In this activity, students explore different methods of decomposing a parallelogram and rearranging the pieces to find its area. Here are two key approaches for finding the area of parallelograms:

• Decompose the parallelogram, rearrange the parts into a rectangle, and multiply the length of the base by the length of a  side to find the area.
• Enclose the parallelogram in a rectangle and subtract the combined area of the extra regions.

Presenting the parallelograms on a grid makes it easier for students to see that the area does not change as they decompose and rearrange the pieces. This investigation lays a foundation for later reasoning about the area of triangles and other polygons.

Monitor for students who use the two key approaches and whose reasoning can highlight the usefulness of using a related rectangle to find the area of a parallelogram.

Some students may begin by counting squares but that strategy is not reinforced here. Encourage students to listen for and try more sophisticated, grade-appropriate methods shared during the class discussion.

In the digital version of the activity, students use an applet with some given rectangles and right triangles to visualize their reasoning (decomposition, rearrangement, and enclosure). The digital version may be helpful for visualizing the rearrangement of pieces to form rectangles.

Invite selected students to share how they found the area of the first parallelogram. Begin with students who decomposed the parallelogram in different ways. Follow with students who enclosed the parallelogram and rearranged the extra right triangles. (For students using the digital version, begin with those who reasoned with the smaller rectangles in the applet. Follow with students who reasoned with the largest rectangle.)

As students share, display and list the strategies for all to see. Restate them in terms of decomposing, rearranging, and enclosing, as needed. The list will serve as a reference for upcoming work. If one of the key strategies is not mentioned, illustrate it and add it to the list.

Use the reflection questions in the launch to help highlight the usefulness of rectangles in finding the area of parallelograms. Consider using the applet to illustrate this point, https://ggbm.at/kj5DcRvn.

While there is not one correct way to find the area of a parallelogram, each parallelogram here is designed to elicit a particular strategy. Monitor for students who use these two main strategies:

• Decomposing and rearranging the parallelograms into a rectangle. Parallelograms A and C encourage this strategy. 
• Enclosing the parallelogram and subtracting the area of the extra pieces. Parallelogram B is not as easy to decompose and rearrange (though some students are likely to try that approach first) and may prompt this strategy. 

The presence of a grid for Parallelograms A and B and its absence for Parallelogram C allow students to reason concretely and abstractly, respectively, about the measurements that they need to find the area (MP2). 

With repeated reasoning, students may begin to see regularity in the segments and measurements that tend to be useful in finding area (MP8). Students formalize this awareness in an upcoming lesson. Highlighting the segments and measurements without referring to bases or heights can help support students’ future work. 

This is the first time that Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or by teacher think-aloud (such as "I notice . . . I wonder . . ."). A typical discussion prompt is “What is the same and what is different?”, comparing their own strategy to the strategies of others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

The goal of this discussion is to highlight strategies and measurements that may be useful for finding the area of parallelograms with different characteristics or when given different information.

For each parallelogram, display 2 or 3 approaches used by previously selected students and share their strategies with everyone. If time allows, invite students to briefly describe their work. If an important strategy is not mentioned, bring it up and illustrate it. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Ask students:

• “How are the strategies the same? How are they different?”
• “Did anyone solve the problem the same way, but would explain it differently?”

After hearing from students about all parallelograms, consider asking the following questions. Provide access to tracing paper or a copy of the drawing in case students wish to verify their thinking by physically cutting and rearranging pieces.  

• “Which strategy—decomposing and rearranging, or enclosing and subtracting—seems more practical for finding the area of a shape similar to Parallelogram B? Why?” (Enclosing and subtracting, because it can be done in fewer steps. Decomposing the figure into small pieces could get confusing and lead to errors.)
• “If you decomposed Parallelogram C into a right triangle and another shape, how do you know that the cut-out piece actually fits on the other side, given that there's no grid to use?” (The two opposite sides of a parallelogram are parallel, so the longest side of the right triangle that is rearranged would match up perfectly with the segment on the other side.)
• “Three measurements are shown for Parallelogram C. Which ones did you use? Which ones did you not use? Why and why not?” (The 4 units and 6 units are side lengths of a rectangle that has the same area as that of the parallelogram. If we decompose the parallelogram with a vertical cut and move the piece on the left to the right to make a rectangle, the 4.5-unit length is no longer relevant.)