This is the most carefully and completely documented use of the "The Handshake Problem" (also known as the "Glass Clink Problem") that you can find. It spans 5 days worth of lessons, homework assignments, enrichment and an assessment.
The beauty of the "handshake problem" is that it can be looked at as a way to represent a problem visually through the use of points and connecting edges, but it can be investigated as a sequence of triangular numbers. This is then be compared to other types of sequences, including square and rectangular numbers.
There is also an investigation into the use of "Gaussien Sums," which is method developed by Karl Friedrich Gauss to add up a series of numbers. The unit also goes into graphing triangular, rectangular and square numbers, as well as arithmetic and geometric progressions.
And yes, this is all accomplished in 5 lessons - yes, 5 lessons! Don't believe me? Purchase this and see for yourself.
This curriculum represents over 25 years of development and refinement. My colleagues and I used this problem as the first unit when teaching 7th grade at the Bank Street School for Children in New York City. Our goal was to asses the skills and learning styles of our students by incorporating content that was relevant and fun. We began by thinking about how we could turn our first class meeting into a learning opportunity, which would allow us to move forward, instead of focusing on the review of low level computational skills, as is found in many of those cruddy textbooks you’re probably using (I’m looking at you, Houghton-Mifflin-McGraw-Hill-Heath....)
This was originally used with 7th graders who were going to be studying basic algebra in a problem solving environment, but I have adapted it for grades 5 - 8 in order to focus on more essential concepts like visual modeling, problem solving with patterns, as well as detours into mathematical progressions, graphing parabolas, using substitution in algebraic equations, as well as organizing data in tables and, if that is not enough, Gaussien sums!
What you’ll find is extensive instructions about each lesson, an activity sheet for that lesson, as well as a homework assignment that follows up on that lesson. You’ll also notice that there are two versions of this activity, because it can be described in two ways: as “handshakes,” or, as I like to do it, “glasses clinking.” Either way, the activities are exactly the same.
I think you’re going to love this curriculum: it is only 5 days, covers a lot of mathematical territory in interesting and unique ways, and will give your students an actual feel for what mathematics is all about.