What is the difference between “area” of a geometric figure and “surface area” of a geometric figure?
Regardless of the length of a circle’s circumference and its diameter (radii), what do all circles have in common?
How do geometric relationships and measurements help us to solve problems and make sense of our world?
How is visualization essential to the study of geometry?
How are the measures of attributes that characterize two- and three-dimensional geometric figures the same and/or different?
Lesson Plan B.5: Angle Relationship
Lesson Seed B.4: Circumference and Area of a Circle
Content Emphasis By Clusters in Grade 7
Progressions from Common Core State Standards in Mathematics
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Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
This unit incorporates students’ prior knowledge of two- and three-dimensional figures and the characteristics of their
angles. It focuses on building new connections between prior knowledge of geometrical figures and a new understanding of circles and angle
relationships in order to solve authentic problems involving area of two-dimensional objects, as well as surface area and volume of three-
At the completion of this unit on the use of properties of operations to generate equivalent expressions, the student will understand that:
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:
Interdisciplinary connections fall into a number of related categories:
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-Life and Mathematical Problems Involving Angle Measure, Area, Surface Area and Volume
area of a circle: A circle is the set of points in a plane that are all the same distance from a given point called the center. The
area of a circle is the number of square units enclosed within its circumference, the length around the circle at the same distance from the center.
circumference of a circle: A circle is the set of points in a plane that are all the same distance from a given point called the center.
The circumference is the length around a circle at the same distance from the center.
radius: The radius of a circle is a line segment that connects the center of a circle to a point on its circumference.
diameter: The radius of a circle is a line segment that connects the center of a circle to a point on its circumference.
Chord: The radius of a circle is a line segment that connects the center of a circle to a point on its circumference.
Center of a Circle: The radius of a circle is a line segment that connects the center of a circle to a point on its circumference.
A circle is the set of points in a plane that are all the same distance from a given point called the center. A circle is named by the
point that is its center, such as circle A.
pi: Pi (π) is the ratio of the circumference of any circle to its diameter. π ≈ 3.14 or
supplementary angles: Two angles whose measures add to 180° are supplementary angles.
complementary angles: Two angles whose measures add to 90° are complementary angles.
The “near parallelogram” diagram offers a visual argument that is useful in development of the area formula for a circle, A = πr²
Vertical angles are formed by two intersecting lines that share a common vertex. Vertical angles are opposite each other.
3 are vertical angles, as are
4. The measures of
vertical angles are congruent
Adjacent angles share a common side and a common vertex, and lie next to each other on their common side.
2 are adjacent angles, as are
Part II – Instructional Connections outside the Focus Cluster
rational numbers: Numbers that can be expressed as an integer, as a quotient of integers (such as 1
/2,4/3, 7, -1
/2, -4/3, –7 ), or as a decimal where the decimal part is either
finite or repeats infinitely (such as 2.75,–2.75 and 3.3333… and –3.3333) are considered rational numbers.
algebraic solution: An algebraic solution is a proof or an answer that uses letters (algebraic symbols) to represent numbers, and
uses operations symbols to indicate algebraic operations of addition, subtraction, multiplication division, extracting roots, and raising to
arithmetic solution: An arithmetic solution is a proof or an answer that uses rational numbers under the operations of addition,
subtraction, multiplication and division. .
inequality terminology: Words or phrases used in the context of problem-solving with inequalities include: more than, less than, at
least, no more than, minimum, maximum, and not equal to.