As mentioned previously, I find it helpful to have a couple of activities in mind in case someone in the family suddenly asks for a game (even if the request is sometimes signaled indirectly by poking a sibling). Usually, this doesn't mean a full and rich exploration of a topic, but it can be a brief introduction or a continuation of something we've played with before.
Here are two more NRICH activities that fit that:
Jate got to play a little with the chain of changes tonight while I was helping Jin with some homework. I didn't have time to make or print pieces, so instead we used our mini-wipe board. I drew a collection of "allowed" shapes (square, triangle, circle, pentagon) in four colors (red, blue, green, black) and asked him to make a 4 step chain from blue square to black circle. Then I asked if he could make a one-step chain from the blue square to the black circle. Then I asked how many of those he could make. After that, he started adding his own shapes to the selections, so he got to set the tone and practice drawing.
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