Mathematical Modeling and Multi-Step Problem Solving

Deconstructing Word Problems means taking a word problem apart into small, easy pieces. Instead of trying to solve everything at once, we read carefully and find the important parts.

When we deconstruct a word problem, we look for what we know, what is happening, and what we need to find out. This helps us choose the right math.

Word problems can feel tricky because they use sentences instead of just numbers. But if we break the problem into steps, we can understand it better and solve it with confidence.

Step 1: Read the whole problem slowly.

First, read all the words. Do not rush. Think about what the story is about.

Step 2: Find the important numbers.

Circle or list the numbers that tell us how many. These are called the given quantities. They are the facts the problem gives us.

Step 3: Find the action words.

Action words tell what is happening in the story. They help us know what kind of math to use.

• added, more, together, in all often mean add
• left, gave away, how many fewer, remain often mean subtract

Step 4: Find the question.

Look for the part that asks something. This tells us the unknown. The unknown is what we need to find.

You can ask yourself:

• What numbers do I know?
• What is happening to those numbers?
• What is the problem asking me to find?

Step 5: Make a plan.

After we know the facts and the question, we decide what to do. Sometimes we only need one step. Sometimes we need two steps.

A helpful way to annotate a word problem

1. Underline the question.
2. Circle the numbers.
3. Box the action words.
4. Say the problem in your own words.

This helps turn a big problem into smaller parts.

Worked Example 1: One-step addition

Problem: Mia has 7 crayons. Her friend gives her 5 more crayons. How many crayons does Mia have now?

Let’s deconstruct it:

• Given quantities: 7 crayons and 5 more crayons
• Action words: gives and more
• Unknown: How many crayons Mia has now

What is happening? Mia is getting more crayons, so we add.

Math:

$$7 + 5 = 12$$

Answer: Mia has 12 crayons.

Worked Example 2: One-step subtraction

Problem: There are 14 birds in a tree. 6 birds fly away. How many birds are left?

Let’s deconstruct it:

• Given quantities: 14 birds and 6 birds
• Action words: fly away and left
• Unknown: How many birds are left

What is happening? Some birds leave, so we subtract.

Math:

$$14 - 6 = 8$$

Answer: There are 8 birds left.

Worked Example 3: Two-step problem

Problem: Leo has 8 toy cars. He gets 4 more toy cars for his birthday. Then he gives 3 toy cars to his brother. How many toy cars does Leo have now?

Let’s deconstruct it:

• Given quantities: 8, 4, and 3
• First action: gets 4 more
• Second action: gives 3
• Unknown: How many toy cars Leo has now

This problem has two parts. First Leo gets more, then he gives some away.

Step 1: Add because he gets more.

$$8 + 4 = 12$$

Step 2: Subtract because he gives some away.

$$12 - 3 = 9$$

Answer: Leo has 9 toy cars now.

Worked Example 4: Find the hidden question carefully

Problem: A class picked 9 red apples and 7 green apples. They ate 5 apples at lunch. How many apples are left?

Let’s deconstruct it:

• Given quantities: 9 red apples, 7 green apples, and 5 apples eaten
• First action: red apples and green apples are together
• Second action: they ate 5 apples
• Unknown: How many apples are left

We cannot subtract 5 right away until we know how many apples there are in all.

Step 1: Find the total apples.

$$9 + 7 = 16$$

Step 2: Subtract the apples they ate.

$$16 - 5 = 11$$

Answer: There are 11 apples left.

How to tell what matters and what does not

Sometimes a word problem has extra words to tell the story. We should pay attention to the parts that help answer the question.

For example, in the sentence, “Sam has 10 stickers in a blue box,” the words blue box may not matter if the question is only about how many stickers Sam has.

Ask yourself:

• Does this number help me answer the question?
• Does this action change the amount?
• What is the last thing the question asks?

Good habits when solving word problems

• Read the problem two times.
• Do not grab numbers too fast.
• Think about what each number means.
• Find out if the story has one step or two steps.
• Check if your answer makes sense.

Check your answer

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. If the question asks, “How many in all?” your answer should tell the total.

Let’s practice thinking

When you see a new word problem, you can say:

1. What do I know?
2. What happened first?
3. What happened next?
4. What am I trying to find?

These questions help your brain organize the problem before doing the math.

Summary

Deconstructing a word problem means breaking it into small parts. We find the given numbers, look for the action words, and identify the unknown. Then we make a plan and solve step by step.

When you slow down, underline the question, and think about what is happening, word problems become much easier. You can do it!

Representing with Bar Models and Tape Diagrams

A bar model or tape diagram is a picture made with rectangles, or bars, to help us understand a word problem.

Each bar can stand for a whole amount or for parts of an amount. Bar models help us see what we know, what we do not know, and which math operation to use.

In 2nd grade, bar models are helpful for part-part-whole problems and comparison problems.

Why use a bar model?

• It helps us organize the story in the problem.
• It helps us see whether to add or subtract.
• It helps us solve problems with more than one step.

1. Part-Part-Whole Models

Sometimes a problem tells us about parts that make one whole.

For example, if a basket has 3 red apples and 5 green apples, the apples are the parts. All the apples together are the whole.

We can show that with one long bar for the whole, or two smaller bars that join to make the whole.

It looks like this:

Part + Part = Whole

\(3 + 5 = 8\)

When you see parts joining together, you often add.

