Measure out a generous 1/2 cup of the starter and discard any remaining starter.
Add a scant 1 cup unbleached all-purpose flour, and 1/2 cup water to the 1/2 cup starter.
Mix the starter, flour, and water, cover, and let the mixture rest at room temperature for approximately 12 hours before repeating.
By the end of day 5, the starter should have at least doubled in volume. You'll see lots of bubbles; there may be some little "rivulets" on the surface, full of finer bubbles. Also, the starter should have a tangy aroma — pleasingly acidic, but not overpowering.
If your starter hasn't risen much and isn't showing lots of bubbles, repeat discarding and feeding every 12 hours on day 6, and day 7, if necessary; as long as it takes to create a risen and bubbly starter.
Once the starter is ready, give it one last feeding.
Discard all but a generous 1/2 cup; feed as usual.
Let the starter rest at room temperature for 6 to 8 hours; it should be active, with bubbles breaking the surface.
Remove however much starter you need for your recipe - typically no more than 1 cup.
If your recipe calls for more than 1 cup of starter, give it a couple of feedings without discarding, until you've made enough for your recipe plus 1/2 cup to keep and feed again.
Transfer the remaining 1/2 cup of starter to its permanent home: a crock, jar whatever you'd like to store it in long-term.
Feed this reserved starter with 1 scant cup of flour and 1/2 cup water, and let it rest at room temperature for several hours, to get going, before covering it.
If you're storing the starter in a screw-top jar, screw the top on loosely rather than airtight.
Store this starter in the refrigerator and feed it regularly; I recommend feeding it with a scant 1 cup flour and 1/2 cup water once a week.