Multi-Digit Multiplication

Study Sheet

Multi-Digit Multiplication

Breaking numbers apart to multiply

The Area Model

Concept
The Big Idea
each piece is a partial product

Split each number into its places and multiply the parts, then add. Drawing it as a rectangle shows why nothing gets missed: the pieces fill the whole area.

Example
Finding 23×423 \times 4

20×4=8020 \times 4 = 80 and 3×4=123 \times 4 = 12. Adding the parts, 80+12=9280 + 12 = 92.

In Plain Terms & More Examples

Concept
In plain terms

Break each number into tens and ones, multiply every piece, then add the pieces up — that is the distributive property, a(b+c)=ab+aca(b+c) = ab + ac. Nothing gets skipped.

Example
Finding 34×634 \times 6

30×6=18030 \times 6 = 180 and 4×6=244 \times 6 = 24. Add them: 180+24=204180 + 24 = 204.

Example
Finding 52×752 \times 7

52×7=(50+2)×7,52 \times 7 = (50+2) \times 7, which equals, which is 7×50+7×2.7 \times 50 + 7 \times 2. Add them: 350+14=364350 + 14 = 364.

Example
A multi-step problem

A school orders 33 boxes of 2424 pencils and 22 boxes of 3636 pencils. Step 1: 3×24=723 \times 24 = 72. Step 2: 2×36=722 \times 36 = 72. Step 3: add the parts, 72+72=14472 + 72 = 144 pencils in all.