Sensim Math · Depth 한국어

3-1 · Plane Figures

Count all shapes hidden in a grid

1.G.A.23.OA.D.9 · take · grade 3

Archetype: Systematically Count Shapes in a Figure · step in a 5-type progression

▶ Practice — 10 problems

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. The top row has 33 unit squares, the middle row has 44 unit squares, and the bottom row has 44 unit squares, all aligned at the left edge (so the top row is missing its rightmost cell, giving a stepped shape). Count every square that can be traced in the figure — the 11-cell squares, the 44-cell (2×22\times2) squares, the 99-cell (3×33\times3) square, and so on.

Show solution

Understand

A stepped figure is built from 11 equal unit squares: the top row has 3 cells (columns 1-3), and the middle and bottom rows each have 4 cells (columns 1-4), all left-aligned. I must count every square of every size that can be traced along the grid lines.

Givens
  • The figure is made of 11 equal unit squares.
  • Top row: 3 unit squares (columns 1-3).
  • Middle row: 4 unit squares (columns 1-4).
  • Bottom row: 4 unit squares (columns 1-4).
  • The top-right cell (row 1, column 4) is missing, giving a stepped outline.
Unknowns
  • The total number of squares of all sizes (1x1, 2x2, 3x3) in the figure.
Constraints
  • A counted square must have all of its cells present in the figure.
  • Squares may overlap and be of different sizes.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

To avoid missing or double-counting, I organize the count by square size (a systematic list): first all 1x1 squares, then 2x2, then 3x3. Splitting by size turns one tricky count into a few easy subproblems, and sketching the grid lets me check which big squares actually fit inside the stepped shape.

Execute

#2 Make a Systematic List 3.OA.D.9
Every unit cell is a 1x1 square. The figure is made of exactly 11 unit cells, so there are 11 squares of size 1x1.
3+4+4=113 + 4 + 4 = 11
Just adding the cells in each row is Grade 3 addition you can do by counting the picture.
#7 Identify Subproblems 1.G.A.2
A 2x2 square needs a full 2-by-2 block of present cells. Using rows 1-2 (top+middle): blocks at columns 1-2 and 2-3 fit, but columns 3-4 needs the missing top-right cell, so that one fails (2 squares). Using rows 2-3 (middle+bottom): both rows are full, so columns 1-2, 2-3, and 3-4 all fit (3 squares).
2+3=52 + 3 = 5
Composing four small squares into one bigger square is exactly what 'compose two-dimensional shapes' means in early geometry.
#1 Draw a Diagram 1.G.A.2
A 3x3 square needs three full rows of three present cells. Columns 1-3 work because all three rows have columns 1, 2, and 3. Columns 2-4 would need the top-right cell, which is missing, so it fails. That gives 1 square of size 3x3.
11
Drawing the 3-by-3 box on the grid shows at a glance that only the left block fits the stepped shape.
#2 Make a Systematic List 3.OA.D.9
Total squares = (1x1 count) + (2x2 count) + (3x3 count). There is no room for a 4x4 square because the figure is only 3 rows tall.
11+5+1=1711 + 5 + 1 = 17
Collecting the size-by-size subtotals into one sum is straightforward Grade 3 addition.
Answer: 17

Review

There must be more small squares than big ones, and indeed 11 > 5 > 1, which fits the picture; the total 17 is larger than the 11 unit cells, as expected once overlapping bigger squares are included. No 4x4 square is possible since the figure is only 3 rows tall, so we have not missed a larger size.

Look for a pattern (tool 5): for a full 3x4 grid the counts would be 12 (1x1) + 6 (2x2) + 2 (3x3) = 20; removing the one missing top-right cell deletes 1 unit square, 1 of the 2x2 squares, and 1 of the 3x3 squares, giving 20 - 3 = 17, confirming the answer.

Standards · min grade 3

  • 3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the per-size subtotals and reasoning about the count systematically.
  • 1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes — Recognizing that 4 or 9 unit squares compose a larger 2x2 or 3x3 square.
💡 Count squares one size at a time, smallest to biggest, and you will never miss one - that's Grade 3 thinking you already have!