Reasoning · Grade 5-1 Rules and Correspondence
Problem
Add lines one at a time and watch the count.
A new line is chopped into pieces by the lines it meets, and each piece of the new line splits one old part into two, so counting the pieces of the new line is the same as counting the new parts.
4.G.A.1Solve An Easier Related ProblemAdding one chord at a time shows that each new region comes from one crossing, plus one for the chord itself.
A chord entering a region and leaving it splits that region in two, so one crossing adds exactly one region.
The regions fill the circle with no gap and no overlap, so the running count is the first region plus one for every split made.
The count is 5 plus the crossings.
Once the count depends on only one thing, how many crossing points there are, the drawing stops being guesswork and becomes a target: make the right number of crossings.
4.OA.C.5Look For A PatternFour lines cross at most 6 times.
Listing pairs in order, always pairing a line with the ones after it, makes sure no pair is counted twice and none is forgotten, which is what fixes the maximum at 6.
4.G.A.2Make A Systematic ListSo the reachable counts run 5 to 11.
Working the rule backwards turns draw until it looks right into make exactly this many crossings, which is something a child can aim at on purpose.
4.OA.C.5Look For A PatternSmall counts use plenty of parallel lines.
The three parallel lines work like a ruler with marks on it, and the slant of the fourth line dials the number of crossings up one at a time from 0 to 3.
4.G.A.1Draw A DiagramLarge counts tilt the lines apart to add crossings.
Parallel lines are the tool for skipping crossings and slanted lines are the tool for adding them, so the drawings can be built to order instead of found by luck.
4.G.A.1Draw A DiagramCounting each drawing matches its target.
Counting the shaded pieces directly does not use the rule that produced the drawing, so when the two agree it is real evidence and not the same mistake made twice.
3.OA.D.9Draw A DiagramEvery time two lines cross inside the circle you get one more piece, so count the crossings, add 5, and you know the answer before you shade anything in!