Algebraic Thinking and Equations

Lesson: The Relational Meaning of the Equal Sign

When you see the equal sign, = does not mean the answer comes next. It means is the same as.

The equal sign shows that the amount on one side is equal to, or balanced with, the amount on the other side.

You can think of the equal sign like a balance scale. If both sides have the same value, the scale stays level. If the sides are not the same, the scale tips.

For example, in \(4 + 3 = 7\), the left side is \(4 + 3\), which is \(7\). The right side is \(7\). Both sides are the same, so the equation is true.

We can also write the same idea in a different order: \(7 = 4 + 3\). This is also true because the left side and the right side are still the same amount.

Main Idea: The equal sign tells us that both sides must have the same value.

• \(8 = 8\) means both sides are the same.
• \(5 + 2 = 6 + 1\) means both sides are worth \(7\).
• \(9 = 4 + 5\) means \(9\) is the same as \(4 + 5\).

Sometimes students think the equal sign means, write the answer now. But in math, it really means is the same as.

That is why an equation like \(3 + 4 = 2 + 5\) is true. The left side is \(7\), and the right side is also \(7\). Both sides match.

How to Check an Equation

1. Look at the left side of the equal sign.
2. Find its value.
3. Look at the right side of the equal sign.
4. Find its value.
5. If both values are the same, the equation is true.

Lets practice with some worked examples.

Worked Example 1

Is this true? \(6 + 2 = 8\)

Left side: \(6 + 2 = 8\)

Right side: \(8\)

Both sides are \(8\), so this equation is true.

We can show it like this:

Worked Example 2

Is this true? \(9 = 4 + 5\)

Left side: \(9\)

Right side: \(4 + 5 = 9\)

Both sides are \(9\), so this equation is true.

This shows that the answer does not have to be only on the right side. The equal sign just means both sides are the same.

Worked Example 3

Is this true? \(7 + 1 = 5 + 2\)

Left side: \(7 + 1 = 8\)

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

The sides are not the same. One side is \(8\), and the other side is \(7\).

So this equation is false.

Worked Example 4

Find the missing number: \(3 + \Box = 8\)

We want both sides to be the same.

The right side is \(8\).

On the left side, \(3 + \Box\) must also equal \(8\).

Since \(3 + 5 = 8\), the missing number is 5.

Lets try one more with the missing number in a different place.

Worked Example 5

Find the missing number: \(10 = \Box + 4\)

The left side is \(10\).

So the right side must also be \(10\).

We need a number that makes \(\Box + 4 = 10\).

Since \(6 + 4 = 10\), the missing number is 6.

Things to Remember

• The equal sign means is the same as.
• It does not just mean the answer comes next.
• You can have numbers or addition on either side of the equal sign.
• Both sides must have the same value.

Look at these equations:

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

In each true equation, both sides match. In the false equation, the sides do not match.

Summary

The equal sign means is the same as. It tells us that the left side and the right side must be equal, like a balanced scale. When you see an equation, check both sides. If both sides have the same value, the equation is true.

Evaluating True and False Equations

In math, the equal sign means "is the same as". It does not just mean “the answer is next.”

When we look at an equation, we check whether the amount on the left side is the same as the amount on the right side. If both sides are the same, the equation is true. If the sides are not the same, the equation is false.

You can think of the equal sign like a balance scale. A balance scale is even only when both sides weigh the same. In an equation, both sides must have the same value.

For example, in the equation \(4+5=9\), the left side is \(4+5\), and the right side is \(9\). Since \(4+5=9\), both sides are the same. The equation is true.

In the equation \(4+5=8\), the left side is \(9\), but the right side is \(8\). Since \(9\neq 8\), the equation is false.

How to tell if an equation is true or false

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

2. Look at the right side of the equal sign.

3. Find out what each side is worth.

4. Ask: Are both sides the same?

If the two sides match, the equation is true. If they do not match, the equation is false.

