Algebraic Thinking and Word Problems

Result Unknown Word Problems

Sometimes a math story tells us what happened at the beginning and what changed, but it does not tell us the ending. Our job is to find the result. This is called a result unknown word problem.

We can solve these story problems by listening for what is happening in the story. If more are joining, we usually add. If some are going away, we usually subtract.

Here are two big clues:

• Add when things come together, are added, or more arrive.
• Subtract when things are taken away, leave, or are used up.

When we solve a result unknown problem, we can ask:

1. What number do I start with?
2. What changed?
3. Do I add or subtract?
4. What is the result?

You can also draw a picture, use counters, or act out the story. These tools help you see the math.

Addition result unknown means we know the start and how many more joined, but we need to find the end.

It can look like this:

$$5 + 3 = \square$$

Subtraction result unknown means we know the start and how many went away, but we need to find the end.

It can look like this:

$$9 - 2 = \square$$

Words that may mean add:

• more
• join
• altogether
• in all

Words that may mean subtract:

• left
• went away
• gave away
• how many now

Let’s try some examples.

Example 1: Easy addition story

Lia has 4 balloons. Her dad gives her 2 more balloons. How many balloons does Lia have now?

Step 1: Start with 4.

Step 2: 2 more balloons are added.

Step 3: Use addition.

$$4 + 2 = 6$$

Answer: Lia has 6 balloons now.

Example 2: Easy subtraction story

There are 7 cookies on a plate. Ben eats 3 cookies. How many cookies are left?

Step 1: Start with 7.

Step 2: 3 cookies went away.

Step 3: Use subtraction.

$$7 - 3 = 4$$

Answer: There are 4 cookies left.

Example 3: A little bigger addition story

8 birds are in a tree. 5 more birds fly into the tree. How many birds are in the tree now?

Step 1: Start with 8 birds.

Step 2: 5 more birds join.

Step 3: Use addition.

$$8 + 5 = 13$$

Answer: There are 13 birds in the tree now.

Example 4: A little bigger subtraction story

12 crayons are in a box. 4 crayons are taken out. How many crayons are in the box now?

Step 1: Start with 12 crayons.

Step 2: 4 crayons are taken away.

Step 3: Use subtraction.

$$12 - 4 = 8$$

Answer: There are 8 crayons in the box now.

How to check your work

• Read the story again.
• Ask, “Did more join, or did some go away?”
• Make sure your answer tells how many are there now.
• If you can, draw it or use objects to see if your answer makes sense.

Try thinking like this:

• “I started with 6. Then 3 more came. So I add.”
• “I started with 10. Then 2 went away. So I subtract.”

Important idea

In a result unknown problem, the box or missing part is at the end.

For example:

$$3 + 4 = \square$$

$$9 - 5 = \square$$

The story gives the start and the change. You find the ending.

Summary

Result unknown word problems ask, “How many are there now?” or “How many are left?” You look at the story to see if things are joining or going away. Then you use addition or subtraction to find the answer.

Change Unknown Word Problems are stories about things being added or taken away, but the part that changed is missing.

We know the starting amount. We know the ending amount. We need to find how many were added or how many were taken away.

This lesson will help you learn how to read the story, think about what changed, and solve for the missing number.

What is a change unknown problem?

In these problems, the story starts with one number and ends with another number.

The middle part is missing. That missing part is the change.

Look at this idea:

Start with some. Then something happens. End with some.

We can show it like this:

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

or

$$\text{start} - \text{change} = \text{end}$$

If more things join, we add.

If some things leave, we subtract.

How to solve change unknown problems

1. Read the story carefully. Ask: What number do I start with?

2. Find the ending number. Ask: How many are there at the end?

3. Think about what happened. Did more come? Or did some go away?

4. Find the missing change. Ask: How many were added or taken away?

Helpful clue words

• Added words: got more, joined, came, added, put in

• Taken away words: left, went away, gave away, lost, took out

You can use objects, fingers, drawings, or a number sentence.

Sometimes it helps to draw circles or use counters.

