I use Essential Math Kindergarten (Singapore Math). To be honest, my 4 year old is flying through it. There isn't really much of anything in it she hasn't picked up through play yet. I'll tell you what the "A" book covers:
Same, Different, and Sets (Most 4 and 5 year old can sort things pretty well and tell you what is different in a picture.)
Sets- Just grouping things that are similar together. They do this with their toys daily.
Counting to 5, Numbers to 5. - Being able to count to 5, identify number 1 through 5, and write the number 1 through 5. - Somehow, my daughter also picked this up on her own.
Numbers to 10- the same as through 5.
Number Order- Connect the dots, writing numbers in order.
Shapes- They pick this up on their own, somehow, too.
Patterns- bear, doll, bear, doll, bear... what comes next? Kids pick this up on their own, too. Give them a little practice and they are pros.
Length- Using one object to measure another. (Ex. This pencil is how many crayons long?) You can do this without a book, and mine seemed to pick it up very quickly.
Size- Big, small. Ordering things by size. They know this already, you'll see them doing it with stuffed animals and cars all the time, anyway.
Weight- Heavy vs. light. Looking at a balance scale and seeing which is heavier. You can do this on a much more practical level without a book.
Capacity- Which object can hold more? You can do this at home, and probably already do with sand, water, beans, whatever.
Equal sets- If you have 4 people, and 6 pieces of chicken, are there an equal number of chicken pieces for each person? No. If you have 4 people and 8 ears of corn, are there an equal number of ears for each person? Yes. See. Easy without a book and most kids know this already.
More and Less- There are 5 apples and 2 bananas. Which group has more? Which group has less? Kids already know this, too.
A book might help you know what they "should" be covering, but the kids already know most of what is in the book. And what is in the book is just as easily taught without the book. Let them help in the kitchen, let them play, and they'll pick up just as much. Of course, my kids happen to LOVE worksheets, so the book is something they enjoy. But it isn't even close to being necessary.