Traveling Networks and Pascal's Triangle: Combinatorics and Patterns

Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
Traveling Networks and Pascal's Triangle: Combinatorics and Patterns
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This is a set of activities designed to introduce students to a technique for finding the number of paths on a matrix from corner to another. In the beginning problems, students are permitted to use any technique they like, including the "brute force" method of tracing each and every path. Fortunately, for the first couple of problems, this is too confusing, but as the grid gets larger and larger, there are more and more paths to trace, which can get very confusing.

From here, a new technique is introduced where you can add the two nodes that approach the new node. This makes finding the number of routes on even the largest grid a matter of adding two numbers on a diagonal.

But the fun does NOT stop there! Spozen we took one of these grids with all the routes labeled and turned it 45 degrees: what we end up with is something called "Pascal's Triangle" (even though it was known for hundreds of years before Pascal, btw.) From here we see some beautiful patterns, including powers of 2 (2 squared, 2 cubed, 2 to the fourth power, etc.) as well as the triangular and rectangular numbers.

But the fun does NOT end there either (can you believe it?) Spozen we were now to take the same exact problem and lift it off the plane and into three-dimensional space? Sure, the brute force method can be used to solve the simplest cases (a cube and a double cube), but what will you do when you have a 3 x 3 x 3 solid? Well, we can use the exact same technique as we used with the 2-D networks, only this time we sum up three numbers instead of 2.

Finally, I offer a challenge problem in three dimensions which, when solved correctly, will result i 1,500 different routes!

Fun for all, includes answer keys with multiple explanations and diagrams with instructions on how to use in your classroom.

Another awesome math teaching material from the labs at SamizdatMath!
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25 pages
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