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# What Is Standard Algorithm Subtraction? Explained For Teachers, Parents and Kids

Here we explain what exactly standard algorithm subtraction is, tell you how it is used and learned in elementary schools and provide some examples of questions involving this method!

Standard algorithm subtraction is just one part of a child’s journey through subtraction. It is also one part of the journey where teachers should not rush to get students to this point.

Our younger learners would be far better off with being exposed to, and having ample time to practice, mental methods of subtraction that have been carefully thought out and sequenced.

It is not until 2nd grade that the ‘formal’ method of subtraction algorithm needs to be introduced.

### What is standard algorithm subtraction?

Standard algorithm subtraction is a ‘formal’ written method for solving questions and problems involving subtraction found within common core state standards. Starting in 2nd grade, students will be subtracting with 1,000 with regrouping.

Standard algorithm subtraction method is first introduced in 2nd grade, but isn’t used until after students have a concrete understanding of regrouping between place values. After it’s introduced, it will continue to be used by students up to 5th grade and beyond.

The standard algorithm method of subtraction relies on students being able to correctly arrange a calculation, such as 653 – 321, into columns so that the operation of subtraction can take place.

An example of this can be found below:

It is worth bearing in mind that before students reach this stage, there are many other mathematical skills that students need to be competent in.

### What is the expanded standard algorithm method?

Before going straight to the standard algorithm method above, a good intermediate way to get students ready for it is to use the expanded standard algorithm method.

‘Expanded’ refers to writing out each digit into its corresponding value.

For example, writing 189 in its expanded form would be 100+80+ 9. In the context of column subtraction it would look like this.

789 – 458:

700  80  9

– 400  50  8

=300  30  1

= 331

There is no requirement for teachers to teach the expanded method but it may be useful when first introducing subtraction problems with multi-digit numbers.

### When do children learn about standard algorithm subtraction in school?

Children will first encounter standard algorithm subtraction in 2nd grade. According to the Common Core State Standards (CCSS), they will be expected to:

• add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction;

In 3rd grade, this progresses to the following:

• fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

When in 4th grade, CCSS math standards suggest students should:

• fluently add and subtract multi-digit whole numbers using the standard algorithm.

For 5th grade, students are introduced to decimals and it is recommended that they are able to:

• subtract decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction;

Depending on the curriculum your child’s school follows, they may be taught slightly differently or concepts may be taught in different grade levels. For example, in Texas, students are introduced to subtracting decimals in the 4th grade.

### How do you arrange and subtract columns?

When you rearrange a written subtraction calculation into columns, it is important to remember some crucial elements that need to be correct.

First and foremost, the order of each digit is crucial. Take 789 – 458. The first number is referred to as the minuend and the second number is referred to as the subtrahend.

It is important that when writing the column method that the minuend is the top number and the subtrahend is the bottom number.

For example, 789 – 458 would give a very different answer to 458 – 789.

Another important part of the arrangement is to ensure the place values all align i.e. the ones place for each number in the subtraction is in the ones column, and the tens place digit of each number is in the tens column.

This is especially true for when you need to subtract numbers that have a different number of digits. Getting this part of the process wrong will definitely result in the procedure delivering an incorrect answer, so it is important students are fluent in base ten and their place value knowledge.

If the calculation was 2,345 – 567, it is imperative that these are arranged correctly, particularly that the 5 is placed in the hundreds column rather than the thousands column, as illustrated below.

2,345 and not as 2,345

– 567                  – 567

If useful, at the beginning, students can place temporary zeros in columns in order to make sure calculations are aligned correctly – in the calculation above we would place a zero in the thousandths column in front of 567.

This is why a prerequisite of standard algorithm subtraction is a strong understanding of place value. This is why the expanded method, as mentioned above, can be a useful strategy to include.

### Regrouping

One particular aspect that students initially find difficult is the concept of regrouping. This occurs when a digit in the subtrahend is greater than the digit that has the same place value in the minuend. For example:

783  –

458

As the 8 is greater than the 3, a regrouping strategy will need to be used. Doing this with place value counters can be useful in showing what is happening mathematically.

This shows 783 and demonstrates that 8 ones cannot be taken away as there are only 3.

Now we have exchanged one of the 10s counters for 10 of the 1s counters. This means that I now have 13 ones and this makes it possible to subtract 8.

You can see that the number currently represented is now 775. I can now continue the calculation by subtracting 50 and subtracting 400.

With 50 having been subtracted.

With the 400 having been subtracted.

This is represented in column subtraction in the following way.

### Subtracting 2 digit numbers from 3 digit numbers

As discussed earlier, a strong understanding of place value is crucial when subtracting numbers with a different number of digits, for example two-digit numbers from three-digit numbers.

We want to ensure that students use the following:

345 – and not as 345

67                       67

Using standard algorithm subtraction helps students to avoid this error as the place value of each digit is made clearer. 345 – 67 written in expanded form would be 300  40  5 – 60 7 =

300 40  5 –

60 7

Getting students to perform some calculations using this method before transitioning to the non-standard method may mitigate this issue. You could also get students to represent the calculation with place value counters.

Please note that this is different from using the manipulative to help them solve the actual calculation.

When students can comfortably write the calculation in the following way:

345 –

67

Then the process of solving the problem becomes the same as solving the calculation with regrouping. Depending where students are, they may need manipulatives to help them solve it.

