This lesson prepares students to apply what they know about the area of parallelograms to reason about the area of triangles.
Highlighting the relationship between triangles and parallelograms is a key goal of this lesson. The activities make use of both the idea of decomposition (of a quadrilateral into triangles) and composition (of two triangles into a quadrilateral). The two-way study is deliberate, designed to help students view and reason about the area of a triangle differently. Students see that a parallelogram can always be decomposed into two identical triangles, and that any two identical triangles can always be composed into a parallelogram (MP7).
Because a lot happens in this lesson and timing might be tight, it is important to both prepare all the materials and consider grouping arrangements in advance.
- Describe (orally and in writing) ways in which two identical triangles can be composed, i.e., into a parallelogram or into a rectangle.
- Show how any parallelogram can be decomposed into two identical triangles by drawing a diagonal, and generalize (in writing) that this property applies to all parallelograms, but not all quadrilaterals.
Let’s compare parallelograms and triangles.
Print pairs of triangles from the blackline master for A Tale of Two Triangles (Part 2). If students are cutting out the triangles, use the first page only. If the triangles are to be pre-cut by the teacher, print the second and third pages. Prepare enough sets so that each group of 3–4 students has a complete set (2 copies each of triangles P–U).
For classes using the digital version of the activity, an applet is provided that can be used in place of, or in addition to, the cut out triangles.
- I can explain the special relationship between a pair of identical triangles and a parallelogram.