Path Counting: Count Every A-to-B Grid Route Without Drawing Them All
You trace routes that may only step right or down, then reveal how each intersection stores the number of ways to reach it. It trains you to turn an exploding list of paths into a calm, additive counting rule.
Drawing Works on a 2×2 — Then a 4×4 Grid Buries Every List
On a 2×2 grid you can sketch all six routes and feel confident. Push to a 4×4 and the same hand-drawing quietly skips one path here and double-counts another there, with no way to tell which. This game breaks the wall by letting you tap each node and watch its count appear as the sum of the node above and the node to its left. Suddenly the impossible list becomes a grid of small additions marching toward B. Seeing Pascal's triangle emerge from your own moves is what turns guesswork into systematic counting.
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
- Follow the movement rule: only right and down paths are allowed.
- Build node counts by adding the value from the left and from above.
- Connect the picture to combinations by arranging right and down steps.
Three Moves to Count Paths: Rule, Nodes, Total
- Check the grid size and remember that moves can only go right or down.
- Think about which previous nodes can reach the current node.
- Add left and above node counts until the B node is reached.
- Choose the total and reveal the grid numbers to check the reasoning.
Five-Minute Teacher Routine: Let the Nodes Speak Before the Formula
Start with a 2x2 or 3x3 grid and draw all paths, then fill each node with the number of ways to reach it. The formula will feel less abrupt.
Path Counting Questions Teachers Hear
Q: Why not let students draw every path?
A: Drawing is helpful for tiny grids, but larger grids grow too quickly. Node recursion is the stable counting method.
Q: When should I introduce the combination formula?
A: Once students can explain the number of right moves and down moves as an arrangement, C(n, k) becomes meaningful.