MathJewels math guide · Reason about networks, routes, vertices, and edges
Routes, Arrows, and Reachability
Read a one-way network, trace every possible route from a start, and distinguish a local incoming arrow from a complete path back to the start.
The big idea
A network is made of points joined by connections. In a one-way network, an arrow tells you which direction a move is allowed. A point is reachable from the start only when there is a complete chain of arrows from the start to that point. One arrow pointing into a point is not enough: the beginning of that arrow must itself be reachable.
Worked example
Suppose the arrows are A → B, A → C, C → D, and X → Y. Starting at A, you can reach A, B, C, and D. You cannot reach X or Y. The arrow into Y does not help, because there is no route from A to X, where that arrow begins.
To make every point reachable, one new arrow is enough. For example, add B → X. The new route A → B → X → Y reaches the formerly separate pair.
How children may show it
Children may shade every reachable point, trace routes with a finger or colored pencil, keep a checklist, or write the points in the order they first discover them. Encourage them to repeat the search until following every arrow from every shaded point produces no new point. That stopping rule turns tracing into a complete argument.
Common mix-up
A child may say that a point is reachable because an arrow enters it. Ask, “Can you reach the place where that arrow starts?” Work backward through incoming arrows until either the chain reaches the designated start or it becomes trapped in a separate part of the network.
Try it together
Draw five labeled circles. Make A the start, connect A → B → C, and connect D → E separately. Shade what is reachable from A. Then add the fewest arrows needed to reach every circle. Compare different valid new arrows and write one complete route from A to E for each choice.