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would you suggest buying to use with Singapore? Are there any that I should absolutely have on hand?

We'll be doing LHFHG this fall.

we have centimeter cubes that have come in very very handy. Other than that we've just used household stuff.... but then again we are only in 2A

You don't have to purchase anything for manipulatives in particular, Carrie has ideas for what to use, sometimes snack items that the kiddo's get to eat! Sometimes blocks, legos, cotton balls, cars, spoons, forks, napkins,Teddy bears, sticks, marbles. I use whatever I can find, substitute and make due. You get more creative as the time goes on.
I happen to have some 'Click-blocks' and 'Counting Bears' that someone gave me and they have come in handy, but they are not a must at all.
That is what I love about HOD it is so flexible and teaches us Mom's to think outside the box, if we don't already utilize that skills much.
Success and hope this helps.