Sometimes you can tell without finishing all the adding

Some equations are true because the numbers are just switched around. For example:

Both sides use the same two numbers, just in a different order. Both sides equal \(9\), so the equation is true.

This is helpful because you do not always need to solve everything to know the sides are the same.

Worked Example 1

Is \(3+4=7\) true or false?

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

Right side: \(7\)

Both sides are \(7\), so the equation is true.

Worked Example 2

Is \(6+2=5+3\) true or false?

Left side: \(6+2=8\)

Right side: \(5+3=8\)

Both sides are \(8\), so the equation is true.

Worked Example 3

Is \(10-3=4+2\) true or false?

Left side: \(10-3=7\)

Right side: \(4+2=6\)

Since \(7\neq 6\), the equation is false.

Worked Example 4

Is \(8+1=10-1\) true or false?

Left side: \(8+1=9\)

Right side: \(10-1=9\)

Both sides are \(9\), so the equation is true.

Important idea: The equal sign can be in the middle

Sometimes students think the answer must always come after the equal sign. But equations can be written in different ways.

These are all true equations:

• \(2+6=8\)

• \(8=2+6\)

• \(2+6=7+1\)

In each one, the two sides are the same amount.

Look for balance

When you see an equation, do not just add the numbers on one side. Be sure to check both sides of the equal sign.

For example, in \(7+1=6+3\):

• Left side is \(8\)

• Right side is \(9\)

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

Tips for success

• Remember: the equal sign means the same as.

• Check the left side and the right side.

• If both sides match, the equation is true.

• If both sides do not match, the equation is false.

• Sometimes you can notice number patterns, like \(4+5=5+4\).

Let’s try a few quick checks

• \(5+5=10\) → true

• \(9-2=6\) → false, because \(9-2=7\)

• \(3+6=4+5\) → true, because both sides are \(9\)

• \(12-4=3+4\) → false, because \(8\neq 7\)

Summary

An equation is true when the value on one side of the equal sign is the same as the value on the other side. An equation is false when the two sides are different.

Always remember that the equal sign means "is the same as". To decide if an equation is true or false, compare both sides and check for balance.

Balancing Equations with Unknowns

Sometimes in math, a number is missing. We call that missing number an unknown.

You might see the unknown as a box, a blank, or a question mark. For example, in \(8 + ? = 15\), the question mark stands for a number we do not know yet.

Our job is to find the number that makes the equation true.

An equation is a math sentence with an equal sign. The equal sign means both sides have the same value. It is like a balance scale. If one side is 15, the other side must also be 15.

For example:

This is true because the left side and the right side are equal.

When there is an unknown, we think: What number makes both sides match?

Important idea: Do not just look at the side with the unknown. Always check that both sides are balanced.

Ways to find the unknown

• Use what you know about addition and subtraction.
• Count on or count back.
• Think about the missing part.
• Check your answer by putting the number back into the equation.

1. Unknown in addition

If part of an addition sentence is missing, ask: What number goes with the known part to make the total?

Example: \(8 + ? = 15\)

We can think: \(8 + 7 = 15\). So the unknown is 7.

You can also think of subtraction:

So the missing number is \(7\).

2. Unknown in subtraction

If a subtraction sentence has a missing number, think carefully about where the unknown is.

• If the unknown is at the end, subtract.
• If the unknown is at the beginning, think: What number minus the other number gives the answer?

Example: \(12 = ? - 4\)

We need a number that becomes 12 after taking away 4.

Think: if we add 4 back to 12, we get the starting number.

So:

The unknown is \(16\).

3. The unknown can be on either side

The equal sign does not mean “the answer comes next.” It means is the same as.

These equations both mean the same thing:

So if you see the unknown before the equal sign or after it, that is okay. We still make both sides equal.

4. Use relational thinking

Relational thinking means looking at the whole equation and asking, “How are the two sides the same?”

For example:

First, find the value of the right side:

Now the equation is:

What number makes 10 become 12? It is 2.

Now both sides equal 12, so the equation is balanced.

