Reasoning · Grade 4-2 One-Stroke Drawing
Problem
The totals 7 and 10 yards are the floor.
Counting roads and multiplying by 1 yard each keeps the units straight and gives a target to aim at before any clever thinking starts.
4.MD.A.2Analyze The UnitsMark the road counts to find the odd dots.
Deciding whether a small count is odd or even is second-grade work, and pairing roads up in and out is exactly the pairing children use to test for even numbers.
2.OA.C.3Draw A DiagramMap (1) has only 2 odd dots.
The number of roads at a dot can just be counted off the picture, one road at a time.
2.OA.C.3Draw A DiagramSo map (1) hits its floor at 7 yards.
Once a route is actually written down and traced with a finger, the claim stops being a theory and becomes something a child can check road by road.
4.MD.A.2Guess And CheckMap (2) has 4 odd dots.
The same road-counting at each dot works on the second picture without any new idea.
2.OA.C.3Draw A DiagramRepeat the cheapest road, 1 yard.
There are only a handful of ways to pair off four dots, so writing them all down and picking the shortest is quick and leaves no doubt.
4.OA.A.3Make A Systematic ListAt most two odd dots may survive, so the cheapest pair of roads is the one to retrace.
Retracing a road adds one to the count at each of its two ends, so it flips both of those dots between odd and even.
Among the pairs that would fix the parity, comparing their lengths two at a time chains into one ranking and the shortest wins.
So map (2) needs 11 yards.
Adding one extra yard to the total road length is arithmetic a fourth grader can do, and tracing the written route proves the 11 yards is really achievable.
4.MD.A.2Guess And CheckCount the roads at every dot: the dots with an odd number are the only ones that can force you to walk a road twice.