If the start is 4 and the end is 7, you can count up: 5, 6, 7. That is 3 more.

So the change is 3.

Worked Example 1: Adding with the change missing

Mia had 3 apples. Then she got some more apples. Now she has 8 apples. How many apples did she get?

Step 1: Start amount = 3

Step 2: End amount = 8

Step 3: She got more, so this is an adding story.

We can write:

$$3 + \Box = 8$$

Now count from 3 up to 8:

5, 6, 7, 8

That is 5 numbers: 4, 5, 6, 7, 8? Wait, let us count carefully from after 3:

4, 5, 6, 7, 8

That is 5 more.

So:

$$3 + 5 = 8$$

Answer: Mia got 5 apples.

Worked Example 2: Taking away with the change missing

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

Step 1: Start amount = 9

Step 2: End amount = 4

Step 3: Some flew away, so this is a taking away story.

We can write:

$$9 - \Box = 4$$

Ask: What number makes this true?

$$9 - 5 = 4$$

Answer: 5 birds flew away.

Worked Example 3: Adding with a bigger start

Noah had 6 toy cars. His friend gave him some more toy cars. Now Noah has 10 toy cars. How many toy cars did his friend give him?

Start with 6. End with 10. More were added.

Write the number sentence:

$$6 + \Box = 10$$

Count up from 6 to 10:

7, 8, 9, 10

That is 4 more.

So:

$$6 + 4 = 10$$

Answer: His friend gave him 4 toy cars.

Worked Example 4: Taking away and checking the answer

A jar had 7 cookies. Some cookies were eaten. Now there are 2 cookies left. How many cookies were eaten?

Start amount = 7

End amount = 2

Some were eaten, so some were taken away.

Write the number sentence:

$$7 - \Box = 2$$

Think: What do I take away from 7 to get 2?

$$7 - 5 = 2$$

So 5 cookies were eaten.

Check:

If there were 7 cookies and 5 were eaten, then 2 are left.

That matches the story.

Ways to solve

• Draw a picture: Draw the starting amount. Cross out some if things went away, or draw more if things were added.

• Use counters: Start with counters. Add or remove until you match the ending amount.

• Count on: For adding stories, count up from the start to the end.

• Think subtraction: To find how many were added, you can find the difference between the end and the start.

Let’s practice thinking

If a story says:

“Sam had 2 balloons. He got some more. Now he has 5 balloons.”

Ask yourself:

• What is the start? 2

• What is the end? 5

• Did he get more or lose some? Got more

• What is missing? How many more

Number sentence:

$$2 + \Box = 5$$

The missing number is 3.

Another practice thought

“There were 8 fish in a tank. Some were moved to another tank. Now 3 fish are left.”

• Start = 8

• End = 3

• Some went away, so subtract.

Number sentence:

$$8 - \Box = 3$$

The missing number is 5.

How to check your work

After you find the missing change, put it back into the number sentence.

If your answer makes the story true, you solved it correctly.

For example:

$$4 + 3 = 7$$

If the story starts with 4 and ends with 7, then adding 3 makes sense.

Or:

$$9 - 2 = 7$$

If the story starts with 9 and ends with 7, then taking away 2 makes sense.

Remember

• A change unknown problem has a missing middle part.

• You know the start and the end.

• You solve to find how many were added or how many were taken away.

• You can use a drawing, counters, fingers, counting, or a number sentence.

Summary

Change unknown word problems tell you how many there were at first and how many there are at the end.

Your job is to find the missing change.

If more joined, use an adding idea: $$\text{start} + \Box = \text{end}$$

If some left, use a taking away idea: $$\text{start} - \Box = \text{end}$$

Read carefully, decide what happened, and find the missing number.

Start Unknown Word Problems

Sometimes in a word problem, we do not know how many things there were at the start.

Something happens next. Maybe more are added, or some are taken away. Then we know the ending number.

A start unknown problem asks: How many were there first?

We can solve these problems by thinking carefully about the story and using a picture, objects, or an equation with a box.

