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Lenape Regional High School District – BOE Approved 2/15/12
Geometry
[Implement start year (2012-2013)]
Eileen Meyers ,[email protected], ext. 8682, Stephanie Rieder-Ritter,[email protected], ext. 8768, Nikki Scott, [email protected], ext.
8808, Joanne Wagner, [email protected], ext. 8887
Unit 5 Circles
Level 2: Approximately 15 days
Stage 1 – Desired Results
Established Goals
2009 NJCCC Standard(s), Strand(s)/CPI #
(http://www.nj.gov/education/cccs/2009/final.htm)
Common Core Curriculum Standards for Math and English
(http://www.corestandards.org/)
st
21 Century Themes
( www.21stcenturyskills.org )
____ Global Awareness
x Financial, Economic, Business and Entrepreneurial Literacy
Civic Literacy
Health Literacy
Environmental Literacy
Quantities N-Q: #1-3
st
Congruence G-CO: #1,12,13
Circles G-C: #1-5
Expressing Geometric Properties with Equations G-GPE: #1,4
Modeling with Geometry G-MG: #1
21 Century Skills
Learning and Innovation Skills:
x Creativity and Innovation
x Critical Thinking and Problem Solving
Communication and Collaboration
Information, Media and Technology Skills:
Information Literacy
Media Literacy
ICT (Information, Communications and Technology) Literacy
Life and Career Skills:
Flexibility and Adaptability
x Initiative and Self-Direction
Social and Cross-Cultural Skills
x Productivity and Accountability
Leadership and Responsibility
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Enduring Understandings:
Students will understand that . . .
EU 1
geometric relationships and definitions can be used to create geometric
figures and solve real world problems.
EU 2
good decision making requires logical reasoning based on known facts.
EU 3
algebraic properties of equality are used in geometry to solve problems
and justify reasoning.
EU 4
a circle is a unique geometric shape that appears in nature and in
everyday objects.
EU 5
there is a specific relationship between the circumference and the
diameter of a circle.
EU 6
relationships exist between angles and arc measurements.
Essential Questions:
EU 1,5
How can we use circles to measure objects?
EU 1,2,3,6
How are the lengths of tangent segments, secant segments, and
chords related and applied?
How are angles and intercepted arcs of circles related and
applied?
EU 1,4,5,6
Why are circles special as a geometric shape?
EU 1,2,3,4,5,6
How do we use circles to model and solve real world situations?
When solving real world problems, what assumptions have to be
made?
EU 2,3,5,6
How do we know which method to use when solving problems involving
circles?
EU 4
Where do we find circles in nature?
EU 4,5
Why are mathematicians still looking for the digits of pi?
EU 5
What is the relationship between the circumference and the diameter of a
circle?
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Knowledge:
Students will know . . .
Skills:
Students will be able to . . .
EU 1,2,3,5
circumference and area can be determined from the diameter of a circle.
EU 1,2,3
solve problems involving segment lengths related to a circle.
EU 1,2,3,4,5,6
relationships exist among the many segments, angles and arcs related to
a circle.
EU 1,2,3,5
determine an appropriate method to use in order to provide the
best possible answer.
calculate the area of a sector of a circle.
EU 4,5
pi is an irrational number that was discovered thousands of years ago and
is still being studied today.
EU 1,2,3,6
solve problems involving relationships between angle measures and arc
measures.
EU 1,2,3,5,6
determine arc lengths.
determine area of sectors.
EU 1,2,3,4,5,6
identify and describe tangents, secants, radii, diameters, arcs, area and
circumference.
EU 5
determine the equation of a circle given its center and radius.
EU 6
construct regular polygons inscribed in a circle.
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Stage 2 – Assessment Evidence
Recommended Performance Tasks:
You are a stained-glass window designer and have been asked by a local church to fix a window that has been vandalized. The original window has a
frame shaped like a semicircle, which has a crack in it and also needs replacing. The semicircle is divided into 4 congruent regions.
What is the perimeter of the window?
What is the area of each section?
If stain glass has a cost of $50/sq.ft., how much is it to replace 3 of the 4 sections?
If trim is used to frame the interior and exterior of the window, how much trim is needed for the repair?
What is the area of the wooden frame?
Wood cost $1.25/sqft, how much does it cost to replace the frame?
Have the students write up a bill for the local church indicating the amount of material needed for each repair and the cost of each repair. Also
indicate the overall cost.
All other specifications and requirements are to be determined by the classroom teacher.
2in
40in
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Circles, circles, circles.
For centuries, artists have used the circle in their designs. Some patterns are crafted intertwining circles; some have disturbed the symmetry of the
circle to create optical illusions. After staring at these patterns for a few second, they seem to pulsate.
Have students use geometric terms to describe how figure A and B are related.
Students will make their own optical illusion art by transforming a target design like Figure A. They are to use geometric terms to describe
their design.
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Other Recommended Evidence:
Test on circle relations: segments, arcs and angles.
Quizzes: finding angle, arc and segment measures.
Assessed Elements from Recommended Performance Task.
Other teacher graded evaluations.
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Stage 3 – Learning Plan
Suggested Learning Activities:
In-class journal to include: definitions, diagrams, theorems, etc. (A)
TI-Nspire Activity: Circles – Angles and Arcs(A, M)
o http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=5047&id=13172

Students will know the definitions of and identify central angles, major and minor arcs, intercepted arcs, and inscribed angles of
a circle.

Students will determine and apply the following relationships:
Two inscribed angles intercepting the same arc have the same measure.
An inscribed angle measure of 90° results in the endpoints of the intercepted arc lying on a diameter.
The measure of an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc.

Students will construct viable arguments and critique the reasoning of others.

Students will look for and express regularity in repeated reasoning.
Class discussion (A,M,T)
Research the fountains of Versailles and determine the cost of restorations on the fountains. (A,M,T)
A piece of circular plate was recently dug up on an island in the Mediterranean. The discoverer of the plate wishes to calculate the diameter of
the original plate. Describe how he could do this. (A,M,T)
Create a new clock design of the Floral Clock in Niagara Falls, Canada, given the circumference. Students need to determine the lengths of the
hands of the clock, the area of each of the 12 sectors in which the letters to spell out Niagara Parks, etc. (A,M,T)
LRHSD (2011) Adapted from ASCD © 2004
Lenape Regional High School District – BOE Approved 2/15/12
Approximate timeline for Level 2: 15 Days
Identify and use parts of circles - day 1
Solve problems involving the circumference of a circle – day 1
Identify central angles, major arcs, minor arcs, and semicircles, and find their measures - day 2
Find arc lengths and find area of a sector– days 3 and 4, day 5 quiz
Recognize and use relationships between arcs and chords - day 6
Find measures of inscribed angles, find measures of angles of inscribed polygons – day 7 and day 8 quiz
Use properties of tangents, solve problems involving circumscribed polygons – day 9
Find measures of angles formed by lines intersecting on or inside a circle – day 10
Find measures of angles formed by lines intersecting outside the circle – day 10 and day 11 quiz
Find measures of segments that intersect in the interior of a circle – day 12
Fnd measures of segments that intersect in the exterior of a circle – day 12
Write the equation of a circle day 13
th
Activities may be incorporated into the timeline on the appropriate days; a unit circle test should be given on the 15 day.
Quiz days can be half a period; this is at the teacher’s discretion.
LRHSD (2011) Adapted from ASCD © 2004