If the whole is known and one part is known, we can find the missing part by subtracting.

For example:

\(8 - 3 = 5\)

2. Comparison Models

Sometimes a problem compares two amounts.

One person may have more, and another may have less. A bar model can show which bar is longer and how much longer it is.

For example, if Mia has 9 stickers and Ben has 6 stickers, Mia has more. We can compare the two bars:

Mia: a bar for \(9\)
Ben: a shorter bar for \(6\)
Difference: the extra part is \(3\)

We can write:

\(9 - 6 = 3\)

When you are finding how many more or how many fewer, you often subtract.

3. How to Draw a Bar Model

You do not need fancy art. Simple rectangles are enough.

1. Read the problem slowly.
2. Circle or say the numbers you know.
3. Ask: Is this a part-part-whole problem or a comparison problem?
4. Draw bars to match the story.
5. Put the numbers you know in the bars.
6. Put a question mark for the part you do not know.
7. Solve.
8. Check if your answer makes sense.

4. Words That Can Help

These words can give you clues, but always think about the story too.

• Part-part-whole clues: all, in all, altogether, total
• Comparison clues: more, fewer, less, how many more

Worked Example 1: Adding Parts to Make a Whole

Jada has 4 blue crayons and 3 yellow crayons. How many crayons does she have in all?

Step 1: Think about the problem.

There are 2 parts: blue crayons and yellow crayons.

Step 2: Draw the bar model.

One part is \(4\). One part is \(3\). Together they make the whole.

$$ 4 + 3 = ? $$

Step 3: Solve.

\(4 + 3 = 7\)

Answer: Jada has 7 crayons in all.

Worked Example 2: Finding a Missing Part

There are 10 birds in a tree. 6 birds are brown. The rest are red. How many red birds are there?

Step 1: Think about the problem.

The whole is \(10\) birds. One part is \(6\) brown birds. The other part is red birds.

Step 2: Draw the bar model.

Whole bar: \(10\)

Part 1: \(6\)

Part 2: \(?\)

Step 3: Solve.

\(10 - 6 = 4\)

Answer: There are 4 red birds.

Worked Example 3: Comparing Two Amounts

Noah has 12 toy cars. Eli has 8 toy cars. How many more toy cars does Noah have than Eli?

Step 1: Think about the problem.

This is a comparison problem. We are finding the difference between \(12\) and \(8\).

Step 2: Draw the bar model.

Noah's bar is longer and shows \(12\).

Eli's bar is shorter and shows \(8\).

The extra part is what we want to find.

Step 3: Solve.

\(12 - 8 = 4\)

Answer: Noah has 4 more toy cars.

Worked Example 4: A Two-Step Problem

Lena picked 5 strawberries in the morning and 7 strawberries in the afternoon. Then she gave 3 strawberries to her brother. How many strawberries does Lena have left?

Step 1: Find the total she picked.

The morning strawberries and afternoon strawberries are two parts.

\(5 + 7 = 12\)

Step 2: Show what happened next.

Lena had \(12\) strawberries. She gave away \(3\). Now we find what is left.

\(12 - 3 = 9\)

Answer: Lena has 9 strawberries left.

Tips for Success

• Draw bars the same way each time so your thinking stays neat.
• Label your bars with numbers.
• Use a ? for the missing amount.
• Ask yourself: Are the bars showing parts of one whole or two amounts being compared?
• If the problem has two steps, solve one part at a time.

Let’s Think About Common Mistakes

Sometimes students add when they should subtract.

If the problem asks how many more, you are usually finding the difference, so subtraction often helps.

Sometimes students forget what the whole means.

In a part-part-whole model, the whole is everything together.

Sometimes students draw bars but do not label them.

Always write the numbers you know. That helps the picture match the story.

Quick Practice Questions

• Sara has 6 shells. Tom has 2 shells. How many shells do they have in all?
• There are 14 balloons. 9 are blue. How many are not blue?
• Ava has 11 marbles. Max has 7 marbles. How many more marbles does Ava have?

Summary

Bar models and tape diagrams are pictures that help us understand word problems.

We use them to show parts and wholes and to compare amounts.

When parts join together, we often add. When we find a missing part or the difference between amounts, we often subtract.

Drawing a simple bar model can help you solve even tricky multi-step problems one step at a time.

Add-To and Take-From Word Problems

Word problems tell a little math story. In add-to and take-from problems, the amount changes because something is added or taken away.

These problems can ask us to find different parts of the story. Sometimes we know the start and what changed, and we need the end. Sometimes we know the end and need to find what happened in the middle. Sometimes we know the change and the end, and we need to find the start.

Let’s learn how to understand these story problems step by step.

1. What is an add-to problem?

An add-to problem is when more join a group.

• Start with some
• More are added
• Now there is a new total

You can think: start + more = end

In math, that looks like:

$$\text{start} + \text{more} = \text{end}$$

2. What is a take-from problem?

A take-from problem is when some go away from a group.

• Start with some
• Some are taken away
• Now there is a new amount left

You can think: start - taken away = end

In math, that looks like:

$$\text{start} - \text{taken away} = \text{left}$$

3. The number we do not know can be in different places

In word problems, the missing number is not always at the end. We must read carefully to see what the question is asking.

There are 3 common missing parts:

• Start unknown: We do not know how many there were at first.
• Change unknown: We do not know how many were added or taken away.
• Result unknown: We do not know how many there are at the end.

