3.) Smart Decomposition (Using Nearby Numbers)
This strategy builds directly on the standard multiplication algorithm. But it uses it more flexibly.
In elementary school, we’re taught to break numbers into place value:
631 × 933 → (600 + 30 + 1)(900 + 30 + 3)
Then apply the distributive property.
But those place value splits are just one option, but they’re not required.
You can break numbers apart in any way that makes the calculation easier.
Sometimes, it’s more efficient to use subtraction instead of addition.
Example:
49 × 7
Instead of:
- (40 + 9) × 7
Try:
- (50 − 1) × 7
Now compute:
- 50 × 7 = 350
- 350 − 7 = 343
Subtracting a small number is often easier than adding several larger ones.
Another example:
98 × 36
Instead of breaking 98 into (90 + 8), use a nearby number:
98 = 100 − 2
Now compute:
(100 − 2) × 36 = 100 × 36 − 2 × 36
- 100 × 36 = 3600
- 2 × 36 = 72
So:
3600 − 72 = 3528
This avoids dealing with less convenient numbers and keeps the calculation clean.
This strategy:
- reduces the number of steps
- keeps numbers “clean”
- lowers the load on working memory
Look for numbers that are:
- close to multiples of 10, 50, or 100
- just above or below a “friendly” number
Examples:
- 99 → 100 − 1
- 48 → 50 − 2
- 19 → 20 − 1
This is really the distributive property in disguise, but used strategically.
Instead of blindly expanding, you reshape the numbers first, then apply the structure in a way that minimizes effort.
The standard algorithm breaks numbers apart by place value. Smart decomposition breaks them apart by convenience.