How Do Addition and Subtraction Work?

Putting Together and Taking Apart

Addition is combining two or more numbers to find their total. If you have 3 apples and get 4 more, you now have 3 + 4 = 7 apples. Subtraction is the opposite — finding what's left when you take some away. If you have 7 apples and eat 2, you have 7 − 2 = 5 left. These two operations are the foundation of all mathematics. Every other operation — multiplication, division, algebra, calculus — builds on them.

Number Lines and Counting

A number line is one of the best tools for understanding both operations. To add, start at the first number and jump right. To subtract, start at the first number and jump left. 5 + 3: start at 5, jump 3 to the right, land on 8. 9 − 4: start at 9, jump 4 to the left, land on 5. Number lines make abstract math visual and concrete.

Regrouping (Carrying and Borrowing)

When adding larger numbers, sometimes a column adds up to 10 or more. In 47 + 35, the ones column gives you 7 + 5 = 12. You write the 2 and carry the 1 to the tens column: 4 + 3 + 1 = 8. Answer: 82. This is called regrouping. Subtraction uses the reverse: in 52 − 28, you can't take 8 from 2, so you borrow 1 ten from the 5 (making it 4) and add 10 to the ones column (making it 12). Now 12 − 8 = 4, and 4 − 2 = 2. Answer: 24.

Mental Math Strategies

Make a ten: To add 8 + 7, think "8 + 2 = 10, and 7 − 2 = 5, so 10 + 5 = 15." Count up for subtraction: For 15 − 9, think "9 + ? = 15" — the answer is 6. Break apart numbers: For 36 + 47, try (36 + 40) + 7 = 76 + 7 = 83. These strategies are faster than counting on fingers and build number sense that helps with every future math topic.

The Relationship Between Addition and Subtraction

Addition and subtraction are inverse operations — they undo each other. If 5 + 3 = 8, then 8 − 3 = 5 and 8 − 5 = 3. This relationship is called a fact family, and understanding it is a breakthrough moment for young learners. It means every addition fact automatically gives you two subtraction facts — and checking subtraction with addition (or vice versa) is one of the best ways to verify your work.

Why Addition and Subtraction Are Foundational

Addition and subtraction are the two operations children use every single day, often without realizing it. Counting change at a store, figuring out how many more pages to read before bedtime, splitting Halloween candy evenly — these are all addition or subtraction in action. These two operations form the bedrock of all mathematics. Every other operation builds on them: multiplication is repeated addition, division is repeated subtraction, and algebra uses them constantly.

Understanding addition and subtraction as related operations — that one undoes the other — is a major cognitive milestone. When a child understands that if 8 + 5 = 13 then 13 − 5 = 8, they've grasped a concept called an inverse relationship that will serve them all the way through calculus.

Where Kids Get Stuck

The most common difficulty is regrouping (sometimes called borrowing or carrying). When subtracting 47 from 83, children must understand that they can "break open" a ten from the tens column to create additional ones. Without base-ten understanding, this procedure feels arbitrary and confusing. Using physical base-ten blocks — trading a tens rod for ten unit cubes — makes the concept concrete.

Another frequent struggle is subtraction across zeros. Problems like 400 − 167 require regrouping from the hundreds place through an empty tens place, and many children either freeze or make errors. Teaching them to think "400 is the same as 399 + 1" or to use a compensation strategy (subtract 167 from 400 by first subtracting 167 from 399 and adding 1) gives them a reliable backup method.

Word problems also create confusion because children must decide which operation to use. The word "more" doesn't always mean add, and "left" doesn't always mean subtract. Teaching children to visualize the situation — draw a picture, act it out, or use objects — is more effective than looking for keyword shortcuts.

Strategies That Build Fluency

Research-backed strategies help children move from counting on fingers to mental math:

• Making ten — To add 8 + 6, think "8 needs 2 more to make 10, take 2 from the 6, so 10 + 4 = 14." This strategy works for any near-ten addition.
• Doubles and near-doubles — 7 + 7 = 14 is easy to remember. So 7 + 8 is just "one more than 14" = 15.
• Counting up for subtraction — Instead of subtracting 83 − 47, count up from 47: "47 to 50 is 3, 50 to 80 is 30, 80 to 83 is 3. So 3 + 30 + 3 = 36."
• Compensation — To add 299 + 48, think "300 + 48 = 348, minus the 1 I added = 347."

Try This at Home

• Dice addition — Roll two dice and add the numbers. For a challenge, use three dice or roll and subtract the smaller from the larger.
• Number bond puzzles — Give your child a target number (say, 20) and ask them to find as many pairs that add up to it as possible.
• Store math — When shopping, ask "If this costs $3.75 and I pay with $5, how much change should I get?"
• Step counting — Count steps on a walk, then subtract different amounts: "We walked 200 steps. If we walk back 85, how many are left?"

If your child is struggling, see our guide: Signs Your Child Is Struggling with Math.

Last reviewed: May 2026