Reasoning · Grade 2-2 Various Patterns and Rules
Problem
In (1), after k groups there are k blacks and 1+2+…+k whites.
Adding the whites one group at a time is just skip-style counting a Grade 2 student can list out.
2.NBT.A.2Make A Systematic ListThe gap is widest just as a white run finishes.
Listing the differences 0,1,3,6,10 shows a clear arithmetic pattern; 10 first appears at the 5th black.
3.OA.D.9Look For A PatternThe gaps between black and white run 0, 1, 3, 6, 10, growing by one more each time, so a gap of 10 first appears at the fifth black counter.
Each new white group is one counter longer than the last, so the amount the gap grows by climbs by exactly one every time.
The gap at any point is all the earlier growths added up, so the running list of gaps is built by adding one growth at a time.
Finding where that first hits 10 gives 5 blacks in (1).
Checking the small steps one counter at a time guarantees we did not skip over the first time the gap is 10.
2.NBT.B.5Solve An Easier Related ProblemIn (2) the gap after the m-th white group is exactly m.
Adding the odd group sizes makes square numbers, and adding the even group sizes is just doubling 1+2+...+(m-1) — both are listing-and-adding a young learner can do.
2.NBT.B.6Make A Systematic ListSo the gap first reaches 10 at m = 10.
The peaks are simply 1,2,3,... — so a gap of 10 happens exactly at the 10th white group.
3.OA.D.9Look For A PatternThe blacks then are 2+4+…+18 = 90.
Summing 2 through 18 is grouping and adding within 100, a Grade 2 skill.
2.NBT.B.6Make A Systematic ListList a few groups, track white-minus-black, and the pattern of the peaks tells you exactly when the gap first hits 10 — only Grade 2-3 counting and adding!