Worked Examples

Example 1: Easy addition unknown

Solve:

Step 1: Ask, “What number added to 5 makes 9?”

Step 2: Count on from 5 to 9: 6, 7, 8, 9. That is 4 numbers.

Answer:

The unknown is 4.

Example 2: Unknown at the beginning of subtraction

Solve:

Step 1: We need the starting number.

Step 2: If taking away 3 gives 11, then add 3 back to 11.

Answer:

The unknown is 14.

Example 3: Unknown on the left side

Solve:

Step 1: Ask, “What number plus 6 equals 13?”

Step 2: Use subtraction.

Answer:

The unknown is 7.

Example 4: Compare both sides

Solve:

Step 1: Find the value on the right side.

Step 2: Now solve:

Step 3: What number goes with 9 to make 12? It is 3.

Answer:

Both sides equal 12, so the equation is balanced.

How to check your answer

1. Put your number into the equation.
2. Solve each side.
3. Make sure both sides are equal.

For example, check \(8 + ? = 15\) with \(7\):

The left side is 15 and the right side is 15, so the answer is correct.

Helpful tips

• If something is missing in an addition equation, you can often subtract to find it.
• If the starting number is missing in a subtraction equation, add to find it.
• Remember that the equal sign means both sides are the same.
• Always check your answer.

Watch out for these mistakes

• Mistake: Thinking the equal sign means “write the answer.”
  Fix: Remember it means “is the same as.”
• Mistake: Only looking at one side of the equation.
  Fix: Compare both sides to keep them balanced.
• Mistake: Using the wrong operation.
  Fix: Ask yourself if you should add or subtract to find the missing number.

Summary

Balancing equations with unknowns means finding the missing number that makes both sides equal.

You can solve these equations by thinking about addition and subtraction, counting on or back, and checking that both sides match.

When you see an equation like \(8 + ? = 15\) or \(12 = ? - 4\), remember: the equal sign means both sides must stay balanced.

Input-Output Function Machines are a fun way to see how numbers change.

Think of a function machine like a special box. A number goes in, the machine follows a rule, and a new number comes out.

For example, if the rule is add 2, then an input of 3 becomes an output of 5 because \(3 + 2 = 5\).

We can show this with a table:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 3 & 5 \\\hline 4 & 6 \\\hline 7 & 9 \\\hline \end{array}$$

Each time, the machine does the same rule. That is very important. A function machine must use one rule again and again.

How to find the rule

1. Look at the input number and the output number.

2. Ask, What changed?

3. Try simple rules like:

   • add a number

   • subtract a number

   • multiply by a number

4. Check the rule with the other pairs in the table.

If the same rule works for all the pairs, then you found the rule.

Main rules you may see

• Add: the output is bigger than the input.

• Subtract: the output is smaller than the input.

• Multiply: the input is repeated in equal groups.

Worked Example 1: Find the rule

Look at this table:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 1 & 4 \\\hline 2 & 5 \\\hline 3 & 6 \\\hline \end{array}$$

Let’s compare each input and output.

• \(1 \to 4\): add 3

• \(2 \to 5\): add 3

• \(3 \to 6\): add 3

The same rule works each time.

Rule: add 3

So if the input is 10, the output is:

$$10 + 3 = 13$$

Worked Example 2: Fill in missing outputs

The rule is subtract 2.

Complete the table:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 5 & ? \\\hline 8 & ? \\\hline 10 & ? \\\hline \end{array}$$

Use the rule on each input.

• \(5 - 2 = 3\)

• \(8 - 2 = 6\)

• \(10 - 2 = 8\)

The completed table is:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 5 & 3 \\\hline 8 & 6 \\\hline 10 & 8 \\\hline \end{array}$$

Worked Example 3: Fill in missing inputs

The rule is add 4.

Find the missing inputs:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline ? & 9 \\\hline ? & 12 \\\hline ? & 15 \\\hline \end{array}$$

If the machine adds 4, then we can work backward by subtracting 4 from each output.

