Areas of Simple Shapes

The Class will use adding machine tape to cut out

• Parallelograms
• Triangles
• Trapezoids

and investigate their areas.

Task 1. Parallelograms

• Figure out a way to cut out parallelograms from the tape that does not require you to measure angles.
• Cut out a parallelogram and then cut it into pieces and re-assemble as a rectangle of the same area.
• From this conjecture a formula for area of parallelograms based on the rectangle area formula.
• Fine point. Cut out a parallelogram where the top side is completely to the right of the base. How can you justify the formula in this case?

Task 2. Triangles

Subtask 2A: Two triangles

• Cut out a triangle ABC so that BC is on one edge of the tape and A is on the other. Then cut a second congruent triangle A'B'C'.
• Make a new "double" figure by attaching the two triangles on a common edge. If you have a formula for the new, double figure, then you have a formula for the area of a triangle also.
• For a right triangle, what kind of very simple figure can you get this way?

Subtask 2B: Cut one triangle along midpoint segment

• Start again with ABC but this time fold to get the midpoints E and F of AB and AC. Cut along EF to get two figures, AEF and BCFE.
• What kind of figures are they?
• Assemble a parallelogram from the two new figures to get a second way of seeing the area formula for triangles.

• From this conjecture a formula for area of parallelograms based on the rectangle area formula.
• Fine point. Cut out a parallelogram where the top side is completely to the right of the base. How can you justify the formula in this case?

Task 3. Trapezoids

• Using the ideas from your previous explorations, find two or three ways to derive and show the area formula for a trapezoid.