Use transformations (addition/subtraction of elements, rotation, shading) - CogAT Grades K 2

SECTION 1

Why Do We Study How Shapes Change?

People have always liked puzzles! A long time ago, teachers wanted to know how kids think. They made picture puzzles to see how well kids can spot patterns. These puzzles show shapes that change in a special way. Your job is to figure out the rule!

Long Ago • People Love Patterns

People noticed patterns in nature, like stripes on a zebra or spots on a ladybug. They started drawing shapes and patterns too.

1950s • Picture Puzzles for Kids

Teachers made special puzzles with shapes. The puzzles helped them see how kids think and solve problems without using words.

Today • The CogAT Test

The CogAT test uses shape puzzles. Kids look at how shapes change and pick the answer that comes next. It is a fun way to show how smart you are!

On the CogAT test, you see shapes that change step by step. The big question is: What is the rule? Once you find the rule, you can pick the right answer!

SECTION 2

The Four Big Ways Shapes Change

Shapes can change in four main ways. We call these changes transformations. That is a big word, but it just means "ways things change." Let's learn each one!

Addition (Adding Parts)

A new shape or line gets added to the picture. Like putting a hat on a snowman!

Subtraction (Taking Parts Away)

A shape or line gets taken away from the picture. Like eating one cookie from a plate!

Rotation (Turning)

The shape turns like a clock hand. It stays the same shape but faces a new way.

Shading (Filling In)

The shape gets colored in or the color changes. White might become gray or black.

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KEY TAKEAWAY

Think of transformations like dress-up! You can add a hat, take away your shoes, turn around to show your back, or change your color by putting on a new shirt. Shapes do the same things!

SECTION 3

See the Transformations!

Let's look at a picture that shows all four transformations. Each row shows one kind of change. Look at the arrows to see how the shape goes from one step to the next!

Row 1 shows addition — a new small shape appears each step. Row 2 shows subtraction — one dot disappears each step. Row 3 shows rotation — the triangle turns. Row 4 shows shading — the rectangle fills in more and more.

Look at each row above. See how the shape changes from left to right? The arrows show you the direction. Each step follows the same rule. That is the key to solving these puzzles!

SECTION 4

How to Find the Rule

On the CogAT test, you don't use math formulas. Instead, you use your eyes and your brain! Here is a simple plan to follow every time you see a transformation puzzle.

The 3-Step Plan

Step 1 — Look: Look at the first shape. Then look at the next shape. What is different?
Step 2 — Name the change: Is something added? Taken away? Turned? Colored in?
Step 3 — Use the rule: Do the same change one more time to find the answer!

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Helpful Hint

Ask yourself: "What stayed the same?" and "What changed?" The things that stayed the same help you see the things that changed!

Sometimes more than one thing changes at the same time! A shape might turn AND get shaded. When that happens, check each change one at a time. Don't rush!

SECTION 5

A Closer Look at Each Transformation

Let's look at each type of transformation more closely. The picture below shows different examples so you can practice spotting them.

This diagram shows four boxes with different transformation examples. Notice how each row has a clear rule. On the test, you need to spot the rule and pick the shape that comes next!

Transformation	What to Look For	Example Clue
Addition	Something new appears	More dots, more lines, a new shape
Subtraction	Something disappears	Fewer dots, missing lines, shape gone
Rotation	Shape turns or flips	Arrow points new way, shape tilts
Shading	Color fills in or changes	White becomes gray, gray becomes black

Quick guide to spotting transformations

SECTION 6

Let's Solve One Together!

Let's pretend we see this puzzle on a test. There are three boxes in a row. Let's figure out what goes in the last box!

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The Puzzle

Box 1: A square with 1 star inside (white, not filled in). Box 2: A square with 2 stars inside (the first star is now gray). Box 3: ???

Solving Step by Step

1
Step 1 — Look at Box 1 and Box 2

Box 1 has 1 white star. Box 2 has 2 stars, and the first star turned gray. Two things changed!

A star was added AND the first star changed shading.
2
Step 2 — Name the changes

Change 1: The number of stars went from 1 to 2. That is addition. Change 2: The first star went from white to gray. That is shading.

