In this activity, students apply what they have learned to find the areas of various triangles. They use reasoning strategies and tools that make sense to them. Students are not expected to use a formal procedure or to make a general argument. They will think about general arguments in an upcoming lesson.

Monitor for different ways of reasoning about the area. Here are some paths that students may take, from more elaborate to more direct:

• Draw two smaller rectangles that decompose the given triangle into two right triangles. Find the area of each rectangle and take half of its area. Add the areas of the two right triangles. (This is likely used for B and D.)

  For Triangle C, some students may choose to draw two rectangles around and on the triangle (as shown here), find half of the area of each rectangle, and subtract one area from the other.

  

• Enclose the triangle with one rectangle, find the area of the rectangle, and take half of that area. (This is likely used for right triangle A.)
• Duplicate the triangle to form a parallelogram, find the area of the parallelogram, and take half of its area. (This is likely used with any triangle.)

The goal of this discussion is for students to see a wide range of ways to reason about the area of triangles.

Invite previously selected students to share their approach and display their reasoning for all to see. Sequence the presentations as shown in the Activity Narrative—starting with the most elaborate (most likely a strategy that involves enclosing a triangle) and moving toward the most direct (most likely duplicating the triangle to compose a parallelogram).

Connect the different responses to the learning goals by asking questions such as:

• “Did anyone else reason the same way?”
• “Did anyone else draw the same diagram but think about the problem differently?”
• “Can this strategy be used on another triangle in this set? Which one?”
• “Is there a triangle for which this strategy would not be helpful? Which one, and why not?”

This activity offers one more lens for thinking about the relationship between triangles and parallelograms. The reasoning here supports students in generalizing the process of finding the area of a triangle in a future lesson. Previously, students duplicated triangles to compose parallelograms. Here they see that a different set of parallelograms can be created from a triangle, not by duplicating it, but by decomposing it. 

Students are assigned a parallelogram to be cut into two congruent triangles. They take one triangle and decompose it into smaller pieces by cutting along a line that goes through the midpoints of two sides. They then use these pieces—a trapezoid and a small triangle—to compose a new parallelogram and reason about its area. Two parallelograms can be created. Monitor for students who create different parallelograms from the same pieces.

Students notice that the height of this new parallelogram is half of the height of the original parallelogram, and the area is also half of the area of the original parallelogram. Because the new parallelogram is composed of the same parts as the remaining (or uncut) large triangle, the area of the triangle is also half of that of the original parallelogram. This reasoning provides another way to understand the formula for the area of triangles.

Of the four given parallelograms, Parallelogram B is likely the most manageable for students. When decomposed, its pieces (each with a right angle) resemble those seen in earlier work on parallelograms. Consider this when assigning parallelograms to students.

The goal of this discussion is to enable students to further explain the relationship between the area of a triangle and a related parallelogram with the same base and height.

Display 2–3 approaches from previously selected students for all to see: Two different parallelograms that could be composed from the trapezoid and small triangle cut out from each Parallelogram A, B, C, and D. Use Compare and Connect to help students compare, contrast, and connect the area of each original parallelogram and the areas of the shapes that compose it. Here are some questions for discussion:

• “What do the new parallelograms have in common?” (They are shorter and smaller than the original parallelogram. They are all composed of a trapezoid and a triangle. Each parallelogram in the new pair has the same area.)
• “How does the area of each new parallelogram compare to the area of the original parallelogram? How do you know?” (It is half the area of the original. The original and new parallelograms have a base that is the same length, but the height of the new parallelogram is half of that of the original.)
• “How does the area of each new parallelogram compare to the area of the large triangle? How do you know?” (They are equal. Each new parallelogram is made of two shapes that can be rearranged to match the large triangle exactly.)

Reiterate that we can establish two things about the area of each newly composed parallelogram:

• It is half of the area of the original parallelogram.
• It is equal to the area of the larger triangle.

As a result, we know that the area of a large triangle (formed by decomposing the parallelogram into two equal triangles) is also half of the area of the original parallelogram.