Reasoning · Grade 4-2 Counting Shortest Routes
Problem
A shortest route is 2 rights and 2 ups.
Naming each crossroads with a pair of numbers turns the picture into something countable, and it makes the "never go backwards" rule obvious: every step must increase one of the two numbers.
4.G.A.1Draw A DiagramA shortest route can only go right and up, because any other step would have to be undone later.
The whole walk is its single steps laid end to end, and a step the wrong way must be paid for by an extra step back.
Any route containing such a pair of wasted steps is longer than one without it, so all of them are ruled out at once.
The right half of the middle street is missing.
The whole puzzle turns on one missing segment, so the safest first move is to point at it and say out loud which two points it would have joined.
4.G.A.1Draw A DiagramTwo of the six orders are ruled out.
Counting what a perfect grid would give and then removing only the spoiled cases is far quicker than testing everything from scratch, and it explains exactly what the gap in the street costs you: two routes.
4.OA.A.3Change Focus Count The ComplementAdding along the map also gives 4.
Every route into a corner has to arrive along one of the streets that end there, so the ways just add up — and the missing street shows up all by itself as a junction that receives from only one direction.
4.OA.C.5Look For A PatternDrawing them shows four distinct routes.
Drawing them in a fixed order — outermost first, then the ones that turn more often — is what guarantees the four pictures are genuinely different and that none has been forgotten.
4.G.A.1Make A Systematic ListShortest means you only ever go right or up — so list the orders of the moves, then throw away the ones that need a street that isn't there.