In this unit, students reason about areas of polygons and surface areas of polyhedra, building on geometric understandings developed in earlier grades.

In grade 3, students found the area of rectangles with whole-number side lengths. They also found the area of rectilinear figures by decomposing them into non-overlapping rectangles and adding those areas. Students used a formula for the area of rectangles in grade 4 and found the area of rectangles with fractional side lengths in grade 5.

In this unit, students extend their reasoning about area to include shapes that are not composed of rectangles. They use strategies such as decomposing and rearranging to find areas of parallelograms and generalize their process as a formula. Their work with parallelograms then becomes the basis for finding the area of triangles. Students see that other polygons can be decomposed into triangles and use this knowledge to find areas of polygons.

Next, students calculate the surface areas of polyhedra with triangular and rectangular faces. They study, assemble, and draw nets of prisms and pyramids and use nets to determine surface areas. Students also learn to use exponents ² and ³ to express surface areas and volumes of cubes and their units.

In many lessons, students engage in geometric work without a context. This design choice is made in recognition of the significant intellectual work of reasoning about area. Later in the unit, students have opportunities to apply their learning in context.

Students will draw on the work here to further study exponents later in grade 6 and to find volumes of prisms and pyramids in grade 7. Their understanding of “two figures that match up exactly” will support their work on congruence and rigid motions in grade 8.

A note about multiplication notation:

Students in grade 6 will be writing algebraic expressions and equations involving the letter . Because is easily confused with the “cross” notation for multiplication, , these materials use the “dot” notation for multiplication. Starting a few lessons into the unit, students will see, for instance, instead of . The notation will be new to many students, so they will need explicit guidance in using it.

A note about tools:

Students are likely to need physical tools to support their reasoning. For instance, they may find that tracing paper is an excellent tool for verifying that figures “match up exactly.” At all times in the unit, each student should have access to a geometry toolkit, which contains tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles. Access to the toolkit also enables students to practice selecting appropriate tools and using them strategically (MP5). In a digitally enhanced classroom, apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as comparing, explaining, and describing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Compare

• Geometric patterns and shapes (Lesson 1).
• Strategies for finding areas of shapes (Lesson 3) and polygons (Lesson 11).
• The characteristics of prisms and pyramids (Lesson 13).
• The measurements and units of 1-, 2-, and 3-dimensional attributes (Lesson 16).
• Representations of area and volume (Lesson 17).

Explain

• How to find areas by composing (Lesson 3).
• Strategies used to find areas of parallelograms (Lesson 4) and triangles (Lesson 8).
• How to determine the area of a triangle using its base and height (Lesson 9).
• Strategies to find surface areas of polyhedra (Lesson 14).

Describe

• Observations about decomposition of parallelograms (Lesson 7).
• Information needed to find the surface area of rectangular prisms (Lesson 12).
• The features of polyhedra and their nets (Lesson 13).
• The features of polyhedra (Lesson 15).
• Relationships among features of a tent and the amount of fabric needed for the tent (Lesson 19).

In addition, students are expected to justify claims about the base, height, or area of shapes; generalize about the features of parallelograms and polygons; interpret relevant information for finding the surface area of rectangular prisms; and represent the measurements and units of 2- and 3-dimensional figures. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson	new terminology
	receptive	productive
6.1.1	area region plane gap overlap
6.1.2	area compose decompose rearrange two-dimensional
6.1.3	shaded strategy
6.1.4	parallelogram opposite (sides or angles)	quadrilateral
6.1.5	base (of a parallelogram or triangle) height corresponding expression represent
6.1.6	horizontal vertical
6.1.7	identical	parallelogram
6.1.8	diagram	base (of a parallelogram or triangle) height compose decompose rearrange
6.1.9	opposite vertex
6.1.10	vertex edge
6.1.11	polygon	horizontal vertical
6.1.12	face surface area	area region
6.1.13	polyhedron net prism pyramid base (of a prism or pyramid) three-dimensional	polygon vertex edge face
6.1.15		prism pyramid
6.1.16	volume appropriate quantity	two-dimensional three-dimensional
6.1.17	squared cubed exponent edge length
6.1.18	value (of an expression)	squared cubed net
6.1.19	estimate description	surface area volume