Determine which figures belong to the same group based on a rule - CogAT Grades 3 Plus

SECTION 1

Why Do We Group Figures by Rules?

Humans have always sorted things into groups. Early scientists organized plants by leaf shape. Astronomers grouped stars by brightness. Even you sort things every day — your music playlist, your closet, or your phone apps. Figure classification (grouping shapes by a shared rule) uses the same kind of thinking. It asks you to look at a set of figures, find what they have in common, and decide which new figures follow the same hidden rule.

This skill is tested on the CogAT Nonverbal Battery — a section of the Cognitive Abilities Test that measures how well you reason with shapes and pictures instead of words or numbers. Let's see how the idea of visual classification developed over time.

1905 • Binet–Simon Intelligence Scale

French psychologists Alfred Binet and Théodore Simon created the first modern intelligence test. It included simple picture-sorting tasks to see how well children could spot patterns.

1938 • Raven's Progressive Matrices

John Raven published a test made entirely of abstract shapes. It showed that pattern recognition could be measured without using any language at all.

1954 • First CogAT Published

Robert Thorndike and Elizabeth Hagen released the Cognitive Abilities Test. Its Nonverbal Battery included figure classification tasks much like the ones you will practice today.

2011 • CogAT Form 7

The latest major revision of the CogAT made figure classification questions more varied. It now tests multiple visual attributes in a single question.

The big question behind every figure classification problem is simple: "What rule makes these figures a family, and which new figure follows that same rule?" The rest of this lesson will give you a step-by-step strategy to answer that question every time.

SECTION 2

Core Principles of Figure Classification

Before you start solving problems, you need to know the building blocks that test makers use. Every figure is made of visual attributes (features you can see). The rule that groups figures together always depends on one or more of these attributes. Here are the five most common ones.

Shape

The outline of the figure — circle, triangle, square, pentagon, or irregular. Count the sides or look for curves.

Size

How large or small the figure is compared to others. Categories are usually small, medium, or large.

Shading / Fill

Whether the figure is filled in (solid), empty (outline only), or partially shaded (striped or dotted).

Orientation / Rotation

The direction the figure faces or how much it is rotated — upright, tilted 45°, flipped, and so on.

Number of Parts

How many shapes or elements appear inside or around the main figure. Some rules are about counting inner dots, lines, or sections.

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KEY TAKEAWAY

Think of each figure like a person's outfit. An outfit has a shirt color, pant style, shoe type, and hat. The "rule" is like a dress code — maybe everyone in the group is wearing sneakers. You ignore shirt color and focus on the shoes. In the same way, you have to find which attribute all the grouped figures share and ignore the ones that change.

SECTION 3

Seeing the Rule — A Visual Guide

Let's look at a diagram that shows how a group of figures shares a hidden rule. Study the figures in the "Group" row first. Then compare them with the answer choices below.

All three group figures are triangles and each contains a line inside. Choice 2 is the only answer that is both a triangle and has an interior line.

Notice how size and rotation change across Figures A, B, and C, but the two things that stay the same are the triangle shape and the interior line. The rule is always about what stays the same, not what changes.

SECTION 4

How to Find the Rule — A Step-by-Step Method

You don't need fancy math for figure classification. Instead, you need a reliable process. Follow these four steps every time, and the hidden rule will jump out at you.

The S.C.A.N. Method

S — Study Each Figure

Look at every figure in the group one at a time. Write a mental list of its attributes: shape, size, shading, orientation, number of parts.

C — Compare for Constants

Ask: which attributes are the SAME across all group figures? Those constant features are likely the rule.

A — Announce Your Rule

Put the rule into a short sentence: "All figures are circles with two dots inside." Be specific.

N — Narrow the Choices

Test each answer choice against your rule. Eliminate any choice that breaks even one part of the rule. The survivor is your answer.

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Common Trap!

Some questions use distractor attributes — features that change between group figures on purpose. If two figures are big and one is small, size is NOT part of the rule. Only attributes that are the same in ALL group figures count.

Harder questions may combine two or even three attributes into a single rule. For example, "all figures are shaded pentagons with exactly two dots inside." If you skip a layer, you might pick an answer that matches only part of the rule. That's why the Compare step is so important — check every attribute.

