Identify patterns across rows and columns in a figure matrix - CogAT Grades 3 Plus

SECTION 1

Where Do Figure Matrices Come From?

Have you ever noticed a pattern in floor tiles or a quilt? People have been fascinated by visual patterns for thousands of years. In the 1900s, psychologists started using grids of shapes to measure how well someone can think without words. These grids are called figure matrices. They are a key part of tests like the CogAT.

1936 • Raven's Progressive Matrices

John C. Raven created a test using grids of shapes. Each grid had a missing piece, and test-takers had to find the pattern to fill it in. This was one of the first nonverbal reasoning tests.

1968 • CogAT Is Born

Researchers at the University of Iowa developed the Cognitive Abilities Test (CogAT). It included figure matrix questions in its Nonverbal Battery to test reasoning without language.

1990s • Matrices Go Digital

Computers allowed researchers to study exactly how people solve matrix puzzles. They found that the best solvers check both rows and columns before picking an answer.

Today • Modern CogAT Testing

Figure matrices are still a major part of the CogAT Nonverbal Battery. Students across the country solve these puzzles to show their reasoning skills.

The big question these puzzles answer is: Can you figure out a rule just by looking at pictures? No reading or math is needed. You only need your eyes and your brain. Let's learn exactly how to do it.

SECTION 2

Core Principles of Figure Matrices

A figure matrix is a grid (usually 3 × 3) filled with shapes. One cell is empty, often the bottom-right corner. Your job is to find the missing figure. To do this, you need to understand a few key ideas.

Row Rule

Look at each row from left to right. The shapes in a row follow a pattern—something changes (or stays the same) in a predictable way.

Column Rule

Look at each column from top to bottom. The same type of pattern should repeat down each column, too.

Attributes Change

Common things that change include shape, size, number, shading (filled, striped, or empty), rotation, and position inside the cell.

Double-Check Both Directions

The correct answer must satisfy the row rule AND the column rule at the same time. If it only works in one direction, it's wrong.

Eliminate Wrong Answers

After you figure out the rules, compare each answer choice. Cross out any choice that breaks a rule. Usually only one choice fits perfectly.

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KEY TAKEAWAY

Think of a figure matrix like a game of Sudoku for shapes. In Sudoku, every row and column must follow a number rule. In a figure matrix, every row and column must follow a shape rule. Find the rule, and you find the missing piece!

SECTION 3

Seeing the Pattern: A Visual Guide

Let's look at a sample figure matrix. In the diagram below, you'll see a 3 × 3 grid. Each row has three shapes, and each column also has three shapes. The bottom-right cell has a question mark—that's the one you need to solve.

In this matrix, the shape changes across each row (circle → square → triangle). The shading changes down each column (empty → striped → filled). The size also increases going down. So the missing piece is a large, filled triangle.

Notice how you can discover the answer two ways. Reading across Row 3, the shape pattern says the next shape should be a triangle. Reading down Column 3, the shading pattern says the shape should be filled, and the size pattern says it should be large. Both directions point to the same answer: a large, filled triangle.

SECTION 4

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). The pattern is a rule that tells you how each attribute changes.

Common Attribute Types

Attribute	What to Look For	Example Change
Shape	Circle, square, triangle, star, arrow, etc.	Circle → Square → Triangle across a row
Size	Small, medium, large	Gets bigger going down a column
Shading	Empty, striped/hatched, solid/filled	Empty → Striped → Filled down a column
Number	How many shapes are in the cell	1 → 2 → 3 across a row
Rotation	Which way the shape is turned	Rotates 90° clockwise each step
Position	Where in the cell the shape sits	Moves from top-left to center to bottom-right

Six attributes commonly tested in CogAT figure matrices

Common Rule Types

Constant — the attribute stays the same across the row or column (e.g., all shapes in Row 1 are circles).
Progression — the attribute changes step by step (e.g., small → medium → large).
All Different — each cell in the row or column has a different value (e.g., one circle, one square, one triangle — in any order).
Overlay or Combination — the first two cells combine in some way to create the third (harder puzzles use this).

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Pro Tip

On the CogAT, most matrices test two or three attributes at once. For example, the shape might change across rows while the shading changes down columns. Train yourself to check multiple attributes every time.

SECTION 5

Step-by-Step Strategy

Having a plan makes these puzzles much easier. Here is a step-by-step strategy you can use on every figure matrix question.

This flowchart shows the seven steps for solving any figure matrix. Start by finding the row rule, then find the column rule, predict your answer, and finally match it to the choices.