• \(9 - 4 = 5\)

• \(12 - 4 = 8\)

• \(15 - 4 = 11\)

So the inputs are 5, 8, and 11.

The completed table is:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 5 & 9 \\\hline 8 & 12 \\\hline 11 & 15 \\\hline \end{array}$$

Worked Example 4: A multiplication rule

Look at this table:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 2 & 6 \\\hline 3 & 9 \\\hline 4 & 12 \\\hline \end{array}$$

Let’s check the pattern.

• \(2 \to 6\): multiply by 3

• \(3 \to 9\): multiply by 3

• \(4 \to 12\): multiply by 3

Rule: multiply by 3

If the input is 5, then:

$$5 \times 3 = 15$$

So the output is 15.

How to make your own input-output table

1. Pick a rule, like add 5.

2. Choose some input numbers.

3. Use the rule on each input.

4. Write the outputs.

Example with the rule add 5:

$$\begin{array}{|c|c|}\hline \text{Input} & \text{Output} \\\hline 1 & 6 \\\hline 2 & 7 \\\hline 6 & 11 \\\hline \end{array}$$

Tips for success

• Always check more than one input-output pair.

• The rule must be the same every time.

• If you know the rule and the input, do the rule to find the output.

• If you know the rule and the output, work backward to find the input.

Watch out for mistakes

• Do not change the rule in the middle.

• Do not guess after looking at only one pair.

• Be careful to add, subtract, or multiply correctly.

Let’s review

An input is the number that goes into the machine.

An output is the number that comes out.

A rule tells how the machine changes the number.

We can use tables to find rules, fill in missing numbers, and predict what comes next.

When you solve input-output problems, remember to ask: What rule changes the input to the output every time?

Identifying Arithmetic Patterns means looking at numbers and noticing how they change or repeat in a way we can understand.

In 3rd Grade, patterns help us become stronger at addition, multiplication, and problem solving. When we find a pattern, we can make smart guesses about what comes next.

In this lesson, we will look for patterns in addition tables and multiplication tables. We will learn how to spot number rules like doubles, counting on, and even-and-odd patterns.

What is a pattern?

A pattern is something that happens again and again in a rule-based way. In math, a pattern tells us how numbers are connected.

For example, in the number list \(2, 4, 6, 8, 10\), the pattern is add 2 each time.

Patterns can help us:

• find missing numbers,
• check if an answer makes sense,
• learn facts faster,
• notice number relationships.

Part 1: Patterns in an addition table

An addition table shows what happens when we add numbers from the top row to numbers from the side column.

Here is a small addition table:

Let’s notice some patterns.

• As you move to the right, each number goes up by \(1\).
• As you move down, each number also goes up by \(1\).
• The table is the same on both sides of the diagonal because \(2+4=4+2\).

The diagonal of doubles

One very important pattern in an addition table is the diagonal of doubles. These are facts where both addends are the same.

Examples of doubles are:

• \(1+1=2\)
• \(2+2=4\)
• \(3+3=6\)
• \(4+4=8\)
• \(5+5=10\)

These doubles line up in a diagonal across the table. They follow a pattern:

This pattern increases by \(2\) each time.

Near doubles are also helpful. If you know \(4+4=8\), then \(4+5\) is just one more, so \(4+5=9\).

Part 2: Patterns in a multiplication table

A multiplication table shows products. A product is the answer to a multiplication problem.

Here is a small multiplication table:

We can find many patterns here too.

• In each row, the numbers grow by the number at the side.
• In the row for \(2\), the products go up by \(2\): \(2, 4, 6, 8, 10\).
• In the row for \(5\), the products go up by \(5\): \(5, 10, 15, 20, 25\).

Even and odd patterns

An even number can be split into 2 equal groups with no leftovers. An odd number has 1 leftover when split into 2 equal groups.