Two rules: add 1 star AND shade the oldest star.
3
Step 3 — Use the rules for Box 3

Add 1 more star: now there are 3 stars. Shade the next star: the second star turns gray too.

Answer: A square with 3 stars. The first two stars are gray, and the third star is white.

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REMEMBER

Sometimes a puzzle has TWO changes happening at the same time. That is okay! Just find each change one at a time, like checking one pocket and then the other.

SECTION 7

Tips and Common Mistakes

Good Habit ✅	Common Mistake ❌
Look at ALL the boxes before picking an answer.	Only looking at the last box and guessing.
Count the shapes carefully in each box.	Forgetting to count and missing added or removed shapes.
Check if something turned or rotated.	Thinking the shape is a new shape when it just turned.
Notice if the shading changed (white, gray, black).	Ignoring shading because it seems like a small change.
Check for TWO changes at the same time.	Only finding one change and missing the second one.

Good habits vs. common mistakes

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KEY TAKEAWAY

Think of yourself as a detective! A detective looks at every clue before solving the mystery. You should look at every box and every detail before picking your answer. Slow and careful wins the puzzle!

SECTION 8

Getting Ready for Harder Puzzles

As you get better at spotting transformations, puzzles will get trickier! Here is what easier puzzles and harder puzzles look like.

Easier Puzzles	Harder Puzzles
Only ONE change happens (like just adding).	TWO or THREE changes happen at the same time.
Big, easy-to-see shapes.	Small details that are tricky to spot.
Shapes go in a simple row (left to right).	Shapes may be in a grid (rows AND columns).
Two answer choices.	More answer choices that look alike.

How puzzles get harder as you grow

Don't worry about the hard ones yet! If you practice the four transformation types now, you will be ready. Each time you solve a puzzle, your brain gets stronger. It is like exercising a muscle!

SECTION 9

Practice Problems

Try these puzzles! Think about what kind of transformation is happening. Then pick or describe the answer.

Question 1: CONCEPTUAL

A picture shows a circle in Box 1. In Box 2, the same circle has a small triangle inside it. What type of transformation happened?

Show answer

This is addition. A triangle was added inside the circle. The circle stayed the same, but a new shape appeared. When something new shows up that was not there before, that is addition!

Question 2: BASIC CALCULATION

Box 1 has 5 dots. Box 2 has 4 dots. Box 3 has 3 dots. How many dots should Box 4 have?

Show answer

Box 4 should have 2 dots. The rule is subtraction — one dot is taken away each time. 5, then 4, then 3, so next is 2. Count the dots in each box to see the pattern going down by one.

Question 3: INTERMEDIATE

An arrow points UP in Box 1. The arrow points RIGHT in Box 2. The arrow points DOWN in Box 3. Which way should the arrow point in Box 4?

Show answer

The arrow should point LEFT. This is rotation! The arrow turns clockwise (like a clock) each step. Up → Right → Down → Left. It makes a quarter turn each time. If there were a Box 5, the arrow would point up again!

Question 4: APPLIED

Box 1 has a white square with 1 star. Box 2 has a gray square with 2 stars. What should Box 3 look like? (Hint: two transformations are happening!)

Show answer

Box 3 should have a black square with 3 stars. Two changes happen each step. First, addition: one star is added each time (1 → 2 → 3). Second, shading: the square gets darker each time (white → gray → black). Always check for more than one change!

Question 5: CRITICAL THINKING

Box 1 has a big white circle with a small black triangle on top. Box 2 has a big gray circle with a small black triangle on the right side. What should Box 3 look like? Name ALL the changes.

Show answer

Box 3 should have a big black circle with a small black triangle on the bottom. There are two transformations. First, shading: the big circle gets darker each step (white → gray → black). Second, rotation: the small triangle moves clockwise around the circle (top → right → bottom). The triangle itself does not change color — it stays black. Great detectives check every single part of the picture!

SUMMARY

What We Learned

Summary

In this lesson, you learned about four types of transformations — ways that shapes change in puzzles. Addition means a new part is added. Subtraction means a part is taken away. Rotation means a shape turns. Shading means the color fills in or changes.

To solve transformation puzzles, use the 3-Step Plan: Look at what changed, name the change, and use the rule to find the answer. Remember, sometimes two changes happen at once! Check every detail like a detective, and you will do great on your CogAT test!