What does a start unknown problem sound like?

• Some birds were in the tree. 3 more birds came. Now there are 8 birds. How many birds were there at first?
• Some cookies were on the plate. 2 were eaten. Now 5 cookies are left. How many cookies were on the plate at first?

In both problems, the beginning number is missing.

How to solve start unknown problems

1. Read the story slowly.
2. Ask: What number is missing? Is it the start, the change, or the end?
3. If the start is missing, think: What happened in the story?
4. Use a drawing, counters, fingers, or an equation with a box.
5. Check that your answer makes the story true.

Start unknown with adding

Sometimes more are added, but we do not know how many there were first.

We can write it like this:

$$\Box + 3 = 8$$

This says, “Some number plus 3 equals 8.”

To find the missing start, we can think, What number and 3 make 8?

That number is 5, because:

$$5 + 3 = 8$$

Start unknown with taking away

Sometimes some things are taken away, but we do not know how many there were at first.

We can write it like this:

$$\Box - 2 = 5$$

This says, “Some number take away 2 equals 5.”

To find the missing start, we can think, What number becomes 5 after 2 are taken away?

That number is 7, because:

$$7 - 2 = 5$$

A helpful way to think

• If the story says more came, think: “What number plus more gives the end?”
• If the story says some went away, think: “What number was there before some were taken away?”

You are finding the number that makes the story true.

Worked Example 1

Some apples were in a basket. 4 more apples were put in. Now there are 9 apples. How many apples were in the basket at first?

Step 1: Find the missing part. The start is missing.

Step 2: Write an equation.

$$\Box + 4 = 9$$

Step 3: Think, “What number plus 4 equals 9?”

We know:

$$5 + 4 = 9$$

So, there were 5 apples at first.

Step 4: Check.

If 5 apples were in the basket and 4 more were added, then there are 9 apples. That is correct.

Worked Example 2

Some frogs sat on a log. 3 frogs jumped away. Now 6 frogs are on the log. How many frogs were on the log at first?

Step 1: The start is missing.

Step 2: Write an equation.

$$\Box - 3 = 6$$

Step 3: Think, “What number take away 3 equals 6?”

$$9 - 3 = 6$$

So, there were 9 frogs at first.

Step 4: Check.

9 frogs, then 3 jump away, leaves 6 frogs. That matches the story.

Worked Example 3

Some children were playing. 5 more children came to play. Now 10 children are playing. How many children were playing at first?

Step 1: Write the equation.

$$\Box + 5 = 10$$

Step 2: Find the missing start.

$$5 + 5 = 10$$

So, 5 children were playing at first.

Step 3: Check the story.

Start with 5, add 5 more, and now there are 10. Yes!

Worked Example 4

Some crayons were in a box. 4 crayons were lost. Now there are 7 crayons left. How many crayons were in the box at first?

Step 1: Write the equation.

$$\Box - 4 = 7$$

Step 2: Think about the missing start.

$$11 - 4 = 7$$

So, there were 11 crayons at first.

Step 3: Check.

If the box started with 11 crayons and 4 were lost, then 7 are left. Correct.

Using a drawing

Draw the ending number first if that helps you.

For example, in $$\Box + 2 = 6$$, draw 6 circles. Then cover or circle 2 that were added. The ones left show the start.

In $$\Box - 2 = 4$$, think about what number had to be there before 2 were taken away. You can start at 4 and count on 2 more: 5, 6. So the start was 6.

Words to listen for

• at first
• started with
• some
• now
• left
• more came
• went away

These words help you know that the beginning number might be missing.

Check your thinking

• Did I find the number at the start?
• Did I use the story words correctly?
• Does my answer make the ending number true?

Summary

In a start unknown word problem, the first number is missing.

You can solve it by using a picture, objects, counting, or an equation with a box like \(\Box + 3 = 8\) or \(\Box - 2 = 5\).

Always check your answer by putting it back into the story.

Compare Word Problems help us find out how many more or how many fewer one group has than another group.

