The Area Model
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.
Multiplying a big number is really several small multiplications added together. When you split into and multiply each piece by , you are using the distributive property — you just have not been told its name yet. The area model makes the pieces visible: every part of the rectangle is one partial product, and nothing can be skipped or counted twice.
and . Adding the parts, .
Forgetting a partial product when a number has three or four digits, or lining columns up wrongly when adding them. Write each partial product on its own line, in place-value order, and check that you have one for every pair of digits.
In Plain Terms & More Examples
Break each number into tens and ones, multiply every piece, then add the pieces up — that is the distributive property, . Nothing gets skipped.
and . Add them: .
, which is . Add them: .
A school orders boxes of pencils and boxes of pencils. Step 1: . Step 2: . Step 3: add the parts, pencils in all.
Two-Digit by Two-Digit
With two two-digit numbers the area model has four pieces: tens tens, tens ones, ones tens, ones ones. Add all four.
Why four? Each number splits into two parts, and every part of the first number has to be multiplied by every part of the second: parts times parts makes pieces. In the rectangle they are the four smaller rectangles. Leave one out and the picture has a hole in it.
, , , . Sum: .
Split: and . Multiply every part by every part:
- (, and tens times tens make hundreds)
Add the four pieces: . Estimate to check: is close to , and is close to that.
is about . An answer near is believable; or is not.
Multiplying only tens by tens and ones by ones. For that gives . Two of the four pieces are missing: and . Put them back and .
Split both numbers into tens and ones, write the four partial products, and add.
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, , , . Total .
Word Problems
A story needs multiplication when it has equal groups: rows with the same number of chairs, boxes with the same number of pencils. The words each, every and per are clues. Multiply the number of groups by the size of one group.
Adding eighteen times would work, but multiplication is exactly that repeated addition done in one step. Drawing the groups as the rows of a rectangle turns the story into the area model, and the area model tells you how to split the numbers.
A hall has rows with chairs in each. How many chairs? chairs.
A baker fills trays with muffins each and sells muffins. How many are left?
Step 1, how many were baked: .
Step 2, take away the ones sold: muffins are left.
Adding when the story needs multiplying. For rows of chairs, cannot be right: just two rows already hold chairs. Ask yourself: are there equal groups? Then multiply.
A sticker book has pages with stickers on each page. How many stickers in all?
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stickers.
Bigger Numbers Times One Digit
Split a three- or four-digit number into thousands, hundreds, tens and ones, multiply each part by the one-digit number, and add. The area model just gets more strips.
Why does this work? is . So groups of are groups of , groups of and groups of . Nothing is added or lost when you split; you have only sorted the same things into piles. The picture shows it: the strips side by side make exactly the big rectangle.
Split: . Multiply each part:
- (, and hundreds stay hundreds)
Add: .
Split: . Multiply each part: , , , . Add: .
Split: ; there are no tens. Multiply: and . Add: . Zero tens times is zero, so that strip has no area and is simply left out.
Gluing the partial products side by side instead of adding them. For : , , , written together as . But the is really and the is really , so the pieces are , and , which add to . An estimate catches it: is about .
Find with partial products. Then estimate to check your answer.
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Split: (no tens). Multiply: , , . Add: . Estimate: , very close.
The Standard Algorithm
The column method multiplies one place at a time, starting with the ones. When a place makes or more, write the ones digit and carry the tens to the next column. Add the carry after you multiply that column.
The column method is the area model squeezed into less space. Instead of writing every partial product on its own line, you add them as you go. A carry is regrouping: ones are tens and ones, so the stays in the ones column and the tens wait at the top of the tens column until the tens have been multiplied.
Ones: . Write , carry tens. Tens: tens, plus the carried is tens. Write . Answer: . Check with partial products: .
Ones: ; write , carry . Tens: , plus is ; write , carry . Hundreds: , plus is ; write . Answer: .
Make one row for each digit of .
Row 1: .
Row 2: . Write a in the ones place first, because this row multiplies by tens.
Add the rows: .
Adding the carry before multiplying. In , some people add the carried to the tens digit first: , then , and they write . The carry is tens that have already been multiplied, so it is added to the product , not to the digit. The correct answer is .
Use the column method, then check with partial products.
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Ones: ; write , carry . Tens: , plus is ; write , carry . Hundreds: , plus is . Answer . Check: .
Why It Works
The area model is a proof that the method is correct. A rectangle wide and tall has area . Cut it into a piece and a piece: the two pieces exactly fill the rectangle, so their areas must add up to the whole, . In symbols this is , the distributive property — the same law that will later let you expand .
Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.