### Standard algorithm subtraction with decimals

When students increase their understanding of decimal place value, they will be expected to use their knowledge of this and of standard algorithm subtraction to solve problems involving decimals. This is usually done in the context of measurements.

It is important that students understand that the principles of place value and how standard algorithm subtraction works does not change, it is only the context (decimals) that changes.

### How does standard algorithm subtraction relate to real life?

Standard algorithm subtraction relates largely to real life as day to day we are continuously using subtraction and addition for larger numbers when planning budgets, checking our bank balance, bills or pay checks.

For students, you could use examples of getting their allowance and seeing how much is left when spending it on candy or a trip to the movies. You can also relate standard algorithm subtraction to calculate the distance left to get to a destination.

### How does standard algorithm subtraction relate to other areas of mathematics?

Standard algorithm subtraction is a fundamental skill for students and may be used in nearly every area of math, particularly within calculations and problem solving with measurement.

### What other methods of subtraction are used in schools?

As mentioned in the introduction, standard algorithm subtraction should not be the go-to strategy that students are taught immediately. There are many other mental math strategies that students need to learn first which they do in the early stages of schooling. Common strategies include:

• Partitioning the subtrahend: This is where the subtrahend is partitioned (broken apart) into its different place values. This is then used for the students to perform two separate subtractions.

For example:

78 – 36 =

78 – 30 and  6.

The 6 can be subtracted from the 8 to get 72. Then 30 can be subtracted from 72 to get 42.

• Bridging to ten: This is linked to partitioning, but the number is not broken up into its expanded form but partitioned in such a way that it would take the minuend to the next multiple of 10.  For example, in 165 – 7 the 7 can be partitioned into 5 and 2. 5 can be subtracted from 165 to make it 160.

Finally, 2 can be subtracted from 160 to make 158. Number lines are particularly useful to scaffold the initial learning of this strategy but students should not rely on them.

• Same difference: This strategy is particularly useful when subtracting a number from a minuend that would require plenty of regrouping. E.g. 2,000 – 256. The same difference principle applies to subtraction in the following way:

Imagine it was 10 – 8. The difference is 2. 2 is also the difference between 9 and 7 and 8 and 6 etc. In all instances, the minuend and subtrahend is being decreased by 1 and this results in the same difference.

With 2000 – 256 we can apply the same principle and decrease both numbers by 1 knowing the difference will remain the same. Now the calculation becomes 1999 – 255 =.

Now students can subtract and rely on subtraction facts that they are much more familiar with. 9 – 5 = 4,  90-50 = 40 and 900 minus 700 is 200. The 1000 will remain the same so the final answer will be 1744. This can be written in the following way:

2000 – 256 = 1999 – 255 = 1744

When students go on to learn the formal method of subtraction, they fall into the habit of using this method to solve every subtraction they come across. Teachers, however, should encourage the use of the formal written method as a fall back, or checking strategy, and prioritize the mental strategies in elementary years.

### Standard algorithm subtraction worked examples

1. 874 – 123 =

A straightforward question where if you don’t ensure that the subtrahend and minuend are correctly positioned will post relative issues. This is the type of question where we should encourage students to attempt it mentally first.

2. 4,783 – 2,349

4,783 –

2,349

Here students will see that they will need to exchange 1 ten for 10 ones. This will give the following:

Next, the procedure can carry on as normal working from the lowest place value to the highest.

There are no additional difficulties once the exchange has taken place.

3. 2,354.43 – 1,789.52 =

The start of this process is straightforward and should be familiar to readers who have read the blog.

Where an additional difficulty may occur is in the next column.

Here the student needs to exchange in the tens column and carry over 1 ten to the units column knowing that because of the previous step, the units column has a value of 3 and not 4.

For the rest of the problem, the same pattern continues.

### Standard algorithm subtraction word problems and answers

1. Harry had $365 in his bank account. His mom let him spend$249 on a new bike. How much was left in his account?

This question is a typical example of a subtraction that demonstrates the structure of taking away. A physical quantity is being taken away from a larger quantity and so arranging this correctly is important.

2. Bill’s book was 561 words long. Ted’s book was 395 words long. How many more words did Bill read than Ted?

This question is an example of the subtraction structure of difference. We are not taking one quantity away from another but rather looking at the difference between the two quantities.

### Standard algorithm subtraction example questions

1. 345 – 234 =

2. 6,473 – 4,287 =

3. 795.32 – 452.12 =

4. Harry had $734 in his bank account. His mom let him spend$375 on a new bike. How much was left in his account?

5. Bill’s book was 1,453 words long. Ted’s book was 954 words long. How many more words did Bill read than Ted?

### FAQs

How do you explain standard algorithm subtraction?

The standard algorithm method is a method of mathematical calculation. Standard algorithm subtraction and standard algorithm addition set a number out into columns depending on the place value of each digit.

What does column mean in math?

In math, a column is the arrangement of numbers one on top of the other.

How do you explain subtraction to a child?

Subtraction can be explained as taking away or finding the difference. For example ‘how much more is 12 than 10?’.

Wondering about how to explain other key math vocabulary to your children? Check out our Primary Math Dictionary, or try these:

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The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.

##### Neil Almond
Woodland AT Primary