Learning about basic shapes is something that students do in kindergarten, but at the second-grade level, they should be starting to recognize the particular and subtle differences between many shapes. Focus on one shape each day, and make it the center of this activity. If today is "triangle day," teach your child about the three kinds of triangles: isosceles, equilateral and scalene. Throughout the day, and during free time, encourage your student to find examples of all three kinds in everyday life, be it around the house, in a magazine or in nature. Use this activity to have a conversation about how geometry is all around us, even if we don't always recognize it.
For this activity, take advantage of the fact that you have access to an entire house. Give your child a series of counting tasks involving household objects: you could ask him to count the number of stairs in the house, apples in the fridge or tiles in the bathroom. Ask him to record his numbers on a piece of paper. Now, use this data to illustrate addition and subtraction. For example, ask him to calculate stairs plus apples. If he cannot do it in his head or on paper, show him that all he has to do is count the stairs again, and keep counting once he moves on to apples.
If your child is more advanced in addition and subtraction, give him a "quick minute" math quiz every day. This involves a test sheet with a large number -- at least 30 -- of simple arithmetic problems on it. Give your child one minute to complete as many as possible, warning him in advance that he should not expect to complete them all. Because your child is being home schooled, he is will not be subject to pressure from other students, allowing the only competition to be with himself. Every day, challenge him to beat his own record.
In a classroom setting, this fun activity might devolve into chaos, but in the quiet of your home, it can work very well. Together, choose a fairy tale that your child likes, and rewrite it by inserting numbers. If you choose "Goldilocks and the Three Bears," for example, insert a specific temperature when the story states how hot or cold the porridge is. This activity should get your child thinking about numbers in a creative way.