4. A good way to solve word problems

1. Read the story slowly.
2. Ask: Is this add-to or take-from?
3. Find what number is missing.
4. Write an equation with a box or a letter for the missing number.
5. Solve the equation.
6. Check: Does the answer make sense in the story?

5. Helpful clue words

Clue words can help, but always read the whole story. Some words may suggest adding or taking away.

• Add-to clues: joined, got more, added, came, bought more
• Take-from clues: gave away, lost, left, ate, went away, took away

Worked Example 1: Add-to, result unknown

Lena has 7 crayons. Her teacher gives her 5 more crayons. How many crayons does Lena have now?

Step 1: This is an add-to problem because she gets more.

Step 2: We know the start is 7. We know 5 more were added. We do not know the end.

Equation:

$$7 + 5 = \Box$$

Solve:

$$7 + 5 = 12$$

Answer: Lena has 12 crayons now.

Check: Starting with 7 and getting 5 more should make a bigger number. 12 is bigger than 7, so the answer makes sense.

Worked Example 2: Take-from, result unknown

There were 14 apples in a basket. Sam ate 3 apples. How many apples are left?

Step 1: This is a take-from problem because some apples were eaten.

Step 2: We know the start is 14. We know 3 were taken away. We do not know how many are left.

Equation:

$$14 - 3 = \Box$$

Solve:

$$14 - 3 = 11$$

Answer: There are 11 apples left.

Check: If apples were taken away, the answer should be smaller than 14. 11 is smaller, so it makes sense.

Worked Example 3: Add-to, change unknown

Maya had 6 stickers. Now she has 13 stickers. How many stickers did she get?

Step 1: This is an add-to problem because her number of stickers got bigger.

Step 2: We know the start is 6. We know the end is 13. We do not know how many were added.

Equation:

$$6 + \Box = 13$$

To solve, think: What number goes with 6 to make 13?

$$6 + 7 = 13$$

Answer: Maya got 7 stickers.

Check: Start with 6. Add 7. You get 13. That matches the story.

Worked Example 4: Take-from, start unknown

Some birds were in a tree. 4 birds flew away. Now 9 birds are still in the tree. How many birds were in the tree at first?

Step 1: This is a take-from problem because birds flew away.

Step 2: We do not know the start. We know 4 were taken away. We know 9 are left.

Equation:

$$\Box - 4 = 9$$

To solve, think: What number minus 4 equals 9?

Or think: if 4 flew away and 9 are left, then the start was:

$$9 + 4 = 13$$

Answer: There were 13 birds in the tree at first.

Check:

$$13 - 4 = 9$$

The answer fits the story.

6. How to model the story

Modeling means showing the math in a way that helps us understand the story.

You can model with:

• Drawings: circles, stars, or simple pictures
• Counters: blocks, beans, or coins
• Number sentences: equations like \(8 + 2 = 10\)
• Acting it out: use objects and move them in or out

For example, for \(5 + 3\), you can draw 5 dots, then 3 more dots, and count all the dots.

For \(12 - 4\), you can draw 12 dots, cross out 4, and count what is left.

7. How to think about unknown numbers

When the missing number is at the end, solve in the usual way.

• \(8 + 6 = \Box\)
• \(15 - 5 = \Box\)

When the missing number is in the middle, think: What number makes the equation true?

• \(9 + \Box = 14\)
• \(17 - \Box = 12\)

When the missing number is at the start, think about what the amount must have been before the change happened.

• \(\Box + 3 = 10\)
• \(\Box - 2 = 7\)

8. Be careful with the question

Sometimes students see the words “more” or “left” and rush. But we must look at what the question asks.

Look at these two different questions:

• Jada had 8 toy cars. She got 4 more. How many does she have now? This asks for the end.
• Jada had 8 toy cars. Now she has 12. How many did she get? This asks for the change.

Both stories are about adding, but the missing number is different.

9. Try these thinking steps

• What happened first?
• Did the amount get bigger or smaller?
• What number do I know at the start?
• What number changed?
• What number do I know at the end?
• Which number is missing?

10. Quick practice ideas

You can practice by making your own little stories.

• Start with 10 blocks. Add 2. How many now?
• Start with 13 blocks. Take away 5. How many left?
• Start with 4 blocks. End with 9 blocks. How many were added?
• Some blocks were there. Take away 3. Now 8 are left. How many were there first?

11. What to remember

• Add-to means more join the group.
• Take-from means some go away.
• The missing number can be the start, the change, or the end.
• Draw, act it out, or write an equation to help solve.
• Always check that your answer matches the story.

Summary

Add-to and take-from word problems are math stories about amounts that change. In add-to problems, more are added. In take-from problems, some are taken away.

The missing number can be in different places, so read the story carefully. Ask yourself what happened, write an equation, solve it, and check your answer.

Put-Together and Take-Apart Word Problems help us think about parts and wholes.

Sometimes a story problem tells us about two parts and asks for the whole. Sometimes it tells us the whole and one part, and we must find the missing part.

These problems are called put-together and take-apart problems.

• Put-together means we join parts to make a whole.
• Take-apart means we know the whole and break it into parts.

In both kinds of problems, the parts and the whole stay the same. Nothing is being added later, and nothing is being taken away later. We are just looking at how the whole and the parts fit together.

