Path Counting: Count Every A-to-B Grid Route Without Drawing Them All

Drawing Works on a 2×2 — Then a 4×4 Grid Buries Every List

Count right-and-down paths from the top-left A to bottom-right B on fresh grid sizes with optional obstacles. Answers come from the path recurrence, and a new task changes the actual diagram.

Right-and-Down Route Counting Builds Grade 5-6 Combinatorial Thinking

Path Counting suits Grades 5-6 enrichment, where learners already extend numerical patterns (5.OA.B.3) and read simple expressions (6.EE.A.2). Here those skills fuse into the addition principle: each node's count is built from two smaller counts, the same recursion behind Pascal's triangle. That prepares students for organizing outcomes with tables and tree diagrams (7.SP.C.8) and, later, the combination formula C(n, k).

Three Combination Skills Practiced on the Grid

• Count routes that move only right or down
• Assign 1 to the start and 0 to blocked cells, then add counts from above and left
• Use dynamic counting to compare the effect of obstacles

Three Moves to Count Paths: Rule, Nodes, Total

Five-Minute Teacher Routine: Let the Nodes Speak Before the Formula

Mark blocked cells as 0 and work from the start in row or column order. A binomial count checks an open rectangle; obstacles require subtracting blocked routes or using the recurrence.