Mathematical Modeling and Proof

Deconstructing Word Problems means taking a story problem apart into small, clear pieces before solving it.

Sometimes a word problem has lots of words, extra details, or numbers that can feel confusing. Good mathematicians do not rush. They slow down, read carefully, and look for what really matters.

When you deconstruct a word problem, you are like a detective. You search for the important clues, ignore extra information, and decide what math action to use.

Why is this helpful? If you understand the problem first, you are more likely to choose the correct numbers and the correct operation.

Here is a simple plan you can use every time.

1. Read the whole problem.
2. Circle or list the important numbers.
3. Underline what the question is asking.
4. Cross out or ignore extra details that do not help.
5. Choose the math action: add, subtract, multiply, or divide.
6. Solve and check if your answer makes sense.

Let’s look at each part more closely.

1. Read the whole problem carefully.

Read all the words once. Then read it again more slowly. On the second reading, look for clues.

2. Find the important numbers.

Not every number in a word problem is needed. Some numbers matter for solving. Some do not.

Ask yourself, “Does this number help answer the question?”

3. Find the question.

The question tells you what you are trying to find. If you do not know what the problem is asking, it is easy to do the wrong math.

You can underline words like how many, how much, how many left, or in all.

4. Ignore extra details.

Some story problems include fun details, like a child’s favorite color, the day of the week, or what someone was wearing. These details may make the story interesting, but they do not help you solve the math.

5. Choose the operation.

• Add when parts are put together or when you find the total.
• Subtract when something is taken away or when you compare to find how many more or fewer.
• Multiply when equal groups are put together.
• Divide when sharing equally or making equal groups.

6. Check your work.

After solving, read the question again. Make sure your answer matches what was asked.

If the question asks, “How many are left?” your answer should tell how many are left, not how many there were at the start.

A helpful annotation strategy

When you annotate a problem, you mark it up to make it easier to understand. You can do this on paper or in your head.

• Circle the numbers you need.
• Underline the question.
• Put a box around clue words like in all, left, each, or shared equally.
• Cross out details that are not needed.

Now let’s practice with worked examples.

Worked Example 1: Finding the total

Lena has 14 stickers. Her friend gives her 8 more stickers. Lena keeps them in a purple box. How many stickers does Lena have now?

Step 1: Important information

• Lena has 14 stickers.
• She gets 8 more stickers.
• The purple box is not important.

Step 2: What is the question?

How many stickers does Lena have now?

Step 3: Choose the operation

She has some stickers, and then she gets more. That means we add.

$$14 + 8 = 22$$

Answer: Lena has 22 stickers now.

Worked Example 2: Finding how many are left

There were 25 apples in a basket. Mr. Lee used 7 apples to make a pie on Saturday. The basket is brown. How many apples are left?

Step 1: Important information

• 25 apples in the basket
• 7 apples were used
• Saturday and the brown basket are extra details

Step 2: What is the question?

How many apples are left?

Step 3: Choose the operation

If some apples were used, we subtract.

$$25 - 7 = 18$$

Answer: There are 18 apples left.

Worked Example 3: Equal groups

A teacher puts 4 pencils on each table. There are 6 tables. The classroom has a big window. How many pencils does the teacher need in all?

Step 1: Important information

• 4 pencils on each table
• 6 tables
• The big window is not important

Step 2: What is the question?

How many pencils are needed in all?

Step 3: Choose the operation

There are equal groups: 4 pencils on each of 6 tables. That means we multiply.

$$4 \times 6 = 24$$

Answer: The teacher needs 24 pencils.

Worked Example 4: Sharing equally

Marcus has 18 crackers. He shares them equally with 3 friends. Marcus is wearing a red hat. How many crackers does each friend get?

Step 1: Important information

• 18 crackers
• Shared equally with 3 friends
• The red hat is not important

Step 2: What is the question?

How many crackers does each friend get?

Step 3: Choose the operation

When sharing equally, we divide.

$$18 \div 3 = 6$$

Answer: Each friend gets 6 crackers.

How to tell if a detail is extra

Ask yourself these questions:

• Does this fact help me find the answer?
• Does this number connect to the question?
• If I remove this detail, can I still solve the problem?

If the answer is yes, that detail is probably extra.