Even numbers are:

• \(2, 4, 6, 8, 10, 12\)

Odd numbers are:

• \(1, 3, 5, 7, 9, 11\)

Now look at multiplication patterns:

• If one factor is \(2\), the product is always even.
• In the \(3\) row, products can alternate between odd and even: \(3, 6, 9, 12, 15\).
• In the \(4\) row, all products are even because they are groups of \(4\).

For example:

• \(3\times1=3\) odd
• \(3\times2=6\) even
• \(3\times3=9\) odd
• \(3\times4=12\) even

So the products in the \(3\) row make an alternating pattern: odd, even, odd, even.

How to identify a pattern

When you look at a table or a list of numbers, ask yourself these questions:

1. What changes each time?
2. Does the number go up by the same amount?
3. Do numbers repeat in a special way?
4. Are the numbers even, odd, or alternating?
5. Do I see doubles or skip-counting?

These questions help you find the rule.

Worked Example 1: Find the addition pattern

Look at the doubles facts:

What pattern do you see?

Step 1: Look at the answers: \(2, 4, 6, 8\).

Step 2: Compare each answer to the next one.

Each time, the answer increases by \(2\).

Answer: The doubles pattern goes up by \(2\) each time.

Worked Example 2: Find a missing number in an addition table

Suppose part of an addition table shows:

What number goes where the question mark is?

Step 1: Find the row and column. It is in the row for \(3\) and the column for \(5\).

Step 2: Add them: \(3+5=8\).

Step 3: Check the pattern. The row goes \(6, 7, 8\), which increases by \(1\).

Answer: The missing number is \(8\).

Worked Example 3: Find the multiplication pattern

Look at this row from a multiplication table:

What is the pattern?

Step 1: Check how the numbers change.

\(8-4=4\), \(12-8=4\), \(16-12=4\).

Step 2: The numbers increase by \(4\) each time.

Step 3: These are products in the \(4\) row: \(4\times1, 4\times2, 4\times3, 4\times4, 4\times5\).

Answer: The pattern is add 4 each time.

Worked Example 4: Even and odd in a multiplication row

Look at the \(3\) row:

What even-and-odd pattern do you see?

Step 1: Name each number as even or odd.

• \(3\) is odd
• \(6\) is even
• \(9\) is odd
• \(12\) is even
• \(15\) is odd
• \(18\) is even

Step 2: Notice the order.

It goes odd, even, odd, even, odd, even.

Answer: The products alternate between odd and even.

Helpful tips

• In an addition table, moving right or down usually adds \(1\) when the headers go up by \(1\).
• The diagonal of doubles is important because both numbers are the same.
• In multiplication rows, products often follow skip-counting patterns.
• Look for even and odd patterns to help you understand multiplication facts.
• If you are not sure, say the facts out loud and listen for the pattern.

Let’s practice thinking

If you see the addition answers \(4, 5, 6, 7\), you can notice they go up by \(1\).

If you see the multiplication answers \(5, 10, 15, 20\), you can notice they go up by \(5\).

If you see \(2, 4, 6, 8\), you can notice they are all even and they increase by \(2\).

Summary

Arithmetic patterns are number patterns that follow a rule. In addition tables, you can look for rows that increase by \(1\), matching facts, and the diagonal of doubles.

In multiplication tables, you can look for skip-counting patterns and even-or-odd patterns. When you identify the rule, you can understand the table better and find missing numbers more easily.

Translating Scenarios into Equations means turning a story or word problem into a math sentence.

Sometimes a problem tells us about something we know and something we do not know yet. An equation helps us show that information clearly.

An equation is a math sentence with an equals sign, like \(x + 4 = 9\). The equals sign means is the same as.

A variable is a letter or symbol that stands for the unknown part. In 3rd grade, you might use a box, a blank, or a letter like \(n\) or \(x\).

When we translate a scenario into an equation, we ask: What is happening in the story? Are things being added, taken away, grouped, or shared equally?

Step 1: Find what you know. Look for the numbers in the story.

