## Unit Overview

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Plan A.1: Areas (Composition/Decomposition)

Lesson Seed A.1: Surface Area of Irregular Figures

Lesson Seed A.3: Area of Composite Shapes

**Download Seeds, Plans, and Resources (zip)**

**Content Emphasis By Clusters in Grade 6**

**Progressions from Common Core State Standards in Mathematics**

### Unit Overview

Students in grade 6 build on their elementary school experiences with area by reasoning about relationships among shapes to determine area, surface area, and volume. Students determine areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms; find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine; reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths; and prepare for future work on scale drawings, constructions, and transformations by drawing polygons on a coordinate plane.

**Teacher Notes:**

- Teachers should prepare students to understand the relationships between two- and three-dimensional figures.
- Through working with manipulatives and authentic situations, teachers should help students see the origin of formulas for area, surface area, and volume.
- Teachers should provide students with opportunities to solve authentic problems by graphing figures on a coordinate plane.

**Enduring Understandings:**

At the completion of this unit on solving real-world and mathematical problems involving area, surface area, and volume, the student will understand that:

- Patterns can be used to predict attributes of design.
- Designs can change position without changing shape.
- Strategies for finding surface area and volume of any three dimensional figure will work for any similar three dimensional figure.
- Change in one linear dimension affects area measurements of polygons and circles, as well as surface area measurements and volume of prisms and cylinders.
- Formulas can be used to determine missing measurements of polygons, pyramids, and prisms.

**Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):
**

**6.G.A.1:**Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

**6.G.A.2:**Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas and to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

**6.G.A.3:**Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

**6.G.A.4:**Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

**Possible Student Outcomes:**

The student will be able to:

- combine triangles to create quadrilaterals, and partition quadrilaterals and other polygons into triangles.
- understand that the base and height of a triangle are also the length and width of a rectangle.
- discover that packing a right rectangular prism with unit cubes results in a volume measurement is the same as multiplying the lengths of the length, width, and height of the prism.
- draw polygons on the coordinate plane, given coordinates for the vertices.
- use nets of three-dimensional figures made up of rectangles and triangles to determine surface area.

**Evidence of Student Learning:**

*The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.*

**Fluency Expectations and Examples of Culminating Standards:**

**Common Misconceptions:**

Students may

- have difficulty distinguishing between area and surface area; and between surface area and volume.
- not be able to clearly envision the complete surface area of three-dimensional figures that are presented or drawn in a two-dimensional format due to:
- lack of spatial understanding that prevents some students from mentally rotating, unfolding, and manipulating three-dimensional objects.
- have difficulty with understanding the concept of perspective that is inherent with three-dimensional shapes.
- have difficulty with understanding the concept of conservation, that different shapes can have equivalent volumes.

**Interdisciplinary Connections:**

Interdisciplinary connections fall into a number of related categories:

*Literacy standards within the Maryland Common Core State Curriculum*

*Science, Technology, Engineering, and Mathematics standards*

*Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.*

**Sample Assessment Items: ***The items included in this component will be aligned to the standards in the unit and will include:*

*Items purchased from vendors*

*PARCC prototype items*

*PARCC public release items*

*Maryland Public release items*

*Formative Assessments*

**Interventions/Enrichments/PD: ***(Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)*

**Vocabulary/Terminology/Concepts: ***This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.*

**
Part I – Focus Cluster
Solve Real-World and Mathematical Problems Involving Area, Surface Area, and Volume **

**NOTE: None of the vocabulary, terminology, and concepts in this cluster are new, nor should they be particularly problematic for instruction of the related standards.**

**Part II – Instructional Connections outside the Focus Cluster**

**NOTE: None of the vocabulary, terminology, and concepts in this cluster are new, nor should they be particularly problematical for instruction of the related standards.**