Use simple operations and relational reasoning to complete a number - CogAT Grades K 2

SECTION 1

Why Do We Find Missing Numbers?

People have been solving puzzles with numbers for a very long time! Long ago, people needed to count things like sheep and apples. Sometimes they knew how many they started with and how many they had at the end, but they needed to figure out what happened in the middle. This is how finding missing numbers began!

Long Ago • Counting with Fingers

People first used their fingers to count things and figure out how many they had.

Ancient Times • Number Symbols

People made up symbols to write numbers down so they could remember them and solve puzzles.

In School • Math Operations

Kids learn to add and subtract. These are called operations. They help us find missing numbers!

Today • CogAT Puzzles

Now you get to solve fun number puzzles on tests like the CogAT. You use what you know about adding and subtracting to find the answer!

So here is the big question: if you see some numbers and one is missing, how do you figure out what it is? That is what this lesson is all about!

SECTION 2

The Big Ideas

To find a missing number, you need to understand a few important ideas. Let's look at each one!

Adding (Putting Together)

When you add, you put groups together to get a bigger number. Like 3 + 2 = 5.

Subtracting (Taking Away)

When you subtract, you take some away to get a smaller number. Like 5 − 2 = 3.

Relational Reasoning

This means looking at how numbers are connected. If you see 2 + ? = 5, you think: what do I add to 2 to get 5?

Patterns

Numbers can follow a pattern. Like 2, 4, 6, ?. The pattern is adding 2 each time, so the answer is 8!

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KEY TAKEAWAY

Think of it like a seesaw! If one side has 3 blocks and the other side has 5 blocks, the seesaw tips. You need to figure out how many blocks to add to make both sides even. That is what relational reasoning is — finding the number that makes everything balance!

SECTION 3

See It in a Picture!

Pictures can help us understand missing numbers. Look at the diagram below. It shows how adding and subtracting are like going forward and backward on a number line!

This number line shows how we start at 3 and jump forward 2 spaces to land on 5. The green arrow shows adding!

When you add, you jump to the right on the number line. When you subtract, you jump to the left. If a number is missing, you can count the jumps to find it!

SECTION 4

How the Math Works

When you see a number puzzle, you use addition or subtraction to find the missing piece. Here are the shapes these puzzles can take.

FIND THE ANSWER

3 + 4 = ?

You know both numbers. Just add them! The answer is 7.

FIND A MISSING PART

? + 4 = 7

You know the answer (7) and one part (4). Think: what plus 4 makes 7? You can subtract: 7 − 4 = 3.

FIND WHAT WAS TAKEN AWAY

8 − ? = 5

You start with 8 and end with 5. How many did you take away? Count: 8 − 5 = 3. So 3 was taken away.

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Remember!

Adding and subtracting are opposites. If adding makes numbers bigger, subtracting makes them smaller. You can use one to check the other!

SECTION 5

Finding Patterns in Numbers

Sometimes the CogAT shows you a row of numbers, and you need to find a pattern. A pattern is a rule that numbers follow. Look at the picture below to see how patterns work!

Three types of number patterns. Pattern 1 adds 2 each time. Pattern 2 subtracts 3 each time. Pattern 3 is trickier — the amount you add grows each time!

To find a pattern, look at two numbers that are next to each other. Ask yourself: did the number go up or down? By how much? Then check if the same thing happens between the next pair. If it does, you found the pattern!

SECTION 6

Let's Solve One Together!

Here is a puzzle: 5 + ? = 9. Let's solve it step by step!

Solving 5 + ? = 9

1
Step 1 — Read the Puzzle

We see 5 + ? = 9. This means: 5 plus some number equals 9. We need to find that missing number!
2
Step 2 — Think About What We Know

We start with 5. We end at 9. We need to figure out how far apart they are.
3
Step 3 — Count Up or Subtract

We can count up from 5: 6, 7, 8, 9. That is 4 jumps! Or we can subtract: 9 − 5 = 4.

9 − 5 = 4
4
Step 4 — Check Our Answer

Let's put 4 back in the puzzle: 5 + 4 = 9. Yes! That is right!

