Problem
Find a rectangle side via one unknown
An 88-inch wire is bent into a rectangle whose long side is 3 times the short side, using all the wire. We need the length of the long side.
Measurement & dataOperations
Your answer
inches
Watch out It's tempting to divide the wire evenly into 4 equal sides (88 ÷ 4 = 22) as if it were a square, but the long side is actually 3 times the short side, not equal to it.
How to solve
Strategy Draw a Diagram — Sketch the rectangle and count the short-side units around the perimeter: a short, a long (3 shorts), and again, giving 8 equal short-side pieces; divide the wire among them.
1STEP 1
Count short-side units in the perimeter
Going around: short + long + short + long. Each long is 3 shorts, so the perimeter is 1 + 3 + 1 + 3 = 8 short-side lengths.
1 + 3 + 1 + 3 = 8 short sides
Measuring the whole border in short-side units turns the wire into 8 equal pieces.
3.MD.D.8Draw A DiagramMeasured all the way around, the rectangle's border is 8 short-side lengths long.
Why?
The border is the four sides joined into a loop, so its total length is those four sides added together.
Why?
Cutting the border into its four sides leaves no gaps and no overlaps, so the four side lengths add back to the whole border.
🧱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?
A rectangle's two short sides match and its two long sides match, so going around passes two equal shorts and two equal longs.
Why?
Folding the rectangle in half lays one short side exactly on the other and one long side exactly on the other, so each pair is equal in length.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Each long side is as long as 3 short sides, so counting around the loop gives 1 + 3 + 1 + 3 = 8 short sides.
Why?
Saying the long side is 3 times the short side means 3 short-side lengths laid end to end.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Find the short side
The 88-inch wire splits into 8 equal short-side pieces: 88 ÷ 8 = 11.
88 ÷ 8 = 11 inches
Sharing the total length equally among the 8 units gives one short side.
3.OA.C.7Guess And Check3STEP 3
Find the long side
The long side is 3 times the short side: 11 × 3 = 33.
11 × 3 = 33 inches
The long side is three short pieces joined, so triple the short side.
3.OA.A.3Draw A DiagramAnswer
33 inches
11 × 3 = 33
Check: short 11, long 33, perimeter = 2 times (11 + 33) = 2 times 44 = 88 inches, matching the wire exactly. The long side being longer than the short fits the picture.
Takeaway
This only needs Grade 3 perimeter sense: the border is 8 short sides, so divide, then triple for the long side!
- Count short-side units in the perimeter
- Find the short side
- Find the long side
Where next?
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