Each Math Research Project outline consists of a *Seed Question*, *Questions to Build On*, *Extensions*, and *Basic Background*. Click here to see a list of available Math Research Project outlines.

The *Seed Question *is a question that appears in a typical math curriculum. *Questions to Build On* are simple extensions, re-interpretations, and generalizations of the* Seed Question *that a student can build a simple math research paper around, with help from a mentor (be it student or teacher).

The *Extension Questions *are bigger, more challenging extensions of the *Seed Question*. If a student can make some headway into understanding these, great! If not, they are good questions to include as part of the “Where do we go from here?” section of the paper.

The *Basic Background *consists of some of the mathematical leg-work that underlies the investigation. It can be worked on simultaneously with the investigation, which gives it context.

### Grid Walking

How many distinct paths are there from (0,0) to (5,3), if movement is only allowed “North” and “East” along the lines y = 0, y = 1, y = 2, and y = 3, and x = 0, x = 1, x = 2, …, x =5?

How many distinct paths are there from (0,0) to (5,4)?

How many distinct paths are there from (0,0) to (6,3)?

How many distinct paths are there from (0,0) to (*m*,n)?

How many distinct paths are there from (0,0) to (5,4) that pass through (2,2)?

**Extension Questions**

How many distinct paths are there from (0,0,0) to (5,4,3)?

How many distinct paths are there from (0,0) to (5,4) if in addition to moving “North” or “East”, you are allowed at most one move “West”?

How many distinct paths are there from (0,0) to (5,4) if in addition to moving “North” or “East”, you are allowed to move along diagonal line segments?

How many distinct paths are there from (0,0) to (5,5) if you are not allowed to cross the line y = x?

**Basic Background**

Derive the formulas for combinations.

Explore the basic features of recursion and some recursive sequences.

Demonstrate some relationships between combinations and Pascal’s triangle.

*Click here to see a list of available Math Research Project outlines.*