We use compare word problems when we look at two groups and ask, “Which group has more?” and “What is the difference?”

For example, if one child has 8 apples and another child has 5 apples, we can compare the groups to see how many more apples one child has.

The big idea: To find how many more or how many fewer, we can subtract.

We can think of it like this:

$$\text{bigger number} - \text{smaller number} = \text{difference}$$

The difference is how many more or how many fewer.

Words that tell us to compare:

• how many more
• how many fewer
• how many less
• more than
• fewer than

When you see these words, stop and ask:

1. What two groups am I comparing?
2. Which group has more?
3. Which group has fewer?
4. How much bigger is the bigger group?

How to solve compare word problems:

1. Read the story carefully.
2. Find the two numbers.
3. Decide which number is bigger.
4. Subtract the smaller number from the bigger number.
5. Answer the question with words.

Let’s practice.

Worked Example 1

Sam has 7 toy cars. Lee has 4 toy cars. How many more toy cars does Sam have than Lee?

First, find the two numbers: 7 and 4.

Next, see which number is bigger. The bigger number is 7.

Now subtract:

$$7 - 4 = 3$$

So, Sam has 3 more toy cars than Lee.

Worked Example 2

Mia has 9 crayons. Ben has 6 crayons. How many fewer crayons does Ben have than Mia?

The two numbers are 9 and 6.

Mia has more, because 9 is bigger than 6.

Subtract:

$$9 - 6 = 3$$

So, Ben has 3 fewer crayons than Mia.

Notice something important: “How many more?” and “How many fewer?” can use the same subtraction sentence.

In this problem, both sentences are true:

• Mia has 3 more crayons than Ben.
• Ben has 3 fewer crayons than Mia.

Worked Example 3

There are 10 birds in a tree. There are 8 birds on the ground. How many more birds are in the tree than on the ground?

The numbers are 10 and 8.

The bigger number is 10.

Subtract:

$$10 - 8 = 2$$

So, there are 2 more birds in the tree than on the ground.

Worked Example 4

Ava read 12 books. Noah read 7 books. How many fewer books did Noah read than Ava?

The numbers are 12 and 7.

The bigger number is 12.

Subtract:

$$12 - 7 = 5$$

So, Noah read 5 fewer books than Ava.

Try thinking with pictures in your mind.

If one group has more, imagine matching one item from the bigger group to one item from the smaller group.

When you finish matching, some items may be left over in the bigger group.

Those extra items show the difference.

For example, compare 6 and 4.

If you match 4 items with 4 items, then 2 items are left over.

That means:

$$6 - 4 = 2$$

So 6 is 2 more than 4, and 4 is 2 fewer than 6.

Helpful tips:

• Look for the words more and fewer.
• Use the bigger number first when you subtract.
• The answer tells the difference between the groups.
• Always answer with words from the story.

Be careful!

Sometimes children see the word more and think they should add. But in compare word problems, “how many more” usually means we are finding the difference between two groups.

If the problem asks how many more one group has than another, compare the numbers and subtract.

Let’s look at one more quick example.

Lila has 11 stickers. Max has 11 stickers. How many more stickers does Lila have than Max?

The numbers are the same.

Subtract:

$$11 - 11 = 0$$

So, Lila has 0 more stickers than Max.

That means they have the same number of stickers.

Summary

Compare word problems ask us to find how many more or how many fewer.

We compare two groups, find the bigger number and the smaller number, and subtract to find the difference.

Remember:

$$\text{difference} = \text{bigger number} - \text{smaller number}$$

If you read carefully, choose the two numbers, and subtract, you can solve compare word problems with confidence.

Understanding the Equal Sign as Equivalence

Today we will learn about the equal sign, written like this: \(=\).

The equal sign does not mean, “the answer is coming.” It means “is the same as” or “has the same value as.”

When we see an equal sign, we can think of a balance. One side and the other side must match. They must be the same amount.

For example, in \(3 + 2 = 5\), the left side is \(3 + 2\). That makes \(5\). The right side is \(5\). Both sides are the same, so the number sentence is true.

