I.3 Multiply 2-digit numbers by 1-digit numbers
What does multiplying a 2-digit number by a 1-digit number mean?
Multiplying a 2-digit number by a 1-digit number means combining equal groups. A 2-digit number has a tens place and a ones place (for example, the number 34 has 3 tens and 4 ones). A 1-digit number has only a ones place (for example, the number 7). When we multiply them, we find the total amount, or product, of those equal groups. This operation is a key building block for understanding larger multiplication problems and real-world math, such as calculating the total cost of multiple items or the total number of objects in several equal sets.
Imagine you have 4 shelves. On each shelf, there are 23 books. To find the total number of books, you would multiply the number of shelves (4) by the number of books per shelf (23). The answer is the total number of books.
Remember, the numbers we multiply are called factors, and the answer is called the product. In 4 × 23 = 92, both 4 and 23 are factors, and 92 is the product.
Understanding place value in 2-digit numbers
Place value is the value of a digit based on its position in a number. In a 2-digit number, the digit on the left is in the tens place, and the digit on the right is in the ones place. This means that the number 57 is actually 5 tens (which is 50) and 7 ones (which is 7). Breaking a number apart by place value is the first step to understanding how multiplication works.
- In the number 42, the digit 4 means 4 tens (40), and the digit 2 means 2 ones (2). So, 42 = 40 + 2.
- In the number 81, the digit 8 means 8 tens (80), and the digit 1 means 1 one (1). So, 81 = 80 + 1.
Understanding that 2-digit numbers are just a sum of tens and ones will help you use the distributive property when you multiply.
Multiplying using the distributive property
The distributive property is a math rule that lets you break apart a difficult multiplication problem into smaller, easier parts. To multiply a 2-digit number by a 1-digit number, you can break the 2-digit number into its tens and ones, multiply each part by the 1-digit number, and then add the two products together.
Step 1: Break 37 into tens and ones: 30 + 7.
Step 2: Multiply each part by 5: (5 × 30) + (5 × 7).
Step 3: Calculate the smaller multiplications: 5 × 30 = 150, and 5 × 7 = 35.
Step 4: Add the two products: 150 + 35 = 185.
So, 5 × 37 = 185.
This strategy works for any 2-digit by 1-digit multiplication problem and helps you understand why the standard method works.
The standard multiplication algorithm (no regrouping)
The standard algorithm is a step-by-step way to multiply. We write the problem vertically, lining up the numbers by place value. We always multiply the 1-digit number by the digit in the ones place first, then by the digit in the tens place. When the product in a place is less than 10, we do not need to regroup.
Step 1: Multiply the ones: 3 × 2 = 6. Write the 6 in the ones place of the answer.
Step 2: Multiply the tens: 3 × 3 = 9. Write the 9 in the tens place of the answer.
Since 9 tens is 90, and 6 ones is 6, the total is 96.
Always start with the ones place. This keeps your work organized and prevents mistakes.
The standard multiplication algorithm (with regrouping)
Sometimes, when you multiply a digit, the product is 10 or greater. When this happens, you must regroup, or carry, the extra tens to the next place value. This is just like addition. You write the ones digit in the answer line and add the tens digit to your next multiplication.
Step 1: Multiply the ones: 4 × 7 = 28. 28 has 2 tens and 8 ones. Write the 8 in the ones place of the answer. Regroup the 2 tens above the tens place (write a small 2 above the 5).
Step 2: Multiply the tens: 4 × 5 = 20. Then, add the regrouped 2: 20 + 2 = 22. Write the 22 next to the 8 in the answer. Since 22 represents 22 tens, which is 220, the total product is 228.
Don't forget to add the regrouped number! A common mistake is to forget to add the tens you carried over.
Multiplying when the 1-digit number is in the tens place
You might also see problems where the 2-digit number is on the bottom. The math works exactly the same way. You still multiply the 1-digit number by each digit of the 2-digit number, starting with the ones place. The order of the factors does not change the product.
Step 1: 6 × 2 = 12. Write 2 in the ones place, regroup the 1 ten.
Step 2: 6 × 4 = 24. Add the regrouped 1: 24 + 1 = 25. Write 25. The product is 252.
Multiplication is commutative, meaning 6 × 42 gives the same product as 42 × 6. You can choose whichever arrangement is easier for you to solve.
Multiplying with larger 2-digit numbers and regrouping twice
Sometimes you will need to regroup more than once. For example, when you multiply the tens place, the product plus the regrouped number might be 10 or greater, requiring you to regroup again into the hundreds place. This is normal and shows you are working with larger numbers.
Step 1: 7 × 9 = 63. Write 3 in the ones place. Regroup the 6 tens (write the small 6 above the 8).
Step 2: 7 × 8 = 56. Add the regrouped 6: 56 + 6 = 62. Write 62. The product is 623.
Notice that 62 represents 62 tens, which is 620, plus the 3 ones makes 623.
When you have a regrouped number, always add it right after you multiply. This keeps your work accurate.
Multiplying numbers with a zero in the ones place
When the 2-digit number has a zero in the ones place (like 20, 30, 40, etc.), the multiplication is straightforward. Any number multiplied by zero in the ones place gives zero, which often means you don't have to regroup at that step.
Step 1: 4 × 0 = 0. Write 0 in the ones place.
Step 2: 4 × 6 = 24. Write 24. The product is 240.
You can also think of this as 4 × 6 tens = 24 tens, which is 240.
Multiplying by a multiple of ten is easy because you can multiply the non-zero digits and then add a zero to the end of the product.
Checking your work with estimation
Estimation is a quick way to see if your answer is reasonable. Before you multiply exactly, you can round the 2-digit number to the nearest ten and multiply by the 1-digit number. The answer you get should be close to your actual product. If it is very different, you might have made a mistake.
Estimate: Round 48 to the nearest ten, which is 50. Then multiply 50 × 3 = 150.
Actual problem:
144 is close to 150, so our answer is reasonable.
Estimation does not give you the exact answer, but it is a great tool to check if your final answer makes sense.
Solving word problems with multiplication
Word problems describe a real-life situation. To solve them, you must read carefully to find the numbers you need and decide what operation to use. Look for clue words like "total," "each," "every," "in all," or "per," which often mean you should multiply.
A school ordered 24 new boxes of markers. Each box contains 8 markers. How many markers did the school order in total?
Step 1: Identify the factors: 24 boxes and 8 markers per box.
Step 2: Set up the multiplication: 24 × 8.
Step 3: Solve:
Step 4: Write the answer with the unit: The school ordered 192 markers in total.
Always label your final answer in a word problem. This shows that you understand what the numbers represent.
Common mistakes to avoid
Even experienced mathematicians make small errors. Knowing the common mistakes can help you avoid them. Always double-check your place value, your regrouping, and your addition.
- Mistake: Forgetting to add the regrouped number. Fix: Write the regrouped number clearly above the tens place and circle it as a reminder to add it after multiplying.
- Mistake: Multiplying the tens place first. Fix: Always start with the ones place to keep your work in the correct order.
- Mistake: Lining up the numbers incorrectly. Fix: Use grid paper or write neatly so the place values are aligned vertically.
- Mistake: Making errors in basic multiplication facts. Fix: Practice your times tables regularly so they become automatic.
Mistakes are learning opportunities. When you find a mistake, figure out why it happened so you won't make it again.
Common Core alignment: CCSS.MATH.CONTENT.4.NBT.B.5 – Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.NBT.B.5. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework. The lesson emphasizes place value understanding, the distributive property, and the standard algorithm with and without regrouping. It is designed to build conceptual understanding before procedural fluency.