In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions. The lesson in this section is optional because it offers additional opportunities to practice standards that are not a focus of the grade.

In this section, students explore strategies for reasoning about the area of two-dimensional figures. The explorations highlight two principles about area:

• Figures that match exactly have equal areas. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
• Area is additive. If a given figure is decomposed into pieces, then the area of the given figure is the sum of the areas of the pieces. If a figure is composed from pieces that don't overlap, the sum of the areas of the pieces is the area of the figure. Rearranging the pieces doesn’t change their areas.

First, students compare the amount of the plane covered by different shapes by thinking about the relationships between the shapes. They also recall their understanding of what area is, refine it, and develop a shared definition.

Next, students use their definition of “area” and strategies, such as composing, decomposing, and rearranging, to reason about areas of tangram shapes. They create figures with certain areas and find the area of given shapes. Finally, students apply these reasoning strategies to find the areas of figures drawn on and off a grid.

A note about terminology: 

In Grade 6, the term “congruent” is not used to describe “two figures that match up exactly.” Instead, these materials use “identical,” “identical copies,” or similar terms. What “identical” means might require clarification (for instance, that it is independent of color and orientation). In Grade 8, students will learn to refer to such figures as “congruent” and to describe congruence in terms of rigid motions (reflections, rotations, and translations).

The term “polygon” is not used with students until it is defined later in the unit.

​In this section, students explore ways to find areas of triangles, generalize their observations as a formula, and use the formula to find the area of any triangle. They also apply their insights regarding triangles and parallelograms to find areas of other polygons.  

Students begin by investigating the relationship between triangles and parallelograms. They see that a parallelogram can always be decomposed into two identical triangles. Likewise, two copies of any triangle can be composed into a parallelogram. 

Next, students learn that triangles also have bases and heights, which correspond to those in a related parallelogram. They observe that the area of a triangle is half that of a related parallelogram that shares the same base and height. Students generalize their observations with the expression and use it to solve problems.

This section introduces students to polyhedra and surface area. Students apply their knowledge about areas of polygons to create nets and find surface areas of three-dimensional figures.

First, students learn that surface area is the number of unit squares that covers all the faces of a three-dimensional figure, without gaps or overlaps. They reason about the surface areas of rectangular prisms and figures built from unit cubes.

Then students explore the characteristics of a polyhedron and develop a working definition for one. They learn that a net is a two-dimensional representation of a polyhedron. Students assemble prisms and pyramids from nets and also visualize the polyhedron that can be assembled from a given net. Later, students create nets for given prisms or pyramids and use the nets to calculate the surface areas.

The section includes an optional lesson to help reinforce students’ understanding of surface area and volume as distinct attributes of three-dimensional figures.

In this section, students reason about areas of parallelograms. They learn about bases and heights and analyze the measurements that can be used to find the area of any parallelogram.

First, students use strategies they learned earlier in the unit to find the areas of given parallelograms. They see that one way to find the area of a parallelogram is by decomposing it and rearranging the pieces into a rectangle. Another way is to enclose it in a rectangle and subtract the areas of the extra pieces.

Along the way, students notice regularity in both the process of finding area (that it can be done using one or more related rectangles) and in the measurements that are useful for finding area (that they are side lengths of the related rectangles). Students make sense of these lengths as “bases” and “heights” of a parallelogram and learn to identify them. Then, students generalize the process of using bases and heights to find areas, express it as a formula, and use the formula to find the area of any parallelogram and to solve problems.

Two drawings of parallelograms on grids. On the left, a triangle in the bottom left and top right corner is in white and the center is colored blue. On the right, the image from the left is repeated, but the triangles in the corners are colored yellow. Arrows are drawn from the triangles to the right. The triangles two triangles are joined on the right to form a rectangle.