Select the Missing Figure That Completes the Matrix - CogAT Grades 3 Plus

SECTION 1

Why Do We Use Figure Matrices?

Have you ever tried to finish a jigsaw puzzle and searched for that one missing piece? Figure matrix questions work in a similar way. You look at a grid of shapes, find the pattern, and pick the missing piece. People have been using puzzles like these for over a hundred years to understand how we think and solve problems.

1936 • Raven's Progressive Matrices

John C. Raven created one of the first matrix-style puzzles. He wanted a test that didn't depend on language so anyone could try it.

1954 • Nonverbal Testing Grows

Researchers started using figure matrices in schools. These puzzles helped measure reasoning skills without words.

1984 • CogAT Introduces Nonverbal Battery

The Cognitive Abilities Test (CogAT) added a Nonverbal Battery. Figure matrices became a key part of this section.

2011 • CogAT 7 Released

The newest version of the CogAT made figure matrices even more important. Millions of students take this test every year.

So why do tests use these puzzles? Because they measure your ability to see patterns and think logically — without needing to read or do math. The big question is: how do you find the right figure when you're staring at a grid of shapes? That's exactly what this lesson will teach you.

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Core Principles of Figure Matrices

A figure matrix is a grid (usually 3 rows × 3 columns) filled with shapes. One cell is empty, and your job is to pick the shape that belongs there. To do that, you need to understand four key ideas.

Row Rules

Look across each row from left to right. The shapes in a row follow the same pattern. If something changes in Row 1, the same kind of change happens in Rows 2 and 3.

Column Rules

Look down each column from top to bottom. Columns follow their own pattern too. Always check both rows AND columns before choosing your answer.

Attributes Change

Pay attention to shape, size, shading, number of objects, orientation (which way it faces), and position. One or more of these attributes changes across the row or column.

Eliminate Wrong Answers

After you find the pattern, check each answer choice. Cross out any choice that breaks a rule. The correct answer follows every rule you found.

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KEY TAKEAWAY

Think of a figure matrix like a tic-tac-toe board where every row and every column follows a recipe. If Row 1's recipe says "add one more dot each time," then Row 2 and Row 3 follow the same recipe. Your job is to figure out the recipe and bake the missing piece!

SECTION 3

Seeing the Pattern: A Visual Walkthrough

Let's look at a simple figure matrix so you can see how patterns work. In the diagram below, notice how each row and each column has a consistent rule for shape and shading.

Each row has the same three shapes (circle, square, triangle). Each row changes shading: Row 1 is filled, Row 2 is striped, Row 3 is empty outlines. The missing figure must be an empty outline triangle.

Here's what to notice. The row rule is about shading — every shape in a row has the same fill. The column rule is about shape — every figure in a column is the same shape. When you combine both rules, only one answer is possible.

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How Figure Matrix Patterns Work

Figure matrices don't use math equations, but they do follow a logical system. Think of each cell in the grid as having a set of attributes (features you can describe). Each attribute follows a rule across rows and down columns. Here are the six most common attributes you'll see.

The Six Key Attributes

Attribute	What to Look For	Example of a Rule
Shape	Circle, square, triangle, star, hexagon, etc.	Each column uses the same shape.
Size	Small, medium, large	Shapes get bigger from left to right.
Shading	Filled, striped, empty	Each row has the same shading.
Number	How many objects are in each cell	The count increases by one across each row.
Orientation	Rotation — which direction a shape points	The arrow rotates 90° clockwise in each column.
Position	Where an element sits inside the cell	A dot moves from top-left to center to bottom-right.

The six attributes that change most often in CogAT figure matrices.

On the CogAT, easy questions usually change only one attribute. Harder questions might change two or even three at the same time. That's why it's important to check each attribute one at a time instead of trying to see the whole pattern at once.

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PRO TIP

When you're stuck, try covering one row with your hand. Study just two rows first. Then uncover the third and see if the pattern holds. This makes hard puzzles feel easier.

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Common Pattern Types You'll See

Now that you know the six attributes, let's look at the most common ways they change. The diagram below shows four classic pattern types you'll run into on the CogAT.

