「the number of」と「a number of」の違い

We had an interesting discussion in class the other day. Somebody had said something like:

× The number of people was doing something.
○ A number of people were doing something.

The person brought up a good point, though.

"Isn't 'number' usually used with a singular noun?" he asked.

He was right that we often use "number" that way, but "number" has a number of meanings.

A number of [plural noun] [plural verb]. <- "number" means "several"
The number of [plural noun] [singular verb]. <- "number" means "quantity"

• the number of ～の数 (definition from Eijiro on the Web)
• a number of 多数の、たくさんの、多くの、いくつかの、いくらかの、複数の、若干の (definition from Eijiro on the Web)

Since "a number" means "several", it's plural, so if you use it as the subject, you need to use a plural verb. "The number" means one quantity, so it should be followed by a singular verb.

A number of leaves on this plant are a bit strange, like this one. The number isn't so large, but I'm still a little worried about it. I don't have a green thumb, so I'm not so sure what to do about it.

• green thumb 園芸｛えんげい｝［植物栽培｛しょくぶつ さいばい｝］の才◆【語源】親指が緑色になるくらいに植物をいじっているところから。(definition from Eijiro on the Web)

Do any of the gardners out there have any advice? I'm sure a number of you kind readers are better with plants than me.