The Written Method: Your Step-by-Step Guide to Mastering Calculations
Core Principles of the Written Method
At its heart, the written method is about organization and process. It transforms a potentially confusing jumble of numbers into a clear, logical sequence. This approach is built on several key principles that ensure accuracy and build mathematical fluency.
First and foremost is the concept of place value. Understanding that the digit 5 in 57 represents 5 tens, not just 5 ones, is fundamental. Written methods like column addition and subtraction rely entirely on this principle. Second is the principle of alignment. Properly aligning numbers by their place value units (ones under ones, tens under tens, etc.) is critical for accurate computation. A misaligned column can lead to a completely incorrect answer. Finally, there is the principle of systematic recording. Each operation, whether it's carrying over a value in addition or borrowing in subtraction, must be clearly written down. This creates a transparent audit trail of your thought process.
Essential Written Methods for Arithmetic
The journey of mastering written methods begins with the four basic operations. Each has its own standard algorithm that, once learned, becomes a reliable tool for life.
1. Column Addition
This is the standard method for adding multi-digit numbers. You write the numbers one on top of the other, ensuring the place values are perfectly aligned in columns. You then add the digits in each column, starting from the rightmost column (ones). If the sum of a column is 10 or more, you carry over the tens digit to the next column on the left.
Example: Add 347 + 286.
| Step | Description | Written Calculation |
|---|---|---|
| 1 | Write the numbers, aligning by place value. | 3 4 7 + 2 8 6 -------- |
| 2 | Add the ones column: 7 + 6 = 13. Write the 3 and carry the 1 to the tens column. | 1 3 4 7 + 2 8 6 -------- 3 |
| 3 | Add the tens column: 1 + 4 + 8 = 13. Write the 3 and carry the 1 to the hundreds column. | 11 3 4 7 + 2 8 6 -------- 3 3 |
| 4 | Add the hundreds column: 1 + 3 + 2 = 6. Write the 6. | 11 3 4 7 + 2 8 6 -------- 6 3 3 |
2. Column Subtraction
Similar to addition, numbers are aligned in columns. The key concept here is borrowing or regrouping. When a digit in the minuend (top number) is smaller than the digit in the subtrahend (bottom number), you borrow 1 from the next higher place value column.
Example: Subtract 512 - 267.
| Step | Description | Written Calculation |
|---|---|---|
| 1 | Write the numbers, aligning by place value. | 5 1 2 - 2 6 7 -------- |
| 2 | In the ones column, 2 is less than 7. Borrow 1 from the tens column. The 1 becomes 0, and the 2 becomes 12. Subtract: 12 - 7 = 5. | 4012 5 - 2 6 7 -------- 5 |
| 3 | In the tens column, 0 is less than 6. Borrow 1 from the hundreds column. The 5 becomes 4, and the 0 becomes 10. Subtract: 10 - 6 = 4. | 40 - 2 6 7 -------- 4 5 |
| 4 | Subtract the hundreds column: 4 - 2 = 2. | 40 - 2 6 7 -------- 2 4 5 |
3. Long Multiplication
This method extends the concept of column addition. You multiply the top number by each digit of the bottom number separately, shifting your results one place to the left each time, and then add all the partial products together.
Example: Multiply 47 x 36.
4 7
× 3 6
--------
2 8 2 (This is 47 x 6)
+1 4 1 0 (This is 47 x 30, notice the zero in the ones place)
--------
1 6 9 2
4. Long Division
This is the most complex of the basic algorithms. It involves a cycle of four steps: Divide, Multiply, Subtract, Bring Down. You divide the divisor into parts of the dividend, step by step.
Example: Divide 672 ÷ 4.
1 6 8
4 ) 6 7 2
-4
--
2 7
-2 4
--
3 2
-3 2
--
0
The process: 4 goes into 6 one time (Divide). Write 1 above the bar. 1 x 4 = 4 (Multiply). 6 - 4 = 2 (Subtract). Bring down the 7 (Bring Down). Repeat the cycle with 27, then with 32.
Applying Written Methods to Algebra and Geometry
As students progress into middle and high school, the principles of the written method extend beyond arithmetic. They form the backbone of solving algebraic equations and geometric problems.
In algebra, solving for an unknown variable, like x, is a written method. The steps are shown clearly to isolate the variable.
Example: Solve for x in 2x + 5 = 13.
- Write the original equation: $2x + 5 = 13$
- Subtract 5 from both sides (the subtraction is shown): $2x + 5 - 5 = 13 - 5$ → $2x = 8$
- Divide both sides by 2: $\frac{2x}{2} = \frac{8}{2}$
- State the solution: $x = 4$
In geometry, calculating the area of a composite shape (a shape made from other simple shapes) requires a written plan. You might break the shape into a rectangle and a triangle, calculate each area using the formulas $A_{rectangle} = l \times w$ and $A_{triangle} = \frac{1}{2} b \times h$, and then add the results together. Writing down each step ensures you don't miss a component and allows for easy verification.
Common Mistakes and Important Questions
Q: Why is my answer wrong even though I followed the steps?
The most common culprits are simple computational errors or misalignment. Double-check your basic facts (like 7 x 8). Ensure all numbers are correctly aligned by their place value. Also, verify that you carried over or borrowed correctly in every step where it was needed. A single small mistake in one column will throw off the entire result.
Q: In the age of calculators, why do we still need to learn written methods?
Written methods build number sense and a deep conceptual understanding of how numbers work. They are not just about getting an answer; they are about understanding the why behind the answer. This foundational knowledge is essential for estimating answers to check if a calculator's result is reasonable, for understanding more advanced math, and for developing logical problem-solving skills that apply far beyond mathematics.
Q: I keep forgetting the steps for long division. Is there a trick to remembering?
Yes! Use a mnemonic device. A very popular one is "Dad, Mom, Sister, Brother" which stands for Divide, Multiply, Subtract, Bring Down. Reciting this in your head as you work through each digit of the dividend can help you maintain the correct order of operations.
The written method is far more than a classroom exercise; it is a fundamental pillar of mathematical literacy. From the simple column addition learned in elementary school to the multi-step algebraic proofs of high school, these methods provide a reliable, transparent, and logical framework for problem-solving. They foster discipline, enhance understanding, and create a tangible record of one's reasoning. Mastering these techniques empowers students to approach complex calculations with confidence, ensuring they have the tools not just to find the right answer, but to comprehend the journey they took to get there.
Footnote
[1] Algorithm: A step-by-step procedure for solving a problem or accomplishing a task. In mathematics, it is a set of defined rules for performing a calculation.
[2] Minuend: The number from which another number (the subtrahend) is to be subtracted. In a - b, a is the minuend.
[3] Subtrahend: The number that is to be subtracted from another (the minuend). In a - b, b is the subtrahend.
[4] Number Sense: A person's general understanding of numbers and operations, and the ability to develop useful strategies for solving mathematical problems.