Use attributes (shape, size, orientation, shading, parts)...

SECTION 1

Why Do We Classify Figures?

Humans have been sorting objects into groups for thousands of years. Think about it — you already classify things every day. You sort your clothes by type, your apps by folder, and your playlist by genre. Figure classification works the same way, except you are sorting shapes and pictures instead of songs or shirts.

On the TACHS test, you will see a group of figures that belong together. Your job is to figure out why they belong together. Then you pick a new figure that fits the same rule. To do this, you need to look at attributes (features or characteristics) of each figure.

Ancient

Early Geometry

Ancient Egyptians and Greeks classified shapes like triangles and circles to build pyramids and study the stars.

1700s

Linnaeus & Classification

Carl Linnaeus created a system to classify living things by their features. This same idea — grouping by shared attributes — applies to figures.

1900s

IQ & Aptitude Tests

Psychologists began using figure classification puzzles to measure reasoning ability. These puzzles test how well you spot patterns.

Today

TACHS & Standardized Tests

Tests like the TACHS use figure classification to see how well students can find rules and apply them to new situations.

The big question figure classification asks is: "What rule connects these figures, and which new figure follows the same rule?" Let's learn the five attributes you need to check every time.

SECTION 2

The Five Key Attributes

Every figure on the TACHS can be described using five main attributes (features you can observe). When you look at a group of figures, check each attribute one at a time. If all the figures in the group share something — like they are all triangles, or all shaded — that shared feature is part of the rule.

Shape

What kind of figure is it? Circle, triangle, square, pentagon, hexagon, star, arrow, or something else? Count the sides and corners.

Size

How big or small is the figure compared to others? Look for large, medium, or small versions of the same shape.

Orientation

Which direction does the figure point or face? Is it rotated, flipped, or tilted? A triangle pointing up is different from one pointing down.

Shading

Is the figure filled in (solid), empty (outline only), half-shaded, or patterned with stripes, dots, or crosshatching?

Parts

Does the figure have extra pieces inside or attached? Look for inner shapes, lines, dots, tails, or divisions.

✦ KEY TAKEAWAY

Think of the five attributes like a checklist at a pet shelter. You might sort dogs by breed (shape), size (big or small), which way they face (orientation), fur color (shading), and whether they wear a collar or tag (parts). Checking each feature one by one helps you figure out the group rule.

SECTION 3

Seeing the Five Attributes

The diagram below shows a single shape changed by each of the five attributes. Study it carefully. Notice how the same basic triangle looks completely different when you change just one attribute at a time.

Each box shows one attribute changed on a triangle. On the TACHS, figures in a group will share one or more of these attributes. The bottom section shows a quick-check example.

In the diagram above, notice how each attribute changes the way the triangle looks. On the TACHS, the figures in a group might share just one attribute (like all being circles) or they might share two or three at the same time (like all being large, shaded triangles). Your job is to check each attribute carefully until you find the rule.

SECTION 4

How the Classification Process Works

Figure classification is not about math formulas. It is about a step-by-step thinking process. Every time you see a classification question, follow these four steps.

The Four-Step Process

Step 1 — Observe. Look at each figure in the given group. Do not rush. What do you notice first?
Step 2 — Compare. Check each of the five attributes across all the figures. Ask: do they share the same shape? Same size? Same shading?
Step 3 — State the rule. Say the rule to yourself in a simple sentence. Example: "All figures are small, shaded circles."
Step 4 — Test the answer choices. Check each answer choice against your rule. The correct answer matches every part of the rule.

⚠️ Common Trap

Sometimes two answer choices match some of the attributes but not all. Always check every attribute before picking your answer. A figure that matches the shape and size but has the wrong shading is not the right answer.

Let's see how this process works with a flowchart.

Follow these four steps every time. Observe, compare attributes, state the rule, then test each answer choice. If a choice does not match, eliminate it and try the next one.

SECTION 5

A Closer Look at Each Attribute

Let's explore each attribute in more detail. Understanding exactly what to look for will help you work faster and more accurately on test day.

Summary of the five attributes with examples of what to look for.

Attribute	What to Look For	Common Variations	Example Rule
Shape	Count the sides and corners. Is it curved or straight-edged?	Circle, triangle, square, rectangle, pentagon, hexagon, star, arrow, heart	"All figures are four-sided shapes."
Size	Compare figures to each other. Are they all the same size, or are some bigger?	Large, medium, small	"All figures are small."
Orientation	Which direction does the figure point? Is it rotated or flipped?	Up, down, left, right, tilted 45°, rotated 90°	"All arrows point to the right."
Shading	Is the figure filled in, empty, or patterned?	Solid (filled), empty (outline), half-shaded, striped, dotted, crosshatched	"All figures are half-shaded."
Parts	Does the figure have extra elements inside or outside?	Inner shape, dividing line, dot in center, tail, border, multiple pieces	"All figures have a dot in the center."

💡 Pro Tip

On harder questions, the rule uses two or three attributes at once. For example: "All figures are large, striped triangles with a dot inside." That's four attributes! Always check all five, even if the first one seems obvious.

All three figures in the group are large, striped circles with a horizontal line through the center. Answer C matches all four attributes. A fails on shading, B fails on shape, and D fails on size.

SECTION 6

Worked Example: Finding the Rule

Let's walk through a complete TACHS-style question together. Imagine you see three figures in a group: a small solid square, a small solid triangle, and a small solid circle. The answer choices are: (A) a large solid square, (B) a small empty hexagon, (C) a small solid pentagon, (D) a small striped circle.

