Reasoning · Grade 2-2 Various Patterns and Rules
Problem
Triangle n adds n new numbers and starts where the last one ended.
Listing the first few triangles shows the growth rule directly, which is the heart of a Grade 4 pattern problem.
4.OA.C.5Make A Systematic ListSo triangle n ends at 1 + n(n+1) ÷ 2.
Adding 1+2+...+n is a triangular number, and matching 11, 16, 22 to the picture confirms the rule.
3.OA.D.9Look For A PatternThe last number on triangle n is 1 plus 2 plus all the way to n, because each triangle adds one more number than the one before.
The triangles use up the counting numbers in order without skipping or repeating, so the last number on triangle n is simply how many numbers have been used so far.
That running sum is quickly found by pairing the first with the last, the second with the second-last, since every pair makes the same total.
Triangle 8 ends at 37, so triangle 9's 4th number is 40.
Once the starting number 37 is known, counting on three more is simple Grade 2 addition.
2.NBT.B.7Look For A PatternTriangle 12 runs 67 to 79, so 72 sits on triangle 12.
Computing a couple of triangle endpoints with simple addition pins down which triangle 72 lives on.
2.NBT.B.7Solve An Easier Related ProblemCounting 67 as first, 72 is the 6th number.
The position is just how far 72 is past the start, a subtract-and-add-one count within 100.
2.NBT.B.5Look For A PatternEach triangle just adds one more number than the one before, so a triangular-number rule tells you exactly where any number lands — Grade 4 pattern thinking!