The four most common pattern types: Progression (size changes), Rotation (turning), Counting (more elements), and Shading Change (fill changes from solid to striped to empty).

Many CogAT questions combine two or more of these pattern types in the same matrix. For example, shapes might get bigger across a row (progression) while also changing from filled to empty (shading change). When you notice one pattern, always ask yourself: "Is there a second pattern hiding here too?"

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Step-by-Step Worked Example

Let's walk through a complete matrix problem. Imagine a 3×3 grid where the bottom-right cell is missing. The shapes involve arrows of different sizes and orientations.

Finding the Missing Arrow

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Step 1 — Scan Row 1

Row 1 has three arrows: a small arrow pointing up, a medium arrow pointing right, and a large arrow pointing down. Notice two things are changing — size increases and the arrow rotates 90° clockwise each time.

Row rule: size goes small → medium → large; rotation goes up → right → down (90° clockwise each step).
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Step 2 — Verify with Row 2

Row 2 has: a small arrow pointing right, a medium arrow pointing down, and a large arrow pointing left. This follows the same two rules! Size still increases, and the arrow still turns 90° clockwise. But notice that Row 2 starts pointing right instead of up.

Confirmed: the row rule works for Row 2 as well.
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Step 3 — Check the Columns

Column 1 has: small up, small right, and something unknown. The size stays the same (small!) and the direction rotates 90° clockwise down the column. So the missing cell in Column 3 Row 3 follows the column rules too.

Column rule: size stays constant; direction rotates 90° clockwise from row to row.
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Step 4 — Apply Both Rules to the Missing Cell

Row 3 starts with a small arrow pointing down, then a medium arrow pointing left. Following the row rule: the last shape should be large (size increases) and should point up (90° clockwise from left is up). Let's double-check with the column: Column 3 has large down, large left — yes, the sizes are all large and the direction rotates 90° clockwise. Large arrow pointing up fits!

Answer: a large arrow pointing up.
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Step 5 — Eliminate Wrong Choices

Look at the answer choices. Any choice that is medium or small is wrong (breaks the size rule). Any choice that doesn't point up is wrong (breaks the rotation rule). Only one choice should be a large arrow pointing up. Select it with confidence!

Always double-check by plugging your answer back into the matrix.

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Best Strategies and Common Mistakes

Strategy (Do This ✓)	Common Mistake (Avoid This ✗)
Check rows AND columns before picking an answer.	Only checking rows and ignoring columns.
Look at one attribute at a time (shape, then size, then shading, etc.).	Trying to see the whole pattern at once — this leads to confusion.
Eliminate answer choices that break any rule.	Picking the first answer that "looks right" without testing it.
Plug your answer back into the matrix to verify.	Forgetting to double-check — careless errors happen on easy problems too!
Look for multiple patterns — harder problems often layer 2 or 3 rules.	Stopping after finding only one pattern and missing the second one.

Strategies vs. common mistakes on figure matrix questions.

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KEY TAKEAWAY

Think of solving a matrix like being a detective. You're collecting clues (patterns in rows and columns) and ruling out suspects (wrong answer choices). The more clues you find, the easier it is to catch the right answer. Never arrest a suspect (pick an answer) with only one clue!

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From Simple Matrices to Advanced Challenges

As you move through the CogAT, the matrices get trickier. Here's how the difficulty ramps up. Understanding these differences will help you prepare for the hardest questions.

Feature	Easier Matrices	Harder Matrices
Number of changing attributes	1 attribute changes (e.g., only shape)	2–3 attributes change at the same time
Pattern type	Simple progression or repetition	Layered rules, e.g., rotation + shading + size
Distractors (wrong answers)	Clearly wrong — different shape or color	Tricky — almost right but one detail is off
Matrix size	2×2 or 3×3 with simple shapes	3×3 with complex, multi-part figures
Missing cell location	Usually bottom-right corner	Can be anywhere in the grid

How figure matrix difficulty increases on the CogAT.

Don't worry if the harder matrices seem overwhelming. The same step-by-step approach works every time. You just need to be more patient and thorough. The skills you build solving these matrices — pattern recognition, logical reasoning, and process of elimination — are useful in math, science, coding, and many other subjects.