A good way to think about these problems is:

$$\text{part} + \text{part} = \text{whole}$$

or

$$\text{whole} - \text{part} = \text{missing part}$$

Parts and Whole

A whole is everything together. A part is one piece of the whole.

For example, if 4 children are wearing red shirts and 3 children are wearing blue shirts, then all the children together make the whole.

$$4 + 3 = 7$$

Here:

• 4 is one part
• 3 is another part
• 7 is the whole

How to Solve These Problems

1. Read the problem slowly.
2. Ask: What are the parts?
3. Ask: What is the whole?
4. Decide what is missing.
5. Choose addition or subtraction.
6. Write an equation.
7. Check if your answer makes sense.

When do I add?

• Add when you know the parts and need the whole.

When do I subtract?

• Subtract when you know the whole and one part and need the missing part.

Clue Words Can Help, But Think About the Story

Some word problems use words like in all, total, or altogether. These often mean we are finding the whole.

Some problems ask, How many are boys? or How many are left in this part? These often mean we know the whole and need a missing part.

But the most important thing is to understand the story. Ask yourself: Am I joining parts, or am I finding a missing part of the whole?

Use a Part-Part-Whole Picture

You can draw a simple model to help:

• One big box for the whole
• Two smaller boxes for the parts

If the parts are 5 and 2, then the whole is 7.

$$5 + 2 = 7$$

If the whole is 7 and one part is 5, then the missing part is 2.

$$7 - 5 = 2$$

Worked Example 1: Find the Whole

There are 6 apples on the table. There are 3 oranges on the table. How many pieces of fruit are on the table?

Step 1: Find the parts.

• 6 apples
• 3 oranges

Step 2: Find the whole.

We need all the fruit together.

Step 3: Write an equation.

$$6 + 3 = 9$$

Answer: There are 9 pieces of fruit on the table.

Worked Example 2: Find a Missing Part

There are 10 birds in a tree. 4 birds are blue. The rest are yellow. How many birds are yellow?

Step 1: Find the whole and the known part.

• Whole: 10 birds
• One part: 4 blue birds

Step 2: Find the missing part.

We need to know how many are yellow.

Step 3: Write a subtraction equation.

$$10 - 4 = 6$$

Answer: There are 6 yellow birds.

Worked Example 3: The Unknown Can Be First

Some red balloons and 5 blue balloons make 12 balloons in all. How many red balloons are there?

Here, the missing part comes first. That is okay.

Step 1: Find what we know.

• One part: 5 blue balloons
• Whole: 12 balloons
• Missing part: red balloons

Step 2: Write an equation with a box for the unknown.

$$\Box + 5 = 12$$

Step 3: Solve by subtracting.

$$12 - 5 = 7$$

So,

$$7 + 5 = 12$$

Answer: There are 7 red balloons.

Worked Example 4: A Bigger Story Problem

At the park, 8 children are on the swings and 7 children are on the slide. How many children are playing on the swings or the slide?

Step 1: Find the parts.

• 8 on the swings
• 7 on the slide

Step 2: Find the whole.

We want the total number of children playing on the swings or the slide.

Step 3: Add.

$$8 + 7 = 15$$

Answer: 15 children are playing on the swings or the slide.

How to Check Your Work

• If you added parts, ask: Does my whole make sense?
• If you found a missing part, add the two parts together to check the whole.

For example, in Example 2, we found 6 yellow birds.

To check:

$$4 + 6 = 10$$

It matches the whole, so the answer makes sense.

Things to Watch Out For

• Do not rush. Read carefully.
• Make sure you know if the problem is asking for a whole or a part.
• If you know both parts, add.
• If you know the whole and one part, subtract.
• The missing number can be at the beginning, middle, or end of the equation.

Try Thinking Like This

• What numbers tell about the parts?
• What number tells about the whole?
• What am I trying to find?

If you can answer those questions, you can solve put-together and take-apart problems.

Summary

Put-together and take-apart word problems are about parts and wholes.

When you know the parts and need the whole, use addition.

When you know the whole and one part and need the missing part, use subtraction.

Always read the story carefully, write an equation, and check your answer.

Comparison Word Problems help us compare two groups.

When we compare, we ask questions like:

• Who has more?
• Who has fewer?
• How many more?
• How many fewer?

These problems are about finding the difference between two numbers.

The difference tells how far apart the numbers are. We usually find the difference by subtracting.

For example, if one child has 9 stickers and another child has 6 stickers, we can compare the amounts:

So, 9 is 3 more than 6. And 6 is 3 fewer than 9.

Important idea: The same comparison can be said in two ways.

• 9 is 3 more than 6.
• 6 is 3 fewer than 9.

Both sentences talk about the same difference: 3.

When you solve comparison word problems, it helps to follow these steps:

1. Read the problem slowly.
2. Find the two groups being compared.
3. Decide which group is bigger and which group is smaller.
4. Look for clue words like more, fewer, or less.
5. Choose the math you need.
6. Write the answer in a complete sentence.

Clue words can help, but always think about what the story is asking.

• How many more? usually means find the difference.
• How many fewer? usually means find the difference.
• More than can mean add when you know the smaller group and the difference.
• Fewer than can mean subtract when you know the bigger group and the difference.

Let’s look at the kinds of comparison problems you may see.

Type 1: Find how many more or fewer

In these problems, you know both amounts. You need to find the difference.

