Operation Foundations and Fact Fluency Within 20

The Meaning of Addition and Subtraction

In math, addition and subtraction help us understand what is happening with numbers in real life.

We use addition when things are joined or put together. The total gets bigger.

We use subtraction when things are taken away, separated, or when we want to compare two amounts. The amount can get smaller, or we can find how many more or how many fewer.

Learning what addition and subtraction mean helps you choose the right operation when you solve a story problem.

1. Addition means joining or putting together

Addition tells us what happens when we combine groups.

• Join: One group comes together with another group.
• Put together: Two parts make one whole.

We use the plus sign \\(+\\) for addition.

Example: If you have 3 apples and get 2 more apples, you join the groups.

$$3 + 2 = 5$$

This means 3 and 2 together make 5.

2. Subtraction means taking away or separating

Subtraction tells us what happens when some things are taken from a group.

• Take away: A group gets smaller.
• Separate: Part of the group moves away from the whole.

We use the minus sign \\(-\\) for subtraction.

Example: If you have 7 cookies and eat 3 cookies, some are taken away.

$$7 - 3 = 4$$

This means 4 cookies are left.

3. Subtraction can also mean comparing

Sometimes subtraction does not mean taking away. Sometimes it means we are finding the difference between two numbers.

When we compare, we ask:

• How many more?
• How many fewer?
• How much bigger?
• How much smaller?

Example: Mia has 8 stickers. Jay has 5 stickers. How many more stickers does Mia have?

$$8 - 5 = 3$$

Mia has 3 more stickers than Jay.

4. Addition and subtraction are connected

Addition and subtraction are like opposites. They are called inverse operations. That means one operation can undo the other.

If you know:

$$4 + 3 = 7$$

Then you also know:

$$7 - 3 = 4$$

and

$$7 - 4 = 3$$

These number sentences are connected because they use the same three numbers.

5. Clue words can help

In story problems, some words can help you think about which operation to use.

Addition clue words:

• in all
• altogether
• together
• more
• join
• sum

Subtraction clue words:

• left
• take away
• fewer
• how many more
• how many less
• difference

Be careful: the words help, but the most important thing is to understand what is happening in the story.

Worked Example 1: Joining groups

Lena has 6 crayons. Her friend gives her 2 more crayons. How many crayons does Lena have now?

Step 1: Ask what is happening. Lena is getting more, so the groups are joining.

Step 2: Write an addition sentence.

$$6 + 2 = 8$$

Answer: Lena has 8 crayons now.

Worked Example 2: Taking away

There are 9 birds in a tree. 4 birds fly away. How many birds are left?

Step 1: Ask what is happening. Some birds leave, so we take away.

Step 2: Write a subtraction sentence.

$$9 - 4 = 5$$

Answer: 5 birds are left.

Worked Example 3: Putting together parts

A toy box has 5 cars and 7 blocks. How many toys are in the box?

Step 1: We are putting two parts together to find the whole.

Step 2: Write an addition sentence.

$$5 + 7 = 12$$

Answer: There are 12 toys in the box.

Worked Example 4: Comparing two amounts

Noah has 14 marbles. Ava has 9 marbles. How many more marbles does Noah have than Ava?

Step 1: We are comparing two amounts.

Step 2: Use subtraction to find the difference.

$$14 - 9 = 5$$

Answer: Noah has 5 more marbles than Ava.

How to decide whether to add or subtract

1. Read the story carefully.
2. Ask yourself, “Are things being joined together?” If yes, use addition.
3. Ask yourself, “Are things being taken away or compared?” If yes, use subtraction.
4. Write a number sentence.
5. Check if your answer makes sense.

Quick practice thinking

• 8 frogs are on a log. 3 more hop up. Is this addition or subtraction? Addition, because more are joining.
• 12 pencils are in a box. 5 are taken out. Is this addition or subtraction? Subtraction, because some are taken away.
• Sam has 10 books. Kim has 7 books. How many more does Sam have? Subtraction, because we are comparing.

Remember

• Addition means join or put together.
• Subtraction means take away, separate, or compare.
• Addition and subtraction are connected.
• Think about the action in the story to choose the correct operation.

Lesson Summary

Addition helps us find a total when groups join or parts are put together. Subtraction helps us find what is left when something is taken away, and it also helps us compare two amounts. When you solve a story problem, think about what is happening in the story. That will help you decide whether to add or subtract.

The Equal Sign Means “Is the Same As”

Today we will learn about the equal sign: \(=\).

Many students think the equal sign means, “Write the answer now.” But the equal sign really means “is the same as” or “has the same value as.”

You can think of the equal sign like a balance scale. Both sides must match. One side cannot be more or less than the other side.

For example, in \(3 + 2 = 5\), the left side is \(3 + 2\), and the right side is \(5\). Since \(3 + 2\) makes \(5\), both sides are the same.

Main Idea: The equal sign shows that the amount on one side is equal to the amount on the other side.

