Add and Subtract Whole Numbers Accurately - ISEE Lower Level Mathematics Achievement

SECTION 1

Why Do We Add and Subtract?

People have been adding and subtracting for thousands of years! Long ago, farmers needed to count how many animals they had. Shopkeepers needed to figure out how much change to give back.

Addition means putting groups together to find a total. Subtraction means taking away or finding the difference between two amounts. These two operations are the building blocks of all math!

3000 BC • Ancient Counting

People in Egypt and Mesopotamia used marks on clay tablets to add and subtract goods like grain and cattle.

500 AD • The Abacus Spreads

The abacus became a popular tool in Asia and Europe. It uses sliding beads to help people add and subtract quickly.

1200 AD • Written Algorithms

Mathematicians began writing numbers in columns. This made it much easier to add and subtract large numbers on paper.

Today • ISEE Test Day

You use these same skills every day — and on the ISEE! No calculator allowed, so your brain is the best tool you have.

On the ISEE, you will see addition and subtraction problems with numbers that can be pretty big. The key question is: How do we line up digits and carry or borrow correctly every time? Let's find out!

SECTION 2

Core Principles of Addition & Subtraction

Before we dive into big problems, let's learn four important ideas. These rules will keep you on track every time!

Place Value Matters

Always line up numbers by their place value: ones under ones, tens under tens, hundreds under hundreds. This keeps everything neat.

Regrouping (Carrying)

When digits in a column add up to 10 or more, you carry the extra to the next column on the left. Think of it like trading 10 ones for 1 ten.

Borrowing (Renaming)

When the top digit is smaller than the bottom digit in subtraction, you borrow 1 from the column to the left. That gives you 10 more in the current column.

Check Your Work

You can check addition by subtracting, and subtraction by adding. On the ISEE, this trick catches careless mistakes!

✦

KEY TAKEAWAY

Think of place value like a parking garage. Cars (digits) need to park on the right floor. Ones go on floor 1, tens on floor 2, and hundreds on floor 3. If you park a car on the wrong floor, the whole building gets confused! Always line up your columns carefully before you start.

SECTION 3

See How Addition Works

Let's look at how we add 367 + 485 step by step. Watch how carrying works in the diagram below!

Start from the ones column on the right. When a column adds up to 10 or more, write the ones digit and carry the 1 to the next column. The answer is 852!

Notice how we worked from right to left. Always start with the ones column first! That way, you know if you need to carry before you add the next column.

SECTION 4

The Rules for Adding and Subtracting

Here are the simple rules you need to remember. They work every single time!

ADDITION RULE

Add each column from right to left. If the sum is 10 or more, write the ones digit and carry 1.

Example: In the ones column, 7 + 5 = 12. Write 2 and carry 1 to the tens column.

SUBTRACTION RULE

Subtract each column from right to left. If the top digit is smaller, borrow 1 from the column to the left.

Borrowing turns the left column's digit down by 1 and gives you 10 more in the current column.

CHECK YOUR ANSWER

Addition check: Sum − One Addend = Other Addend. Subtraction check: Difference + Subtrahend = Minuend.

If 367 + 485 = 852, then 852 − 485 should equal 367. If it does, you got it right!

💡

ISEE Test Tip

On the ISEE, answer choices are listed from least to greatest. If your answer is not one of the choices, you made a mistake somewhere. Go back and check your carrying or borrowing!

SECTION 5

See How Subtraction with Borrowing Works

Now let's see subtraction in action! We'll solve 704 − 258. This one is tricky because it has a zero — but you've got this!

When you need to borrow and the next column is 0, keep going left until you find a column that is not 0. Then borrow through the zeros. The answer is 446!

⚠

Watch Out for Zeros!

Zeros in subtraction can be tricky. If you need to borrow from a column that has a 0, you must keep moving left until you find a digit you can borrow from. This is called borrowing across zeros. It shows up often on the ISEE!

