Check solutions by substituting back into the relationship - CogAT Grades K 2

SECTION 1

Why Do We Check Our Answers?

People have been checking their work for a very long time! When you solve a puzzle, you want to know if you got the right answer. Checking your answer means going back and making sure it fits. Think of it like trying a puzzle piece — you put it in and see if it works!

Long Ago • Counting with Stones

People used stones to count things. They would count again to make sure they got the right number.

In School • Teachers Teach Checking

Teachers started showing kids how to check their math. If you add 2 + 3 and get 5, you can check by counting on your fingers!

Today • Checking on Tests

On tests like the CogAT, you can check your answer by putting it back into the problem to see if it makes sense.

So how do we check if an answer is right? We put it back! If the answer fits, we know we did a great job. If it does not fit, we try again. This is called substituting back.

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What Does It Mean to Check Your Answer?

When you find an answer, you need to make sure it is correct. Substituting means putting your answer back into the problem. If everything works out, your answer is right!

Find the Pattern

Look at the numbers. What rule connects them? Maybe they go up by 2, or one side equals the other.

Pick an Answer

Choose the answer you think is right. This is your guess!

Put It Back In

Place your answer back into the problem. Does it follow the same rule or pattern?

Does It Fit?

If your answer makes the pattern work, you are right! If not, try a different answer.

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KEY TAKEAWAY

Think of it like a lock and a key. Your answer is the key. You put the key in the lock (the problem). If the lock opens, the key is right! If it does not open, you need a different key.

SECTION 3

See How Checking Works

This picture shows two examples. The top row shows a correct answer (4) that works. The bottom row shows a wrong answer (5) that does not fit.

Look at the picture above. In the top row, we think the answer is 4. We put 4 back in: 3 + 4 = 7. That matches! So 4 is right. In the bottom row, we try 5. But 3 + 5 = 8, and 8 is not 7. So 5 is wrong. Checking saves us from picking the wrong answer!

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The Simple Rule for Checking

On the CogAT, you will see number puzzles. Some puzzles use addition. Some use subtraction. Some use patterns. No matter what, you can always check by putting your answer back in!

THE CHECKING RULE

Put your answer in → Do the math → Does it match?

If both sides are the same, your answer is correct. If they are different, try again!

ADDITION CHECK

2 + ? = 5 → Try 3 → 2 + 3 = 5 ✓

We guessed 3. We added 2 + 3 and got 5. That matches, so 3 is right!

PATTERN CHECK

2, 4, 6, ? → Try 8 → Does 8 follow +2? Yes! ✓

The pattern goes up by 2 each time. 6 + 2 = 8. So 8 fits the pattern!

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Remember!

You do not need big math to check. Just put the answer in and see if it makes sense. If it feels wrong, it probably is!

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Different Kinds of Number Puzzles

On the CogAT, you will see different kinds of number puzzles. Let us look at each kind and learn how to check your answer for each one!

This picture shows three kinds of puzzles you might see. Missing Number problems ask you to fill in a blank. Number Pattern problems ask what comes next. Balance problems ask you to make both sides the same.

No matter which type of puzzle you see, the trick is the same. Put your answer in and see if everything works. If it does, you are done!

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Let's Solve One Together!

Let's work through a problem step by step. We will solve it and then check our answer by putting it back in.

Problem: 5 + ? = 8. What number goes in the box?

1
Step 1 — Read the Problem

We see 5 + ? = 8. We need to find the number that goes where the ? is.
2
Step 2 — Think About It

What do I add to 5 to get 8? I can count up from 5: 6, 7, 8. That is 3 jumps!

I think the answer is 3.
3
Step 3 — Put It Back In (Substitute!)

Now I put 3 where the ? was. The problem becomes: 5 + 3 = ?
4
Step 4 — Do the Math

5 + 3 = 8. Is that what the problem said? Yes! The problem said it should equal 8, and it does!

8 = 8 ✓ The answer 3 is correct!

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Great Job!

