Cut the corners out of a square to make an octagon. It's pretty easy to see that the corners should be cut off at a 45-degree angle, but how far from the corners should the cuts be made? Algebra holds the answer. The problem is hard enough that it requires a little thought, but requires no special knowledge beyond middle school algebra and the Pythagorean theorem--which is usually a topic of middle school algebra. Once you get the proportions correct and cut off the corners, you can paint the octagon like a stop sign--the most common octagon in the modern world. There is a similar problem with making hexagons from squares.
Minimize the material used to build an object, or conversely, maximize the volume of an object that's constructed from parts cut from a standard sized sheet. For a box, cut out four corners and fold up the sides. For a cylinder, cut out a rectangle that makes a tube and a circle for the bottom. Use algebra to find the dimensions that will give the maximum volumes. Graph the equation for volume and look for the peak of the curve.
Combine resistors into a network of resistors that have a novel value. Two resistors R1 and R2 can be combined to produce new values. Connected in series, the total resistance Rt = R1 + R2. Connected in parallel, 1/Rt = 1/R1 + 1/R2. If you have dozens of resistors, you can combine them in a network to make any resistance desired. Compute the value with algebra and check it by building the network and checking it with an ohmmeter.
Scale up, or down, recipes using algebra. For example, if a recipe makes 48 cookies, a little algebra can tell you how to adjust the recipe to give everyone in your class exactly three cookies. This project is especially interesting if you can actually bake the cookies--as in a home school setting. It's also an excellent time to demonstrate the advantages of the metric system. Metric recipes are much easier to change.