Clue words can help, but think carefully

Some words often point to an operation.

• in all, total, altogether often mean add
• left, remain, fewer often mean subtract
• each, equal groups often mean multiply
• shared equally, split into groups often mean divide

But do not use clue words only. Always think about the story and what is happening.

Using more than one representation

Good problem solvers show their thinking in different ways. This helps them check their work.

You can represent a word problem by using:

• Words: tell what is happening
• Numbers: write the equation
• Drawings: sketch groups, objects, or boxes

For example, in the pencil problem, you could draw 6 tables with 4 pencils on each table. Then you could count them or write:

$$4 + 4 + 4 + 4 + 4 + 4 = 24$$

That matches:

$$4 \times 6 = 24$$

Debugging a mistake

Sometimes students choose the wrong operation because they use the wrong clue. Let’s see an example.

Problem: Sara had 12 balloons. 5 popped. How many balloons are left?

A student writes:

$$12 + 5 = 17$$

This answer is not correct because balloons popped. The number of balloons went down, not up.

The correct operation is subtraction.

$$12 - 5 = 7$$

Correct answer: 7 balloons are left.

This is called debugging your work. You look back, find the mistake, and fix it.

Precise thinking matters

When you solve a word problem, try to explain why you chose your operation.

For example:

“I used subtraction because 7 apples were used, so the number of apples became smaller.”

This kind of clear explanation shows strong math thinking.

Try this thinking on your own

1. What numbers matter?
2. What is the question asking?
3. Are there extra details?
4. What operation matches the story?
5. Does my answer make sense?

Summary

Deconstructing word problems means breaking a story problem into important parts. First, find the needed numbers and the question. Next, ignore extra details, choose the right operation, and solve carefully.

Strong mathematicians do not just find answers. They also explain their thinking, use drawings or equations, and check for mistakes.

Selecting the Optimal Representation means choosing the best way to show and solve a math problem.

In 3rd Grade math, we do not always solve every problem the same way. Sometimes it is fastest to think in your head. Sometimes it helps to use objects you can move. Sometimes a picture helps. Sometimes writing the steps in order is best.

Good mathematicians ask, "What tool will help me understand this problem best?" That is called choosing the best representation.

Here are four helpful ways to represent a problem:

• Mental strategy: solving in your head
• Physical manipulative: using counters, cubes, or other objects
• Visual tape diagram: drawing bars to show parts and wholes or equal groups
• Formal algorithm: writing steps in order to solve

Let’s learn when each one is a smart choice.

1. Use a mental strategy when the numbers are friendly and you can solve quickly in your head.

Mental math works well when you can count on, make a ten, double, halve, or use facts you already know.

• Example: \(25 + 25\)
• Example: \(30 + 8\)
• Example: \(45 - 5\)

If the problem is simple and you can see the answer without drawing or writing much, mental math may be the best choice.

2. Use physical manipulatives when you need to touch and move objects to understand the problem.

This is helpful when the problem is about counting, making equal groups, sharing, or showing parts.

You might use:

• counters
• connecting cubes
• buttons
• beans
• small blocks

Manipulatives are useful when a problem feels hard to picture in your head.

3. Use a tape diagram when a picture of bars can help you see the math.

A tape diagram is a simple drawing made of rectangles. It helps show:

• parts and a whole
• equal groups
• comparison

For example, if one bar is split into 3 equal parts, it can show 3 groups. If two smaller bars join to make one long bar, it can show addition. If one part is missing, it can help show subtraction.

4. Use a formal algorithm when writing the steps clearly is the easiest way to solve.

An algorithm is a set of math steps written in order. In 3rd Grade, this might mean writing addition, subtraction, multiplication, or division carefully on paper.

This is often best when the numbers are larger or when mental math is harder.

How do I choose the best representation?

Ask yourself these questions:

1. Are the numbers easy enough to do in my head?
2. Do I need to move objects to understand the problem?
3. Would a bar picture help me see parts, groups, or comparison?
4. Would writing the steps be the clearest way?

There is not always only one right choice. Sometimes more than one representation can work. But one may be faster, clearer, or easier to understand.

Worked Example 1: Choose a mental strategy

Problem: Mia has \(40\) stickers. Her friend gives her \(10\) more. How many stickers does she have now?

