Mental Math: Advanced Subtraction

As our children grow and develop their math skills, the mental math strategies grow with them.

The basics of mental math don’t change:

• Use friendly numbers.
• Estimate and adjust the answer.

But we have new ways to help children do math in their heads as the numbers get bigger and the problems more challenging.

For example, how might kids figure out a multi-digit subtraction like 67 − 38?

First, we need to adjust our mindset…

Subtraction is Not Take-Away

The big idea of subtraction is comparison, finding the difference between our two numbers.

Children first learn subtraction as “take away,” removing a portion of the initial quantity. But “take away” is merely an application, not the main idea. In truth, we compare the amount we had at first to the amount removed, and the difference between these quantities is what remains.

But subtraction doesn’t actually change the numbers. The original amount remains itself, and the subtrahend (the amount subtracted) hasn’t gone anywhere. Subtraction merely connects these two numbers in relationship with their difference.

The two mental math strategies in this post emphasize the core meaning of subtraction as comparing two numbers.

Find the Distance: Adding Up

To calculate 67 − 38, we want to find how far apart the numbers are. We can visualize this difference on the number line as the distance between the numbers.

One way to calculate this distance is to ask ourselves, “How much would I need to add to get from 38 to 67?”

This method emphasizes the inverse relationship between addition and subtraction. They are two ways to look at the same mathematical relationship, like two sides of the same coin.

Adding 2 gets us up to a friendly 40, and then we need 27 more:

67 − 38 = 2 + 27

Comparison: Constant Difference

Remember the model of subtraction as comparing the lengths of two blocks, where the empty space is the difference between them.

Can you see that we could add any number of blocks to the front of both lines without changing that difference?

On a number line, consider the difference “67 – 38” as a solid block. We can slide that block up or down the number line without changing its value.

67 − 38
= 68 − 39
= 69 − 40
= 70 − 41, etc.

Or 67 − 38
= 66 – 37
= 65 − 36
= 64 − 35, etc.

We slide our constant difference to whatever makes a friendly-number calculation. In this case, adding two to both numbers lets us subtract a friendly multiple of ten:

67 − 38
= (67 + 2) − (38 + 2)
= 69 − 40

Playful Practice Makes It Feel Natural

Many people find these advanced mental math strategies confusing at first. But if you practice them with your students, they will soon become valuable problem-solving tools.

Choose a strategy to practice, and take turns making up problems for each other to solve. You don’t need a long session. Even just two or three practice problems a day builds up your mental math muscles.

Read the Whole Series

Check out all the posts in my Mental Math Series:

• Mental Math Is the Key to Algebra
• Three Basic Principles
• Early Addition
• Early Subtraction
• Early Multiplication
• Early Division
• Advanced Addition
• Advanced Subtraction
• Coming Soon: Advanced multiplication strategies…