The missing number is 4

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Pro Tip!

Always check your answer by putting it back into the puzzle. If both sides match, you got it right!

SECTION 7

Helpful Strategies

Different puzzle types call for different tricks. Here is a chart that shows you which strategy to use!

Puzzle Type	What It Looks Like	Strategy to Use
Missing Answer	3 + 4 = ?	Just add or subtract the two numbers.
Missing Part	? + 4 = 7	Use the opposite operation. Subtract: 7 − 4 = 3.
Number Pattern	2, 5, 8, ?	Find what changes between each pair. Then do the same thing to the last number.
Balancing	3 + 2 = 1 + ?	Figure out each side. 3 + 2 = 5. So 1 + ? = 5, which means ? = 4.

Common puzzle types and strategies

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KEY TAKEAWAY

Think of addition and subtraction like a door that swings both ways. If you go through the addition door to get from 3 to 7, you can go back through the subtraction door (7 − 3) to find how far you went. Knowing this trick helps you solve any missing-number puzzle!

SECTION 8

Getting Ready for Bigger Puzzles

The skills you are learning now will help you with even bigger math puzzles later. Let's see how today's skills connect to what comes next!

What You Learn Now	What Comes Later
Finding a missing number in 3 + ? = 7	Solving equations like x + 3 = 7 in older grades
Spotting patterns like 2, 4, 6, 8	Understanding skip counting and multiplication
Using opposite operations to check answers	Fact families and inverse operations
Balancing both sides: 2 + 3 = 1 + ?	Understanding equality and the equal sign

How today's skills build into bigger math

Every time you solve a missing-number puzzle, you are building a strong math brain. These same thinking skills will help you in school for years to come!

SECTION 9

Practice Problems

Now it's your turn! Try these puzzles. Start with the easy ones and work your way up. You can do it!

Question 1: CONCEPTUAL

If you have 4 apples and someone gives you more apples, will you have more or fewer than 4 apples?

Show answer

You will have more than 4 apples! When you add something to a number, the number gets bigger. This is the key idea behind addition. No matter how many apples someone gives you, as long as they give you at least one, you will have more than you started with.

Question 2: BASIC CALCULATION

What is the missing number? 2 + ? = 6

Show answer

The missing number is 4. To find it, you can subtract: 6 − 2 = 4. You can also count up from 2 to 6: that is 3, 4, 5, 6 — four jumps! To check, put the answer back: 2 + 4 = 6. It works!

Question 3: INTERMEDIATE

What number comes next in this pattern? 1, 3, 5, 7, ?

Show answer

The answer is 9. Look at each pair of numbers: 1 to 3 is +2, 3 to 5 is +2, 5 to 7 is +2. The pattern is adding 2 each time. So 7 + 2 = 9. These are the odd numbers!

Question 4: APPLIED

You have some stickers. Your friend gives you 3 more stickers. Now you have 8 stickers. How many stickers did you start with? Write the number sentence and solve it.

Show answer

The number sentence is ? + 3 = 8. You started with 5 stickers. To find this, subtract: 8 − 3 = 5. Check it: 5 + 3 = 8. That matches! This is a real-life example of finding a missing part using subtraction.

Question 5: CRITICAL THINKING

Look at this puzzle: 4 + 3 = ? + 5. What is the missing number?

Show answer

The missing number is 2. First, figure out the left side: 4 + 3 = 7. Now you know both sides must equal 7. So ? + 5 = 7. To find the missing number, subtract: 7 − 5 = 2. Check: 2 + 5 = 7. Both sides equal 7, so they balance! This kind of puzzle uses relational reasoning because you compare both sides.

SUMMARY

What We Learned

Summary

In this lesson, you learned how to find missing numbers using addition and subtraction. You discovered that relational reasoning means looking at how numbers are connected. You can use a number line to count jumps, and you can use opposite operations to find missing parts.

You also learned to spot number patterns by looking at what changes between each pair of numbers. Always remember to check your answer by putting it back into the puzzle. If both sides match, you know you got it right! Keep practicing and your number skills will grow stronger every day.