Let’s look at it in different ways.

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

All of these are true because the two sides have the same value.

What makes an equation true?

An equation is a math sentence with an equal sign. An equation is true when both sides are the same.

An equation is not true when the two sides are different.

Look at these:

• \(7 = 7\) is true.
• \(4 + 1 = 5\) is true.
• \(8 = 3 + 4\) is not true, because \(3 + 4 = 7\), not \(8\).

You do not always solve only the left side.

Sometimes there are numbers and operations on both sides of the equal sign. That is okay. We check whether both sides are the same.

For example:

$$5 + 1 = 4 + 2$$

The left side is \(6\). The right side is also \(6\). So this equation is true.

Think balance!

If one side has \(6\), the other side must also have \(6\). That is what the equal sign tells us.

Worked Example 1

Is this true or false?

$$2 + 3 = 5$$

Step 1: Find the left side. \(2 + 3 = 5\).

Step 2: Look at the right side. It is \(5\).

Step 3: Compare both sides. \(5 = 5\).

Answer: True. The equal sign shows both sides are the same.

Worked Example 2

Is this true or false?

$$7 = 3 + 3$$

Step 1: Find the right side. \(3 + 3 = 6\).

Step 2: Compare both sides. The left side is \(7\), but the right side is \(6\).

Answer: False. \(7\) is not the same as \(6\).

Worked Example 3

Find the missing number:

$$4 + 2 = \Box + 3$$

Step 1: Find the left side. \(4 + 2 = 6\).

Step 2: The right side must also equal \(6\).

Step 3: Think: what number plus \(3\) equals \(6\)?

$$3 + 3 = 6$$

Answer: The missing number is \(3\).

Now the equation is

$$4 + 2 = 3 + 3$$

Both sides are \(6\), so it is true.

Worked Example 4

Find the missing number:

$$9 - 4 = \Box$$

This is a familiar kind of equation. First find \(9 - 4\).

$$9 - 4 = 5$$

So the missing number is \(5\).

Now let’s look at a different kind:

$$9 - 4 = \Box + 1$$

Step 1: Find the left side. \(9 - 4 = 5\).

Step 2: The right side must also be \(5\).

Step 3: Think: what number plus \(1\) equals \(5\)?

The missing number is \(4\).

Now the equation is

$$9 - 4 = 4 + 1$$

Both sides equal \(5\).

Helpful tips

• The equal sign means the same as.
• Look at both sides of the equal sign.
• Ask, “Do both sides have the same value?”
• If there is a missing number, make both sides balance.

Common mistake

Sometimes students see this:

$$8 = \Box + 3$$

and they want to put \(11\) in the box because they add \(8 + 3\).

But that is not what the equal sign means.

We need the box plus \(3\) to be the same as \(8\).

So we ask, “What number plus \(3\) equals \(8\)?”

The answer is \(5\).

$$8 = 5 + 3$$

Try thinking with a balance

If one side has \(8\), the other side must also have \(8\). That is the job of the equal sign.

Summary

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

In any equation, the left side and the right side must have the same value. You can check both sides, and you can find missing numbers by making the equation balance.

Determining True and False Equations

In math, an equation is a number sentence with an equal sign: \(=\).

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

When we look at an equation, we ask: Are both sides equal?

• If both sides have the same value, the equation is true.
• If both sides do not have the same value, the equation is false.

Think of the equal sign like a balance scale. Both sides must match.

For example, in \(5 + 2 = 7\), the left side is \(5 + 2\), which equals \(7\). The right side is \(7\). Since both sides are \(7\), the equation is true.

In \(6 + 1 = 10\), the left side is \(7\), but the right side is \(10\). Since \(7\) and \(10\) are not the same, 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. Find its value.
3. Look at the right side of the equal sign.
4. Find its value.
5. Compare both sides.

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

Worked Example 1

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

First, find the left side:

Now look at the right side:

Both sides are \(7\). So this equation is true.

Worked Example 2

Is \(9 - 2 = 6\) true or false?

Find the left side:

Now look at the right side:

The sides are \(7\) and \(6\). They are not the same. So this equation is false.

Worked Example 3

Sometimes the number comes first, like this: \(8 = 5 + 3\).

Find the left side:

Find the right side:

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

This is important: the equal sign can be in the middle, and the equation can still be true.

Worked Example 4

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

Find the left side:

Find the right side:

The sides are \(6\) and \(5\). They are not the same. So the equation is false.

What to remember about the equal sign

• The equal sign means the same as.
• You can have numbers on both sides of the equal sign.
• You should check both sides, not just one side.
• Addition and subtraction can both be used in equations.

Helpful strategy

If an equation looks tricky, solve each side one at a time.

For example, with \(7 + 1 = 10 - 2\):

Since both sides are \(8\), the equation is true.