Step 2: Find what you do not know. This unknown amount can be shown with a letter or symbol.

Step 3: Find the action words. These words tell which math operation to use.

• add, more, in all, together often mean addition
• left, fewer, gave away, minus often mean subtraction
• groups of, times often mean multiplication
• shared equally, split into equal groups often mean division

Step 4: Write the equation. Put the unknown, the numbers, and the operation into a math sentence.

Step 5: Check your equation. Ask yourself, “Does this equation match the story?”

Let’s look at some examples.

Example 1: Addition

Lena has 5 stickers. Her friend gives her some more stickers. Now she has 12 stickers. How many stickers did her friend give her?

We know:

• She started with 5 stickers.
• She got some more.
• Now she has 12 stickers.

The unknown is how many she got. Let that be \(s\).

The story becomes:

This equation says: 5 stickers plus some more stickers equals 12 stickers.

Example 2: Subtraction

There were 18 birds in a tree. Some birds flew away. Now there are 11 birds left. How many birds flew away?

We know:

• There were 18 birds at first.
• Some flew away.
• 11 birds are left.

The unknown is how many flew away. Let that be \(b\).

The story becomes:

This equation says: 18 birds minus the birds that flew away equals 11 birds.

Example 3: Multiplication

There are 4 bags. Each bag has 6 marbles. How many marbles are there in all?

We know:

• There are 4 equal groups.
• Each group has 6 marbles.
• We want the total.

The unknown is the total number of marbles. Let that be \(m\).

The story becomes:

This equation says: 4 groups of 6 marbles equals the total number of marbles.

Example 4: Division

24 cookies are shared equally among 6 children. How many cookies does each child get?

We know:

• There are 24 cookies total.
• They are shared equally among 6 children.
• We want to know how many each child gets.

The unknown is how many cookies each child gets. Let that be \(c\).

The story becomes:

This equation says: 24 cookies divided into 6 equal groups equals the number each child gets.

Be Careful! Sometimes the unknown is at the beginning, middle, or end of the equation.

For example:

Some number plus 7 equals 15.

Or:

20 minus some number equals 8.

The letter can go wherever the unknown belongs in the story.

Helpful Thinking Questions

1. What numbers do I see?
2. What is the unknown amount?
3. What words tell me the operation?
4. What does the equals sign show?
5. Does my equation match the story?

Words to Watch For

• Addition: more, in all, total, together
• Subtraction: left, remain, fewer, took away
• Multiplication: groups of, each, times
• Division: shared equally, each group, split

Let’s translate one more story together.

Jada has 3 rows of chairs. There are 5 chairs in each row. How many chairs are there?

The words 3 rows and 5 in each row tell us there are equal groups. That means multiplication.

If \(c\) is the total number of chairs, the equation is:

Another one:

Malik had 14 crayons. He gave 6 crayons to his sister. How many crayons does he have now?

The words gave 6 tell us to subtract.

If \(r\) is the number of crayons left, the equation is:

Summary

To translate a scenario into an equation, read the story carefully and find the numbers, the unknown, and the action.

Use a letter or symbol for the unknown. Then write a math sentence that matches the story.

Remember: the equals sign means both sides are the same. A good equation shows the story clearly and correctly.

Two-Step Problem Solving Framework means solving a word problem in two parts.

Sometimes a problem does not ask for the answer right away. First, you must find a hidden question. Then you use that answer to solve the final question.

This is called a two-step problem because you do Step 1 and then Step 2.

Why is this important?

Some word problems give extra details that must be used in order. If we try to do everything at once, we can get confused. A two-step plan helps us think clearly.

How to solve a two-step word problem

1. Read the whole problem carefully.
2. Ask: “What is the problem really asking me to find?”
3. Look for the hidden question. Ask: “What do I need to know first?”
4. Solve Step 1. Find the intermediate answer.
5. Solve Step 2. Use the answer from Step 1 to find the final answer.
6. Check your work. Make sure your answer makes sense.

