Subtraction as Difference Formula

Subtraction as difference is understanding subtraction as finding the gap or difference between two quantities, rather than just 'taking away.' This.

The Formula

difference=larger−smaller\text{difference} = \text{larger} - \text{smaller}

When to use: How much taller is a 6-foot person than a 4-foot person? The difference is 2 feet.

Quick Example

12−8=412 - 8 = 4 means 12 is 4 more than 8, or 8 needs 4 to reach 12.

Notation

The −- sign in a difference context reads as 'how far from' rather than 'take away'

What This Formula Means

Understanding subtraction as finding the gap or difference between two quantities, rather than just 'taking away.' This comparison model asks 'how many more?' or 'how far apart?'

How much taller is a 6-foot person than a 4-foot person? The difference is 2 feet.

Formal View

d(a,b)=∣a−b∣,  the unsigned difference satisfying d(a,b)=d(b,a)≥0d(a, b) = |a - b|, \; \text{the unsigned difference satisfying } d(a,b) = d(b,a) \geq 0

Worked Examples

Example 1

easy
Rosa has 8 grapes. Tom has 5 grapes. How many MORE grapes does Rosa have than Tom?

Answer

3 more grapes

First step

1
We want to find the difference between 8 and 5.

Full solution

  1. 2
    Write: 8−5=?8 - 5 = ?
  2. 3
    Count up from 5 to 8: 6, 7, 8 — that is 3 steps.
  3. 4
    Rosa has 3 more grapes than Tom.
Subtraction as difference compares two amounts to see how much more one has than the other. The difference between 8 and 5 is 3.

Example 2

medium
A red tower has 11 blocks. A blue tower has 6 blocks. How many fewer blocks does the blue tower have?

Example 3

easy
You have 5 stickers. Your friend has 2. How many more do you have?

Common Mistakes

  • Adding the two amounts because both are present - a 'how many more' question subtracts to find the gap.
  • Subtracting in the wrong order so the gap comes out negative - take larger minus smaller for the distance.
  • Thinking nothing was removed so it cannot be subtraction - comparison is still subtraction.

Why This Formula Matters

Many word problems compare rather than remove, and children who only know 'take away' freeze on them. The difference model also grounds distance on a number line and the meaning of a−ba - b for any two numbers. Recognizing it by "Am I finding the gap between two amounts rather than removing one?" — rather than by familiar numbers — is what lets a student tell it apart from subtraction (take away) and addition and comparison (which is more) in a mixed problem set.

Frequently Asked Questions

What is the Subtraction as Difference formula?

Understanding subtraction as finding the gap or difference between two quantities, rather than just 'taking away.' This comparison model asks 'how many more?' or 'how far apart?'

How do you use the Subtraction as Difference formula?

How much taller is a 6-foot person than a 4-foot person? The difference is 2 feet.

What do the symbols mean in the Subtraction as Difference formula?

The −- sign in a difference context reads as 'how far from' rather than 'take away'

Why is the Subtraction as Difference formula important in Math?

Many word problems compare rather than remove, and children who only know 'take away' freeze on them. The difference model also grounds distance on a number line and the meaning of a−ba - b for any two numbers. Recognizing it by "Am I finding the gap between two amounts rather than removing one?" — rather than by familiar numbers — is what lets a student tell it apart from subtraction (take away) and addition and comparison (which is more) in a mixed problem set.

What do students get wrong about Subtraction as Difference?

The procedure for subtraction as difference is the easy part; the trap is adding the two amounts because both are present. Asking "Am I finding the gap between two amounts rather than removing one?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Subtraction as Difference formula?

Before studying the Subtraction as Difference formula, you should understand: subtraction.

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