Try thinking about these

• \(6 + 2 = 8\)
• \(12 - 5 = 6\)
• \(4 + 4 = 9 - 1\)
• \(7 = 10 - 3\)

Ask yourself each time: Does the left side have the same value as the right side?

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, solve the left side, solve the right side, and compare them. If they match, it is true. If they do not match, it is false.

Fact Families and Inverse Operations

In math, some numbers belong together. They can make a little team called a fact family.

A fact family uses the same 3 numbers to make addition and subtraction facts. This helps us see that addition and subtraction are connected.

When two operations help undo each other, they are called inverse operations. In 2nd grade, this means addition and subtraction are opposites. If you add, you can subtract to go back.

For example, if you know that \(3 + 4 = 7\), then you also know that \(7 - 4 = 3\). The subtraction fact helps you check the addition fact.

What is a fact family?

A fact family is made with 3 numbers:

• 2 smaller numbers
• 1 larger number

The 2 smaller numbers are added to make the larger number. Then the larger number can be subtracted to find the smaller numbers again.

If the numbers are \(2\), \(5\), and \(7\), they make this fact family:

These 4 number sentences all use the same 3 numbers. That is why they are a fact family.

Why are there 4 facts?

There are usually 2 addition facts and 2 subtraction facts.

• In addition, you can switch the two smaller numbers.
• In subtraction, you start with the largest number and subtract one smaller number to get the other smaller number.

So a fact family often looks like this:

You do not need to remember letters. Just remember:

• Add the two smaller numbers to get the bigger number.
• Subtract from the bigger number to go back.

Addition and subtraction are inverse operations

Inverse operations are operations that undo each other.

If you put together with addition, subtraction can take apart.

Look at this:

This means 6 and 3 together make 9.

Now subtract:

We started with 9, took away 3, and got back to 6.

That is why addition and subtraction are called inverse operations. One can undo the other.

How to find a fact family

1. Look for the 3 numbers.
2. Find the largest number.
3. Use the two smaller numbers in addition.
4. Use the largest number in subtraction.

Let’s practice this step by step.

Worked Example 1

Use the numbers \(1\), \(8\), and \(9\).

Step 1: Find the largest number. It is \(9\).

Step 2: Add the two smaller numbers.

Step 3: Write the subtraction facts using 9.

So the fact family is:

Worked Example 2

Use the numbers \(4\), \(6\), and \(10\).

The largest number is \(10\).

The addition facts are:

The subtraction facts are:

All 4 facts belong in the same family because they use the same 3 numbers.

Worked Example 3

Suppose you know:

What other facts are in this fact family?

The 3 numbers are \(7\), \(5\), and \(12\).

Switch the addends to make the second addition fact:

Now use the largest number, 12, to write subtraction facts:

So the full fact family is:

Worked Example 4

Look at this subtraction fact:

Can we find the whole fact family?

Yes. The 3 numbers are \(13\), \(9\), and \(4\).

The largest number is \(13\), so that will be the total in addition and the starting number in subtraction.

The addition facts are:

The subtraction facts are:

How fact families help you check your work

Fact families are helpful because they let you check whether an answer makes sense.

If you solve:

You can check with subtraction:

If the subtraction works, your addition answer is probably correct.

You can also check subtraction with addition.

If you solve:

Check by adding:

This shows that subtraction and addition are partners.

Tips to remember

• A fact family uses 3 numbers.
• The largest number is the answer in addition.
• The largest number comes first in subtraction.
• Addition and subtraction undo each other.

Watch out for these mistakes

• Do not use a number that is not in the family.
• Do not start subtraction with a small number when writing the fact family.
• Make sure all 4 facts use the same 3 numbers.

For example, with \(3\), \(2\), and \(5\), this is correct:

But this is not in the fact family:

That does not follow the fact family pattern we use in 2nd grade, because we subtract from the largest number.

Let’s think together

If the numbers are \(11\), \(3\), and \(8\), what is the largest number?

The largest number is \(11\).

So the addition facts are:

And the subtraction facts are:

Summary

A fact family is a group of 4 facts made from the same 3 numbers.

The two smaller numbers are added to make the largest number. Then the largest number is used in subtraction to find the smaller numbers again.

Addition and subtraction are inverse operations because they undo each other.

When you know one fact in the family, you can use it to find the other facts. This helps you solve problems and check your answers.

Finding Missing Addends means figuring out the number that is missing in an addition sentence.

For example, in \(5 + \square = 9\), we know one part is 5 and the whole is 9. We need to find the missing part.

This is an important skill because addition and subtraction work together. If we know the whole and one part, we can find the missing part.

Let’s learn how to do it step by step.

What is a missing addend?

An addend is a number we add. In the equation \(3 + 4 = 7\), the numbers 3 and 4 are the addends.

Sometimes one addend is missing, like this:

$$6 + \square = 10$$

The box is the missing addend. We need to figure out what number makes the addition sentence true.

Two helpful ways to find the missing addend

• Count up from the number you know to the whole.
• Subtract the number you know from the whole.