The numbers are friendly: \(40\) and \(10\).

This is a good time to use mental math.

We can think:

$$40 + 10 = 50$$

So Mia has 50 stickers.

Why was mental math the best choice? The numbers were easy to add in your head. We did not need objects or a drawing.

Worked Example 2: Choose physical manipulatives

Problem: There are \(12\) cubes. Put them into \(3\) equal groups. How many cubes are in each group?

This problem is about sharing equally. It can help to use cubes or counters.

Start with \(12\) cubes. Make \(3\) groups. Put one cube in each group until all cubes are used.

Each group gets \(4\) cubes.

So:

$$12 \div 3 = 4$$

Why were manipulatives the best choice? We could move the cubes and check that the groups were equal.

Worked Example 3: Choose a tape diagram

Problem: A ribbon is \(18\) inches long. Sara uses \(7\) inches. How many inches are left?

This is a part and whole problem. A tape diagram can help.

Draw one long bar for \(18\). Split it into two parts: one part is \(7\), and the other part is the missing amount.

We can write:

$$7 + \Box = 18$$

or

$$18 - 7 = 11$$

So 11 inches are left.

Why was a tape diagram the best choice? The bar helps us see the whole and the missing part.

Worked Example 4: Choose a formal algorithm

Problem: The school library has \(246\) books in one room and \(137\) books in another room. How many books are there in all?

These numbers are bigger. A formal algorithm is a clear choice.

Write the numbers in columns:

$$\begin{array}{r} 246 \\ +137 \\ \hline 383 \end{array}$$

So there are 383 books in all.

Why was a formal algorithm the best choice? The numbers are large enough that writing the steps helps us stay organized.

Let’s compare choices

Look at each kind of problem and think about the best tool.

• Easy number facts → mental strategy
• Equal sharing or building groups → physical manipulatives
• Missing part, whole, or comparison → tape diagram
• Larger numbers with more steps → formal algorithm

Important idea: The best representation helps you understand the problem, not just get an answer.

If a problem feels confusing, try a different representation. For example, if mental math feels hard, draw a tape diagram or use objects. If objects take too long, a written method may be better.

Strong mathematicians are flexible. They can choose a tool, test it, and switch if needed.

Practice thinking

When you see a problem, try saying:

• “I can do this in my head.”
• “I should use counters to make equal groups.”
• “A tape diagram will help me see the parts.”
• “I should write the steps carefully.”

That is what selecting the optimal representation means.

Summary

Math problems can be shown in different ways. You can use mental math, manipulatives, a tape diagram, or a formal algorithm.

The best choice depends on the problem. Choose the representation that makes the math clear, organized, and easy to understand.

Algorithmic Debugging means looking at a finished math problem, finding where a mistake happened, and fixing it.

Sometimes a student gets a wrong answer even though some of the work is correct. A good mathematician does not just say, “It is wrong.” A good mathematician asks, “Which step went wrong?”

This is called debugging. It is like being a math detective!

Why is this important?

• It helps us understand math better.
• It helps us learn from mistakes.
• It helps us check our work carefully.

When we debug a math problem, we go step by step.

1. Look at the problem.
2. Read each step in order.
3. Find the first step that is not correct.
4. Explain what the mistake is.
5. Fix the work from that step on.

Remember: the first wrong step is the most important one. After that, other steps may also be wrong because they used the mistake.

Let’s learn how to do this with examples.

Example 1: Addition

A student solved:

$$\begin{array}{r}27\\+15\\\hline 32\end{array}$$

Let’s debug it.

Step 1: Add the ones. The ones are \(7\) and \(5\).

$$7+5=12$$

That means we write down the \(2\) in the ones place and regroup the \(1\) ten.

Step 2: Add the tens. The tens are \(2\) tens, \(1\) ten, and the regrouped \(1\) ten.

$$2+1+1=4$$

So the correct answer is:

$$\begin{array}{r}27\\+15\\\hline 42\end{array}$$

What was the mistake? The student forgot to regroup after adding \(7+5\).

First wrong step: Adding the ones place without regrouping correctly.

Example 2: Subtraction

A student solved:

$$\begin{array}{r}52\\-28\\\hline 30\end{array}$$

Let’s check step by step.

