded layers. Students must pick the pattern that shows where every hole will appear once the paper is completely unfolded.

Today, paper-folding questions appear on the CogAT to see how well you can think in pictures. The big question is: can you look at a folded and punched piece of paper and predict what it looks like once it is completely unfolded? Let's learn how.

SECTION 2

Core Principles of Paper Folding

Before you try any folding-and-punching problem, you need to understand a few key ideas. These ideas will guide you every time you see a new question.

The Fold Line

A fold line, or crease, acts like a mirror. Whatever is near the crease on one side gets reflected onto the other side once the paper is folded.

Folding Stacks Layers

Each time you fold the paper, you stack layers on top of one another. One fold creates 2 layers, two folds create 4 layers, and so on.

One Punch, Many Holes

When a hole is punched through the folded stack, it goes through every layer at once. A single punch can turn into several holes once the paper is unfolded.

Holes Mirror Across Every Fold

Each hole reflects across every crease it crosses. To find all the holes, mirror the punch across the most recent fold first, then the fold before that.

Unfold in Reverse Order

Undo the folds in the opposite order they were made. The last fold you made in your mind should be the first crease you "undo."

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KEY TAKEAWAY

Think of making a paper snowflake. You fold a square piece of paper in half a couple of times, snip or punch a few small shapes near the folded edges, and then unfold it. The finished snowflake has far more holes and notches than you actually cut — because every snip passed through several layers at once. A paper-folding question works the same way: figure out how many layers were punched and where each hole lands once everything is unfolded.

SECTION 3

Seeing the Fold in Action

The diagram below shows a square piece of paper being folded in half and punched on the left, and the fully unfolded result on the right. Study how one punch becomes two holes, mirrored across the fold line.

The square paper is folded in half along the dashed crease, then a hole is punched through both layers. When the paper is unfolded, the single punch becomes two holes, mirrored across the fold line at equal distances from the crease.

When you look at a folded-and-punched shape, imagine picking it up and unfolding it one crease at a time — starting with the very last fold you made. Every time you undo a fold, any hole near that crease gets a mirror-image twin on the other side.

SECTION 4

How Paper Folding Works Step by Step

There is no complicated math formula for paper-folding questions. Instead, you use a clear step-by-step process. Think of it as a mental recipe.

The 4-Step Folding Process

Step 1 — Count the folds. How many times was the paper folded? Each fold is a possible mirror line for the holes.
Step 2 — Picture each crease. Note where every fold line sits on the original square — vertical, horizontal, or diagonal.
Step 3 — Find the hole in the folded shape. Locate exactly where the punch is and how close it sits to each crease.
Step 4 — Unfold one crease at a time, last fold first. Reflect the hole across the most recent fold to add a new hole, then reflect all the holes across the next fold back, until the paper is fully unfolded.

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Quick Tip

Use elimination! Count how many folds were made. If there were 2 folds and the hole is not sitting on a crease, the correct answer must show exactly 4 holes arranged symmetrically. Any answer with the wrong number of holes, or holes that aren't the same distance from a fold line, is wrong.

Remember, every hole must stay the same distance from the fold line as its mirror twin. If a hole is punched close to one crease, its reflection should be just as close to that same crease on the other side.

SECTION 5

Common Fold Patterns You Will See

Not every folding question uses the same number of folds or the same fold direction. The CogAT may show a single fold, two folds, or a diagonal fold. The diagram below shows the most common patterns and roughly how many holes to expect.

Three common patterns are shown: a single fold (2 mirrored holes), two folds (4 holes, one per quadrant), and a diagonal fold (2 holes mirrored across the diagonal). A hole punched exactly on a crease will not double when that particular fold is undone.

When you see a folding question on the test, the first thing to do is count how many folds were made. This tells you roughly how many holes to expect. Two folds usually mean four holes — unless the punch sits exactly on one of the creases, in which case that reflection lands on top of itself and you'll see fewer holes.

SECTION 6

Worked Example: Folding and Punching a Square

Imagine a square piece of paper is folded in half vertically (left onto right), then folded in half again horizontally (top onto bottom). A hole is punched near the corner formed by the two open edges (not the corner where the creases meet). Which unfolded pattern is correct?

Two Folds, One Punch, Four Holes

1
Step 1 — Identify the folds

Fold 1 is vertical (left half onto right half). Fold 2 is horizontal (top half onto bottom half). Together, these two folds create 4 stacked layers.

2 folds = 4 layers
2
Step 2 — Locate the hole

The hole is punched near the corner made by the open edges — the corner farthest from where the two creases meet.

Hole = near the open-edge corner
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Step 3 — Unfold the last fold first

Undo the last fold first (the horizontal fold). Reflecting the hole across that crease creates a second hole directly above or below the first, at the same distance from the horizontal crease.

1 hole becomes 2 holes
4
Step 4 — Unfold the first fold

Now undo the first fold (the vertical fold). Reflecting both holes across that crease doubles them again, placing one hole near each of the four corners of the fully unfolded square.

Final answer: 4 holes, one near each corner

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Check Your Work

After picking an answer, check the hole count first. If you counted 2 folds, the answer should show 4 holes (unless the punch was on a crease). Then check that each hole is the same distance from its fold line as its mirror twin.