Both ways help you find the same answer.

Method 1: Count Up

Start with the number you know. Count up until you reach the total. Then count how many numbers you said.

Example:

$$7 + \square = 11$$

Start at 7 and count up to 11:

8, 9, 10, 11

We counted 4 numbers, so the missing addend is 4.

$$7 + 4 = 11$$

Method 2: Subtract

If you know the whole and one part, you can subtract to find the missing part.

Example:

$$7 + \square = 11$$

Take the whole and subtract the part you know:

$$11 - 7 = 4$$

So the missing addend is 4.

That means:

$$7 + 4 = 11$$

Think about parts and whole

In an addition sentence, the two addends are the parts. The answer is the whole.

If one part is missing, use the whole and the other part to find it.

For example:

$$9 + \square = 13$$

• Known part: 9
• Whole: 13
• Missing part: ?

You can count up from 9 to 13 or subtract \(13 - 9\).

Worked Example 1

Find the missing addend:

$$4 + \square = 6$$

Count up: Start at 4. Count to 6.

5, 6

That is 2 counts, so the missing addend is 2.

$$4 + 2 = 6$$

Check: Does \(4 + 2\) equal 6? Yes!

Worked Example 2

Find the missing addend:

$$8 + \square = 12$$

Subtract:

$$12 - 8 = 4$$

So the missing addend is 4.

$$8 + 4 = 12$$

Check: Does \(8 + 4\) equal 12? Yes!

Worked Example 3

Find the missing addend:

$$\square + 5 = 14$$

The missing number is at the beginning, but we solve it the same way.

Subtract:

$$14 - 5 = 9$$

So the missing addend is 9.

$$9 + 5 = 14$$

Check: Does \(9 + 5\) equal 14? Yes!

Worked Example 4

Find the missing addend:

$$10 + \square = 18$$

Count up: Start at 10 and count to 18.

11, 12, 13, 14, 15, 16, 17, 18

We counted 8 numbers, so the missing addend is 8.

$$10 + 8 = 18$$

A quick way to think

When you see a missing addend problem, ask:

• What is the whole?
• What part do I already know?
• How much more do I need?

This helps you decide whether to count up or subtract.

Tips to help you

• If the numbers are close together, counting up can be easy.
• If you know your subtraction facts, subtracting can be fast.
• Always check your answer by putting the number back into the addition sentence.

Let’s look at a few more quick checks

$$3 + \square = 7$$

Count up from 3: 4, 5, 6, 7. That is 4 counts, so the answer is 4.

$$\square + 6 = 15$$

Subtract: \(15 - 6 = 9\), so the answer is 9.

$$11 + \square = 11$$

If the number stays the same, the missing addend is 0.

$$11 + 0 = 11$$

Summary

A missing addend is a number missing from an addition sentence. To find it, you can count up from the known part to the whole, or you can subtract the known part from the whole.

Remember: addition and subtraction are a team. If you know one part and the whole, you can find the missing part.

Counting On and Counting Back helps us add and subtract small numbers in our heads. We do not always need to start at 1. We can start at a number we already know and move forward or backward.

When we count on, we move forward to add. When we count back, we move backward to subtract.

This is a smart math strategy because it is faster and helps us solve problems within 20.

Counting on means starting with a number and saying the next numbers to add more.

• If we add, we count forward.
• We usually start with the larger number.
• Then we count on by the smaller number.

For example, in \(8 + 3\), start at 8 and count on 3 numbers:

\(9, 10, 11\)

So, $$8 + 3 = 11$$

Counting back means starting with a number and saying the numbers before it to take some away.

• If we subtract, we count backward.
• Start with the number you have first.
• Then count back by the number being taken away.

For example, in \(12 - 2\), start at 12 and count back 2 numbers:

\(11, 10\)

So, $$12 - 2 = 10$$

How to count on

1. Look at the two numbers.
2. Start with the larger number.
3. Count forward the smaller number of steps.
4. The number you land on is the answer.

How to count back

1. Start with the first number.
2. Count backward the number of steps you are taking away.
3. The number you land on is the answer.

You can use your fingers, a number line, or your voice in your head to help count the steps.

Worked Example 1: Easy counting on

Solve \(6 + 2\).

Start at 6. Count on 2 steps: \(7, 8\).

So, $$6 + 2 = 8$$

Worked Example 2: Counting on with the larger number

Solve \(4 + 7\).

It is easier to start with 7 because 7 is larger.

Start at 7. Count on 4 steps: \(8, 9, 10, 11\).

So, $$4 + 7 = 11$$

Worked Example 3: Easy counting back

Solve \(9 - 3\).

Start at 9. Count back 3 steps: \(8, 7, 6\).

So, $$9 - 3 = 6$$

Worked Example 4: Counting back within 20

Solve \(15 - 4\).

Start at 15. Count back 4 steps: \(14, 13, 12, 11\).