SECTION 6

Worked Example: Step by Step

Let's solve a full ISEE-style problem together. Follow along with each step!

Problem: A school library had 1,253 books. The school donated 478 books to another school. Then 165 new books arrived. How many books does the library have now?

Solving the Library Problem

1
Step 1 — Understand the Problem

We start with 1,253 books. We subtract 478 (donated books) and then add 165 (new books).
2
Step 2 — Subtract 1,253 − 478

Line up the numbers by place value. Ones: 3 − 8. We can't do that, so borrow from tens. Tens: 5 becomes 4, and ones becomes 13. Now 13 − 8 = 5. Tens: 4 − 7. Borrow from hundreds. Hundreds: 2 becomes 1, tens becomes 14. Now 14 − 7 = 7. Hundreds: 1 − 4. Borrow from thousands. Thousands: 1 becomes 0, hundreds becomes 11. Now 11 − 4 = 7.

1,253 − 478 = 775
3
Step 3 — Add 775 + 165

Line up the numbers. Ones: 5 + 5 = 10. Write 0, carry 1. Tens: 7 + 6 + 1 = 14. Write 4, carry 1. Hundreds: 7 + 1 + 1 = 9. Write 9.

775 + 165 = 940
4
Step 4 — Check Your Answer

To check subtraction: 775 + 478 should equal 1,253. Let's see: 5 + 8 = 13, write 3 carry 1. 7 + 7 + 1 = 15, write 5 carry 1. 7 + 4 + 1 = 12, write 2 carry 1. Write 1. That gives 1,253. ✓ It checks out!

The library now has 940 books.

SECTION 7

Common Mistakes & How to Avoid Them

Even great math students make mistakes sometimes. Here are the most common errors — and how to beat them!

Mistake	Why It Happens	How to Fix It
Forgetting to carry	You rush and skip writing the carried 1 above the next column.	Always write the small 1 above the next column. Don't keep it in your head!
Forgetting to reduce after borrowing	You borrow 1 from the left column but forget to subtract 1 from that column.	Cross out the old digit and write the new smaller digit right away.
Misaligned columns	Digits are not lined up by place value, so you add or subtract the wrong digits.	Use the scratch space on the test. Write numbers neatly in columns.
Borrowing across zeros incorrectly	When you hit a 0, you stop instead of continuing to borrow from the next column.	Keep moving left until you find a non-zero digit. Then borrow step by step back to the right.

✦

KEY TAKEAWAY

Think of checking your work like proofreading a story. You wouldn't turn in an essay without reading it once more, right? Always take a few seconds to check your answer by doing the opposite operation. It's the fastest way to catch mistakes on the ISEE!

SECTION 8

Building Toward Bigger Math

Adding and subtracting whole numbers is the foundation for almost everything else in math. Once you master these skills, bigger topics become much easier!

What You Know Now	What Comes Next
Adding whole numbers with carrying	Adding decimals (like $12.75 + $8.50)
Subtracting whole numbers with borrowing	Subtracting fractions and mixed numbers
Multi-step word problems	Solving missing-number equations like ? + 45 = 120
Checking work with the opposite operation	Using estimation to check if your answer makes sense

On the ISEE, you might also see problems where you add or subtract to find perimeter, work with money, or solve data problems using bar graphs and tables. Your addition and subtraction skills are the key to all of those!

SECTION 9

Practice Problems

Time to practice! Try each problem on scratch paper before looking at the answer. Remember: no calculator, just like on the real ISEE!

Question 1: CONCEPTUAL

When you add 56 + 37 and the ones column gives you 13, what do you do next?

A. Write 13 in the ones column
B. Write 3 in the ones column and carry 1 to the tens column
C. Write 1 in the ones column and carry 3 to the tens column
D. Start over from the tens column

Show answer (B)

B is correct. When a column adds up to 13, you write the ones digit (3) in that column and carry the tens digit (1) to the next column on the left. Choice A is wrong because you can't fit a two-digit number in one column. Choice C mixes up which digit to write and which to carry. Choice D makes no sense — you always move left, never start over.

