Determine which figures belong to the same group based on a rule - CogAT Grades K 2

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Why Do We Group Things Together?

People have always liked to sort things into groups. Think about how you put your toys away. You might put all the cars in one box and all the blocks in another box. That is grouping! When we group things, we use a rule to decide what goes together.

Long Ago • People Sorted Food

Long ago, people sorted berries by color. Red berries went in one pile. Blue berries went in another.

In School • Kids Learn to Sort Shapes

Teachers started asking kids to sort shapes by looking at things like size, color, and number of sides.

Today • The CogAT Test

Today, tests like the CogAT ask you to look at shapes and find which ones follow the same rule. This shows how well you can think and solve puzzles!

So the big question is: How do we figure out the rule that makes some shapes belong together? Let's learn how!

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The Big Ideas About Grouping

When you see shapes on a test, you need to find what they have in common. Common means "the same." Here are the big ideas that help you group shapes together.

Shape

Is it a circle, a square, a triangle, or something else? Shapes that look the same might belong together.

Size

Is it big or small? Sometimes the rule is about how big or how small the figures are.

Color or Shading

Is the shape dark, light, striped, or empty inside? The fill of a shape can be the rule!

Number of Parts

How many shapes or dots are inside? Counting the parts can help you find the rule.

Direction or Position

Is the shape pointing up, down, left, or right? Where it sits or points can be the rule too.

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KEY TAKEAWAY

Think of it like sorting your crayons. You could sort them by color (all reds together), by size (big crayons and little crayons), or by whether they are sharp or flat. The same idea works with shapes on a test!

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See How Grouping Works

Let's look at a picture that shows how shapes can belong to the same group. In this picture, the top row shows shapes that follow a rule. Your job is to find which shape in the bottom row follows that same rule.

The top row shows three cyan circles. The rule is: circle shape and cyan color. Answer C is a cyan circle, so it belongs to the group!

Did you see the rule? All the shapes in the group are circles and they are all the same color. Choice C is a circle and it is the same color. So C belongs!

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How to Find the Rule Step by Step

Here is a simple way to find the rule. You can follow these steps every time you see a grouping question.

The 3-Step Trick

Step 1 — Look: Look at the shapes in the group. What do you see? Say out loud what each shape looks like.
Step 2 — Compare: Ask yourself, "What is the SAME about all of them?" Check shape, size, color, shading, and direction.
Step 3 — Match: Look at the answer choices. Find the one that follows the same rule you found.

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Helpful Tip!

Sometimes two things might look the same, but only one follows all parts of the rule. For example, a shape might be the right color but the wrong shape. Always check everything!

Let's think about what kinds of rules you might see. The rule could be about just one thing (like all circles) or about two things (like all big circles). The harder questions use rules with two or even three things at once.

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Different Kinds of Rules

Let's look at different types of rules you might see on the test. Each kind of rule asks you to pay attention to something different about the shapes.

This diagram shows four types of rules. The shape rule groups all circles. The shading rule groups all filled-in shapes. The size rule groups all big shapes. The number rule groups shapes with 2 dots inside.

Rule Type	What to Look For	Example
Shape	Are they all circles? All squares? All triangles?	Three triangles of different sizes
Size	Are they all big? All small? All the same size?	Three small shapes
Shading	Are they all filled in? All empty? All striped?	Three shapes with stripes
Number	Do they all have the same number of dots or parts inside?	Three shapes, each with 3 dots
Direction	Are they all pointing the same way?	Three arrows all pointing up

Common types of grouping rules on the CogAT

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Let's Solve One Together!

Let's walk through a problem step by step. Imagine you see three shapes in a group: a big black square, a big black triangle, and a big black circle. Now you need to pick which answer choice belongs with them.

Example: Big Black Shapes

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Step 1 — Look at the Group

The group has a big black square, a big black triangle, and a big black circle. Let's look carefully at each one.
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Step 2 — Ask: What Is the Same?

Are they the same shape? No — one is a square, one is a triangle, one is a circle. So the rule is NOT about shape. Are they the same size? Yes! They are all big. Are they the same color? Yes! They are all black (filled in).

The rule is: big AND filled in black.
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Step 3 — Check the Answer Choices

Choice A: A small black circle. It is black but NOT big. ✗ Choice B: A big white square. It is big but NOT black. ✗ Choice C: A big black diamond. It IS big AND it IS black. ✓ Choice D: A small white triangle. It is not big and not black. ✗

The answer is C! The big black diamond follows both parts of the rule: big and black.

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Remember!

Some rules have two parts. In this problem, the answer had to be big AND black. If it was only one of those things, it did not belong!

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What Makes These Questions Easy or Hard?

Some grouping questions are easier, and some are harder. Let's look at what makes them different so you know what to expect.