So, $$15 - 4 = 11$$

Helpful tips

• For addition, think: start big, count on.
• For subtraction, think: start there, count back.
• Say one number for each step.
• Be careful not to count the starting number as a step.

Look at \(10 + 3\). Start at 10. Then count on: \(11, 12, 13\). The answer is 13.

Look at \(13 - 2\). Start at 13. Then count back: \(12, 11\). The answer is 11.

Watch out for this mistake: If you solve \(7 + 2\), do not say "7, 8." That is only 1 step. Start at 7, then count on 2 steps: \(8, 9\). So the answer is 9.

Why this strategy works

Addition means putting more on, so we move forward. Subtraction means taking away, so we move backward.

Counting on and counting back help you solve math facts quickly and understand what numbers do.

Summary

• Count on to add small numbers.
• Count back to subtract small numbers.
• Start with the larger number when adding.
• Count one step for each number being added or taken away.

With practice, you will be able to add and subtract within 20 more quickly and confidently.

Making Ten to Add is a smart way to add numbers in your head.

When we make ten, we break apart one number so the first number can become 10. Then the addition is easier, because adding to 10 is quick and friendly.

This strategy works best when one number is close to 10, like 8, 9, 7, or 6.

For example, in \(8 + 5\), the 8 needs 2 more to make 10. We can break the 5 into 2 and 3. Then we add in a new order:

$$8 + 5 = 8 + 2 + 3 = 10 + 3 = 13$$

We did not change the total. We only broke apart the 5 to make the problem easier.

Why make ten?

• 10 is an easy number to work with.
• It helps us add faster in our heads.
• It helps us see number parts clearly.

How to make ten

1. Look at the first number.
2. Ask: How many more does it need to make 10?
3. Break apart the second number.
4. Use one part to make 10.
5. Add the leftover part.

Here are some helpful facts for making 10:

• \(9\) needs \(1\)
• \(8\) needs \(2\)
• \(7\) needs \(3\)
• \(6\) needs \(4\)
• \(5\) needs \(5\)

If you know these pairs, making ten gets much easier.

Worked Example 1

Find \(9 + 4\).

First, ask: How many does 9 need to make 10? It needs 1.

Now break apart 4 into 1 and 3.

$$9 + 4 = 9 + 1 + 3 = 10 + 3 = 13$$

So, \(9 + 4 = 13\).

Worked Example 2

Find \(8 + 6\).

The 8 needs 2 more to make 10.

Break apart 6 into 2 and 4.

$$8 + 6 = 8 + 2 + 4 = 10 + 4 = 14$$

So, \(8 + 6 = 14\).

Worked Example 3

Find \(7 + 5\).

The 7 needs 3 more to make 10.

Break apart 5 into 3 and 2.

$$7 + 5 = 7 + 3 + 2 = 10 + 2 = 12$$

So, \(7 + 5 = 12\).

Worked Example 4

Find \(6 + 8\).

This time, 6 needs 4 more to make 10.

Break apart 8 into 4 and 4.

$$6 + 8 = 6 + 4 + 4 = 10 + 4 = 14$$

So, \(6 + 8 = 14\).

Let’s look closely at what is happening.

In \(8 + 5\), we can think of the 5 as two parts: \(2\) and \(3\).

Then the 8 takes the 2 to become 10.

The 3 is left over.

So the problem becomes:

$$10 + 3$$

That is much easier to solve.

Try thinking with number bonds

A number bond shows how a number can be split into parts.

For \(8 + 5\), the number bond for 5 is:

• \(5 = 2 + 3\)

We picked 2 because 8 needs 2 to make 10.

What if the bigger number is second?

That is okay. You can still make ten.

In \(5 + 9\), you can think about 9 first, because it is close to 10.

The 9 needs 1 more to make 10. Break apart 5 into 1 and 4.

$$5 + 9 = 9 + 1 + 4 = 10 + 4 = 14$$

So, it does not matter which addend comes first. You can use the number that is easiest to make into 10.

Tips for success

• Look for a number close to 10.
• Ask how many more it needs to become 10.
• Break apart the other number.
• Make 10 first.
• Add what is left.

Common mistake to avoid

Sometimes a student knows that 8 needs 2 to make 10, but then forgets to split the other number correctly.

For example, in \(8 + 5\), if you use 2 from the 5, there are 3 left, not 2 left.

That is why:

$$8 + 5 = 10 + 3 = 13$$

not

$$10 + 2$$

Always check the leftover part carefully.

Practice your thinking

• \(9 + 6\): 9 needs 1, so \(6 = 1 + 5\), and \(10 + 5 = 15\)
• \(8 + 3\): 8 needs 2, so \(3 = 2 + 1\), and \(10 + 1 = 11\)
• \(7 + 6\): 7 needs 3, so \(6 = 3 + 3\), and \(10 + 3 = 13\)

Summary

Making ten means breaking apart one addend so the other addend can become 10.

