Finding Area by Rearrangement
Finding Area by Rearrangement introduces the principle that area is preserved when a shape is cut into pieces and rearranged without gaps or overlaps. Covered in Illustrative Mathematics Grade 6, Unit 1: Area and Surface Area, Grade 6 students use this property to find the area of irregular shapes by transforming them into simpler shapes like rectangles or triangles whose areas they know. This approach visually proves why the formulas for parallelograms and triangles are correct.
Key Concepts
Property The area of a shape remains the same when it is cut into pieces and rearranged into a new shape without any gaps or overlaps. This method transforms a complex shape into a simpler one, like a rectangle or square, to make calculating the area easier.
Examples A parallelogram with base $b=8$ and height $h=5$ can be rearranged into a rectangle. By cutting a triangle from one side and moving it to the other, we form a rectangle with dimensions $8 \times 5$. The area is $8 \times 5 = 40$ square units. A trapezoid with bases $b 1=6$ and $b 2=10$ and height $h=4$ can be transformed. By making a cut through the midpoints of the non parallel sides and rotating the top section, a parallelogram is formed with base $b 1+b 2=16$ and height $\frac{h}{2}=2$. Or, more simply, it can be rearranged into a rectangle with length $\frac{b 1+b 2}{2} = 8$ and width $h=4$, giving an area of $8 \times 4 = 32$ square units.
Explanation This skill involves visually decomposing a shape and then recomposing the pieces into a more familiar figure. It's a powerful reasoning tool that shows area is conserved even when a shape's form changes. Instead of just adding or subtracting areas, you are transforming the shape itself. This method is particularly useful for deriving area formulas for shapes like parallelograms and trapezoids by relating them back to rectangles.
Common Questions
What is the rearrangement principle for area?
If a shape is cut into pieces and rearranged without gaps or overlaps, the total area stays the same. This lets you transform complex shapes into simpler ones.
How do you use rearrangement to find the area of a parallelogram?
Cut a right triangle from one end of the parallelogram and move it to the other end. The shape becomes a rectangle, which is easy to calculate using length × width.
Why does rearrangement preserve area?
No material is added or removed — the pieces are just moved. As long as there are no gaps or overlaps, the total area is unchanged.
Where is finding area by rearrangement in Illustrative Mathematics Grade 6?
This concept is in Unit 1: Area and Surface Area of Illustrative Mathematics Grade 6.
Can rearrangement be used for triangles too?
Yes. Two congruent triangles can be rearranged into a parallelogram, showing that the triangle area is half the parallelogram area.