Multiplying 3-digit numbers

What is this skill?

Fifth graders are expected to multiply larger whole numbers smoothly: three digits by two digits, such as 346 × 27. Nothing new is being invented here. It is the same place-value splitting your child used in fourth grade, now with more pieces. Each digit of the second number multiplies the whole first number, and the results are added up.

For 346 × 27, the 7 ones give 346 × 7 = 2,422. The 2 tens are really 20, so they give 346 × 20 = 6,920. Added together, 2,422 + 6,920 = 9,342. The standard method stacks these two partial products one above the other and carries digits along the way. A rectangle cut into boxes (an area model) shows the same pieces side by side. Most classrooms start with the boxes and finish with the stacked method once the idea is solid.

Why it matters

This is the last time children are asked to multiply whole numbers by hand in elementary school, and the skills built here get reused straight away. Decimal multiplication is whole-number multiplication with a decimal point put back afterward. Multiplying fractions needs fast products. Volume multiplies three numbers. A child who still stumbles on carrying will feel every one of those topics as hard for the wrong reason.

It also trains a habit that matters more than any single answer: estimating first. If 346 × 27 is close to 350 × 30, the answer should be somewhere around ten thousand. A result of 934 or 93,420 is then easy to spot as wrong without anyone telling the child so.

Common mistakes

• Forgetting the placeholder zero. When multiplying by the tens digit, the answer is really in tens, so the ones place of that row stays empty. Skipping it makes the second row ten times too small.
• Multiplying only the digits that line up, instead of every digit of one number by every digit of the other.
• Carrying a digit and then forgetting to add it, or adding it before multiplying instead of after.
• Lining numbers up unevenly, so columns of ones and tens end up mixed together. Graph paper fixes this faster than any lecture.
• Losing track in a number with a zero inside it, like 508. The zero still holds a place and still has to be handled.

How to help at home

• Ask for an estimate before the exact answer. Rounding each number to a friendly one (346 to 350, 27 to 30) takes ten seconds and catches most slips.
• Let your child draw the box model on scrap paper: split each number into hundreds, tens and ones, multiply each box, then add. It is slower, but it shows why the stacked method works.
• Use real totals: 24 rows of 36 seats, 28 days of 15 minutes of reading, 52 weeks of 14 pages.
• When an answer is wrong, point to the estimate and ask whether the two agree, rather than just saying it is wrong. The check is a skill your child gets to keep.
• Keep sessions short. Four or five careful problems teach more than twenty rushed ones.

Worked examples

These are real questions from the Tutorsaurus multiplying 3-digit numbers practice, with the answers worked out.

1. Example 1

   60 × 18 = ?

   Answer: 1080. 60 × 10 = 600 and 60 × 8 = 480. Add: 600 + 480 = 1080.

2. Example 2

   926 × 27 = ?

   Answer: 25002. 926 × 20 = 18520 and 926 × 7 = 6482. Add: 18520 + 6482 = 25002.

3. Example 3

   586 × 47 = ?

   Answer: 27542. 586 × 40 = 23440 and 586 × 7 = 4102. Add: 23440 + 4102 = 27542.