Easier Questions	Harder Questions
The rule is about just ONE thing (like all circles)	The rule is about TWO or THREE things at once
The shapes look very different from the wrong answers	The wrong answers look almost right — they trick you!
The shapes use simple features like color	The shapes use tricky features like number of parts or direction
All group shapes look exactly the same	Group shapes look different but share a hidden rule

Easy vs. Hard Grouping Questions

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KEY TAKEAWAY

Think of it like a club with a secret password. The shapes in the group all know the password. You have to figure out what the password is and then find the answer choice that also knows the password. Sometimes the password is easy ("be a circle") and sometimes it is tricky ("be a small, striped shape pointing up")!

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Building Bigger Thinking Skills

Grouping shapes is a skill that helps you with lots of things! When you get really good at finding rules, you can solve harder puzzles too. Let's see how this connects to bigger ideas.

What You Learn Now	What It Helps You Do Later
Finding what is the SAME about shapes	Sorting information into categories in science and reading
Looking at more than one feature at a time	Solving math problems with more than one step
Checking your answer against the rule	Checking your work in all subjects
Noticing small differences between shapes	Paying close attention to details in everything you do

How grouping skills grow with you

As you grow, you will see harder questions. They might show you groups where the rule is about patterns that change or about how shapes are turned or flipped. But don't worry — the same steps always work. Look, compare, and match!

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Practice Problems

Now it's your turn! Try these five problems. Think about the 3 steps: Look, Compare, Match. Take your time and check each answer choice.

Question 1: CONCEPTUAL

A group has a red circle, a red square, and a red triangle. What is the rule for this group?

Show answer

The rule is that all the shapes are red. The shapes are all different (circle, square, triangle), so the rule is NOT about what shape they are. The sizes could be different too. The one thing they all share is their color — red. So if you needed to pick a shape that belongs, you would pick any shape that is red.

Question 2: BASIC

A group has three big shapes: a big circle, a big square, and a big diamond. The answer choices are: (A) a small circle, (B) a big triangle, (C) a small square, (D) a small diamond. Which one belongs in the group?

Show answer

The answer is (B) a big triangle. The rule is that all the shapes are big. The shapes in the group are all different (circle, square, diamond), so the rule is about size, not shape. Choices A, C, and D are all small, so they do not follow the rule. Only choice B is big, so it belongs.

Question 3: INTERMEDIATE

A group has three shapes: a small striped circle, a small striped square, and a small striped triangle. The answer choices are: (A) a big striped circle, (B) a small solid circle, (C) a small striped diamond, (D) a big solid square. Which one belongs?

Show answer

The answer is (C) a small striped diamond. This rule has TWO parts: the shapes must be small AND they must be striped. Choice A is striped but big (wrong size). Choice B is small but solid, not striped (wrong shading). Choice D is big and solid (wrong on both). Only choice C is both small and striped, so it follows both parts of the rule.

Question 4: APPLIED

A group shows: a white square with 2 dots inside, a white circle with 2 dots inside, and a white triangle with 2 dots inside. The answer choices are: (A) a white diamond with 3 dots, (B) a black circle with 2 dots, (C) a white hexagon with 2 dots, (D) a white square with 1 dot. Which one belongs?

Show answer

The answer is (C) a white hexagon with 2 dots. The rule has TWO parts: the shape must be white AND it must have exactly 2 dots inside. The shapes in the group are all different, so shape type does not matter. Choice A is white but has 3 dots (wrong number). Choice B has 2 dots but is black (wrong color). Choice D is white but has only 1 dot (wrong number). Only choice C is white with 2 dots.

Question 5: CRITICAL THINKING

A group shows: a large black circle with an arrow pointing up, a large black square with an arrow pointing up, and a large black triangle with an arrow pointing up. The answer choices are: (A) a large black diamond with an arrow pointing down, (B) a small black hexagon with an arrow pointing up, (C) a large white star with an arrow pointing up, (D) a large black star with an arrow pointing up. Which one belongs?

Show answer

The answer is (D) a large black star with an arrow pointing up. This rule has THREE parts: the shape must be large, it must be black, and it must have an arrow pointing up. Choice A is large and black but the arrow points down (wrong direction). Choice B is black with an up arrow but it is small (wrong size). Choice C is large with an up arrow but it is white (wrong color). Only choice D matches all three parts: large, black, and arrow pointing up. When a rule has many parts, you have to check every single part!

SUMMARY

Let's Review What We Learned!

Summary

In this lesson, you learned how to find the rule that makes shapes belong to the same group. You can look at shape, size, color or shading, number of parts, and direction to figure out what the shapes have in common.

Remember the 3-step trick: Look at all the shapes in the group, Compare them to find what is the same, and Match the answer choice that follows the rule. Sometimes the rule has one part, and sometimes it has two or three parts. Always check every part of the rule before you pick your answer!