Mathematics is a branch of logic, and anything that builds logic skills builds math skills. Tangrams are puzzles that have been around forever, and they utilize general logic and geometry. I have a book that has reprints of very old Chinese puzzles. The seven shapes are cut from a square, and it seems simple to make various patterns out of them, but it's not. I bought a set of plastic tangrams with a puzzle book, which the kids really enjoyed, but there's no reason why you can't make them out of thin foam sheets, as I did when we used them in a class that I taught to a group of homeschoolers. Here is a pattern, and there are lots of online puzzle sheets to print out, or make your own. There are also plenty of online interactive tangrams, but that doesn't have the same kinetic learning component as the actual physical things. Some of the Chinese tangrams use a more complicated puzzle set, and are also lots of fun.
Another simple thing to make are sets of attribute blocks. Here is a video of a teacher showing how she uses them in her classroom. They are also simple to make from four colors of foam sheets. The four attributes commonly seen are shape, size, color, and thickness -- for this last one you'd have to glue two sheets together or buy the thicker stuff. They are used in Venn diagram activities and pattern-string activities -- check the picture below. Again, you can make your own. If you have a small dry erase board, you can draw patterns or make sets and have your child model them, or make her own with the same type of pattern but different attributes. Just play around with them for a while -- ideas will come.
And speaking of pattern strings, if you've got some chunky beads and some yarn, or some colored paper clips, it's the same as the expensive colored links that are sold as a math manipulative. Make a pattern that is an addition series (2 reds, 4 yellows, 6 blues, etc.) or a multiplication series (1, 4, 9, 16, etc.) and then have your child guess the pattern -- then have your child make one and you guess. Or you can just use the beads to teach with arrays, like this web site discusses.
Another thing we love are pattern blocks. These are just fun! I don't know if they would be as successful made from foam -- the painted wood is very satisfying to play with, and my kids have spent hours with them.
One more free hands-on idea: Wrap-Ups are for practicing your arithmetic skills -- check the picture to see what they look like. Each "key" focuses on a different group of arithmetic facts. The one I've done here is "times four" -- the numbers that are multiplied are on the left, and the products are scrambled up on the right. You take the string and start at the number on the top left and wrap it to the answer on the right, then go to the second on the left, and it's product, and on down. Then you turn it over and on the back there is are raised grooves for where the string should go, and if your string matches, you've got them all correct. These could be made easily from cardboard, or even better, foam core board -- notched ten times on each side. You'd just have to wrap the string correctly, then turn it over and draw in the pattern the string should make.
So, with all the money you've saved on those you can now afford to buy some things I think are essential (and would be difficult to make): Cuisenaire rods, Base Ten Number Concept Sets, and a balance like in the Weigh To Learn Math kit. All of these are described and there are photos in the Fun Math class and Cool Math class links under Homeschool Resources, and they're available at Rainbow Resource.
And a word on why this is important. Too many children are pushed too quickly past using manipulatives, which support the concrete thinking stage that young children rely on. Teachers seem to think it's a waste of time, since what is tested is that they can write a 7 in the answer box if asked what 3 plus 4 is. But children who don't spend time with manipulatives often end up with a shaky sense of number.We need to step back from having them memorize which symbols go with other symbols (3+4=7) and even step back from *** + **** = *******. Manipulatives help with understanding quantities, but there's a deeper, much-neglected stage, in which a child ponders on the properties of numbers themselves. Four has vastly different properites than three (the first composite number, the second square number) but all most children learn is that it's next in line after three. When a child plays around with math manipulatives, he comes to grasp that certain numbers can do certain things, like be divided in halves or thirds, or make perfect squares, or build three dimensional shapes. This is the basis for a solid future in math (and the tool that helps the most with this, in my opinion is Cuisenaire rods).
You can see my math power points and class transcripts for more on this, but here's a favorite quote:
“Mathematics, like
music, needs to be expressed in
physical actions and human
interactions before its symbols can evoke
the silent patterns of mathematical ideas
(like musical notes), simultaneous
relationships (like harmonies) and expositions or proofs (like melodies).” Richard Skemp, The Psychology of Learning Mathematics