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Practice Problems

Question 1: CONCEPTUAL

In a 3×3 figure matrix, you notice that every shape in Row 1 is a circle, every shape in Row 2 is a square, and Row 3 has a triangle in the first two cells. The bottom-right cell is missing. What shape should go there, and why?

Show answer

The missing shape is a triangle. The row rule is that each row contains the same shape. Row 1 is all circles, Row 2 is all squares, and Row 3 follows the same pattern with all triangles. Since the first two cells of Row 3 are triangles, the third cell must also be a triangle to keep the pattern consistent.

Question 2: BASIC CALCULATION

A 3×3 matrix shows dots inside each cell. Row 1 has 1, 2, and 3 dots. Row 2 has 2, 3, and 4 dots. Row 3 has 3, 4, and a missing cell. How many dots should be in the missing cell? Describe the row rule and the column rule.

Show answer

The missing cell should have 5 dots. The row rule is that the dot count increases by 1 from left to right (for example, Row 3 goes 3 → 4 → 5). The column rule is that the dot count also increases by 1 from top to bottom (Column 3 goes 3 → 4 → 5). Both rules agree, confirming the answer is 5 dots.

Question 3: INTERMEDIATE

A matrix has two patterns happening at once. Across each row, the shape rotates 90° clockwise. Down each column, the shading changes from filled to striped to empty. The missing cell is in Row 3, Column 3. Row 3, Column 1 has an upward-pointing filled arrow. Row 3, Column 2 has a right-pointing striped arrow. What does the missing figure look like?

Show answer

Wait — let's re-read the rules. The rotation rule says the shape turns 90° clockwise across each row: up → right → down. The shading rule says shading changes down each column, but across Row 3 we also see filled → striped → empty. So the missing figure is a downward-pointing empty (outline-only) arrow. It must point down (rotation rule) and be empty (shading rule). Both patterns must be satisfied at the same time.

Question 4: APPLIED

You're looking at a 3×3 matrix on the CogAT. The answer choices are: (A) a small filled circle, (B) a large empty triangle, (C) a large filled triangle, (D) a large empty circle, (E) a small empty triangle. You've figured out that the row rule requires a large shape and the column rule requires an empty triangle. Which answer do you choose, and how did you eliminate the others?

Show answer

The correct answer is (B), a large empty triangle. Here's how to eliminate: Choice (A) fails because it's small, not large, and it's a circle, not a triangle. Choice (C) is a large triangle but it's filled, not empty — that breaks the column rule. Choice (D) is large and empty but it's a circle, not a triangle. Choice (E) is an empty triangle but it's small, not large. Only Choice (B) satisfies both rules: it's large (row rule) and an empty triangle (column rule).

Question 5: CRITICAL THINKING

A 3×3 matrix changes three attributes at the same time: shape (circle → square → triangle across rows), size (small → medium → large down columns), and shading (filled → striped → empty across rows). The missing cell is in Row 2, Column 2 — the center of the grid. Describe how you would figure out the answer, and explain what the missing figure looks like.

Show answer

First, break the problem into one attribute at a time. Shape rule across Row 2: circle → ? → triangle, so the middle cell must be a square. Size rule down Column 2: since size increases down columns and this is the middle row, the size is medium. Shading rule across Row 2: filled → ? → empty, so the middle cell must be striped. Putting it all together, the missing figure is a medium striped square. By tackling each attribute separately, you avoid confusion even when three rules are layered together.

SUMMARY

Lesson Summary

Summary

Figure matrix questions ask you to find the missing piece in a grid of shapes by spotting the patterns. Every matrix has row rules and column rules that control how the shapes change. The six key attributes to watch are shape, size, shading, number, orientation, and position. Always check one attribute at a time, look at both rows and columns, and use process of elimination to rule out wrong answer choices.

The four most common pattern types are progression, rotation, counting, and shading change. Harder questions layer multiple patterns together, but the same step-by-step strategy works every time: scan, identify, verify, apply, and eliminate. Practice these steps and you'll tackle any figure matrix with confidence!