Basic Puzzles
Construct a simple puzzle for your toddler that will help him learn basic concepts of matching. Find a piece of thick cardboard and two or three large cookie cutters. Trace around the cookie cutters and carefully cut them out with a utility knife, being careful not to tear the surrounding cardboard. Help your child put the cut-out shapes back into their places. Try constructing this puzzle using foam board glued onto the cardboard if you want to add to the longevity of the toy.
Use your toddler's original artwork to create a basic puzzle. Glue one of your toddler's pictures onto a piece of sturdy cardboard. Cut it into two or three large interlocking pieces and help your toddler learn to put the pieces together. Give your child some space so she can try to put it together on her own. Toddlers learn how the pieces fit together using a process of trial and error.
Everyday Object Puzzles
Use common household objects and discarded cereal boxes to make a number of fun puzzles for your toddler. Choose an empty cereal box and cut off the front panel. Use a marker to draw interlocking shapes on the cardboard. Cut these into pieces and allow your toddler to put the cereal box puzzle together. Use larger pieces for younger children and cut them into smaller pieces as children grow older. Due to the wide variety of cereal boxes available in stores, you can create an extensive puzzle collection. Store pieces in a Ziploc bag to keep them together when not in use.
Create a paper plate pizza puzzle to foster imaginary play and to teach beginning fractions. Give your child a white paper plate and allow him to decorate it to look like a pizza. Cut the plate into four to eight pizza slices (triangular wedges). Help your child take the pizza apart and serve it before putting it back together again. Show your child what 1/2 or 1/4 of a pizza looks like. While your toddler might not understand the mathematical reasoning behind fractions, this can be a good way to start to concretely show these types of numeric concepts.