The half square triangle unit is probably the most commonly used patch in a quilt block -- well it is second only to the square. It is made simply with two triangles, filling up half the square each -- hence the name. Take a quick look through this website, or in any quilting magazine, and you will start to notice that this half square triangle, regardless of its size, is present in more quilts than not.
Star on Star
Free Quilt Block Pattern
The Star on Star quilt block is a wonderful (and easy) block to introduce the three-quarter square triangle patch. There are three triangles that make up the square in this quilt patch, but the name is a little misleading (at least to me) in that it seems to suggest that these triangles are all actually quarter square triangles, when in fact it is comprised of only 2 quarter square triangles as well as a half square triangle.
Now all that being said, this really is a simple block to construct, even though it sounds complicated from my description. This block would be a great teaching tool for a beginner class.
The sample quilts illustrated give you some ideas on how to set the finished quilt blocks, but I am sure you can come up with several interesting layouts of your own.