Y.3 Add a one-digit number to a two-digit number - without regrouping
What is addition?
Addition is a mathematical operation that combines two or more numbers into a single total, called the sum. When we add, we are putting groups together or counting how many we have in all.
- If you have 3 apples and get 2 more apples, you add 3 + 2 to find you now have 5 apples.
- If there are 4 children on the playground and 1 more child joins, there are 4 + 1 = 5 children in all.
The plus sign (+) tells us to add numbers together. The equals sign (=) means "is the same as" or "the total is."
Understanding place value: tens and ones
Place value tells us what each digit in a number means, based on its position. In a two-digit number, the digit on the left is in the tens place and tells how many groups of ten we have. The digit on the right is in the ones place and tells how many single ones we have.
- The number 24 has 2 tens and 4 ones. We can show this as 20 + 4.
- The number 56 means 5 groups of ten and 6 extra ones, or 50 + 6.
- We can use base-ten blocks: a "rod" or "long" represents a ten, and a "unit" or "cube" represents a one.
Knowing place value is like knowing that 2 dimes and 4 pennies are worth 24 cents. The dimes are the tens, and the pennies are the ones.
What does "without regrouping" mean?
In addition, "without regrouping" means that when we add the digits in each place value column, the sum in the ones place is 9 or less. This sum does not create a new group of ten that needs to be carried over, or "regrouped," to the tens place. The tens place value stays the same.
- Adding 13 + 5: The ones digits are 3 and 5. 3 + 5 = 8, which is less than 10. No regrouping is needed. The sum is 18.
- Adding 42 + 7: The ones digits are 2 and 7. 2 + 7 = 9. This sum does not make a new ten, so we do not regroup. The sum is 49.
You can think of "no regrouping" as a simple case where everything stays in its own place. The ones stay in the ones column, and the tens stay in the tens column.
The step-by-step process for adding without regrouping
To correctly add a one-digit number to a two-digit number without regrouping, follow a clear, two-step process focused on place value.
- Identify the tens and ones: Look at the two-digit number. Write it as tens and ones. For example, 36 is 3 tens and 6 ones.
- Add the ones only: Take the ones digit from the two-digit number and add it to the one-digit number. Keep the tens digit the same.
- Write the total: Combine the original tens with the new sum of the ones to make the final two-digit answer.
- Step 1: 24 has 2 tens and 4 ones. We are adding 3 ones.
- Step 2: Add the ones: 4 ones + 3 ones = 7 ones.
- Step 3: Keep the 2 tens. Combine: 2 tens and 7 ones = 27.
- Answer: 24 + 3 = 27.
Always start with the ones place. The tens place only changes if we regroup, and in this lesson, it never does.
Using models to understand the process
We can use physical or drawn models to make the addition process visible and concrete. These tools help us see why the tens digit stays the same.
Let's solve 31 + 8 using blocks.
- Show 31: Draw 3 rods (tens) and 1 unit (one).
- Show +8: Draw 8 more units.
- Combine the units: Move the 1 unit and the 8 units together. Count them: 1 + 8 = 9 units.
- Look at all the blocks: You still have 3 rods and now 9 units.
- The model shows 3 tens and 9 ones, which is 39.
Let's solve 45 + 4 using a number line.
- Start at the larger number, 45, on the number line.
- The one-digit number is 4, so we make 4 jumps to the right (forward).
- Jump 1: 46, Jump 2: 47, Jump 3: 48, Jump 4: 49.
- We land on 49. 45 + 4 = 49.
Models prove that we are only increasing the ones. The tens remain untouched because we never make enough ones to trade for a new ten.
Writing the equation vertically and horizontally
Addition problems can be written in two main ways: horizontally (across) and vertically (up and down). It is important to set up vertical equations correctly so the place values line up.
- The problem is written in a line: 17 + 2 = 19.
- We solve it by thinking: "17 has 1 ten and 7 ones. 7 ones + 2 ones = 9 ones. 1 ten and 9 ones is 19."
- The problem is written with the numbers stacked by place value.
- Always write the two-digit number on top. Write the one-digit number underneath it, making sure it is in the ones column. Draw a line below.
- Add the column of ones (7 + 2 = 9). Write the 9 below the line in the ones column.
- Bring down the ten from the top number (1) into the tens place of the answer.
