Making Sense of Area Formulas.

  The other day in Geometry, one of my students asked how they got the area formula for a kite and a rhombus because essentially the formulas are the same, yet the shapes are slightly different.  Another student asked about the area formula of a trapezoid and where it came from.

So out came the paper and the scissors to cut out shapes and how how they related.  The kids watched every step and I saw lots of 'Aha' moments sparkle across the room as light bulb after light bulb exploded.

For both the kite and the rhombus, I drew a rhombus inside a regular square, cut the rhombus out before flipping the cut pieces around to cover the rhombus.

The students saw that the diagonals were actually the height and base or length and width of a square.  They saw then  the cut pieces made a second rhombus and that is where the divide by 2 or multiply by 1/2 came from.

When I repeated it with a kite inside a rectangle, they were like wow.  They saw how the cut pieces when place over the kite, produced a second kite of exactly the same size.  They also saw the diagonals on the kite also represented the height and base of a rectangle and the rectangle produced two kites thus the division by two in the formula.

As for the trapezoid, I placed two right next to each other, one right side up, one flipped so the two of them together formed a rectangle. 

I labeled various parts as base 1 and base 2 along with height.  One young man, declared you found the area by adding the length of the bases together to form one side multiplying that number by the height.  Since the trapezoid was half of the rectangle, you had to divide the answer by two. 

I think next year, I'm going to create a hands on activity where students get to explore area formula's so they derive them by themselves.  This will help them gain a better understanding of the how the formulas come to be.

Let me know what you think.  I'd love to hear.