Worked Example 1

Lena has 12 crayons. Max has 8 crayons. How many more crayons does Lena have than Max?

Step 1: Find the bigger number and the smaller number.

• Bigger number: 12
• Smaller number: 8

Step 2: Subtract to find the difference.

Answer: Lena has 4 more crayons than Max.

We can also say: Max has 4 fewer crayons than Lena.

Worked Example 2

There are 15 birds in one tree and 9 birds in another tree. How many fewer birds are in the second tree?

The second tree has 9 birds. The first tree has 15 birds. We compare them by subtracting:

Answer: There are 6 fewer birds in the second tree.

Type 2: Find the bigger group

In these problems, you know the smaller amount and how many more the bigger amount has.

To find the bigger group, add.

Worked Example 3

Noah has 7 toy cars. Mia has 5 more toy cars than Noah. How many toy cars does Mia have?

Noah has 7. Mia has 5 more than 7, so we add:

Answer: Mia has 12 toy cars.

Here is another way to think about it:

• Start with the smaller group: 7
• Add the difference: 5
• Find the bigger group: 12

Type 3: Find the smaller group

In these problems, you know the bigger amount and how many fewer the smaller amount has.

To find the smaller group, subtract.

Worked Example 4

A basket has 14 apples. A bag has 6 fewer apples than the basket. How many apples are in the bag?

The basket has the bigger amount: 14.

The bag has 6 fewer, so subtract:

Answer: There are 8 apples in the bag.

Let’s practice how to think about comparison problems.

If the problem asks how many more or how many fewer, compare the two groups and find the difference.

If the problem tells you one group has more than another, you may need to add to find the bigger group.

If the problem tells you one group has fewer than another, you may need to subtract to find the smaller group.

Helpful question to ask yourself: Am I finding the difference, the bigger group, or the smaller group?

Let’s look at a model that can help.

Suppose Sam has 10 blocks and Eva has 7 blocks.

• Sam: 10 blocks
• Eva: 7 blocks

The extra part is the difference.

So Sam has 3 more blocks than Eva.

You can imagine lining up the blocks. The blocks that do not match are the difference.

Watch out for these mistakes:

• Do not just pick the numbers and add every time.
• Do not forget to find which group is bigger.
• Read the question at the end carefully.
• If it says how many fewer, the answer is still the difference.

Try this thinking:

• If I know both groups, I can subtract to compare.
• If I know the smaller group and how many more, I can add.
• If I know the bigger group and how many fewer, I can subtract.

Mini Check

1. Sara has 11 flowers. Ben has 4 flowers. How many more flowers does Sara have?

Subtract:

Sara has 7 more flowers.

2. Tom has 6 marbles. Lee has 3 more marbles than Tom. How many marbles does Lee have?

Add:

Lee has 9 marbles.

3. A jar has 13 buttons. A box has 5 fewer buttons than the jar. How many buttons are in the box?

Subtract:

The box has 8 buttons.

Summary

Comparison word problems help us see how two amounts are alike or different.

We often find the difference by subtracting.

Sometimes we use the difference to find a missing bigger amount by adding, or a missing smaller amount by subtracting.

Always read carefully and ask: What am I finding?

Strategizing Two-Step Word Problems means figuring out that a story problem needs two actions to solve. You cannot jump to the end right away. First, you find the hidden question. Then you use that answer to solve the final question.

Sometimes a word problem asks one question, but there is really another question hiding inside it. Good math thinkers stop, read carefully, and ask, “What do I need to know first?”

In this lesson, you will learn how to find the hidden question, choose the right operations, and solve two-step word problems one step at a time.

What is a two-step word problem?

A two-step word problem is a problem that needs two math steps. You might:

• add, then subtract
• subtract, then add
• add two groups, then compare
• find a total, then find how many are left

You use the answer from the first step to help with the second step.

How to solve a two-step word problem

1. Read the whole problem slowly.
2. Ask: What is the final question?
3. Ask: What do I need to know first?
4. Solve the hidden question.
5. Use that answer to solve the final question.
6. Check: Does my answer make sense?

Look for clue words

Clue words can help, but always read the whole story because some problems use more than one operation.

• Add: in all, altogether, total, joined, more
• Subtract: left, remain, how many more, gave away, fewer

Important idea: The final question is not always the first thing you solve.

For example, if the problem asks, “How many are left?” you may first need to find out how many there were in all.

Strategy: Find the hidden question

Here are some helpful questions to ask yourself:

• Do I need to find a total first?
• Do I need to find how many are left first?
• Do I need to compare two amounts after finding one of them?
• Is there a number I do not know yet that will help me answer the end?

If the answer is yes, then that is your hidden question.

Worked Example 1: Add, then subtract

Lia has 12 stickers. Her friend gives her 5 more stickers. Then Lia gives 3 stickers to her brother. How many stickers does Lia have now?

Step 1: Find the hidden question.

How many stickers does Lia have after getting 5 more?

$$12 + 5 = 17$$

Step 2: Solve the final question.

Now subtract the 3 stickers she gave away.

$$17 - 3 = 14$$

Answer: Lia has 14 stickers now.

Check: She got more first, so the number went up. Then she gave some away, so the number went down. That makes sense.

Worked Example 2: Add two groups, then compare

There are 8 red balloons and 7 blue balloons. How many balloons are there altogether? Then 6 balloons float away. How many balloons are left?

