Explain why the scratch arithmetic algorithm works.
The scratch arithmetic algorithm is a method used to perform addition and subtraction using scratch paper or any writing surface. It is often used in elementary school to teach basic arithmetic operations. Now, let's understand why the scratch arithmetic algorithm works step by step.
First, let's start with addition using the scratch arithmetic algorithm. We will take the example of adding two numbers: 324 + 187.
1. Write down the two numbers, one above the other, with their place values aligned:
324
+ 187
2. Start from the rightmost (ones) place and add the digits in that position (4 + 7 = 11). Write down the sum below the line:
324
+ 187
----
11
3. Now, move to the next place value (tens) and add the digits in that position (2 + 8 = 10). Write down the sum below the line:
324
+ 187
----
1011
4. Finally, move to the leftmost (hundreds) place and add the digits in that position (3 + 1 = 4). Write down the sum below the line:
324
+ 187
----
1011
The final result is 511, which is the sum of 324 and 187.
Now, let's explain why the scratch arithmetic algorithm works for addition:
- The algorithm works because it follows the principles of place value. Each digit in a number has a specific place value, such as ones, tens, hundreds, etc.
- Starting from the rightmost digit, we add the digits in each place value. If the sum is less than 10, we write it as is. If it is greater than 10, we write down the rightmost digit and carry over the leftmost digit to the next place value.
- This process continues until we have added all the digits in each place value column, resulting in the correct sum.
Now, let's briefly explain the scratch arithmetic algorithm for subtraction:
1. Write down the two numbers, one above the other, with their place values aligned.
2. Begin subtracting from the rightmost digit. If the top digit is smaller than the bottom digit, borrow from the next higher place value.
3. Subtract the corresponding digits, and if necessary, subtract the borrowed value from the next higher place value.
4. Continue subtracting, moving from right to left, until all digits have been subtracted.
5. Write down the final result below the line.
The scratch arithmetic algorithm for subtraction works similarly to the addition algorithm but with the additional step of borrowing when necessary.
In conclusion, the scratch arithmetic algorithm works because it follows the principles of place value and ensures that the correct addition or subtraction is performed at each place value column. This method is reliable and easy to understand, especially for beginners learning arithmetic operations.