Let's talk about each step in more detail. In Steps 1–3, you look at the rows. Ask yourself: what stays the same and what changes as I go from left to right? Say the rule out loud or in your head, like "the shape rotates 90 degrees clockwise." Then check that this rule works in every row.

In Steps 4–5, you do the same thing for columns. Go from top to bottom. Often, a different attribute changes in the column direction. For instance, the shape might change across rows while the shading changes down columns.

In Step 6, you combine both rules to predict what the missing figure should look like. Then in Step 7, you compare your prediction to the answer choices and pick the best match.

SECTION 6

Worked Example: Solving a Matrix

Let's walk through a complete example using our strategy. Imagine a 3 × 3 matrix where:

	Column 1	Column 2	Column 3
Row 1	1 small empty circle	2 small empty circles	3 small empty circles
Row 2	1 medium striped square	2 medium striped squares	3 medium striped squares
Row 3	1 large filled triangle	2 large filled triangles	?

The matrix we need to solve

Finding the Missing Figure

1
Step 1 — Scan Row 1

Row 1 has 1 circle, then 2 circles, then 3 circles. The shape stays the same (circle). The size stays the same (small). The shading stays the same (empty). Only the number changes: 1 → 2 → 3.

Row rule: the number increases by 1
2
Step 2 — Verify the Row Rule

Check Row 2: 1 square → 2 squares → 3 squares. Yes, the number increases by 1. Check Row 3: 1 triangle → 2 triangles → ?. The row rule should give us 3 in the missing cell.

Row rule confirmed: number goes 1 → 2 → 3
3
Step 3 — Scan Column 1

Column 1 has: small empty circle, then medium striped square, then large filled triangle. Three attributes change going down. Shape: circle → square → triangle. Size: small → medium → large. Shading: empty → striped → filled.

Column rules: shape, size, and shading each change down each column
4
Step 4 — Predict the Answer

The missing cell is in Row 3, Column 3. From the row rule, we need 3 shapes. From Row 3, the shape is a triangle. From Row 3, the size is large and the shading is filled. So we need 3 large filled triangles.

Prediction: 3 large filled triangles
5
Step 5 — Verify with Columns

Column 3 currently has: 3 small empty circles, 3 medium striped squares, and ?. Shape goes circle → square → triangle ✓. Size goes small → medium → large ✓. Shading goes empty → striped → filled ✓. Number stays 3 throughout Column 3 ✓. Everything checks out!

Answer: 3 large filled triangles ✓

SECTION 7

Common Traps and How to Avoid Them

Test makers design wrong answer choices to trick you. If you know the traps, you can avoid them. Here are the most common mistakes and how to beat them.

Trap	What Happens	How to Avoid It
Only checking rows	You find a choice that fits the row, but it breaks the column rule.	Always check BOTH rows and columns before choosing.
Ignoring one attribute	You match the shape correctly but miss a size or shading change.	Go through the attribute checklist: shape, size, number, shading, rotation, position.
Rushing	You pick the first answer that looks close without verifying.	Predict your answer BEFORE looking at the choices.
Distractor answers	A wrong choice has the right shape but wrong shading or size.	Compare each choice attribute by attribute. Eliminate any choice that fails even one rule.

Common traps in figure matrix questions

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KEY TAKEAWAY

Think of it like ordering a pizza. If you ask for large, pepperoni, and thin crust, you won't accept a pizza that is large and pepperoni but thick crust. Every attribute must match. If even one thing is wrong, it's not your answer.

SECTION 8

Beyond the Basics: Advanced Pattern Types

As you get better, you'll see harder patterns on the CogAT. These advanced patterns go beyond simple progressions. Here's how basic patterns compare to advanced ones.

Feature	Basic Patterns	Advanced Patterns
Number of attributes	1–2 attributes change	3 or more attributes change at once
Rule type	Simple progression (A → B → C)	Overlay, XOR, or distribution of 3 rules
Overlay / XOR	Not used	Cell 1 and Cell 2 combine to make Cell 3
Example	Shape changes: circle → square → triangle	Shapes in cells 1 and 2 overlap; only non-shared parts appear in cell 3

Basic vs. advanced figure matrix patterns

An overlay pattern means you stack or combine shapes from two cells to get the third. An XOR pattern (short for "exclusive or") means you keep only the parts that appear in one cell but not both. Don't worry if these seem tricky now. The more matrices you practice, the easier they become.

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Looking Ahead

On the CogAT, higher-level questions often combine simple rules in creative ways. If you master the basics of checking rows and columns for each attribute, you'll have a strong foundation for even the hardest puzzles.