This problem already shows the two parts clearly.

Step 1: Find the total number of balloons.

$$8 + 7 = 15$$

Step 2: Subtract the balloons that float away.

$$15 - 6 = 9$$

Answer: There are 9 balloons left.

Hidden question: How many balloons were there altogether before some floated away?

Worked Example 3: Subtract, then add

Ben had 18 toy cars. He gave 4 toy cars to his cousin. Later, he got 3 new toy cars for his birthday. How many toy cars does Ben have now?

Step 1: Find the hidden question.

How many toy cars did Ben have after giving some away?

$$18 - 4 = 14$$

Step 2: Add the new toy cars.

$$14 + 3 = 17$$

Answer: Ben has 17 toy cars now.

Check: The number went down when he gave cars away. Then it went up when he got more. That makes sense.

Worked Example 4: A hidden comparison question

Mia picked 9 flowers in the morning and 6 flowers in the afternoon. She used 5 flowers to make a gift. How many flowers does she have left?

This final question asks how many are left. But first we need to know how many flowers she picked in all.

Step 1: Find the total flowers picked.

$$9 + 6 = 15$$

Step 2: Subtract the flowers she used.

$$15 - 5 = 10$$

Answer: Mia has 10 flowers left.

How to model the problem

Math modeling means showing the story with numbers, words, or a simple plan. You can model a two-step problem like this:

• Words: First find the total. Then find how many are left.
• Number sentence: \\(9 + 6 = 15\\), then \\(15 - 5 = 10\\)
• Quick plan: total first  subtract second

You do not have to solve it all in your head at once. Breaking it into two smaller parts makes it easier.

Tips for success

• Circle or say the final question.
• Underline the numbers you need.
• Ask, “What do I need to know first?”
• Solve one step at a time.
• Write both number sentences.
• Check that your answer matches the story.

Common mistakes to avoid

• Doing only one step. If the story changes two times, you probably need two steps.
• Using the wrong first step. Stop and think about the hidden question.
• Forgetting what the question asks. The answer should match the final question, not just the first part.

Try thinking like this

If you read, “Sam has 7 marbles. He gets 8 more. Then he loses 2. How many marbles now?” you can say:

• First, I need to know how many after he gets more.
• Then, I can find how many after he loses 2.

That means:

$$7 + 8 = 15$$

$$15 - 2 = 13$$

Summary

Two-step word problems need two math actions. First, find the hidden question that must be solved before the final answer. Then use that answer in the second step. When you read carefully, solve one part at a time, and check your work, you can solve tricky story problems with confidence.

Evaluating the Reasonableness of Answers means asking, “Does my answer make sense?”

Sometimes we solve a math problem and get an answer, but we still need to check it. A good math thinker does not stop after writing a number. A good math thinker asks if the number fits the story.

When we check if an answer is reasonable, we compare it to what we know about the problem. We can use an estimate, think about the size of the numbers, and ask if the answer is too big, too small, or just right.

Why is this important?

• It helps us catch mistakes.
• It helps us understand the problem better.
• It helps us know if our answer fits real life.

How to check if an answer is reasonable

1. Read the problem again. Ask: What is happening in the story?
2. Think about what kind of answer makes sense. Should the answer be bigger or smaller? Should it be close to 10, 20, 50, or another number?
3. Estimate. Make a quick, close guess.
4. Compare. Is your answer close to your estimate?
5. Ask: Does my answer match the story?

Clue words to help you think

• If you add, the answer should usually be bigger.
• If you subtract, the answer should usually be smaller.
• If the story is about sharing or making equal groups, your answer should fit that idea.
• You cannot have things like negative apples or more items than the story allows.

Use estimation to check your answer

An estimate is a number that is close, but not exact. Estimating helps us decide if an answer makes sense.

For example, if you add \(18 + 21\), you might think:

\(18\) is close to \(20\), and \(21\) is close to \(20\).

So the estimate is about:

$$20 + 20 = 40$$

The exact answer is:

$$18 + 21 = 39$$

Since \(39\) is close to \(40\), the answer is reasonable.

Ask yourself these helpful questions

• Is my answer too big?
• Is my answer too small?
• Should my answer be more or less than the numbers in the problem?
• Does my answer fit the story?
• Is my answer close to my estimate?

Worked Example 1: One-step addition

Lena has \(23\) stickers. Her friend gives her \(14\) more stickers. How many stickers does Lena have now?

First, solve the problem:

$$23 + 14 = 37$$

Now check if \(37\) is reasonable.

Estimate:

\(23\) is about \(20\), and \(14\) is about \(10\).

$$20 + 10 = 30$$

Our exact answer is \(37\). That is close to \(30\), so it makes sense.

We can also think: Lena got more stickers, so the answer should be bigger than \(23\). Since \(37\) is bigger than \(23\), the answer fits the story.

Worked Example 2: One-step subtraction

There are \(45\) birds in a park. Then \(12\) birds fly away. How many birds are left?

First, solve the problem:

$$45 - 12 = 33$$

Now check if \(33\) is reasonable.

Estimate:

\(45\) is about \(50\), and \(12\) is about \(10\).

$$50 - 10 = 40$$

The exact answer is \(33\). That is fairly close to \(40\), so it is reasonable.