We can also turn it around: \(5 = 3 + 2\). This is also true because \(5\) and \(3 + 2\) are the same amount.

Main Idea: The equal sign shows that the two sides are equal.

• Left side = right side
• Same value = same value
• Same amount = same amount

Let’s look at some true number sentences:

• \(4 = 4\)
• \(2 + 3 = 5\)
• \(5 = 2 + 3\)
• \(6 - 1 = 3 + 2\)

In each one, both sides have the same value.

Now let’s look at some number sentences that are not true:

• \(4 = 5\)
• \(2 + 1 = 5\)
• \(7 = 3 + 1\)

These are not true because the two sides are not the same.

Sometimes a number is missing. We can find the missing number by making both sides the same.

This is an important way to think in math. We do not just “do the problem and write the answer.” We check that both sides match.

Worked Example 1

Find the missing number in \(3 + 2 = \Box\).

First, add the left side: \(3 + 2 = 5\).

So the box must be \(5\).

Both sides are the same, so it is true.

Worked Example 2

Find the missing number in \(5 = \Box + 1\).

We need the right side to equal \(5\).

What number plus \(1\) makes \(5\)? It is \(4\).

The left side is \(5\). The right side is also \(5\). They match.

Worked Example 3

Is this true or false?

Let’s check both sides.

• Left side: \(2 + 2 = 4\)
• Right side: \(1 + 3 = 4\)

Both sides are \(4\), so this is true.

Worked Example 4

Find the missing number in \(6 - 2 = \Box + 1\).

First, solve the left side: \(6 - 2 = 4\).

Now we need the box plus \(1\) to make \(4\).

That means the box is \(3\).

Now both sides equal \(4\), so the number sentence is true.

Helpful Ways to Think

• The equal sign means is the same as.
• There is a left side and a right side.
• Both sides must have the same value.
• The missing number makes both sides match.

You can ask yourself:

• What is the value on the left side?
• What is the value on the right side?
• Are they the same?

Try These Ideas

Look at \(7 = 5 + 2\). Is it true? Yes, because both sides are \(7\).

Look at \(4 + 1 = 6\). Is it true? No, because \(4 + 1 = 5\), not \(6\).

Look at \(8 - 3 = 2 + \Box\). The left side is \(5\). So the box must be \(3\), because \(2 + 3 = 5\).

Summary

The equal sign, \(=\), means “is the same as.” It shows that the amount on one side matches the amount on the other side.

When you see an equal sign, do not think, “Here comes the answer.” Instead think, “Are both sides the same?”

If both sides have the same value, the number sentence is true. If they do not match, it is not true.

True and False Equations

Sometimes we see a number sentence with an equal sign, like \(3+2=5\). We can ask, Is it true or false?

A true equation is a number sentence that is correct. Both sides have the same value.

A false equation is a number sentence that is not correct. The two sides do not have the same value.

The equal sign means is the same as. It does not mean “the answer is next.” It means the amount on one side is the same as the amount on the other side.

Look at this:

$$3+2=5$$

On the left side, \(3+2=5\). On the right side, there is \(5\). Both sides are 5, so this equation is true.

Now look at this:

$$4+1=5+2$$

On the left side, \(4+1=5\). On the right side, \(5+2=7\). Since 5 is not the same as 7, this equation is false.

How to check if an equation is true or false

1. Look at the numbers on the left side of the equal sign.

2. Find the value on the left side.

3. Look at the numbers on the right side of the equal sign.

4. Find the value on the right side.

5. Ask: Are they the same?

   • If yes, the equation is true.

   • If no, the equation is false.

Important idea

The equal sign is like a balance. If both sides are the same, it is balanced and true. If both sides are not the same, it is not balanced and is false.

Worked Example 1

Is this equation true or false?

$$2+3=5$$

Left side: \(2+3=5\)

Right side: \(5\)

Both sides are 5.

Answer: This equation is true.

Worked Example 2

Is this equation true or false?

$$6=4+1$$

Left side: \(6\)

Right side: \(4+1=5\)

6 is not the same as 5.

