Problem
Rectangle side and perimeter from equal squares
Five identical small rectangles tile a big rectangle 20 cm tall. Three stand upright on top, two lie flat below. The two rows must be the same width. Find the width of the big rectangle.
Reasoning · Grade 3-1 Shapes and Length
Your answer
Watch out The 20 cm looks like one rectangle's height, but two rows stack — it is the long side plus the short one.
How to solve
Strategy Draw a Diagram — Drawing and labeling the short side and long side of one small rectangle shows how the upright pieces and the lying pieces stack and line up. Reading the height and matching the two row widths gives two simple number facts that pin down the side lengths; a quick guess-and-check confirms the only pair that works.
1STEP 1
Read the height from the two stacked rows
Down the right edge, long + short = 20.
b + a = 20
Stacking one tall piece on one flat piece, the two heights simply add up to the whole height.
2.MD.B.5Draw A Diagram2STEP 2
Match the widths of the top and bottom rows
Equal widths give short × 3 = long × 2.
3 × a = 2 × b
Both rows reach across the same big rectangle, so they have to be the same length.
3.OA.A.4Identify SubproblemsBoth rows stretch across the same big rectangle, so the two row widths have to be equal.
Why?
Each row is its small rectangles laid side by side with no gap and no overlap, so a row's width is those side lengths added up.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Two lengths that both span the same rectangle are a visible statement of equality, which can then be worked on from both sides at once.
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
3STEP 3
Find the side lengths
Solving both gives short 8, long 12.
a + 3/2a = 20 → 5/2a = 20 → a = 8, b = 12
Only one pair of sides makes both the height add to 20 and the two row widths match, so guess-and-check lands quickly on 8 and 12.
3.OA.D.8Guess And Check4STEP 4
Compute the width of the large rectangle
The width, read along the top row, is 3 × 8 = 24 cm.
3 × 8 = 24 cm
Once the small rectangle is 8 by 12, the big width is just three short sides lined up.
3.OA.A.3Identify SubproblemsAnswer
24 cm
3 × 8 = 24
The big rectangle is 24 cm wide and 20 cm tall, so it is a bit wider than tall, matching a figure that is roughly square but slightly wide. Its area 24 x 20 = 480 should equal 5 small rectangles: 5 x (8 x 12) = 5 x 96 = 480. It checks out.
Takeaway
When two rows have to be the same length, just match '3 short sides = 2 long sides' and the sizes pop right out!
- Read the height from the two stacked rows
- Match the widths of the top and bottom rows
- Find the side lengths
- Compute the width of the large rectangle