SECTION 9

Practice Problems

Try these five problems on your own. For each one, think about the row rule and the column rule before picking your answer. Then check the answer and explanation below.

Question 1: CONCEPTUAL

In a 3 × 3 figure matrix, you notice that every row contains one circle, one square, and one triangle. Row 3 already has a circle in Column 1 and a triangle in Column 2. What shape goes in Column 3 of Row 3?

Show answer

The answer is a square. The row rule is "all different" — each row must contain one of each shape. Since Row 3 already has a circle and a triangle, the only shape missing is a square. You can verify by checking the column: if Column 3 in the other rows has a circle and a triangle, then a square fits the column rule too.

Question 2: BASIC CALCULATION

A matrix has these shapes across Row 1: a star with 4 points, a star with 5 points, a star with 6 points. Row 2 follows the same number pattern but with arrows instead of stars. Row 2 shows an arrow with 1 head, an arrow with 2 heads, and what? How many heads does the last arrow have?

Show answer

The answer is 3 heads. The row rule is a progression: the number of features increases by 1 each step. In Row 1, the points go 4 → 5 → 6 (increase by 1). In Row 2, the heads go 1 → 2 → 3 (increase by 1). Notice that the actual starting number doesn't need to be the same across rows — the rule (adding 1) is what matters.

Question 3: INTERMEDIATE

In a matrix, Row 1 has: a small empty hexagon, a medium empty hexagon, a large empty hexagon. Column 1 has: a small empty hexagon, a small striped pentagon, a small filled square. What goes in the cell at Row 2, Column 2 (the center cell)?

Show answer

The answer is a medium striped pentagon. The row rule tells you that size increases from left to right (small → medium → large). Column 1 tells you the column rule: shape changes (hexagon → pentagon → square), and shading changes (empty → striped → filled). For Row 2, Column 2, we need the medium size (from the row rule's second position) combined with the striped shading and pentagon shape (from Row 2's attributes). This gives a medium striped pentagon.

Question 4: APPLIED

You see a 3 × 3 matrix on the CogAT. In Row 1, a black arrow points up, then right, then down. In Row 2, a gray arrow points up, then right, then down. In Row 3, a white arrow points up, then right, then ?. You must pick from five choices: (A) white arrow pointing left, (B) white arrow pointing down, (C) gray arrow pointing down, (D) white arrow pointing up, (E) black arrow pointing down. Which is correct?

Show answer

The answer is (B), a white arrow pointing down. The row rule is that the arrow rotates 90° clockwise each step: up → right → down. This rule holds in Rows 1 and 2 and predicts "down" for Row 3's last cell. The column rule is that color changes down each column: black → gray → white. Column 3 needs white. Choice (B) is the only option that is both white and pointing down. Choice (A) is wrong direction. Choice (C) is wrong color. Choice (D) is wrong direction. Choice (E) is wrong color.

Question 5: CRITICAL THINKING

In a tricky matrix, each cell contains two overlapping shapes. Row 1 has: (circle + square), (square + triangle), (triangle + circle). Row 2 has: (star + diamond), (diamond + heart), (heart + star). Row 3 has: (cross + arrow), (arrow + moon), (?). What two shapes should appear in the missing cell? Also explain the pattern rule.

Show answer

The answer is (moon + cross). The pattern rule is a rotation of three elements within each row. In Row 1, the three shapes are circle, square, and triangle. Each cell pairs two of them, and the pairing shifts: (A+B), (B+C), (C+A). In Row 2: star, diamond, heart follow the same shifting pattern: (S+D), (D+H), (H+S). In Row 3: cross, arrow, moon shift the same way: (cross+arrow), (arrow+moon), (moon+cross). You can also verify by columns: Column 1 always has the first and second elements, Column 2 always has the second and third, and Column 3 always has the third and first.

SUMMARY

Lesson Summary

Summary

A figure matrix is a grid of shapes where one cell is missing. To find the missing piece, you look for row rules (patterns going left to right) and column rules (patterns going top to bottom). The key attributes to watch for include shape, size, number, shading, rotation, and position. Common rule types are constant (stays the same), progression (changes step by step), and all different (each cell in a row or column has a unique value).

Always follow the seven-step strategy: scan rows, name the row rule, verify it, scan columns, name the column rule, predict the answer, and then match your prediction to the answer choices. Remember that the correct answer must satisfy both the row rule and the column rule. Watch out for traps like distractor answers that match one rule but break another. With practice, you'll spot patterns quickly and confidently on the CogAT!