Multi-digit multiplication step by step: partial products vs the standard algorithm

Two things make multi-digit multiplication much easier.

• Times tables up to 9 × 9. Every step comes back to a single-digit fact. If 7 × 8 still takes a while, everything else slows down. Our post on times tables practice that sticks has a plan.
• Multiplying by tens. They should know that 40 × 20 is 4 × 2 with two zeros on the end, so 40 × 20 = 800, and be able to say why.

Partial products split each number into tens and ones, multiply every piece by every piece, and add it all up. For 46 × 23, split 46 into 40 and 6, and 23 into 20 and 3.

Draw a box with two columns (40 and 6) and two rows (20 and 3). Fill each square:

• 40 × 20 = 800
• 6 × 20 = 120
• 40 × 3 = 120
• 6 × 3 = 18

Then add the four pieces: 800 + 120 + 120 + 18 = 1058.

The box is a picture of area. A rectangle 46 wide and 23 tall, cut into four smaller rectangles. That is why teachers like it: a child can see that every part of one number gets multiplied by every part of the other.

The standard algorithm does the same multiplying, but packs it into fewer lines. You multiply the top number by one digit of the bottom number at a time.

Start with a one-digit multiplier

Practice with a single digit on the bottom first, like 238 × 4.

1. 1Ones: 8 × 4 = 32. Write 2, carry 3 above the tens.
2. 2Tens: 3 × 4 = 12, plus the carried 3 makes 15. Write 5, carry 1 above the hundreds.
3. 3Hundreds: 2 × 4 = 8, plus the carried 1 makes 9. Write 9.
4. 4The answer is 952.

Then a two-digit multiplier

Now 46 × 23. Multiply 46 by the 3 first: 46 × 3 = 138. That is the first line.

Next, multiply 46 by the 2. But that 2 is in the tens place, so it is really 20. Write a 0 in the ones place of the second line first, as a placeholder. Then 46 × 2 = 92 goes in front of it, giving 920.

Add the two lines: 138 + 920 = 1058. Same answer as the box.

Look at the box again. The bottom row of the box was 40 × 3 and 6 × 3, which is 120 + 18 = 138. That is the first line of the standard algorithm.

The top row was 40 × 20 and 6 × 20, which is 800 + 120 = 920. That is the second line.

So the standard algorithm is partial products with each row of the box already added up. Showing your child this link is one of the most useful things you can do. It turns two separate tricks into one idea.

Once two-digit problems are comfortable, try three digits by two. First line: 247 × 6 = 1482. Second line: 247 × 30 = 7410. Add them: 1482 + 7410 = 8892.

Estimate first: 247 is about 250 and 36 is about 40, and 250 × 40 = 10000. The real answer should be a bit under that, because we rounded both numbers up. 8892 fits.

• Forgetting the placeholder zero. Without it, the second line shifts right and the answer is far too small. For 46 × 23 you get 138 + 92, which is nowhere near the estimate.
• Adding the carry before multiplying. In 238 × 4, the tens step is 3 × 4 then add 3, not (3 + 3) × 4.
• Mixing up carries from the two lines. On paper, cross out the carries from the first line before starting the second, so old carries do not get added in again.
• Adding the partial products carelessly. The multiplying can be perfect and the final addition still slips. It is regular column addition with carrying, so the same care applies.

Whichever one their teacher is asking for right now. Many schools teach partial products first and move to the standard algorithm in fourth or fifth grade. If your child is still on the box method, there is no rush. Understanding why it works is what makes the standard algorithm stick later.

If you want a quick reference, the multi-digit multiplication practice page sums up the steps and mistakes on one screen.

On the MathPad pad, multiplication is laid out in the standard form with a row for each partial product and the placeholder zero already in place. Carries are optional boxes, the way most kids write them small, and Practice mode can turn every real carry into a box the pad points to, with a hint, for kids who keep dropping them.