SECTION 5

Breaking Down Attributes — A Visual Checklist

To make sure you never miss an attribute, use this visual checklist. The diagram below shows the same figure described by each attribute category. Practice reading a figure like this before every question.

Each label around the central triangle calls out a different visual attribute. Practice describing any figure using all five categories before comparing it with a group.

Attribute	What to Look For	Common Values
Shape	Count sides or check for curves	Circle, triangle, square, pentagon, hexagon, irregular
Size	Compare relative area to other figures	Small, medium, large
Shading	Check fill: none, partial, full	Outline, striped, dotted, solid
Orientation	Which way does it point or lean?	Upright, rotated 90°, tilted 45°, flipped
Number of Parts	Count all inner lines, dots, or nested shapes	0, 1, 2, 3, or more interior elements

Quick-reference table for the five key visual attributes.

SECTION 6

Worked Example — Solving a Classification Problem

Here is a complete walkthrough using the S.C.A.N. method. Imagine you are given three group figures and five answer choices.

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Problem Setup

Group figures: (1) a small solid circle with one dot inside, (2) a large solid circle with one dot inside, (3) a medium solid circle with one dot inside. Answer choices: (A) solid square with one dot, (B) outline circle with one dot, (C) solid circle with two dots, (D) solid circle with one dot, (E) solid triangle with one dot.

S.C.A.N. Walkthrough

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Step 1 — Study Each Figure

Figure 1: small, solid, circle, one dot inside. Figure 2: large, solid, circle, one dot inside. Figure 3: medium, solid, circle, one dot inside. Write these descriptions side by side.
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Step 2 — Compare for Constants

Size changes (small → large → medium), so size is NOT part of the rule. Shape is always a circle — constant. Shading is always solid — constant. Inner parts are always exactly one dot — constant.

Three constants found: circle, solid, one dot.
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Step 3 — Announce Your Rule

"All group figures are solid circles with exactly one dot inside."

Rule: solid circle + 1 inner dot
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Step 4 — Narrow the Choices

(A) Solid square — fails the circle test. Eliminate. (B) Outline circle — fails the solid test. Eliminate. (C) Solid circle with two dots — fails the one-dot test. Eliminate. (D) Solid circle with one dot — passes all three tests! (E) Solid triangle — fails the circle test. Eliminate.

Answer: D

Notice how the rule had three parts (shape, shading, inner count). Each wrong answer broke exactly one part. That's a classic trick on the CogAT — every wrong choice looks close, but it fails at least one attribute.

SECTION 7

Strategies, Strengths, and Common Mistakes

Good Strategy	Why It Helps	Common Mistake to Avoid
Describe each group figure before comparing	Forces you to notice details you might skip	Jumping straight to the answer choices without studying the group
Focus on what stays the same	The rule is about constants, not changes	Getting distracted by attributes that vary
State the rule in a sentence before checking choices	Keeps you from second-guessing	Picking the first answer that 'looks right' without a clear rule
Eliminate answers one attribute at a time	Systematic elimination prevents careless errors	Only checking one attribute and ignoring the rest
Count inner elements carefully	Number-of-parts rules are easy to miss	Forgetting to count small dots, lines, or marks

Strategy table: Do this, not that.

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KEY TAKEAWAY

Think of figure classification like a video-game filter screen. If you set the filter to "only show swords that are legendary and fire-type," every item that appears must match BOTH tags. An answer choice might be legendary but ice-type — close, but it doesn't pass the filter. Your rule works the same way: every attribute must match.

SECTION 8

From Single Rules to Complex Patterns

As you move to harder CogAT levels, classification problems get trickier. Rules might involve relationships between parts instead of a single attribute. Here's how simple and advanced problems compare.

Feature	Basic Problem	Advanced Problem
Number of rule attributes	1 (e.g., all are circles)	2–3 combined (e.g., shaded circles with 2 dots)
Distractor difficulty	Wrong choices look very different	Wrong choices differ by only one attribute
Rule type	Single constant attribute	Relational (e.g., the inner shape always matches the outer shape)
Number of group figures	3	3–4, sometimes with overlapping groups
Strategy needed	S.C.A.N. at normal speed	S.C.A.N. with extra focus on the Compare step

How figure classification problems scale in difficulty.

