Learn Italian: Hanno fatto uno sconto su questa pezza di lana. - They gave a discount on this bolt of wool.

In Italian, the standard phrase for giving a discount is fare uno sconto (literally "to make a discount"). Using dare here would sound like a direct, unnatural translation from English.

What is the difference between a pezza and a pezzo?

While pezzo is the general word for "piece" (like a piece of pizza or a puzzle piece), pezza specifically refers to textiles. It can mean a patch, a cloth, or, as in this sentence, an entire "bolt" or roll of fabric.

Since this lesson is about the preposition a, why is it di lana rather than a lana?

Because lana (wool) is the material the fabric is made of, which requires the preposition di (or sometimes in). You would use a to describe the fabric's pattern or features, such as una pezza a righe (a striped bolt of cloth) or a quadri (checkered).

How could we use the rate-marking a to talk about buying this wool?

You use a to state both the price and the unit of measurement. For example, you might buy it a venti euro al metro (at twenty euros per meter). The first a introduces the price, and al (a + il) shows the distributive rate.