Multi-digit multiplication
The facts you know stop at ten times ten. How do you multiply numbers that have several digits each?
Where you are 3 inside
What this area is about
Every large product of whole numbers is built from small ones. Split the factors by place value, multiply the parts using basic facts and shifts of place, and add the pieces. The distributive property is what guarantees the pieces add up to the whole product.
How these fit together
The building block. Multiplying by tens and hundreds gets products such as 40 \times 600 from a basic fact, here 4 \times 6 = 24, and a shift of places.
The method that shows why. Partial products splits the factors by place value, multiplies every pair of parts, as the pieces of an area model, and adds the results.
The short record. The standard multiplication algorithm does the same work with one row for each digit of the bottom factor, carrying into the next column as it goes.
A path through it
Multiplying by tens and hundreds and partial products come first, in either order; the standard algorithm comes last, since each of its rows is a shortened partial product. All of it starts from the multiplication facts.
Finish here and you are ready for long division and for estimating by rounding to check that a product makes sense.