Question 2: BASIC CALCULATION

What is 2,847 + 1,365?

A. 3,102
B. 4,112
C. 4,212
D. 4,312

Show answer (C)

C is correct. Ones: 7 + 5 = 12. Write 2, carry 1. Tens: 4 + 6 + 1 = 11. Write 1, carry 1. Hundreds: 8 + 3 + 1 = 12. Write 2, carry 1. Thousands: 2 + 1 + 1 = 4. The answer is 4,212. Choice A (3,102) results from forgetting to carry. Choice B (4,112) comes from a carrying error in the hundreds place. Choice D (4,312) comes from adding incorrectly in the hundreds column.

Question 3: INTERMEDIATE

What is 5,003 − 1,748?

A. 3,255
B. 3,265
C. 3,355
D. 4,255

Show answer (A)

A is correct. This problem has two zeros, so you must borrow across them. Follow the borrowing chain step by step. Step 1: The ones column needs to subtract 8 from 3, which is impossible, so you need to borrow — but tens is 0. Step 2: The tens column needs to borrow — but hundreds is also 0. Step 3: Borrow from thousands: thousands goes from 5 to 4, and hundreds goes from 0 to 10. Step 4: Hundreds lends 1 to tens: hundreds goes from 10 to 9, and tens goes from 0 to 10. Step 5: Tens lends 1 to ones: tens goes from 10 to 9, and ones goes from 3 to 13. Now subtract column by column: ones 13 − 8 = 5, tens 9 − 4 = 5, hundreds 9 − 7 = 2, thousands 4 − 1 = 3. Answer: 3,255. Choice B (3,265) has an error in the tens column. Choice C (3,355) has an error in the hundreds column. Choice D (4,255) forgot to reduce the thousands after borrowing.

Question 4: APPLIED

Maria had $1,500 in her savings. She spent $387 on a bike and $245 on gear. How much money does she have left?

A. $768
B. $868
C. $968
D. $1,113

Show answer (B)

B is correct. First, find the total spent: $387 + $245 = $632. Then subtract from $1,500: $1,500 − $632 = $868. Choice A ($768) subtracts incorrectly in the tens place during the final step. Choice C ($968) makes a carrying error during the addition step. Choice D ($1,113) only subtracts the cost of the bike ($1,500 − $387 = $1,113) and forgets to subtract the cost of the gear.

Question 5: CRITICAL THINKING

A number plus 2,479 equals 6,102. What is the number?

A. 3,623
B. 3,633
C. 4,623
D. 8,581

Show answer (A)

A is correct. If ? + 2,479 = 6,102, then ? = 6,102 − 2,479. Ones: 2 − 9 is impossible. The tens digit is 0, so go to hundreds. Hundreds (1) becomes 0 and lends to tens: tens (0) becomes 10, then lends to ones: tens becomes 9, ones becomes 12. Ones: 12 − 9 = 3. Tens: 9 − 7 = 2. Hundreds: 0 − 4 is impossible. Borrow from thousands: thousands (6) becomes 5, hundreds becomes 10. Hundreds: 10 − 4 = 6. Thousands: 5 − 2 = 3. Answer: 3,623. Choice B (3,633) has an error in the tens subtraction. Choice C (4,623) forgets to borrow from thousands. Choice D (8,581) added instead of subtracting.

SUMMARY

What You've Learned

Summary

You learned how to add whole numbers by lining up digits by place value and carrying when a column adds up to 10 or more. You also learned how to subtract whole numbers by borrowing when the top digit is smaller than the bottom digit — even when you have to borrow across zeros.

Remember these ISEE tips: always start from the right (ones column), write neatly in columns, and check your work using the opposite operation. If your answer doesn't match any of the choices, go back and look for a carrying or borrowing mistake. You've got this — great job!