In the ones place, we need to do \(2-8\). We cannot do that without regrouping.

So we regroup one ten from the \(5\) tens.

Then \(52\) becomes \(4\) tens and \(12\) ones.

Now subtract the ones:

$$12-8=4$$

Subtract the tens:

$$4-2=2$$

So the correct answer is:

$$\begin{array}{r}52\\-28\\\hline 24\end{array}$$

What was the mistake? The student did not regroup before subtracting.

First wrong step: The ones place subtraction.

Example 3: Multiplication with equal groups

A student solved:

There are \(4\) bags. Each bag has \(3\) apples. The student wrote:

$$4\times 3=7$$

Let’s debug it in two ways.

Way 1: Repeated addition

$$3+3+3+3=12$$

Way 2: Equal groups picture in words

• Bag 1: 3 apples
• Bag 2: 3 apples
• Bag 3: 3 apples
• Bag 4: 3 apples

Counting all the apples gives \(12\).

So:

$$4\times 3=12$$

What was the mistake? The multiplication fact was incorrect.

First wrong step: Writing \(4\times 3=7\).

This example shows that debugging can mean checking a fact with another model, like repeated addition or equal groups.

Example 4: A two-step check

A student solved:

$$\begin{array}{r}36\\+27\\\hline 513\end{array}$$

Let’s look carefully.

First add the ones:

$$6+7=13$$

Write the \(3\) in the ones place and regroup \(1\) ten.

Then add the tens:

$$3+2+1=6$$

So the correct answer is:

$$\begin{array}{r}36\\+27\\\hline 63\end{array}$$

What did the student do? The student put both digits, \(1\) and \(3\), at the bottom instead of regrouping the \(1\) ten.

First wrong step: After finding \(6+7=13\), the student wrote the digits in the wrong places.

How to spot mistakes

Here are some helpful questions to ask when you debug:

• Did I start in the correct place?
• Did I add or subtract the ones correctly?
• Did I regroup when I needed to?
• Did I use the correct math fact?
• Does the answer make sense?

Make sense check is a powerful tool. For example, if you add \(27+15\), the answer should be more than \(27\). If you subtract \(52-28\), the answer should be less than \(52\). This helps us notice strange answers.

Try this thinking

If a student says:

$$45+14=49$$

You can debug it like this:

• Ones: \(5+4=9\). That part is correct.
• Tens: \(4+1=5\). The tens should be \(5\), not \(4\).

So the first wrong step is the tens place. The correct answer is \(59\).

Important idea

Sometimes part of the work is right. We do not throw all of it away. We find the exact step where the mistake begins.

That is what expert mathematicians do. They look closely, think clearly, and explain their reasoning.

Summary

Algorithmic debugging means checking a completed math problem step by step to find the first wrong step.

We read the work carefully, explain the mistake, and then fix the answer. We can check with place value, regrouping, math facts, repeated addition, and make-sense thinking.

When you debug, you are not just fixing an answer. You are understanding how the math works.

Constructing Mathematical Arguments means telling why your math answer makes sense, not just writing the answer.

In 3rd Grade math, a strong mathematical argument is like being a math detective. You look at the numbers, think carefully, and explain your ideas so another person can understand them.

When you make a mathematical argument, you use:

• Math words like sum, difference, product, equal, denominator, and numerator
• Evidence from numbers, pictures, or models
• Clear sentences that explain why your answer is correct

Good mathematicians do not only say, “The answer is 12.” They also say, “The answer is 12 because I made 3 equal groups of 4, and 3 groups of 4 is 12.”

Why is this important?

• It helps you understand your own thinking.
• It helps others understand your idea.
• It helps you notice mistakes and fix them.
• It shows that your answer is not just a guess.

Parts of a strong math argument

1. State your answer.
2. Tell how you know.
3. Use math words.
4. Show evidence with numbers, drawings, equations, or models.

You can use sentence starters like these:

• I know this because...
• My model shows...
• The equation tells me...
• These groups are equal, so...
• The denominator tells how many equal parts...
• The product is...

Math words that help explain

• Sum: the answer to an addition problem
• Difference: the answer to a subtraction problem
• Product: the answer to a multiplication problem
• Equal: the same amount
• Numerator: the top number in a fraction; it tells how many parts we have
• Denominator: the bottom number in a fraction; it tells how many equal parts make the whole

You can prove your thinking in more than one way.