We can also think: some birds flew away, so the answer should be less than \(45\). Since \(33\) is less than \(45\), it makes sense.

Worked Example 3: Two-step problem

A class has \(16\) red crayons and \(17\) blue crayons. The students use \(9\) crayons. How many crayons are left?

Step 1: Find how many crayons there are at first.

$$16 + 17 = 33$$

Step 2: Subtract the crayons that were used.

$$33 - 9 = 24$$

So, \(24\) crayons are left.

Now check if \(24\) is reasonable.

Estimate the first part:

\(16\) is about \(20\), and \(17\) is about \(20\).

$$20 + 20 = 40$$

Then subtract about \(10\):

$$40 - 10 = 30$$

Our estimate is about \(30\). Our exact answer is \(24\). That is close enough to make sense.

We can also think about the story. At first there were more than \(30\) crayons because \(16 + 17 = 33\). After using \(9\), there should be fewer than \(33\). Since \(24\) is fewer than \(33\), the answer fits.

Worked Example 4: Catching an unreasonable answer

Sam has \(27\) toy cars. He gives \(8\) toy cars to his cousin. He says he has \(35\) toy cars left. Is that reasonable?

No, that answer is not reasonable.

Why not?

Sam gave away toy cars, so he should have fewer than \(27\), not more.

Let’s solve it:

$$27 - 8 = 19$$

Estimate:

\(27\) is about \(30\), and \(8\) is about \(10\).

$$30 - 10 = 20$$

The exact answer \(19\) is close to \(20\), so \(19\) is reasonable.

The answer \(35\) is not close to \(20\), and it does not fit the story. So \(35\) is unreasonable.

Tips for multi-step word problems

• Check each step, not just the final answer.
• After adding, ask if the total should be bigger.
• After subtracting, ask if the amount should be smaller.
• Use quick estimates to see if you are close.
• Think about the story from beginning to end.

What reasonable answers look like

• They are close to your estimate.
• They match what is happening in the story.
• They are not too big or too small.
• They use the numbers in a way that makes sense.

Let’s practice thinking

If a problem says a child had \(12\) books and got \(15\) more, would an answer of \(5\) make sense?

No. The child got more books, so the answer should be more than \(12\).

If a problem says there were \(39\) cookies and \(6\) were eaten, would an answer of \(90\) make sense?

No. Some cookies were eaten, so the answer should be less than \(39\).

Summary

After you solve a problem, always stop and check. Ask yourself, “Does my answer make sense?”

Use an estimate, think about whether the answer should be bigger or smaller, and make sure the answer matches the story. That is how you evaluate the reasonableness of an answer.

Constructing Mathematical Justifications means showing and telling how you know your answer is correct.

In 2nd Grade math, it is not enough to only write the answer. A strong math explanation tells what you did, why you did it, and how the numbers match the story.

When we solve a word problem, we can justify our thinking with:

• math words like add, subtract, total, difference, left, more, fewer, equal
• equations like \(7 + 5 = 12\)
• drawings or diagrams like circles, boxes, number lines, or tape drawings

A mathematical justification is like saying, “Here is my proof.”

Why do we justify our thinking?

Justifying helps us slow down and think carefully.

It also helps other people understand our idea. If a friend or teacher reads our work, they should be able to follow our steps.

Good justifications also help us check for mistakes. If our picture, equation, and words all match, our answer is more likely to be correct.

What should a strong math justification include?

You can remember this simple plan:

1. Tell what the problem is asking.
2. Show the math you used.
3. Explain why that math makes sense.
4. Write the answer with a label.

For example, if a problem is about apples, the answer should say apples, not just a number.

Math words that help explain thinking

These words can help you build a strong explanation:

• add or plus: putting groups together
• subtract or minus: taking away or finding how many are left
• total: all together
• difference: how many more or fewer
• equal: the same amount
• left: what remains after taking away
• more: a bigger amount
• fewer: a smaller amount

Good math explanations often start like this:

• I added because...
• I subtracted because...
• First, I...
• Then, I...
• My equation is...
• My drawing shows...
• So the answer is...

Using equations to justify

An equation is a math sentence. It helps show exactly what happened in the problem.

If there are two steps, you may need two equations.

For example:

First step: $$8 + 4 = 12$$

Second step: $$12 - 3 = 9$$

These equations show that we first put groups together, then took some away.

Using drawings to justify

Drawings help us see the story in the problem.

You can use:

• circles or dots for objects
• boxes for groups
• a number line for counting on or back
• a simple tape drawing with parts and a whole

Your drawing does not have to be fancy. It only needs to clearly match the problem.

How to justify a one-step problem

Let us use this problem:

Lena has 6 stickers. Her friend gives her 3 more stickers. How many stickers does Lena have now?

Here is a strong justification:

What is the problem asking? It is asking how many stickers Lena has now.

What math should we use? We add because she got 3 more stickers.

Equation: $$6 + 3 = 9$$

Drawing: 6 dots and 3 more dots make 9 dots.

Answer: Lena has 9 stickers.

This is a justification because it uses words, an equation, and a reason.

How to justify a two-step problem

Some word problems have more than one action. We need to explain each step in order.

Look for words that tell what happens first and next.

Helpful order words are:

• first
• then
• next
• after that
• now

When we justify a two-step problem, we can say:

1. First, I ...
2. Then, I ...
3. So ...

Worked Example 1: Add to find a total

Problem: There are 5 birds in a tree. 4 more birds land in the tree. How many birds are in the tree now?