Then you add the leftover part to 10.

This strategy helps you add within 20 quickly and clearly.

When you see an addition problem like \(8 + 5\), think: What do I need to make 10?

Decomposing to Subtract Through Ten

Sometimes subtraction is easier when we first go to 10. This is called decomposing to subtract through ten.

Decomposing means breaking a number into parts. When we subtract through ten, we break apart the number we are taking away. First, we subtract enough to get to 10. Then we subtract the rest.

This strategy helps because 10 is a friendly number. It is easy to work with in your head.

Let’s look at the idea with a number sentence like \(13 - 5\).

We want to start at 13 and take away 5. But instead of taking away all 5 at once, we can break 5 into two parts.

Ask: How much do I subtract from 13 to get to 10?

Since \(13 - 3 = 10\), we break 5 into \(3\) and \(2\).

Now subtract in two steps:

So, \(13 - 5 = 8\).

Here is the step-by-step plan:

1. Look at the first number.
2. Ask, “How far is it from this number down to 10?”
3. Break apart the number you are subtracting.
4. Subtract to 10 first.
5. Subtract the rest.

Why does this work?

If you subtract the same total amount, the answer stays the same. You are still taking away the whole number. You are just doing it in smaller parts.

For example, taking away 6 is the same as taking away 4 and then 2, because \(4 + 2 = 6\).

Let’s practice with worked examples.

Example 1: \(12 - 4\)

First, go from 12 down to 10.

That takes away 2, so break 4 into \(2\) and \(2\).

Answer: \(12 - 4 = 8\)

Example 2: \(15 - 7\)

First, go from 15 down to 10.

That takes away 5, so break 7 into \(5\) and \(2\).

Answer: \(15 - 7 = 8\)

Example 3: \(14 - 6\)

First, go from 14 down to 10.

That takes away 4, so break 6 into \(4\) and \(2\).

Answer: \(14 - 6 = 8\)

Example 4: \(18 - 9\)

First, go from 18 down to 10.

That takes away 8, so break 9 into \(8\) and \(1\).

Answer: \(18 - 9 = 9\)

Let’s notice a pattern.

• From 11 to 10, subtract 1.
• From 12 to 10, subtract 2.
• From 13 to 10, subtract 3.
• From 14 to 10, subtract 4.
• From 15 to 10, subtract 5.
• From 16 to 10, subtract 6.
• From 17 to 10, subtract 7.
• From 18 to 10, subtract 8.
• From 19 to 10, subtract 9.

This helps you know how to break apart the number you are subtracting.

For example:

• In \(16 - 7\), it takes 6 to get to 10, so 7 becomes \(6 + 1\).
• In \(17 - 8\), it takes 7 to get to 10, so 8 becomes \(7 + 1\).

Another way to think about it

You can imagine hopping backward on a number line.

For \(13 - 5\):

• Hop back 3 to land on 10.
• Then hop back 2 more.
• You land on 8.

This is the same as decomposing 5 into \(3\) and \(2\).

Helpful questions to ask yourself

• What number am I starting with?
• How many do I need to subtract to get to 10?
• How can I break apart the number I am subtracting?
• What is left to subtract after I reach 10?

Be careful!

• Do not break apart the first number. Break apart the number you are taking away.
• Make sure the two parts still add up to the whole number you are subtracting.
• Always subtract to 10 first, then subtract the rest.

Let’s check one more.

Suppose you have \(16 - 7\).

From 16 to 10 is 6. So break 7 into \(6\) and \(1\).

That means \(16 - 7 = 9\).

Summary

Decomposing to subtract through ten means breaking apart the number you subtract. First, subtract enough to get to 10. Then subtract the rest. This makes subtraction within 20 easier and faster.

Doubles and Near Doubles are helpful addition tricks that make math faster and easier.

A double is when you add the same number to itself. For example, \(4+4\) is a double. When you know your doubles facts, you can use them to solve other addition problems.

A near double is an addition fact with numbers that are almost the same. They are usually just 1 apart, like \(6+7\) or \(8+9\). You can use a doubles fact you already know to solve it.

Learning doubles and near doubles helps you add within 20 in your head. It is a great way to become quicker and more confident with math facts.

Let’s start with some doubles facts.

• \(1+1=2\)
• \(2+2=4\)
• \(3+3=6\)
• \(4+4=8\)
• \(5+5=10\)
• \(6+6=12\)
• \(7+7=14\)
• \(8+8=16\)
• \(9+9=18\)
• \(10+10=20\)

These are important facts to know. If you remember them, you can solve many other problems.

How to use doubles

When both addends are the same, just think of the double.

For example, in $$7+7=14$$ you are adding 7 two times. That is the double of 7.

How to use near doubles

If the numbers are almost the same, first find the double you know. Then adjust by 1.