Sometimes you can use:

• a number sentence
• a drawing
• equal groups
• a number line
• words

Using more than one way makes your argument even stronger.

Worked Example 1: Addition argument

Problem: Mia says that \(7 + 5 = 12\). How can she explain why?

Answer: Mia is correct.

Argument: I know that \(7 + 5 = 12\) because when I start with 7 and count on 5 more—8, 9, 10, 11, 12—I land on 12. The sum is 12.

Another way to show it:

$$7 + 5 = 12$$

You could also draw 7 dots and then 5 more dots, and count all 12 dots.

Worked Example 2: Multiplication argument

Problem: Sam says that \(3 \times 4 = 12\). How can he defend his answer?

Answer: Sam is correct.

Argument: I know that \(3 \times 4 = 12\) because it means 3 equal groups of 4. If I add 4 three times, I get:

$$4 + 4 + 4 = 12$$

So the product is 12.

Picture idea:

• Group 1: 4 stars
• Group 2: 4 stars
• Group 3: 4 stars

There are 12 stars altogether, so the equation is true.

Worked Example 3: Subtraction argument

Problem: Ava says the answer to \(15 - 6\) is 9. How can she explain it?

Answer: Ava is correct.

Argument: I know that \(15 - 6 = 9\) because subtraction finds the difference. If I take 6 away from 15, 9 are left.

Check: I can add the difference and the number taken away:

$$9 + 6 = 15$$

Since \(9 + 6 = 15\), the difference of \(15 - 6\) must be 9.

Worked Example 4: Fraction argument

Problem: Leo says that the shaded part is \(\frac{3}{4}\) because 3 out of 4 equal parts are shaded. How can he explain this clearly?

Answer: Leo is correct if the whole is split into 4 equal parts and 3 of those parts are shaded.

Argument: The fraction is \(\frac{3}{4}\). The numerator, 3, tells how many parts are shaded. The denominator, 4, tells that the whole is split into 4 equal parts. Since 3 of the 4 equal parts are shaded, the shaded part is \(\frac{3}{4}\).

How to make your argument better

• Say more than just the answer.
• Use the correct math word.
• Show your work with a drawing, equation, or model.
• Make sure your explanation matches the numbers.
• Read your sentence and ask: “Did I explain why?”

Example of a weak answer:

“It is 8 because I know it.”

Example of a strong answer:

“The answer is 8 because I added 5 and 3. The sum of \(5 + 3\) is 8.”

When you disagree in math

Sometimes two students get different answers. In math, we can be respectful and use evidence.

You can say:

• I disagree because...
• Can you show your model?
• I got a different answer. Here is my equation...
• Let’s check which answer matches the picture.

This is part of constructing a mathematical argument too. You are using math facts, models, and clear words to explain your thinking.

Try this thinking pattern

1. What is the problem asking?
2. What is my answer?
3. How can I prove it?
4. What math words should I use?
5. Can I show it another way?

Summary

Constructing mathematical arguments means explaining why an answer is correct. A strong argument includes the answer, math words, and proof from numbers, equations, pictures, or models.

When you explain your reasoning, you become a stronger mathematician. You are not only finding answers. You are showing that your thinking makes sense.

Estimating for Reasonableness means making a quick, close guess before or after solving a math problem.

We use estimating to ask, "Does my answer make sense?"

This is a very helpful math habit. Good mathematicians do not just find an answer. They also stop and check if the answer seems reasonable.

One easy way to estimate is to use round numbers. Round numbers are numbers that are easier to work with in your head, like 10, 20, 50, or 100.

When we estimate, we are not trying to get the exact answer. We are trying to get an answer that is close enough to help us check our work.

Why do we estimate?

• To make a quick guess before solving
• To check if an answer is too big or too small
• To catch mistakes
• To build strong number sense

How to estimate for reasonableness

1. Look at the numbers in the problem.
2. Round each number to a nearby easy number.
3. Do the math with the rounded numbers.
4. Compare your estimate to your exact answer.
5. Ask, "Is my exact answer close to my estimate?"

Rounding to a nearby ten

Many 3rd Grade problems can be estimated by rounding to the nearest ten.

