Multi-digit Multiplication
Multiply large numbers (like 34 × 27) step by step using place value to break the work into smaller, easier pieces.
Reading is good — doing is better. Practice Multi-digit multiplication as an interactive lesson.
Practice freeDefinition
Multi-digit multiplication means multiplying numbers that each have more than one digit. Instead of doing it all at once, we use place value to split the problem into smaller multiplications and then add the results together. The most common method taught in 4th grade is the standard algorithm, where you multiply the top number by each digit of the bottom number one at a time, remembering to shift one place to the left for each new row.
Remember the rule
Multiply → Shift → Add. Multiply the top number by each digit of the bottom number (ones first, then tens), shift each new row one place left, then add all the rows together.
Key words
- Digit
- A single number symbol: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. The number 34 has two digits.
- Factor
- A number you are multiplying. In 34 × 27, both 34 and 27 are factors.
- Product
- The answer you get after multiplying. 34 × 27 = 918, so 918 is the product.
- Place Value
- The value of a digit based on where it sits in a number. In 27, the 2 is in the tens place (worth 20) and the 7 is in the ones place (worth 7).
- Regrouping (Carrying)
- When a multiplication or addition gives you 10 or more, you move the extra amount to the next place to the left. Also called carrying.
- Partial Product
- One of the smaller answers you get before adding everything together. In the area model, each box holds a partial product.
- Standard Algorithm
- The step-by-step written method most people learn for multiplication, working digit by digit from right to left.
- Area Model
- A rectangle drawn on paper and split into sections to show partial products visually — great for understanding why the algorithm works.
Worked examples
32 × 3
→ 96. Multiply 3 × 2 = 6 (ones), then 3 × 3 = 9 (tens). Write 96. · This is one-digit times two-digit — a warm-up to show the idea of working place by place.
45 × 20
→ 900. Multiply 45 × 2 = 90, then attach a zero because you are really multiplying by 20 (two tens). Answer: 900. · Multiplying by a multiple of 10 always adds a zero at the end of your answer.
34 × 27
→ 918. Step 1: 34 × 7 = 238. Step 2: 34 × 20 = 680 (write 34 × 2 = 68, then add a zero to get 680). Step 3: 238 + 680 = 918. · The second partial product (680) is shifted one place left because the 2 in 27 really means 20, not 2.
56 × 43
→ 2,408. Step 1: 56 × 3 = 168. Step 2: 56 × 40 = 2,240. Step 3: 168 + 2,240 = 2,408. · Check by estimating: 60 × 40 = 2,400 — very close, so 2,408 is reasonable.
125 × 4
→ 500. Multiply 4 × 5 = 20 (write 0, carry 2). 4 × 2 = 8 + 2 carried = 10 (write 0, carry 1). 4 × 1 = 1 + 1 carried = 2. Answer: 500. · Regrouping (carrying) is important when any digit multiplication gives you 10 or more.
213 × 32
→ 6,816. Step 1: 213 × 2 = 426. Step 2: 213 × 30 = 6,390. Step 3: 426 + 6,390 = 6,816. · Even with three-digit numbers, the same Multiply → Shift → Add pattern works perfectly.
Common mistakes
- Forgetting to shift the second row one place to the left. If you multiply by the tens digit but do not shift, your partial product is 10 times too small.
- Skipping the zero placeholder when multiplying by a tens digit (writing 68 instead of 680 for 34 × 20).
- Regrouping errors — forgetting to add the carried number after multiplying, or adding it before multiplying.
- Adding the partial products incorrectly at the end. Take time to line up the digits carefully by place value before adding.
- Mixing up the order and multiplying the tens digit first. Always start with the ones digit on the bottom number to keep things consistent.
FAQs
Why do we shift the second row over when using the standard algorithm?
Because the digit you are multiplying by is in the tens place, which means it is really worth 10 times more. Shifting left one space is the same as multiplying by 10, so the place value stays correct.
What is an easy way to check if our answer makes sense?
Round each factor to the nearest ten and multiply those rounded numbers in your head. If your answer is close to that estimate, you are probably right. For example, 34 × 27 is close to 30 × 30 = 900, and our answer 918 is close to 900, so it looks good.
Is the area model the same as the standard algorithm? Which one should we use?
They give the same answer but look different. The area model shows each partial product in a box, making it easier to see why the method works. The standard algorithm is faster once you understand the concept. Many teachers want kids to know both.
What do we do when there is a zero in the middle of a number, like 203 × 4?
Multiply the zero just like any other digit: 4 × 0 = 0. Write the 0 in that spot and keep going. Do not skip it or the digits will end up in the wrong places.
My child keeps getting the wrong answer but shows the right steps — what is happening?
The most common culprits are small addition mistakes when adding partial products, or forgetting to add a carried number. Have your child slow down and double-check each carry mark by writing it small above the next column.
Does multiplication order matter? Is 34 × 27 the same as 27 × 34?
Yes, the answer is identical either way (both equal 918). This is called the Commutative Property of Multiplication. Sometimes flipping the numbers makes the work slightly easier.
Want the whole picture for your child?
Every K–6 subject, an AI tutor that teaches step by step, unlimited practice, and a reward world.
Start a 3-day free trial