Describe why the Scratch Addition Algorithm works?
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The Scratch Addition Algorithm is a method used to add two multi-digit numbers by carrying over numbers from one place value to another. It works because it is based on the principles of place value and regrouping.
To better understand why it works, let's break down the algorithm into steps:
Step 1: Write down the two numbers to be added vertically, aligning them according to their place values (ones, tens, hundreds, and so on).
For example, let's add 345 and 278:
345
+ 278
Step 2: Start from the rightmost column (ones place) and add the digits in that column. In this case, 5 + 8 equals 13. Since 13 is a two-digit number, we write down the ones digit (3) in the ones place and carry over the tens digit (1) to the next column.
345
+ 278
------
3
Step 3: Move to the next column (tens place) and add the digits in that column along with the carried-over digit. In this case, 4 + 7 + 1 (carried-over) equals 12. Write down the ones digit (2) in the tens place and carry over the tens digit (1) to the next column.
345
+ 278
------
23
Step 4: Continue this process for each column, adding the digits along with any carried-over digits. In this case, 3 + 2 + 1 (carried-over) equals 6, which we write down in the hundreds place.
345
+ 278
------
623
Step 5: Finally, if there are no more columns to add, you have the sum of the two numbers.
The Scratch Addition Algorithm works because it combines the individual place values, taking into account any carried-over digits. It follows the basic principles of arithmetic, such as adding digits and carrying over when the sum exceeds 9. By breaking down the addition into smaller steps, it makes the process easier to understand and perform mentally or on paper.