For example, in \(7+8\), the numbers are close. You might know \(7+7=14\). But \(7+8\) has one more than \(7+7\), so the answer is one more than 14.

$$7+8=15$$

You can also think of \(8+8=16\). Since \(7+8\) is one less than \(8+8\), the answer is one less than 16.

Both ways work.

Steps for solving a near double

1. Look at the two numbers.
2. Ask, “Are they the same or almost the same?”
3. Use a doubles fact you know.
4. Add 1 or subtract 1 if needed.

Worked Example 1: A doubles fact

Solve \(5+5\).

The numbers are the same, so this is a double.

$$5+5=10$$

Answer: 10

Worked Example 2: A near double with 1 more

Solve \(6+7\).

The numbers are 1 apart. Use the double \(6+6=12\).

Since \(6+7\) is 1 more than \(6+6\), add 1.

$$6+7=12+1=13$$

Answer: 13

Worked Example 3: A near double with 1 less

Solve \(8+9\).

The numbers are 1 apart. Use the double \(9+9=18\).

Since \(8+9\) is 1 less than \(9+9\), subtract 1.

$$8+9=18-1=17$$

Answer: 17

Worked Example 4: Another near double

Solve \(4+5\).

Use the double \(4+4=8\).

\(4+5\) is 1 more than \(4+4\), so add 1.

$$4+5=8+1=9$$

Answer: 9

Tips to remember

• If the numbers are the same, use a double.
• If the numbers are 1 apart, use a near double.
• You can start with the smaller double or the bigger double.
• Then change the answer by 1.

Let’s compare.

• \(3+3=6\) is a double.
• \(3+4=7\) is a near double because it is 1 more.
• \(9+9=18\) is a double.
• \(9+10=19\) is a near double because it is 1 more.

Why this strategy helps

Your brain does not have to start from the beginning every time. If you know a doubles fact, you can use it to solve a new fact quickly.

That means fewer facts to memorize and more facts you can figure out by thinking.

Quick practice to try in your head

• \(2+2=4\)
• \(2+3=5\)
• \(7+7=14\)
• \(7+8=15\)
• \(10+10=20\)
• \(9+10=19\)

Summary

A double is when you add the same number to itself, like \(6+6\).

A near double is when the numbers are almost the same, like \(6+7\) or \(8+9\).

Use the doubles fact you know, then change the answer by 1. This strategy helps you add quickly and accurately within 20.

Fact Fluency: Addition Within 20 means being able to solve addition facts quickly and correctly in your head.

We are not just guessing or memorizing without understanding. We use smart strategies to help our brains learn the facts.

When you practice these strategies, addition within 20 gets faster and easier.

What does “within 20” mean? It means the total is 20 or less. For example, \(8+5=13\) and \(9+9=18\).

Let’s learn some helpful ways to add.

1. Count on from the bigger number

When you add two numbers, start with the bigger number in your head. Then count on the smaller number.

This is faster than counting both groups from 1.

• For \(7+3\), start at 7 and count on 3 more: 8, 9, 10.
• So, \(7+3=10\).

2. Use doubles

Doubles are facts where the same number is added to itself.

• \(1+1=2\)
• \(2+2=4\)
• \(3+3=6\)
• \(4+4=8\)
• \(5+5=10\)
• \(6+6=12\)
• \(7+7=14\)
• \(8+8=16\)
• \(9+9=18\)
• \(10+10=20\)

If you know doubles, many other facts become easier.

3. Use near doubles

Near doubles are addition facts that are close to a double.

For example, \(6+7\) is close to \(6+6\).

Since \(6+6=12\), then \(6+7\) is just 1 more, so it equals 13.

4. Make 10

Making 10 is a very helpful strategy because 10 is easy to work with.

Some number pairs that make 10 are:

• \(1+9=10\)
• \(2+8=10\)
• \(3+7=10\)
• \(4+6=10\)
• \(5+5=10\)

If one part of your addition can make 10, use that first.

For example, in \(8+5\), the 8 needs 2 more to make 10. Break apart the 5 into 2 and 3.

Then:

$$8+5 = 8+2+3 = 10+3 = 13$$

This strategy helps you add in your head.

5. Turn-around facts

Numbers can switch places and still have the same total.

$$3+8=8+3$$

Both equal 11.

This is helpful because if you know one fact, you also know its turn-around fact.

So if you know \(4+9=13\), then you also know \(9+4=13\).

Worked Example 1: Count on

Solve \(9+2\).

Start with the bigger number, 9. Count on 2 more: 10, 11.

So, $$9+2=11$$

Worked Example 2: Use doubles

Solve \(7+7\).

This is a double. Double 7 is 14.

So, $$7+7=14$$

Worked Example 3: Use near doubles

Solve \(8+9\).

This is close to the double \(8+8\).

We know:

$$8+8=16$$

But \(8+9\) is 1 more than \(8+8\), so:

$$8+9=17$$

Worked Example 4: Make 10

Solve \(6+8\).

The 6 needs 4 more to make 10. Break apart the 8 into 4 and 4.

