Building New Algorithms from Existing Ones

Most algorithms are not invented from nothing. They are assembled from smaller algorithms whose behavior is already known and trusted. If you have a reliable max finder, you already trust that it checks every number in a list and returns the largest. Starting there saves work: you don't have to reinvent the logic for scanning a list, and you don't introduce new bugs into a part that already works. You only have to think hard about the new part.

Combination can mean chaining pieces together or modifying one for a new purpose. A max finder plus a swap, repeated, becomes selection sort: find the largest remaining value, swap it into place, repeat on a shrinking list. A sum algorithm becomes an average algorithm by adding one new step, a count of how many numbers were added, then dividing the sum by that count at the end. A max finder plus a MOD test becomes a largest-even-number finder: check each number with MOD 2 to filter for evens, then run the max logic on just those.

This matters because Big Idea 3 is built on the idea that variables, loops, and procedures are reusable building blocks, not one-off tricks. Once an algorithm is trusted, it becomes a unit you can call on inside something bigger, the same way a procedure call hides its own steps. But assembly is not automatic safety: the old pieces being correct does not mean the new combination is correct, which is why testing comes back into the picture even for proven parts.