Reasoning · Grade 4-2 Counting Shortest Routes
Problem
Shortest means only 5 rights and 4 ups.
Shortest means never walking backwards, so all the walking is bookkeeping: 5 steps right and 4 steps up, whatever order they come in.
4.OA.A.3Draw A DiagramCut the walk into two halves at the bookstore.
A compulsory stop is a doorway that every route has to pass through, so counting before the doorway and after the doorway covers everything exactly once.
4.OA.A.3Identify SubproblemsCutting the walk in two at the bookstore lets the two halves be counted separately and multiplied.
How you reached the bookstore puts no limit on how you leave it, so every first half can be joined to every second half.
Every route passes the bookstore exactly once, so cutting there names each route by exactly one pair of halves.
Home to bookstore gives 20 routes.
You can only walk into a corner from the left or from below, so the ways of getting there are just the ways of getting to those two neighbours added together — no route is counted twice and none is missed.
4.OA.C.5Look For A PatternBookstore to school gives 3 routes.
With only three steps in total, deciding the route is just deciding which of the three steps is the up-step — and there are three places it could be.
4.OA.C.5Make A Systematic ListMultiplying gives 20 × 3 = 60.
Every first half pairs up with every second half, and that is exactly what multiplication counts — equal groups, one group for each way of finishing.
3.OA.A.1Identify SubproblemsA stop you must make splits the trip in two — count each half on its own, then multiply the two counts together.