• If the ones digit is 0, 1, 2, 3, or 4, round down.
• If the ones digit is 5, 6, 7, 8, or 9, round up.

Examples:

• 23 rounds to 20
• 47 rounds to 50
• 61 rounds to 60
• 88 rounds to 90

Estimate first, then solve

Suppose you are solving \(38 + 21\).

First estimate:

\(38\) is close to \(40\), and \(21\) is close to \(20\).

So the estimate is

$$40 + 20 = 60$$

Now solve exactly:

$$38 + 21 = 59$$

The exact answer, \(59\), is close to the estimate, \(60\). That means the answer is reasonable.

Worked Example 1: Addition

Find \(27 + 34\). Then check if the answer is reasonable.

Step 1: Estimate

Round to the nearest ten:

• \(27 \rightarrow 30\)
• \(34 \rightarrow 30\)

Estimate:

$$30 + 30 = 60$$

Step 2: Solve exactly

$$27 + 34 = 61$$

Step 3: Compare

\(61\) is very close to \(60\), so \(61\) is a reasonable answer.

Worked Example 2: Subtraction

Find \(72 - 29\). Then check if the answer is reasonable.

Step 1: Estimate

Round to the nearest ten:

• \(72 \rightarrow 70\)
• \(29 \rightarrow 30\)

Estimate:

$$70 - 30 = 40$$

Step 2: Solve exactly

$$72 - 29 = 43$$

Step 3: Compare

\(43\) is close to \(40\), so the answer is reasonable.

Worked Example 3: Catching a mistake

A student solved \(46 + 33\) and got \(19\).

Does that answer make sense?

Step 1: Estimate

• \(46 \rightarrow 50\)
• \(33 \rightarrow 30\)

Estimate:

$$50 + 30 = 80$$

Step 2: Compare

The estimate is about \(80\), but the student got \(19\).

\(19\) is not close to \(80\). That tells us the answer is not reasonable.

Let's solve it correctly:

$$46 + 33 = 79$$

Now the exact answer, \(79\), is close to the estimate, \(80\). This makes sense.

Worked Example 4: Estimating in a word problem

A box has \(58\) crayons. Another box has \(24\) crayons. About how many crayons are there altogether?

Step 1: Estimate

• \(58 \rightarrow 60\)
• \(24 \rightarrow 20\)

Estimate:

$$60 + 20 = 80$$

So there are about \(80\) crayons.

Step 2: Exact answer

$$58 + 24 = 82$$

Step 3: Compare

\(82\) is close to \(80\), so the exact answer is reasonable.

Helpful thinking questions

When you finish a problem, ask yourself:

• Is my answer close to my estimate?
• Should the answer be bigger or smaller than the numbers I started with?
• Does my answer make sense for the story?

For example:

• If you add, your answer is usually bigger.
• If you subtract, your answer is usually smaller.

Tips for estimating well

• Use numbers that are easy to work with in your head.
• Rounding to the nearest ten is often enough.
• Your estimate does not need to be exact.
• Use the estimate to check your work, not replace your work.

What estimating helps you notice

• If you added when you should have subtracted
• If your answer is far too large
• If your answer is far too small
• If you made a mistake while solving

Summary

Estimating for reasonableness means making a quick, close guess using easy numbers.

You can round numbers, solve with the rounded numbers, and then compare that estimate to your exact answer.

If your exact answer is close to your estimate, it is probably reasonable. If it is very far away, you should check your work again.

Perseverance in Non-Routine Puzzles means sticking with a puzzle even when the answer does not pop out right away.

In math, some problems are non-routine. That means there is not one easy, familiar step to follow. You may need to try ideas, look for patterns, make a list, draw a picture, and check your work.

Good mathematicians do not just guess and hope. They work carefully, one step at a time. They know that if one idea does not work, they can try another.

This lesson will help you learn how to keep going when a puzzle feels tricky.

Why perseverance matters

Sometimes a puzzle can feel confusing at first. That is normal. A hard start does not mean you cannot solve it.

Perseverance means:

• trying more than once,
• changing your strategy if needed,
• checking what you already know,
• and not giving up too soon.

When you persevere, you learn more than just one answer. You learn how to think.

Helpful strategies for tricky puzzles

Here are some smart ways to solve non-routine puzzles.