17
+ 2
----
19
In vertical addition, lining up the ones digits is the most important rule. If they are not lined up, you will add the wrong place values and get an incorrect answer.
Recognizing when regrouping is not needed
A key skill is looking at a problem and knowing immediately that it is a "without regrouping" problem. This saves time and helps you choose the correct strategy.
Look only at the ones digit of the two-digit number and the one-digit number you are adding. Add them together in your head. If the sum is 9 or less, you will not regroup.
- Example 1: 23 + 4 → Look at 3 and 4. 3 + 4 = 7. 7 is less than 10. No regrouping.
- Example 2: 50 + 9 → Look at 0 and 9. 0 + 9 = 9. 9 is less than 10. No regrouping.
- Example 3: 68 + 1 → Look at 8 and 1. 8 + 1 = 9. No regrouping.
- 28 + 5 → Look at 8 and 5. 8 + 5 = 13. 13 is 10 or more. This WOULD require regrouping (which is not part of this lesson).
If the ones digit of the two-digit number is small (like 0, 1, 2, 3) or the one-digit number is small, you likely will not regroup. But always do the quick mental check to be sure.
Applying the skill in word problems
Word problems require us to find the mathematics hidden in a story. We must read carefully, decide what operation to use, and then solve using our skills.
- Read: Read the entire problem slowly.
- Find: Find the numbers you need to work with.
- Question: Find the question being asked. What do you need to find out?
- Plan: Decide if you are putting groups together (addition).
- Solve: Write the equation and solve it using the without-regrouping method.
- Check: See if your answer makes sense with the story.
- Problem 1: Sam has 25 crayons. He finds 3 more under his desk. How many crayons does he have now?
Think: He starts with 25 and gets more, so we add. 25 + 3.
Solve: 5 ones + 3 ones = 8 ones. Keep the 2 tens. Answer: 28 crayons. - Problem 2: There are 41 students on the bus. At the next stop, 2 students get on. How many students are on the bus now?
Think: There were 41, then more joined, so we add. 41 + 2.
Solve: 1 one + 2 ones = 3 ones. Keep the 4 tens. Answer: 43 students.
Words like "more," "join," "find," "in all," and "total" often tell us to add. The numbers in the problem will tell you if regrouping is needed.
Checking your work for accuracy
Good mathematicians always check their answers. Checking helps you find and fix mistakes, leading to greater confidence and accuracy.
- Use the inverse operation: Subtraction is the opposite of addition. To check 32 + 6 = 38, subtract the one-digit number you added from your answer: 38 - 6. You should get back to the original two-digit number, 32.
- Use a model: Draw a quick sketch of base-ten blocks for your answer. For 32 + 6 = 38, draw 3 rods and 8 units. Circle 2 of the units and the 6 extra units you added. See that they make the 8 units. Verify you have 3 rods.
- Estimate: Ask if your answer is reasonable. If you add a small number (like 1-4) to a two-digit number, the tens digit should not change. If your answer for 27 + 2 was 49, you would know immediately that a ten changed incorrectly.
You solved: 54 + 5 = 59.
Check with subtraction: 59 - 5 = 54. ✓ The answer is correct.
Getting into the habit of checking your work is one of the most powerful tools you can have in mathematics. It turns "being done" into "being confident."
Building a foundation for future math
Mastering addition without regrouping is not an isolated skill. It is the essential first step in a sequence of learning that builds all the way to advanced mathematics.
- Next Step - Regrouping: Once you are an expert at adding when the ones sum is 9 or less, you are ready to learn what happens when the ones sum is 10 or more. You will learn to "carry the ten."
- Larger Numbers: The same place-value rules apply to adding hundreds and thousands. You will always add the ones column first, then the tens, then the hundreds.
- Algebra Readiness: The habit of combining like terms (ones with ones, tens with tens) directly relates to combining variables in algebra (x with x, y with y).
- Real-Life Applications: You use this skill when counting money (adding coins to dollars), measuring (adding inches to a foot), telling time, and much more.
Think of math as building a tower. This lesson is a strong, solid block at the bottom. Every block you place on top depends on this one being perfectly straight and secure.
Common Core alignment: CCSS.MATH.CONTENT.1.NBT.C.4 – Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.1.NBT.C.4. Use it for whole-class instruction, independent practice, or homework.
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