Reasoning · Grade 5-1 Area of Figures
Problem
The unprinted width in (1) is 8 cm.
A straight side is exactly as long as the pieces it is chopped into, so adding the labelled pieces recovers a length the drawing never printed.
3.MD.D.8Draw A DiagramThe wrapping rectangle is 80 cm².
Length times width gives the area of any rectangle, and the smallest rectangle that boxes a figure in is always the easiest thing to measure first.
4.MD.A.3Change Focus Count The ComplementWrapping the figure in the smallest rectangle that holds it turns the area into an easy product minus the notches.
The figure and the notches together fill that rectangle with no gap and no overlap, so their areas add back to it.
Knowing the rectangle and the notches, subtracting returns the figure, because taking away undoes putting together.
Removing the two notches gives 48 cm².
When a figure is a rectangle with bites taken out, the area of the whole is the area of the parts added together, so the shaded part is the big area minus the bites.
3.MD.C.7Change Focus Count The ComplementCutting it into three bars also gives 48 cm².
Splitting a rectilinear figure into non-overlapping rectangles and adding their areas must give the same total as subtracting the holes, so agreement between the two routes is a real check.
3.MD.C.7Identify SubproblemsThe whole rectangle in (2) is 108 cm².
A hole strictly inside a shape does not disturb its outline, so the outer area can be computed first and corrected afterwards.
4.MD.A.3Change Focus Count The ComplementThe L-shaped hole is 27 cm².
An L is two rectangles pushed together, and one straight cut separates them, so its area is just the sum of two easy products.
3.MD.C.7Identify SubproblemsSubtracting gives 81 cm².
Cutting the same L along a different line has to give the same area, so two matching answers make an arithmetic slip very unlikely.
3.MD.C.7Change Focus Count The ComplementA bumpy shape is just a plain rectangle with a few rectangles missing — find the big one, take the holes away, and the hard shape disappears!