Overview
Read
Next Steps
Read
Subtraction Using Mental Strategies
This lesson teaches mental strategies for subtracting numbers without writing out every step, including counting up, compensation with friendly numbers, breaking numbers apart by place value, and using number line jumps, so students can subtract quickly and check their answers with addition.
Why Mental Strategies Matter
Written subtraction with regrouping, like the method covered in Subtraction with Regrouping Up to 1000, is a reliable way to subtract any two numbers. But mental strategies let you solve many problems faster, estimate answers before checking your work, and build a stronger sense of how numbers relate to each other. They are also a great way to double check an answer you found on paper.
Strategy 1: Counting Up
Counting up turns a subtraction problem into a series of small addition jumps. Instead of asking "what is \( 84 - 27 \)?", you ask "how far is it from 27 to 84?" You count up in friendly chunks and add the jumps together.
For example, to find \( 84 - 27 \):
- From 27, jump to 30: that is \( +3 \)
- From 30, jump to 80: that is \( +50 \)
- From 80, jump to 84: that is \( +4 \)
Add the jumps: \( 3 + 50 + 4 = 57 \). So \( 84 - 27 = 57 \).
This same jumping idea appears again with larger numbers in Subtracting by Jumping Up to 1000, where the jumps are drawn on a number line up to the thousands.
Strategy 2: Compensation with Friendly Numbers
Compensation means adjusting one of the numbers to a friendly value, doing the easier subtraction, and then fixing the answer to make up for the change. This works well when a number is just a little away from a multiple of ten or a hundred.
For example, to find \( 63 - 29 \), notice that 29 is close to 30. Subtract the friendly number first: \( 63 - 30 = 33 \). Since you subtracted 1 too many, add it back: \( 33 + 1 = 34 \). So \( 63 - 29 = 34 \).
The same idea works with hundreds. For \( 500 - 198 \), round 198 up to 200: \( 500 - 200 = 300 \). You subtracted 2 too many, so add 2 back: \( 300 + 2 = 302 \). So \( 500 - 198 = 302 \).
Strategy 3: Breaking Numbers Apart by Place Value
Another mental strategy is to split the number being subtracted into hundreds, tens, and ones, then subtract each part in order. This connects to ideas from Subtraction with Tens and Ones, where numbers are separated into place value groups before subtracting.
For example, to find \( 456 - 213 \), break 213 into \( 200 + 10 + 3 \):
- \( 456 - 200 = 256 \)
- \( 256 - 10 = 246 \)
- \( 246 - 3 = 243 \)
So \( 456 - 213 = 243 \). This method avoids regrouping entirely because each part is subtracted separately.
Checking Your Answer with Addition
Subtraction and addition undo each other, so you can check any mental subtraction by adding the difference back to the number you subtracted. If \( a - b = c \), then \( c + b \) should equal \( a \). Using the example above, \( 243 + 213 = 456 \), which confirms the answer is correct.
Choosing the Best Strategy
There is no single "correct" mental strategy for every problem. Look at the numbers first: if one number is very close to a multiple of ten or a hundred, compensation is often fastest. If the numbers are far apart, counting up in big jumps works well. If the numbers have several digits with little regrouping needed, breaking apart by place value keeps things simple. Estimating first, as shown in Subtraction using Estimation, can also help you judge whether your mental answer is reasonable before you commit to it.
Practice Tip
Start by practicing with two digit numbers before moving to numbers up to 1000, such as those explored in Subtracting with Digits Up to 1000. The more you practice spotting friendly numbers and comfortable jump sizes, the faster mental subtraction becomes.