Helpful clue words

These words can help, but always read the whole problem.

• Add: in all, altogether, total, sum, more
• Subtract: left, remain, fewer, how many more, difference
• Multiply: groups of, each, equal groups
• Divide: shared equally, split, each group

The hidden question

A hidden question is a question the problem does not say directly, but you must answer it first.

For example, if a problem asks how many apples are left after some are eaten, you may first need to find how many apples there were in all.

So the hidden question comes before the final question.

Think of it like a path:

First answer this: What do I need to know first?

Then answer this: Now what can I find?

Example 1: Add, then subtract

Mia has 12 stickers. Her aunt gives her 5 more stickers. Then Mia gives 4 stickers to her friend. How many stickers does Mia have now?

Step 1: Find the hidden question.

How many stickers does Mia have after her aunt gives her more?

$$12 + 5 = 17$$

Now Mia has 17 stickers.

Step 2: Find the final answer.

Mia gives away 4 stickers.

$$17 - 4 = 13$$

Answer: Mia has 13 stickers now.

How we know: We could not subtract 4 until we knew the new total first.

Example 2: Multiply, then add

There are 3 bags of marbles. Each bag has 4 marbles. Sam finds 2 more marbles on the floor. How many marbles are there in all?

Step 1: Find the hidden question.

How many marbles are in the 3 bags?

$$3 \times 4 = 12$$

There are 12 marbles in the bags.

Step 2: Find the final answer.

Add the 2 marbles Sam found.

$$12 + 2 = 14$$

Answer: There are 14 marbles in all.

Example 3: Add, then compare

Lena read 8 pages on Monday and 7 pages on Tuesday. Her brother read 10 pages total. How many more pages did Lena read than her brother?

Step 1: Find the hidden question.

How many pages did Lena read in all?

$$8 + 7 = 15$$

Lena read 15 pages.

Step 2: Find the final answer.

How many more than 10 is 15?

$$15 - 10 = 5$$

Answer: Lena read 5 more pages than her brother.

Example 4: Multiply, then subtract

A teacher has 5 boxes of pencils. Each box has 6 pencils. She gives 8 pencils to students. How many pencils are left?

Step 1: Find the hidden question.

How many pencils are there at first?

$$5 \times 6 = 30$$

There are 30 pencils at first.

Step 2: Find the final answer.

She gives away 8 pencils.

$$30 - 8 = 22$$

Answer: 22 pencils are left.

How to spot a two-step problem

• The problem has more than one action.
• You need to find one amount before you can answer the question.
• The final question depends on another answer first.

Try asking yourself these questions:

• What is the final question?
• What information do I already know?
• Is there something I need to find first?
• What operation should I use first: add, subtract, multiply, or divide?
• What operation should I use next?

Using a box for the hidden answer

Sometimes it helps to write the hidden answer in a box or blank.

Example:

First: $$9 + 6 = \Box$$

Then: $$\Box - 5 = 10$$

This shows that the answer from the first step is used in the second step.

Be careful!

• Do not just use the numbers in the order you see them.
• Do not choose an operation only because of one clue word.
• Always think: What do I need to know first?

Check your answer

After solving, read the problem again.

• Did you answer the final question?
• Did you use the answer from Step 1 in Step 2?
• Does your answer make sense?

For example, if someone starts with 17 stickers and gives away 4, the answer should be less than 17. So 13 makes sense.

Summary

A two-step word problem has two parts. First, find the hidden question. That gives you the intermediate answer. Then use that answer to solve the final question.

When you slow down, look for what you need to know first, and solve one step at a time, two-step problems become much easier.

Two-Step Word Problems

Sometimes a math story has two things to do. You cannot solve it in just one step. You need to do one step first, and then do another step.

These are called two-step word problems.

For example, you might need to add first and then take away. Or you might need to take away first and then add.

When you solve a two-step word problem, it helps to go slowly and think about the story.