Step 1: Tell what the problem asks.

The problem asks for the total number of birds in the tree now.

Step 2: Choose the math.

I should add because 4 more birds joined the 5 birds.

Step 3: Write the equation.

$$5 + 4 = 9$$

Step 4: Show a drawing.

I can draw 5 circles for the first birds and 4 circles for the new birds. Altogether there are 9 circles.

Step 5: Write the answer.

There are 9 birds in the tree now.

Complete justification: I added because 4 more birds landed in the tree. My equation is \(5 + 4 = 9\). My drawing shows 5 birds and 4 more birds, which makes 9 birds. So there are 9 birds in the tree now.

Worked Example 2: Subtract to find how many are left

Problem: Omar had 12 crayons. He gave 5 crayons to his sister. How many crayons does he have left?

Step 1: Tell what the problem asks.

The problem asks how many crayons are left.

Step 2: Choose the math.

I should subtract because Omar gave some away.

Step 3: Write the equation.

$$12 - 5 = 7$$

Step 4: Show a drawing.

I can draw 12 crayons and cross out 5. Then 7 remain.

Step 5: Write the answer.

Omar has 7 crayons left.

Complete justification: I subtracted because Omar gave away 5 crayons. My equation is \(12 - 5 = 7\). My drawing shows 12 crayons with 5 crossed out, so 7 are left. Therefore, Omar has 7 crayons left.

Worked Example 3: Two-step problem

Problem: Mia picked 7 flowers. Then she picked 6 more flowers. She gave 4 flowers to her mom. How many flowers does Mia have now?

Step 1: Find the first action.

First, Mia picked 7 flowers and then 6 more flowers. That means I add.

$$7 + 6 = 13$$

Now Mia has 13 flowers.

Step 2: Find the second action.

Then she gave 4 flowers away. That means I subtract.

$$13 - 4 = 9$$

Step 3: Write the answer.

Mia has 9 flowers now.

Complete justification: First, I added because Mia picked 6 more flowers, so \(7 + 6 = 13\). Then I subtracted because she gave away 4 flowers, so \(13 - 4 = 9\). My drawing could show 13 flowers with 4 crossed out. So Mia has 9 flowers now.

Worked Example 4: Compare amounts and explain why

Problem: Noah has 15 blocks. Ava has 9 blocks. How many more blocks does Noah have than Ava?

Step 1: Tell what the problem asks.

The problem asks for the difference between 15 and 9.

Step 2: Choose the math.

I should subtract because the question says how many more.

Step 3: Write the equation.

$$15 - 9 = 6$$

Step 4: Show a drawing.

I can draw 15 blocks for Noah and 9 blocks for Ava. The extra blocks Noah has are 6.

Step 5: Write the answer.

Noah has 6 more blocks than Ava.

Complete justification: I subtracted because the problem asks how many more blocks Noah has than Ava. My equation is \(15 - 9 = 6\). My drawing shows that after matching 9 blocks with 9 blocks, 6 blocks are extra. So Noah has 6 more blocks than Ava.

How to make your explanation even better

Sometimes students only write a number. For example, they write 9. That is not a full justification.

A better explanation includes the reason and the math.

Compare these:

• Too short: 9
• Better: \(5 + 4 = 9\)
• Best: I added 5 and 4 because 4 more birds landed in the tree. My equation is \(5 + 4 = 9\). So there are 9 birds in the tree now.

The best answer helps someone else understand exactly what happened.

Check your justification

After you solve, ask yourself these questions:

• Did I answer the question that was asked?
• Did I use the right math operation: add or subtract?
• Did I show my equation?
• Did I explain why I added or subtracted?
• Did my drawing match my equation?
• Did I write the answer with a label?

If you can say yes to these questions, your justification is strong.

A simple sentence frame you can use

If writing is hard, you can use this pattern:

I ______ because ______. My equation is ______. My drawing shows ______. So the answer is ______.

Example:

I added because 3 more stickers were given. My equation is \(6 + 3 = 9\). My drawing shows 6 stickers and 3 more stickers. So the answer is 9 stickers.

Common mistakes to watch for

• Using the wrong operation
  If the story says some were given away, you usually subtract.
• Forgetting a step
  In a two-step problem, make sure you solve both parts.
• No explanation
  Do not only write the number. Tell how you got it.
• No label
  Write 9 flowers, not just 9.
• Drawing does not match the math
  Check that your picture shows the same numbers as your equation.

Let’s practice thinking like a math explainer

If a problem says, Sam has 8 toy cars. He gets 2 more. Then he gives 3 away. A strong thinker says:

• First I add: \(8 + 2 = 10\)
• Then I subtract: \(10 - 3 = 7\)
• So Sam has 7 toy cars

A strong justification says even more:

First, I added because Sam got 2 more toy cars, so \(8 + 2 = 10\). Then, I subtracted because he gave away 3 toy cars, so \(10 - 3 = 7\). Therefore, Sam has 7 toy cars.

Summary

Constructing mathematical justifications means explaining your answer with words, equations, and drawings.

A strong justification tells what the problem asks, what math you used, why you used it, and what the answer means.

When you solve word problems, try to use math words like add, subtract, total, difference, and left.

Remember: in math, it is important to not only get the answer, but also to show and tell how you know.