What is the Vein In This Granite?

I - was trying to explain to my children some geology today. When perhaps I should not have. Because I actually know virtually nothing.

And that became obvious.

I was talking to them about these granite rocks. I thought they were granite.

Explaining that they came from a cooling of a molten mass. That the crystals within the mass were the result of slow cooling.

Now these rocks occasionally had quartz looking veins running through them. Sometimes as much as three inches or so thick.

Now that stuff seemed to have larger crystals than the surrounding stuff I've labelled as 'granite'.

So I belatedly realised I wasn't making sense. Any molten material running into cracks in some other material - which is what those veins are, right? - has to cool fairly quickly, surely? Because surrounded by so much mass of cooler stuff.

But if it cooled quicker than the surrounding stuff it should be finer grained. Right?

Can anyone guess at what I was probably looking at?

The 'granite' was disintegrating all over the place. The ground carpeted in a coarse 'sand' of particles from it. Is that normal with granite?

And how do you get larger crystals in a vein than in the surrounding rock?