SECTION 7

Common Mistakes and How to Avoid Them

Even strong students fall into certain traps on folding-and-punching questions. Knowing these traps ahead of time can save you from picking the wrong answer.

Common Mistake	Why It Happens	How to Fix It
Unfolding in the wrong order	You reflect across the first fold before undoing the last one, which misplaces the holes.	Always undo the folds in reverse order — the last fold made should be the first crease you unfold in your mind.
Mirroring across only one fold line	You reflect the hole once and stop, forgetting there was more than one crease.	Check how many folds were made. Mirror the hole across every single fold line, one crease at a time.
Expecting only one hole	You forget that the punch went through several stacked layers of paper.	Count the layers: each fold doubles the number of layers, unless the hole sits right on a crease. The number of holes usually matches the number of layers punched.
Placing holes at the wrong distance	You draw the mirrored hole too close to or too far from the fold line.	A hole and its mirror image must always be the exact same distance from the fold line, just on opposite sides.

Common mistakes on paper-folding questions

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KEY TAKEAWAY

Think about hole-punching a stack of index cards instead of loose paper — the punch goes through every card in the stack at once, leaving a hole in the exact same spot on each one. Folded paper works the same way: the crease just determines where each layer's matching hole ends up once you spread everything back out flat.

SECTION 8

Connecting to Harder Spatial Questions

Paper-folding questions on the CogAT are one type of spatial reasoning challenge. As you get better, you may see harder versions of these ideas. The table below compares what you are learning now with more advanced tasks.

What You Learn Now	What Comes Next
Fold a square in half or into quarters, then punch one hole.	Fold in several directions, including diagonals, and punch more than one hole in different spots.
Track where 2 or 4 holes will land after unfolding.	Track 8 or more holes across multiple overlapping fold lines.
Work with a plain square sheet of paper.	Work with irregular shapes, or shapes that already have notches cut out before folding.
Choose the correct pattern from 4–5 answer choices.	Explain or sketch the reasoning behind exactly where each mirrored hole belongs.

Current skills vs. advanced spatial reasoning

The good news is that practicing folding-and-punching problems now builds a strong foundation. Skills like mental rotation and spatial visualization grow stronger every time you work through one of these problems. These skills are also used in science, engineering, art, and even medicine.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A square sheet of paper is folded in half once, then a single hole is punched through both layers. When the paper is fully unfolded, how many holes will you see, and how are they positioned?

Show answer

You will see 2 holes. Because the paper was folded in half, the punch passed through 2 layers, creating 2 holes. The two holes will be mirror images of each other across the fold line — the same distance from the crease, on opposite sides.

Question 2: BASIC

Paper is folded in half twice, forming a small square with 4 layers. One hole is punched through all four layers, near a corner. How many holes will appear when the paper is completely unfolded, and where will they be?

Show answer

4 holes will appear, one in each quarter of the square. Because both fold lines act as mirrors, the hole reflects across each crease, creating a symmetric hole in every quadrant of the unfolded square — each one the same distance from the center creases as the original punch.

Question 3: INTERMEDIATE

A square paper is folded once along its diagonal (corner to corner), forming a triangle. A hole is punched near one of the two acute-angle corners of the triangle, close to the folded edge (the diagonal). When unfolded, where will the second hole appear relative to the first?

Show answer

The second hole will appear on the opposite side of the diagonal fold line, mirrored at the same distance from the diagonal. Since the fold was made along the diagonal, the two holes are reflections of each other across that line, forming a pattern that is symmetric about the diagonal of the square.

Question 4: APPLIED

You want to make a paper design with exactly 4 evenly spaced holes near the center when unfolded. You start with a square piece of paper. How many times should you fold it in half, and where should you punch the hole?

Show answer

Fold the paper in half twice — first vertically, then horizontally — creating 4 layers. Punch the hole near the corner where the two creases meet (the center of the original square). Because that corner corresponds to the middle of the paper, unfolding reflects the hole across both fold lines, producing 4 holes clustered evenly around the center.

Question 5: CRITICAL THINKING

Two answer choices are being considered for a question that involved two folds. Choice A shows 4 holes, one near each corner of the square. Choice B shows only 2 holes, both near the center. Which choice could be correct, and what should you check to decide?

Show answer

Either choice could be correct, depending on where the hole was punched in the folded shape. If the hole was punched away from both creases, unfolding reflects it across each fold line separately, producing 4 separate holes — matching Choice A. If the hole was punched exactly on one of the fold lines, reflecting it across that particular crease lands it right back on itself, so only 2 distinct holes appear near the center — matching Choice B. The strategy is to check whether the punch touches a crease: holes on a fold line don't double when you unfold across that fold, while holes off the crease always do.

SUMMARY

Lesson Summary

Summary

Paper-folding questions test your spatial reasoning — the ability to picture how a square piece of paper looks after it is folded, punched with a hole, and then unfolded again. Every fold line acts like a mirror, and folding stacks the paper into layers. A single punch goes through all the layers at once, so it can turn into several holes once everything is unfolded.

To solve these problems, follow a clear process: count the folds, locate the hole, and unfold one crease at a time, last fold first, mirroring the hole across each fold line as you go. Use elimination to rule out wrong answers — especially any choice with the wrong number of holes or holes that aren't equally spaced from a fold line. With practice, your brain gets faster at "seeing" the fold, and these questions become much easier!