Chop off the tips of the asparagus and reserve. If the stalks are woody, peel the lower third and chop off the base. Then cut the stalks into large pieces.
Add the potatoes and leek to a large pot of salted boiling water. When you can, with a bit of effort, penetrate a potato with a fork, add the asparagus stalks. Place the tips in a colander or strainer and set over the boiling water.