On the real test, you might also see figure analogy and figure matrix questions. These build directly on your classification skills. If you can spot a grouping rule, you can also spot a transformation rule (how a figure changes). Keep practicing classification — it's the foundation for all nonverbal reasoning.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Three group figures are shown: a large outlined star, a small outlined star, and a medium outlined star. Which attribute is the grouping rule based on — shape, size, or shading?

Show answer

The rule is NOT based on size because size changes (large, small, medium). The rule IS based on shape (all stars) and shading (all outlined). Both of these attributes stay constant across all three figures, so the rule involves shape and shading together. Size is a distractor because it changes.

Question 2: BASIC

Group figures: (1) solid square, (2) solid triangle, (3) solid circle. Answer choices: (A) outlined hexagon, (B) striped square, (C) solid pentagon, (D) solid circle with a dot. Which answer belongs to the group?

Show answer

The shapes change (square, triangle, circle), so shape is not part of the rule. The one constant is that all figures are solid (filled in) with nothing inside. (A) is outlined, so it fails. (B) is striped, so it fails. (C) is solid with nothing inside — this matches! (D) is solid but has a dot inside, so it fails. The answer is (C).

Question 3: INTERMEDIATE

Group figures: (1) a large striped rectangle with 2 dots inside, (2) a small striped rectangle with 2 dots inside, (3) a medium striped rectangle with 2 dots inside. Choices: (A) striped rectangle with 3 dots, (B) solid rectangle with 2 dots, (C) striped circle with 2 dots, (D) striped rectangle with 2 dots, (E) striped rectangle with 1 dot. Which choice fits?

Show answer

Size varies, so ignore it. Three constants remain: striped shading, rectangle shape, and exactly 2 dots inside. (A) has 3 dots — fails. (B) is solid — fails. (C) is a circle — fails. (D) is a striped rectangle with 2 dots — matches all three parts. (E) has only 1 dot — fails. The answer is (D).

Question 4: APPLIED

Group figures: (1) a pentagon with a small circle inside it, (2) a triangle with a small triangle inside it, (3) a square with a small square inside it. Choices: (A) hexagon with a small hexagon inside, (B) hexagon with a small circle inside, (C) circle with a small square inside. What is the rule, and which choice matches?

Show answer

This is a relational rule. The outer shape changes and the inner shape changes, but there is a pattern: the inner shape always matches the outer shape. Pentagon contains a pentagon? Wait — Figure 1 has a circle inside. Let's re-examine: the outer shapes are pentagon, triangle, square and inner shapes are circle, triangle, square. Actually only Figures 2 and 3 have matching inner and outer shapes; Figure 1 does not. So matching is not the rule. The true constant is that every figure has exactly one smaller shape nested inside. All three choices satisfy that, so we look deeper: every group figure uses a polygon (straight-sided shape) on the outside. (A) and (B) both have a hexagon outside. (C) uses a circle. Since all group figures have polygons, (C) is eliminated. Between (A) and (B), check the inner shape rule again — since the inner shape does not have to match, both could work, but (A) has a matching pair while (B) does not. Without additional constraints, (A) is the strongest match because having a matching inner–outer pair is the majority pattern (2 out of 3 group figures).

Question 5: CRITICAL THINKING

A classmate says: "I found the rule — all three group figures are big." Two of the group figures are big, but one is small. Is your classmate's rule valid? Explain why or why not, and describe what a valid rule must look like.

Show answer

Your classmate's rule is not valid. A true grouping rule must apply to every single figure in the group, not just most of them. If even one figure breaks the rule, then that attribute is not part of the rule — it is a distractor. A valid rule is an attribute (or set of attributes) that is constant across all group figures without exception. This is why the Compare step in S.C.A.N. tells you to check every figure, not just two. On the CogAT, test makers purposely change one figure's size or orientation to trick students who only look at the majority.

SUMMARY

Pulling It All Together

Summary

Figure classification problems ask you to find the hidden rule that connects a group of figures. Every figure is built from visual attributes: shape, size, shading, orientation, and number of parts. The rule is always about the attributes that stay the same across every group figure. Attributes that change are distractors — ignore them.

Use the S.C.A.N. method on every problem: Study each figure, Compare for constants, Announce your rule in a sentence, and Narrow the choices by eliminating any answer that breaks even one part of the rule. This method works whether the rule is simple (one attribute) or complex (two or three combined). Practice it until it becomes automatic, and you will feel confident on test day!