Reasoning · Grade 4-2 Counting Large and Small Figures
Problem
Sort the squares by side length.
Sorting first is the standard cure for double counting: once every square has exactly one bin to fall into, adding the bins can never overcount.
4.OA.C.5Make A Systematic ListA tiny board shows the pattern 9, 4, 1.
A 3 by 3 board is small enough to check by eye, so the rule found there is not a guess - it is something a young solver has actually verified.
4.OA.C.5Solve An Easier Related ProblemCount where the top-left corner can sit.
Counting positions as rows times columns is exactly the array picture of multiplication learned in Grade 3 - each choice across can be paired with each choice down.
3.OA.A.1Draw A DiagramA square of a given size is fixed by where its top-left corner sits, so counting the allowed corner positions counts the squares.
Sliding a square to a new corner position keeps its side length, so every position gives a genuine square of the same size.
How far the corner may slide sideways does not depend on how far it may slide down, so the two counts multiply.
By size that is 36, 25, 16, 9, 4, 1.
Each bigger square has less room to slide, so the counts must shrink - and shrinking exactly through the square numbers is the pattern the small boards already showed.
4.OA.C.5Look For A PatternAdding them gives 91.
Six two-digit numbers added in friendly pairs is a short, checkable calculation, and the pairing can be redone in a different order as an instant check.
4.NBT.B.4Make A Systematic ListCount by size, not by eye: a square of each size can only slide into a neat rectangle of positions, so the totals are just 6x6 + 5x5 + 4x4 + 3x3 + 2x2 + 1x1.