E7 — Decomposing Shapes into Equivalent Fractional Areas

It takes 3 rhombi to cover the hexagon and therefore the area of the rhombus is 1/3 of the area of the hexagon. The hexagon may be labeled 3/3 and an equation might be 1/3 + 1/3 + 1/3 = 3/3, or 3 x 1/3 = 3/3. Students may approach this question in a variety of ways though, and it is important to allow that because there are some important big ideas about geometry that will likely emerge. Some students may draw 3 diagonals, each connecting opposite vertices and effectively making 6 triangles, and then imagine a 2:1 relationship of the triangle to the rhombus. If the triangle is 1/6 of the hexagon, then the area of the rhombus is 1/3 of the area of the hexagon. A possible equation for this strategy might be 2 x 1/6 = 1/3, or 1/6 + 1/6 = 1/3. Others may try to draw 3 rhombi from prior knowledge gained in the earlier grades with pattern blocks. As they draw, note whether they use a line segment of the side of one rhombus simultaneously as the side of another. Early in development students do not. They draw 3 rhombi with no shared boundaries. If you see this, it is important to ask about the area between the lines. This space between the rhombi is a part of the area of the hexagon, so this drawing is not a justification that the rhombus is 1/3 of the hexagon. Facilitate a discussion on this and allow for disequilibrium. The big idea involved here is that one line segment can serve as a boundary of two spaces, simultaneously. Support children to consider a way to decompose the hexagon by drawing only three line segments — the three cuts divide the space into thirds.