Most people don’t want to just “wing” math without a curriculum, although you certainly can do this–at least until the whole numbers and order of operations are mastered. There are tons of options available. Personally, I love Ray’s Arithmetic series. You can buy the hardcover versions or you can download the books via Internet Archive. Practical Arithmetic is the one for the upper elementary age child. Intellectual Arithmetic is for kids a little bit younger. If you have a younger child who really wants to begin math work beyond just playing with actual objects, which is what is recommended in Ray’s, you can use Primary Arithmetic as a guide for that. Most families won’t get into Higher Arithmetic, but it’s useful to know that it goes into a little extra detail beyond Practical and is a great refresher before tackling Algebra. If your child is a late bloomer (meaning puberty hasn’t set in by the time you finish most other arithmetic programs), Higher Arithmetic is a great option. The main down side of using these books is that they are OLD. You will likely have to explain some vocabulary, you will also likely have to write problems out on paper or on the board for your child to work through, and the books won’t lie flat on the table. Even with these detractors, this is a solid math program. If the idea of having to write out problems and work so many word problems is just too daunting for you, a good substitute for these texts, with updated vocabulary, can be found in the Abeka worktexts for grades 4 through 7. I never recommend Abeka’s higher math programs, but their arithmetic ones are quite good. You don’t need anything but the worktexts!
If your student needs additional practice in some area of math, try Math Mammoth’s Blue Series to plug whatever holes you see. She has also taken the concepts found in the Blue Series and reordered them into grade level versions of her work. Another wonderful supplement to any math program is the Math Card Games set from RightStart. I love RightStart for the younger kids. I began using this way back before they had leveled curriculum available. It’s even easier to implement nowadays, but it does take quite a bit of time each day. As far as manipulatives go, you can manage with household items but the AL Abacus from RightStart and the blocks from Math-U-See are some of the best things since sliced bread. When thinking of manipulatives, remember that the younger the child, the more concrete the manipulative needs to be. Once they can actually sort cookies onto plates, then they can just THINK about sorting cookies onto plates, which is more abstract. Finally, they will be able to do this sort of thing with just the numbers that represent actual objects.
For higher math, I like Foerster’s Algebra I and Algebra II (it really does get a student ready for calculus). For geometry, I like the work of Larson, Boswell, and Stiff. These three joined up with a fourth author to write a Middle School Math program in the early 2000s that is wonderful as well, which is a secular text that was created before the advent of Common Core. I especially like how this series explains the notetaking process for math, which isn’t always intuitive. Many of the above texts are now out of print but are readily available through used curriculum sellers, including Amazon. Sometimes you will be able to find the corresponding teacher’s manuals; other times, you won’t be as lucky. If this is a problem for you–and it certainly was for me since I’m only fairly competent up to the middle of Algebra–you will need to go with one of the more homeschool-friendly options (of these, my favorite choice is VideoText) or hire a tutor if you do not have a math person in the family who is willing and able to teach these courses. It’s worth your time to figure out how you are going to help your student learn these important subjects because if they don’t and they go to college, they will end up having to take the prerequisite courses there, paying full college tuition, but not earning college credit, and having to take the course at the college pacing, which is twice the speed of a high school course. As stated on the page that deals with the mistakes you will make, this one can be costly, but it’s not the end of the world if it happens to your child. Some kids just won’t “get” higher math until they are older. I definitely understood Algebra much better when I taught it to my older sons than I did in high school or college–my graphing skills still leave a lot to be desired though!
I have much more to say about math but the above three paragraphs contain my all-time favorite things. There are lots of good things out there for doing math. Remember to consider the age of the child, his or her specific needs, and how they think! The other thing to remember is that there is no real reason for it to take 6 to 8 years to learn everything there is to know about arithmetic–except keeping publishing companies in business, that is. The higher mathematics topics of Algebra, Geometry, and Calculus don’t need to be subdivided into so many courses either. They can be, but they don’t HAVE to be. Just think about that for a bit. I’ll be back with more to add to this page. Promise!