Problem
Counting non-square rectangles
A 7-by-4 rectangle is divided into squares. A rectangle counts only if all four sides lie on drawn lines. Squares are excluded. Count the non-square rectangles.
Reasoning · Grade 3-1 Counting Large and Small Shapes
Your answer
Watch out Counted as a full grid the total balloons — most horizontal lines stop partway, so few sides are usable.
How to solve
Strategy Make a Systematic List — A rectangle is fixed by choosing its left and right vertical edges and its top and bottom horizontal edges. So I list the available vertical lines and horizontal lines, then go through every pair-of-verticals with every pair-of-horizontals in order, keeping only those whose whole border is drawn and whose width is different from its height. Breaking the figure into its three columns keeps the listing organized.
1STEP 1
Name the drawn lines
Four vertical lines, but only two horizontals cross the whole width.
Listing the real lines first stops me from 'inventing' a side that the picture does not actually draw.
3.G.A.1Draw A Diagram2STEP 2
Count rectangles inside the left column
The left column gives heights 2, 3, 4 — 6.
3 + 2 + 1 = 6
A 1-wide strip cut into 4 equal pieces lets me just count tall slices, like choosing two of the five horizontal lines.
2.G.A.2Make A Systematic List3STEP 3
Count rectangles that use the left + middle columns together (width 3)
Spanning left and middle, width 3 gives 3.
3 rectangles: 3×2, 3×2, 3×4
Only lines that reach all the way across can be a side, so the middle's height-1 and height-3 cuts cannot be used here.
3.G.A.1Identify Subproblems4STEP 4
Count rectangles that use the middle column alone (width 2)
The middle column alone, minus its squares, leaves 1.
1 rectangle: 2×4
Width 2, height 2 is square, so the only non-square here is the full tall 2-by-4 column.
3.G.A.1Make A Systematic List5STEP 5
Count rectangles that reach into the right 4x4 square (width 6 or 7)
The wide ones reaching the right side are 2.
2 rectangles: 6×4, 7×4
Inside the big right square no lines are drawn, so wide rectangles can only start at x=0 or x=1 and must be the full height 4.
3.G.A.1Identify Subproblems6STEP 6
Add up all the cases
Altogether 6 + 3 + 1 + 2 = 12.
6 + 3 + 1 + 2 = 12
Because each case used a different pair of side-lines, no rectangle is counted twice, so I can simply add the case totals.
2.OA.B.2Make A Systematic ListBecause each case used a different pair of side lines, no rectangle appears in two cases and the case totals simply add.
Why?
A rectangle is fixed by which two vertical lines and which two horizontal lines make its sides, so it names one case and no other.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The cases cover every rectangle and never overlap, so their counts can be added with nothing lost and nothing doubled.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Answer
12 rectangles
6 + 3 + 1 + 2 = 12
Twelve non-square rectangles is sensible: the figure clearly also contains 7 squares (four 1x1, two 2x2, one 4x4), and counting both squares and non-squares the total stays small, matching a hand check of the picture. Each counted rectangle was checked to have all four sides on real lines, and none was a square.
Takeaway
Pick a left line, a right line, a top line, and a bottom line that are really drawn — that makes one rectangle. List them in order and skip the squares!
- Name the drawn lines
- Count rectangles inside the left column
- Count rectangles that use the left + middle columns together (width 3)
- Count rectangles that use the middle column alone (width 2)
- Count rectangles that reach into the right 4x4 square (width 6 or 7)
- Add up all the cases