How to solve a two-step word problem

1. Read the story carefully.
2. Ask: What happens first?
3. Solve the first part.
4. Ask: What happens next?
5. Solve the second part.
6. Check: Does my answer make sense?

Words that can help

• Add words: in all, altogether, now has, got more, joined
• Take away words: left, gave away, ate, went away, how many remain

Be careful. The most important thing is not just one word. The most important thing is what happens in the story first and next.

Let’s look for the two steps

If a story says:

"Mia had 4 apples. Her dad gave her 3 more. Then she ate 2 apples."

We can ask:

• What happened first? She got 3 more apples, so we add.
• What happened next? She ate 2 apples, so we take away.

So the steps are:

$$4 + 3 = 7$$

$$7 - 2 = 5$$

Mia has 5 apples left.

You can also show the steps with a number sentence

Sometimes we write both steps like this:

$$4 + 3 = 7$$

$$7 - 2 = 5$$

This shows the answer from the first step becomes the starting number for the second step.

Worked Example 1

Ben has 5 toy cars. His friend gives him 2 more toy cars. Then Ben gives 1 toy car to his brother. How many toy cars does Ben have now?

Step 1: Ben gets 2 more, so add.

$$5 + 2 = 7$$

Step 2: Ben gives 1 away, so take away.

$$7 - 1 = 6$$

Answer: Ben has 6 toy cars now.

Worked Example 2

There are 9 birds in a tree. 3 birds fly away. Then 2 more birds come to the tree. How many birds are in the tree now?

Step 1: 3 birds fly away, so take away.

$$9 - 3 = 6$$

Step 2: 2 birds come, so add.

$$6 + 2 = 8$$

Answer: There are 8 birds in the tree now.

Worked Example 3

Lila has 6 crayons. She gets 4 more crayons. Then she gives 3 crayons to her friend. How many crayons does Lila have left?

Step 1: She gets 4 more, so add.

$$6 + 4 = 10$$

Step 2: She gives 3 away, so take away.

$$10 - 3 = 7$$

Answer: Lila has 7 crayons left.

Worked Example 4

There are 10 cookies on a plate. Sam eats 2 cookies. Then Mom puts 5 more cookies on the plate. How many cookies are on the plate now?

Step 1: Sam eats 2, so take away.

$$10 - 2 = 8$$

Step 2: Mom puts 5 more, so add.

$$8 + 5 = 13$$

Answer: There are 13 cookies on the plate now.

How to know which step comes first

Read the story in order. Do not just grab numbers and start. Ask yourself:

• What is the starting number?
• What happens first?
• What happens after that?

Let’s try this thinking:

"Noah had 8 stickers. He gave 2 away. Then he got 3 more."

• Start with 8.
• First, give away 2: $$8 - 2 = 6$$
• Next, get 3 more: $$6 + 3 = 9$$

Noah has 9 stickers.

A helpful way to model the story

You can use circles, marks, or drawings to help.

For 7 balloons, then 2 more, then 1 pops:

• Start with 7 marks
• Add 2 marks
• Cross out 1 mark

Then count what is left.

This is a good way to check your math.

Common mistakes to watch out for

• Doing only one step. Remember, the story has two parts.
• Using the wrong order. Do what happens first in the story.
• Forgetting the new number. Use the answer from step 1 in step 2.

Let’s practice thinking

Problem: Ava has 3 flowers. She picks 4 more flowers. Then she gives 2 flowers to her grandma. How many flowers does Ava have now?

• Start with 3.
• First she picks 4 more: $$3 + 4 = 7$$
• Then she gives away 2: $$7 - 2 = 5$$

Ava has 5 flowers now.

Remember

• A two-step word problem has two math actions.
• You may add and then subtract.
• You may subtract and then add.
• Read carefully and solve in order.

Quick summary

Two-step word problems are math stories with two parts. First, find what happens at the beginning. Next, solve what happens after that. Use the answer from the first step to solve the second step.

If you read carefully, work in order, and check your answer, you can solve two-step word problems step by step.