Reasoning · Grade 3-1 Filling and Joining Shapes
Problem
Twelve squares in fours means always three pieces.
Counting the squares and dividing by 4 tells us right away that exactly three tiles fill the shape every time, which keeps the search short.
3.OA.A.1Identify SubproblemsBecause the shape holds twelve squares and each tile covers four, every filling uses exactly three tiles.
The tiles cover the shape with no gap and no overlap, so their squares add back to exactly the shape's twelve squares.
Asking how many fours fit into twelve is a division, and a remainder of zero shows nothing is left dangling.
The jutting square needs a piece that bends down — few candidates.
Starting at the most cramped place is the smart move: the hardest corner has the fewest choices, so it splits the work into a few clear cases.
1.G.A.2Identify SubproblemsFilling the rest for each gives 3 + 3 + 2.
Sorting the fillings by which set of tiles they use keeps the list orderly, so we can see the count add up to 8 without missing or repeating any.
1.G.A.2Make A Systematic ListThat totals 8, matching the eight blank frames.
The blank grids on the page match our count of 8, a nice check that we found them all and none twice.
3.MD.C.6Draw A DiagramCount the 12 squares: since every tile covers 4, each filling needs just 3 tiles (12 / 4 = 3). Start at the tricky square on top, then list fillings in order to find all 8!