Adding Whole Numbers Using the Standard Method

Hello. In the previous lesson, you compared whole numbers by looking at place value from left to right. Addition uses the same place-value structure in a different way: instead of deciding which number is larger, you combine the amounts in matching places.

In this lesson, you will use the standard written method for addition. You will line up digits correctly, add from ones toward greater place values, and regroup whenever a column contains ten or more of a unit. This method works for two-digit numbers, larger whole numbers, and even several addends.

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Keep each place value in its own column

In an addition problem, the numbers being added are called addends. Their answer is the sum.

The standard written method puts the addends in vertical columns. The essential rule is:

Line up digits by place value, not by where they happen to appear on the page.

Ones must be below ones, tens below tens, hundreds below hundreds, and so on. It is usually safest to begin by lining up the rightmost digits, because they are the ones digits.

For example, to add $47$ and $32$, write:

$$\begin{array}{r}47\\ +32\end{array}$$

The 7 and 2 are both ones. The 4 and 3 are both tens.

Now add one column at a time, beginning with the smallest place value, the ones place.

1. Ones: $7+2=9$. Write 9 in the ones place.
2. Tens: $4+3=7$. Write 7 in the tens place.

$$\begin{array}{r}47\\ +32\\ 79\end{array}$$

So,

$$47+32=79$$

Starting at the right matters because a regrouping in one place affects the column immediately to its left.

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Regrouping: exchanging ten of one unit for one of the next unit

A digit in a single column must represent fewer than ten of that column’s unit:

• 10 ones have the same value as 1 ten.
• 10 tens have the same value as 1 hundred.
• 10 hundreds have the same value as 1 thousand.

Regrouping records this exchange. It is also often called “carrying,” but regrouping helps explain what is actually happening.

Consider:

$$58+67$$

First, line up the places:

$$\begin{array}{r}58\\ +67\end{array}$$

Step 1: Add the ones

$$8+7=15$$

Fifteen ones cannot all stay in the ones place. Regroup them as 1 ten and 5 ones.

• Write the 5 in the ones place.
• Write the regrouped 1 above the tens column. It means one ten, not one one.

Step 2: Add the tens

There are 5 tens, 6 tens, and the extra 1 ten:

$$1+5+6=12$$

Twelve tens regroup as 1 hundred and 2 tens. Write 2 in the tens place and 1 in the hundreds place.

$$\begin{array}{rrr}& 1& 1\\ & 5& 8\\ +& 6& 7\\ 1& 2& 5\end{array}$$

Therefore,

$$58+67=125$$

The two small 1s above the problem have different meanings:

• the 1 above the tens column is 1 ten;
• the 1 above the hundreds column is 1 hundred.

Their digit is the same, but their place values are different.

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A three-digit example with regrouping twice

The place-value chart below shows the same method for $269+148$.

Let’s follow the calculation carefully.

$$\begin{array}{rrrr}& 1& 1& \\ & 2& 6& 9\\ +& 1& 4& 8\\ & 4& 1& 7\end{array}$$

Ones column

$$9+8=17$$

Write 7 ones. Regroup 10 of those ones as 1 ten, written above the tens column.

Tens column

Now include the regrouped ten:

$$1+6+4=11$$

Write 1 ten. Regroup the other 10 tens as 1 hundred, written above the hundreds column.

Hundreds column

Now include the regrouped hundred:

$$1+2+1=4$$

Write 4 hundreds. The result is:

$$269+148=417$$

A reliable way to speak through any problem is:

1. Name the place: “ones,” “tens,” or “hundreds.”
2. Add every digit in that column, including any regrouped amount above it.
3. If the total is 10 or more, write the digit for the current place and regroup the rest into the next place.

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Zeros, different digit lengths, and a new place in the answer

A zero holds a place even though it represents no amount of that unit. This is especially important in written addition.

Consider:

$$308+759$$

Line up the numbers by place value:

$$\begin{array}{rrrr}& 3& 0& 8\\ +& 7& 5& 9\end{array}$$

Now calculate from right to left.

• Ones: $8+9=17$. Write 7; regroup 1 ten.
• Tens: $1+0+5=6$. Write 6.
• Hundreds: $3+7=10$. Write 0 hundreds; regroup 1 thousand.

The result has a new thousands place:

$$\begin{array}{rrrr}& 3& 0& 8\\ +& 7& 5& 9\\ 1& 0& 6& 7\end{array}$$

So,

$$308+759=1,067$$

The zero in 308 is essential: it tells us there are zero tens. It must remain in the tens column rather than being skipped.

When the addends have different numbers of digits, still line up the rightmost digits. For example:

$$\begin{array}{rrr}& 4& 6\\ 1& 2& 3\\ 1& 6& 9\end{array}$$

This represents:

$$46+123=169$$

The 46 has no hundreds digit, which is equivalent to having 0 hundreds.

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Adding more than two addends

The same method works when there are three or more addends. Add all the digits in each column, along with any regrouped amount.

For example:

$$\begin{array}{rrrc}& 2& 4& 6\\ & 1& 8& 3\\ +& 2& 7& 5\end{array}$$

In the ones column:

$$6+3+5=14$$

Write 4 ones and regroup 1 ten.

In the tens column:

$$1+4+8+7=20$$

Write 0 tens and regroup 2 hundreds.

In the hundreds column:

$$2+2+1+2=7$$

So:

$$246+183+275=704$$

Notice that the regrouped amount can sometimes be 2 rather than 1. It represents 2 of the next larger unit.

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Preventing the most common errors

The written method is dependable when you protect place value at every step.

Common error	Why it causes trouble	Better habit
Misaligning digits	It combines unlike units, such as tens with ones.	Line up rightmost digits first.
Starting with the leftmost column	A regrouped amount from the right may be missed.	Begin with the ones column.
Forgetting the small regrouped digit	The next column’s total becomes too small.	Before adding a column, check above it.
Treating a regrouped 1 as one one	Its value changes with its position.	Say its value aloud: “one ten” or “one hundred.”
Dropping zeros	A digit may shift into the wrong place.	Keep zeros in the number and maintain each column.

A quick reasonableness check can catch a major mistake. Round each addend to a nearby easy number and compare your exact answer with the estimate.

For example:

$$398+607$$

is close to

$$400+600=1,000$$

The exact answer is $1,005$, which is close to the estimate. An answer such as $105$ or $10,050$ would signal a place-value error.

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Key takeaways

The standard written method for whole-number addition is a place-value method.

• Align digits vertically by ones, tens, hundreds, and larger places.
• Start with the rightmost, smallest place value.
• Add every digit in a column, including any regrouped amount.
• When a column has 10 or more units, regroup 10 of them as 1 unit in the place to the left.
• Zeros and correctly aligned columns matter, especially when the addends have different lengths.
• Estimate with rounded numbers to check whether a sum is reasonable.

Next, you will use place value in the opposite direction for subtraction, including the related idea of regrouping a larger unit into ten smaller units.