1. Understand the problem.
   Ask yourself: What is the puzzle asking? What facts do I know?
2. Try a simple case.
   Start small or use easy numbers first.
3. Make a list.
   Write choices in order so you do not miss any.
4. Draw a picture or model.
   A picture, table, or diagram can make the puzzle easier to see.
5. Look for a pattern.
   See whether something repeats or grows in a regular way.
6. Check your work.
   Make sure your answer fits all the clues.

Important idea: Trial-and-error can be helpful, but it should be smart trial-and-error. That means you try ideas in an organized way, not in a messy way.

For example, if you are testing numbers, do not pick random numbers. Try them in order: 1, 2, 3, 4, and so on. That is called systematic listing.

Worked Example 1: Systematic listing

Sam has 10 stickers. He wants to put them into 2 groups. Each group must have at least 1 sticker. What different pairs of group sizes can he make?

Let the two groups be Group A and Group B. We can make an organized list.

• 
  and 9
• 2 and 8
• 3 and 7
• 4 and 6
• 5 and 5

After 5 and 5, the pairs would start repeating in a different order, like 6 and 4. So we stop.

Answer: The different pairs are
and 9, 2 and 8, 3 and 7, 4 and 6, and 5 and 5.

What helped? We made a list in order. That helped us find all the answers without missing any.

Worked Example 2: Use clues and keep checking

Three friendsLia, Ben, and Rosaeach have a different favorite fruit: apple, banana, and grape.

Clues:

• Lia does not like apple.
• Ben does not like banana.
• Rosa likes grape.

Who likes each fruit?

Start with the strongest clue: Rosa likes grape.

Now grape is taken, so Lia and Ben must have apple and banana.

Lia does not like apple, so Lia must like banana.

That means Ben must like apple.

Answer:

• Lia  banana
• Ben  apple
• Rosa  grape

What helped? We used one clue at a time and checked that the answer matched all the clues.

Worked Example 3: Look for a pattern

A row of shapes follows a pattern:

triangle, square, triangle, square, triangle, square, ...

What is the 9th shape?

The pattern repeats every 2 shapes:

• 1st: triangle
• 2nd: square
• 3rd: triangle
• 4th: square
• 5th: triangle
• 6th: square
• 7th: triangle
• 8th: square
• 9th: triangle

Answer: The 9th shape is a triangle.

What helped? We noticed the repeating pattern and used it to continue the list.

Worked Example 4: Smart trial-and-error

Mia is thinking of a number. It is more than 10 and less than 20. It is even. The digits add to 7. What is the number?

First, list the even numbers between 10 and 20:

12, 14, 16, 18

Now check the digit sums:

• : \(1 + 2 = 3\)
• : \(1 + 4 = 5\)
• : \(1 + 6 = 7\)
• : \(1 + 8 = 9\)

Only 16 has digits that add to 7.

Answer: The number is 16.

What helped? We did not guess wildly. We listed all possible even numbers, then checked each one.

When you get stuck

Getting stuck is part of solving puzzles. If you feel stuck, try one of these moves:

• Read the problem again slowly.
• Circle or say the important facts.
• Draw a picture.
• Make a chart or list.
• Try one idea and see what happens.
• Look for something that repeats.
• Ask, “Does my answer fit all the clues?”

Math habits that help

Strong math thinkers use good habits. These habits make hard puzzles easier.

• Be organized. Keep your list neat.
• Be careful. Check each clue.
• Be patient. Some puzzles take time.
• Be flexible. If one plan does not work, try a new one.

A simple puzzle routine

You can remember these steps:

1. Read the puzzle.
2. Plan a strategy.
3. Try it carefully.
4. Check your answer.

This routine can help with many different puzzles, even new ones.

Mini practice ideas

Try asking yourself these questions when you solve:

• Can I make an ordered list?
• Can I draw it?
• Do I see a pattern?
• Did I test every choice?
• Does my answer match all the rules?

Summary

Non-routine puzzles do not always have an obvious first step, but you can still solve them.

Use perseverance and smart strategies like systematic listing, trial-and-check, drawing models, and pattern finding.

Most of all, remember: if the puzzle feels hard, that does not mean stop. It means think, try, check, and keep going.