I created these pictures to give my students the opportunity to explore the areas of triangles, parallelograms and trapezoids in reference to the area of rectangles. I made copies of these two slides from my lesson presentation for the students to write on and they explored the areas of the shapes by drawing on the dot paper, counting square units, cutting off parts of the polygons and moving them to different places, and making conjectures about how the areas of the figures were related and why. This was a very engaging activity and the students used many different strategies to find the areas of these figures and these explorations formed the foundation for understanding the formulas for finding the areas of these figures. I was very pleased to see that students continued to use ideas from these pictorial models to help them with areas of polygons that were not displayed on dot paper (for example, triangle from left side of parallelogram is moved to the right side to form a rectangle with the same area).
The students had lots of interesting ideas for finding the area of Triangle F. I was amazed and thrilled when several students, connected the area of the obtuse triangle to the area of a paralleogram, instead of a rectangle, which was much easier for students in the class to see.
I use this slide to summarize the exploration. I highlighted part of each of the formulas in red to help students see the connection between all of the formulas: base x height. Even the formula for trapezoid can be understood in this way if we talk about finding the average base x height.
