Reasoning · Grade 3-2 Finding Sums Using Products
Problem
Each new odd wraps one L-shaped layer round the square.
The L-shaped layers are exactly the extra cells needed to grow a square by one, so the number of layers equals the side of the square.
3.OA.D.9Look For A PatternEach odd number is exactly the L-shaped layer of cells needed to grow a square by one, so the count of odd numbers is the square's side.
The layers stack with no gap and no overlap, so all of them together are exactly the finished square.
One odd number builds one layer and one layer adds one to the side, so odd numbers and side length match up exactly.
Six odds fill 6 × 6 = 36.
Six odd numbers means six L-layers, which is a square six cells on a side.
3.OA.D.9Draw A Diagram33 is the 17th odd number.
Listing a few odds shows each one jumps by 2, so reaching 33 takes 17 of them — the same count as the square's side.
3.OA.A.3Solve An Easier Related ProblemSo the sum is 17 × 17 = 289.
The count of odd numbers is the side of the square, so squaring 17 gives the whole sum at once.
3.OA.C.7Look For A PatternAdding odd numbers from 1 builds a perfect square — just count how many you added and multiply that by itself. Pure Grade 3 multiplication!