Reasoning · Grade 5-1 Equal-Area Transformation (2)
Problem
Give the two parallel sides and the height names.
Naming the three lengths once means every expression can be read as a short formula instead of a long phrase, which makes two expressions easy to compare at a glance.
6.G.A.1Organize Information In More WaysPicture 1 glues two copies: method C, expression ii.
This is where the usual trapezoid formula comes from, so it is worth spotting: the mysterious " ÷ 2" is just undoing the second copy you added.
6.G.A.1Draw A DiagramPicture 2 reshapes into a parallelogram: D, expression i.
Compare with picture 1: there the base stayed and the height doubled the count of trapezoids; here the base doubles and the height halves. Same product, opposite bookkeeping — which is exactly why (i) and (ii) look so alike.
6.G.A.1Identify SubproblemsPicture 3 cuts two triangles: A, expression iv.
Both triangles are squeezed between the same pair of parallel lines, so both have height h — the slant of the diagonal does not change that, and it is the fact that makes this the simplest method of the four.
6.G.A.1Identify SubproblemsPicture 4 rebuilds two rectangles: B, expression iii.
It is worth checking the corner pieces really do fill the gap: the half-height cut is exactly the midline, so each leftover triangle is the same size as the notch it has to fill on the row below.
6.G.A.1Draw A DiagramWith numbers all four expressions agree.
Putting numbers in is the quickest way to test whether two expressions are the same rule dressed differently, and it catches a wrong match immediately — a mismatched expression would not come out to 42.
6.EE.A.4Solve An Easier Related ProblemSplitting the product keeps the value, so it always holds.
This is the distributive property read backwards: two lots of "something × h/2" collapse into one, which is exactly the algebra behind gluing the two rectangles of picture 4 into the single parallelogram of picture 2.
6.EE.A.3Organize Information In More WaysThe four expressions must always agree, because each one is just a different way of cutting up the very same trapezoid.
Every method covers the trapezoid with no gap and no overlap, so every method's pieces add back to the same area.
The expressions differ only by pulling a shared factor in or out of a sum, which never changes what the sum comes to.
Cutting a shape up and moving the pieces never changes how much space it covers, so four different pictures have to give four expressions that all mean the same thing.