Answer: This equation is false.

Worked Example 3

Is this equation true or false?

$$3+4=2+5$$

Left side: \(3+4=7\)

Right side: \(2+5=7\)

Both sides are 7.

Answer: This equation is true.

Worked Example 4

Is this equation true or false?

$$8-3=2+2$$

Left side: \(8-3=5\)

Right side: \(2+2=4\)

5 is not the same as 4.

Answer: This equation is false.

Tips to help you

• Always check both sides of the equal sign.

• Do not just look at one side.

• The equal sign means same as.

• If both sides match, it is true.

• If both sides do not match, it is false.

Let’s think about a word problem

Mia has 2 red blocks and 3 blue blocks. That is 5 blocks. Sam has 1 red block and 4 blue blocks. That is also 5 blocks.

We can write:

$$2+3=1+4$$

Left side: 5

Right side: 5

Both sides are the same, so the equation is true.

Another word problem

Leo has 4 toy cars and gets 2 more. That makes 6. Ava has 3 toy cars and gets 1 more. That makes 4.

We can write:

$$4+2=3+1$$

Left side: 6

Right side: 4

The sides are not the same, so the equation is false.

Summary

A true equation has the same value on both sides of the equal sign. A false equation does not. To decide if an equation is true or false, find the value on each side and compare them. If they are the same, it is true. If they are different, it is false.

Balancing Equations means making both sides of a number sentence have the same value.

An equation is a math sentence with an equal sign, like \(3+3=4+\square\).

The equal sign \(=\) means is the same as. It does not mean “the answer comes next.” It means the amount on the left side is the same as the amount on the right side.

When we balance an equation, we find the missing number so both sides match.

Think of a balance scale. If both sides have the same weight, the scale is level. In math, if both sides have the same number, the equation is balanced.

How to balance an equation

• Look at the left side and find its total.
• Look at the right side and see what number is missing.
• Choose the number that makes both sides equal.
• Check your work by counting both sides.

Here is an example:

$$3+3=4+\square$$

First, find the left side: \(3+3=6\).

Now the right side must also equal \(6\). We already have \(4\), so we need \(2\) more.

So the missing number is \(2\).

$$3+3=4+2$$

You can use counting on to help.

If you have \(4\) and need to get to \(6\), count on: 5, 6. That is 2 more.

You can also use subtraction to help.

If one side is \(6\) and the other side already has \(4\), then \(6-4=2\). So the missing number is \(2\).

Worked Example 1

$$2+1=1+\square$$

Left side: \(2+1=3\).

Right side starts with \(1\). What number with \(1\) makes \(3\)?

\(1+2=3\), so the missing number is \(2\).

$$2+1=1+2$$

Worked Example 2

$$5=3+\square$$

The left side is \(5\).

The right side has \(3\) and a missing number. We need the right side to be \(5\).

\(3+2=5\), so the missing number is \(2\).

$$5=3+2$$

Worked Example 3

$$4+2=5+\square$$

Left side: \(4+2=6\).

Right side starts with \(5\). What number makes \(6\)?

\(5+1=6\), so the missing number is \(1\).

$$4+2=5+1$$

Worked Example 4

$$7=\square+4$$

We need a number that works with \(4\) to make \(7\).

Count on from \(4\): 5, 6, 7. That is 3 more.

So the missing number is \(3\).

$$7=3+4$$

Things to remember

• The equal sign means both sides are the same.
• The missing number can be on either side of the equation.
• Add to find the total.
• Count on or subtract to find the missing part.
• Always check that both sides have the same value.

Let’s check one more together

$$1+4=2+\square$$

Left side: \(1+4=5\).

Right side starts with \(2\). What do we add to \(2\) to make \(5\)?

\(2+3=5\), so the missing number is \(3\).

$$1+4=2+3$$

Summary

Balancing equations means making both sides of the equal sign have the same amount.

To solve, find the total on one side, then find the missing number that makes the other side match.

If both sides are equal, the equation is balanced.