TACHS Figure Classification — Step by Step

Step 1 — ObserveLook at the three figures in the group. Figure 1 is a small solid square. Figure 2 is a small solid triangle. Figure 3 is a small solid circle. They are three different shapes.

Step 2 — Compare AttributesShape: Square, triangle, circle — these are different. So shape is NOT the rule. Size: All three are small — this could be part of the rule. Orientation: None are rotated in a special way — not relevant here. Shading: All three are solid (filled in) — this is probably part of the rule. Parts: None of them have extra parts inside — they are all plain.

Step 3 — State the RuleThe rule is: "All figures are small and solid (filled in)." The shape does not matter.

Rule: small + solid

Step 4 — Test the Choices(A) Large solid square — size is wrong (large, not small). ✗ (B) Small empty hexagon — shading is wrong (empty, not solid). ✗ (C) Small solid pentagon — small ✓, solid ✓. This matches! (D) Small striped circle — shading is wrong (striped, not solid). ✗

Answer: (C) small solid pentagon

✦ WHY IT WORKS

Notice that the shapes were all different. That was a clue that shape was not part of the rule. When an attribute changes from figure to figure, it is usually not part of the rule. When an attribute stays the same across all figures, it is probably part of the rule.

SECTION 7

Strategies and Common Mistakes

Strategies vs. common mistakes on figure classification questions.

Good Strategy	Common Mistake
Check all five attributes before deciding.	Picking the first answer that looks similar without checking every attribute.
Notice what is the SAME across all group figures.	Focusing on what is different between the figures (differences are distractors).
Eliminate answers that fail even one part of the rule.	Keeping an answer that matches most (but not all) attributes.
State the rule out loud or in your head before looking at answers.	Jumping straight to the answer choices without forming a clear rule.
Look for hidden parts like inner shapes, dots, or dividing lines.	Ignoring small details like a tiny dot or line inside a figure.

✦ KEY TAKEAWAY

Think of figure classification like a game of "Guess Who?" You flip down faces that don't match a feature. In the same way, you eliminate answer choices that don't match every attribute in your rule. One wrong feature means that choice is out.

SECTION 8

Simple Rules vs. Complex Rules

Easy TACHS questions usually have rules based on just one attribute. Harder questions combine two, three, or even four attributes into a single rule. Here is how they compare.

Simple vs. complex classification rules on the TACHS.

Feature	Simple Rule (Easier)	Complex Rule (Harder)
Number of attributes	1 attribute (e.g., all circles)	2–4 attributes (e.g., all large, striped circles with a dot)
What changes	Most attributes vary between figures	Very few attributes vary — most stay the same
Trickiness	Easy to spot the rule quickly	Distractors match some but not all attributes
Best strategy	Quick scan for the one shared feature	Careful checklist of all five attributes

As you practice more, you will get faster at spotting multi-attribute rules. The five-attribute checklist (shape, size, orientation, shading, parts) works no matter how hard the question is. Stick to the process and you'll do great.

🚀 Looking Ahead

Some advanced tests add even trickier attributes like symmetry (is the figure the same on both sides?), the number of enclosed regions, or the relationship between figures (like one shape always sitting inside another). If you master the five core attributes now, those extras will feel natural later.

SECTION 9

Practice Problems

Try these five practice problems. For each one, use the four-step process: observe, compare attributes, state the rule, then test the choices.

PROBLEM 1 — CONCEPTUAL

A group contains three figures: a large empty triangle, a large empty square, and a large empty circle. Which attribute do they all share? (A) Same shape (B) Same size and shading (C) Same orientation (D) Same number of sides

PROBLEM 2 — BASIC CALCULATION

A group contains three small, solid pentagons. The answer choices are: (A) a small, solid hexagon (B) a small, solid pentagon (C) a large, solid pentagon (D) a small, empty pentagon. Which one belongs in the group?

PROBLEM 3 — INTERMEDIATE

A group has three figures: a medium striped arrow pointing right, a medium striped arrow pointing left, and a medium striped arrow pointing up. The answer choices are: (A) a medium solid arrow pointing down (B) a medium striped triangle pointing right (C) a small striped arrow pointing down (D) a medium striped arrow pointing down. Which one belongs?

PROBLEM 4 — APPLIED

A group contains: a large solid circle with a vertical line through the center, a large solid square with a vertical line through the center, and a large solid triangle with a vertical line through the center. The choices are: (A) a large solid hexagon with a horizontal line through it (B) a large solid diamond with a vertical line through it (C) a small solid pentagon with a vertical line through it (D) a large empty star with a vertical line through it. Which one belongs?

PROBLEM 5 — CRITICAL THINKING

A group contains three figures. Figure 1 is a small dotted triangle pointing up with a circle inside. Figure 2 is a large dotted hexagon pointing up with a circle inside. Figure 3 is a medium dotted square pointing up with a circle inside. Which attributes are part of the rule, and which are NOT? Choose the answer that correctly identifies the rule. (A) Same shape, same shading, circle inside (B) Same shading, pointing up, circle inside (C) Same size, same shading, pointing up (D) Same shape, same size, circle inside

SUMMARY

Summary: Classifying Figures by Attributes

Figure classification on the TACHS asks you to find a rule that connects a group of figures. To find the rule, check five key attributes: shape (what kind of figure — circle, triangle, etc.), size (large, medium, or small), orientation (which direction it points), shading (solid, empty, striped, or dotted), and parts (extra elements inside or attached to the figure).

Use the four-step process every time: observe the group, compare attributes to find what stays the same, state the rule in a simple sentence, then test each answer choice against your rule. Attributes that stay the same are part of the rule. Attributes that change across the group are not. Eliminate any answer that fails even one part of the rule, and you'll find the correct match.