Download Geometry /Unit M: Circles Unit Overview Math Florida Standards

Geometry /Unit M: Circles
Math Florida Standards
Unit Overview
Content Standards
In this unit, students prove basic theorems about circles, such as a tangent line is perpendicular to a
radius, inscribed angle theorem, and theorems about chords, secants, and tangents, dealing with
segment lengths and angle measures. They study relationships among segments on chords, secants,
and tangents as an application of similarity.
Textbook Resources
Pearson Prentice Hall Geometry
Chapter 12
Sections: 12.1, 12.2, 12.3, 12.4
MAFS.912.G-C.2.5
MAFS.912.G-C.1.2
MAFS.912.G-C.1.4
MAFS.912.G-C.1.3
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers can
implement with their students, and rubrics that help the
teacher interpret students' responses. Teachers using MFAS ask
students to perform mathematical tasks, explain their
reasoning, and justify their solutions. Rubrics for interpreting
and evaluating student responses are included so that teachers
can differentiate instruction based on students' strategies
instead of relying solely on correct or incorrect answers. The
objective is to understand student thinking so that teaching can
be adapted to improve student achievement of mathematical
goals related to the standards. Like all formative assessment,
MFAS is a process rather than a test. Research suggests that
well-designed and implemented formative assessment is an
effective strategy for enhancing student learning.
Mathematics Formative Assessment System Tasks
Standards for Mathematical
Practice
MAFS.K12.MP.6.1
MAFS.K12.MP.5.1
MAFS.K12.MP.7.1
Other Resources
Classroom Resources/ Manipulatives
GeoGebra
Lessons/ Activities
Setting up sprinklers
Seeking circle segments
Off on a tangent
Locating a warehouse
Interactive Digital Demonstrations:
Inscribed angle
Central angle theorem
Video Resources:
Virtual Nerd
You Tube
Khan Academy
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry /Unit M: Circles
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was
taught within these standards. Examples:



3.0
Compare two circles with different radii and the area of their defined sectors.
Students will investigate the relationship between central, inscribed and circumscribed angles.
Students will investigate the relationships between inscribed/circumscribed circles and incenters/circumcenters of a triangle.
 Students will utilize various geometric tools to construct a tangent line.
The student will:

2.0
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian
measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (MAFS.912.G-C.2.5)
 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and
circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius
intersects the circle. (MAFS.912.G-C.1.2)
 Construct a tangent line from a point outside a given circle to the circle. (MAFS.912.G-C.1.4)
 Students will construct inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a
circle (MAFS.912.G-C.1.3)
The student will recognize or recall specific vocabulary, such as:
1.0
The student will perform basic processes, such as:
A. Recall properties of similarity.
B. Understand the properties when right triangles are inscribed in a circle.
C. Understand why all circles are similar and be able to describe a sector of a circle in fraction form.
D. Be able to classify an angle as inscribed, circumscribed or central.
Describe a tangent line and be able to utilize the geometric tools.
With help, partial success at 2.0 content but not at score 3.0 content

Inscribed angles, Radii, Chords, Circumscribed angles, Diameter, Tangent, Radians, Intercepted arc, Sector, Arc length
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry /Unit M: Circles
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
MAFS.9-12.C-C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed and circumscribed angles;
inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
 Inscribed angles on a diameter are right angles.
 The measure of a central angle is equal to the measure of the intercepted arc.
 The measure of an inscribed angle is half the measure of its intercepted arc.
 The measure of the circumscribed angle is half the measure of the difference of the intercepted arcs.
 The radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Know
Do
Declarative knowledge: Facts, vocab., information
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Comprehension
 Be able to describe the relationships
between inscribed angles, radii, and chords
 Given the measure of intercepted arc(s), calculate the measure of the missing angle(s).
of a circle using the relationship between
 Identify central angles, inscribed angles, circumscribed angles, tangent line and chords on a circle from a drawing.
central, inscribed, and circumscribed angles;
 Construct and explain examples of central angles, inscribed angles, circumscribed angles, tangent line and chords on
inscribed angles on a diameter are right
a circle
angle.
Key Vocabulary such as,
 Radii, chords, inscribed angles, circumscribed
angles, tangent, point of tangency
Prerequisite skills: What prior knowledge (skills from previous grade levels) do students need to have mastered to be successful with this standard(s)?
 7.G.4-Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between
circumference and area of a circle.
 G.CO.1-Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and
distance around a circular arc.
 G.C.1-Prove that all circles are similar
Learning Goal(s)/Objective(s): (Based on standards above)
 I can describe the relationship among inscribed angles, radii and chords within a circle.
 I can identify and describe the relationship between central, inscribed and circumscribed angles.
 I understand and can identify that inscribed angles on a diameter are right angles.
 I understand and can identify a tangent line to a circle because it is perpendicular at the point the radius intersects the circle.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry /Unit M: Circles
Sample Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
Common Core State Standard: MAFS.9-12.C-C.1.2
Learning Progression
Sample Tasks
Score
1.
2.
3.
Find the measure of ∠2
Justify your reasoning
Write a conjecture, relating ∠2 to the given angle
I can…
4.0
3.5

Make and justify conclusions about the relationships between angles
and their intercepted arcs
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
1. Find the measure of each variable
I can…
3.0
2.5

Given the measure of intercepted arc(s), calculate the measure of the
missing angle(s).
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can identify the following vocabulary:
Inscribed angles, Radii, Chords, Central angle, Inscribed angles
2.0
Circumscribed angles, Tangent, Point of tangency
I can…
Identify the intercepted arc(s) of any angle related to a circle.
1.0
Using the diagram below:
1a. Classify ∠𝐹𝐴𝐵 as inscribed, central or circumscribed.
b. What is its intercepted arc?
2a. Classify ∠𝐷𝐸𝐹 as inscribed, central or
circumscribed.
b. What is its intercepted arc?
I need prompting and/or support to complete 2.0 tasks.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry /Unit M: Circles
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS.912.G-C.2.5
Design Question 1; Element 1
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius,
and define the radian measure of the angle as the constant of proportionality; derive the formula for the area
of a sector.
CCSS Mathematical Practice(s)
4. Model with Mathematics.
CCSS Mathematical Content Standard(s)
Design Question 1; Element 1
Marzano’s Taxonomy
3.0 (Analysis)
Teacher Notes
The direction of motion of the arm is not relevant to this task.
Questions to develop mathematical thinking,
possible misconceptions/misunderstandings,
how to differentiate/scaffold instruction,
anticipate student problem solving strategies
Task
*These tasks can either be teacher created or
modified from a resource to promote higher
order thinking skills. Please cite the source for
any tasks.
A robotic arm replaced a human arm after an accident. The first step in rehabilitation is range of motion. The
arm moved 54 degrees from the shoulder. If the longest fingertip traveled 38 inches, how long is the robotic
arm?
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015