Then:

$$6+8 = 6+4+4 = 10+4 = 14$$

So, \(6+8=14\).

Tips to build fact fluency

• Look for facts you already know, like doubles.
• Start with the bigger number and count on.
• Try to make 10 when you can.
• Use turn-around facts to help you remember.
• Practice a little at a time every day.

Let’s think about a few more facts:

• \(5+6\): Use near doubles. \(5+5=10\), so \(5+6=11\).
• \(9+4\): Make 10. Give 1 from the 4 to the 9. Then \(10+3=13\).
• \(3+7\): This is a make-10 fact. \(3+7=10\).
• \(10+5\): Start at 10 and count on 5. The answer is 15.

Why this matters

When you know addition facts quickly, bigger math problems are easier. You can spend more time thinking and less time counting.

Fact fluency grows with understanding and practice. The more you use these strategies, the more the facts will stick in your brain.

Summary

To add within 20, use smart strategies like count on, doubles, near doubles, make 10, and turn-around facts.

You do not need to count every number from the beginning. Use what you know to help solve new facts quickly.

With practice, you can become fast and confident with addition within 20.

Fact Fluency: Subtraction Within 20

Today we will learn how to subtract within 20 quickly and correctly. Fact fluency means you know the facts well enough to answer without getting stuck.

When we subtract, we start with a whole number and take some away. The answer is called the difference.

Subtraction and addition are connected. They are like a team. If you know an addition fact, you can use it to help with subtraction.

For example, if you know that \(8 + 5 = 13\), then you also know:

$$13 - 5 = 8$$

$$13 - 8 = 5$$

This is called using an inverse, or an opposite operation. Addition puts parts together. Subtraction takes a part away.

Ways to get better at subtraction facts within 20

• Use what you know about addition. Think: “What number goes with this number to make the whole?”
• Count back. Start at the first number and count backward.
• Think about making 10. Facts with 10 are helpful and easy to remember.
• Use doubles you know. If you know \(6 + 6 = 12\), that can help with subtraction too.

1. Use addition to help subtraction

Suppose you see \(15 - 7\). You can ask, “\(7 + ? = 15\)”

If you know \(7 + 8 = 15\), then:

$$15 - 7 = 8$$

This is a fast way to solve subtraction facts.

2. Count back carefully

You can also subtract by counting back. Start at the bigger number and move backward the number of steps you are taking away.

For \(12 - 3\): start at 12 and count back 3 numbers.

$$12 \rightarrow 11 \rightarrow 10 \rightarrow 9$$

So:

$$12 - 3 = 9$$

This works well when you subtract a small number.

3. Use 10 to help

The number 10 is a friendly number. If a subtraction fact goes to 10, it can help you solve other facts.

For example, in \(14 - 6\), you might think:

“I know \(6 + 4 = 10\), and then 4 more makes 14, so \(6 + 8 = 14\).”

That means:

$$14 - 6 = 8$$

4. Use fact families

A fact family is a group of math facts that use the same three numbers.

Look at 4, 9, and 13.

• \(4 + 9 = 13\)
• \(9 + 4 = 13\)
• \(13 - 4 = 9\)
• \(13 - 9 = 4\)

If you know one or two facts in the family, you can figure out the others.

Worked Examples

Example 1: Easy count back

Solve \(11 - 2\).

Start at 11. Count back 2 steps:

$$11 \rightarrow 10 \rightarrow 9$$

So:

$$11 - 2 = 9$$

Example 2: Use addition to help

Solve \(13 - 5\).

Ask: “\(5 + ? = 13\)”

We know:

$$5 + 8 = 13$$

So:

$$13 - 5 = 8$$

Example 3: Use a fact family

Solve \(17 - 9\).

Think: “\(9 + ? = 17\)”

We know:

$$9 + 8 = 17$$

So:

$$17 - 9 = 8$$

Example 4: A harder one within 20

Solve \(18 - 11\).

Think: “\(11 + ? = 18\)”

We know:

$$11 + 7 = 18$$

So:

$$18 - 11 = 7$$

You can also count back 11, but using addition is often faster.

Tips for becoming fluent

• Practice a few facts every day.
• Say the fact family out loud.
• Look for facts you already know, like facts with 10.
• Use addition to check your subtraction answer.

Check your answer

After you subtract, you can check with addition.

If you found that \(16 - 7 = 9\), check by adding:

$$9 + 7 = 16$$

Since the addition fact is true, the subtraction fact is correct too.

Let’s remember

• Subtraction means taking away.
• The answer in subtraction is called the difference.
• Addition and subtraction help each other.
• You can solve subtraction facts by counting back or by thinking of the matching addition fact.
• Knowing subtraction facts within 20 helps you do math faster and with more confidence.

Brief Summary

Subtraction within 20 becomes easier when you use what you know about addition. You can count back, use 10, and use fact families to find answers quickly. The more you practice, the more fluent you will become.