Did you see what we did? We found an answer (3), then we put it back in (5 + 3), and we checked if it matched (= 8). That is how checking works every time!

SECTION 7

When Checking Helps and When It's Tricky

When Checking Is Easy	When Checking Is Harder
Simple adding: 2 + ? = 6	Longer patterns: 2, 5, 8, 11, ?
Small numbers you can count	Bigger numbers that are harder to add
Only one step to check	More than one step to check
You can use your fingers to count	Numbers are too big for fingers

Easy vs. harder checking situations

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HELPFUL HINT

Even when checking is harder, it is still worth doing! Think of it like tasting your food while cooking. A quick taste tells you if you need to add more salt. A quick check tells you if you need a different answer!

✓ Tip 1: Always read the problem again before checking.
✓ Tip 2: Use your fingers to count if the numbers are small.
✓ Tip 3: If your answer does not work, do not worry! Just try a different one.

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Checking Now and Later

Right now, you are learning to check with small numbers and simple patterns. As you grow up, you will use this same skill with bigger problems! Let us see how checking grows with you.

What You Do Now	What Big Kids Do
Check: 2 + 3 = 5	Check: 12 + 15 = 27
Find missing number: ? + 1 = 4	Find missing number: ? × 3 = 12
Simple patterns: 2, 4, 6, ?	Harder patterns: 3, 6, 12, 24, ?
Count on fingers to check	Use paper and pencil to check

How checking grows with you

The idea is the same every time. Put your answer back in and see if it works. This is a skill you will use forever!

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Practice Problems

Now it is your turn! Try each problem. Find the answer, then check it by putting it back in.

Question 1: CONCEPTUAL

If someone says 2 + 3 = 6, how would you check if that is true? What would you do?

Show answer

You would add 2 + 3 yourself. Count: 2, then 3 more: 3, 4, 5. You get 5, not 6. So 2 + 3 = 6 is wrong! The right answer is 2 + 3 = 5. You checked by doing the math yourself.

Question 2: BASIC CALCULATION

Solve: 1 + ? = 6. Then check your answer by putting it back in.

Show answer

Count up from 1 to 6: you need 5 more. So ? = 5. Now check: put 5 back in. 1 + 5 = 6. Yes! 6 = 6. The answer 5 is correct because when you substitute 5 back in, both sides are equal.

Question 3: INTERMEDIATE

Look at this pattern: 3, 5, 7, ?. Someone says the answer is 8. Check if they are right.

Show answer

First, find the rule. 3 to 5 is +2. 5 to 7 is +2. The rule is add 2 each time. Now check: 7 + 2 = 9, not 8. So 8 is wrong! The correct answer is 9. When we put 9 back in the pattern (3, 5, 7, 9), each number goes up by 2. It fits!

Question 4: APPLIED

You have some apples. Your friend gives you 4 more. Now you have 7 apples. How many did you start with? Check your answer!

Show answer

This is like ? + 4 = 7. Count back from 7: take away 4. That gives you 3. So you started with 3 apples. Check: put 3 back in. 3 + 4 = 7. Yes! 7 = 7. You started with 3 apples. The check works because substituting 3 back in gives us the right total.

Question 5: CRITICAL THINKING

Look at these two sides of a balance: one side has 3 + 2, and the other side has 1 + ?. What number makes them balance? Show how you check.

Show answer

First, figure out the left side: 3 + 2 = 5. Now the right side needs to equal 5 too. So 1 + ? = 5. Count up from 1 to 5: that is 4. So ? = 4. Check: put 4 back in. Left side: 3 + 2 = 5. Right side: 1 + 4 = 5. Both sides are 5, so they balance! Your answer of 4 is correct because substituting it back makes both sides equal.

SUMMARY

What We Learned

Summary

Today you learned how to check your answers by putting them back into the problem. This is called substituting. You find an answer, put it where the question mark is, and do the math. If both sides match, you are right!

This works for missing number problems, number patterns, and balance problems. Remember: always check before you pick your final answer on the CogAT! It is like trying on a shoe to see if it fits. If it fits, you are good to go!