Group
#: Group I
Subject
: Mathematics  Geometry  Circles
Grade/Level: Sequential III
Commencement
content standard from MST
STANDARD 3  Mathematics
Students
will understand mathematics and become mathematically confident by conununicating
and reasoning mathematically, by applying mathematics in realworld settings.
and by solving problems through the integrated study of nwnber systems,
geometry, algebra, data analysis, probability and trigonometry.
Benchmark
standards:
Content
standards

Mathematical
Reasoning: Students use mathematical reasoning to analyze mathematical
situations, make conjectures, gather evidence and construct an argument.

Modeling/Multiple
Representation: Students use ,mathematical modeling/multiple representation
to provide a means of presenting, interpreting, communicating, and
connecting mathematical information and relationships.

Measurement:
Students use measurements in both metric and English measure to provide
a major link between the abstractions of mathematics and the real
world in order to describe and compare objects and data.

Patterns/Functions:
Students use patterns and functions to develop mathematical power,
appreciate the true beauty of mathematics, and construct generalizations
that describe patterns simply and efficiently.
Performance
standards
construct
simple logical arguments
follow
and judge the validity of logical arguments
represent
problem situations symbolically by using algebraic expressions, sequences,
tree diagrams, geometric figures, and graphs.
use geometric
relationships in relevant measurement problems involving geometric concepts.
derive
and apply formulas relating angle measure and arc degree measure in a
circle
prove
and apply theorems related to lengths of segments in a circle
use algebraic
and geometric iteration to explore patterns and solve problems use computers
and graphing calculators to analyze mathematical phenomena
Content
standards

Students will be able to use Geometers Sketchpad to
construct sketches with circles, lines, segments and angles.

Students will use Geometers Sketchpad to measure arcs
and segment length.

Students will define and identify properties associated
with circles: radius, diameter, chord, tangent, secant, central angle,
inscribed angle, arc measure, minor arc, major arc, semicircle.

Students will find and be able to state and apply
the relationships involving a diameter drawn perpendicular to a chord.

Students will find and be able to state and apply
relationships between angles and arc measures: central angle, inscribed
angle, angles formed by: 2 chords intersecting in a circle, chord
and tangent intersecting at point of tangency, 2 secants intersecting
outside a circle, tangent and secant intersecting outside a circle,
2 tangents intersecting outside a circle.

Students will find and be able to state and apply
relationships among line segments, chords and arcs: congruent chords
intercept congruent arcs, congruent chords are equidistant from the
center, arcs between parallel chords are congruent, diameter is longest
chord in a circle, perpendicular bisectors of chords intersect at
the center.

Students will find and be able to state and apply
relationships involxing tangents: a tangent is perpendicular to a
radius drawn to point of tangency, tangent segments are congruent.
Performance measures

Students
will be required to keep a folder, which includes printouts of all
activities they complete and a summary list of the properties, relationships
and theorems they have verified.

The
attached rubric will be used to assess the sketches and the students'
proficiency wit GSP.
The enabling
activities include 2 days of assessment activities which will be graded.
Enabling
Activities:
Describe
each day's activity separately or holistically plan for ten days of work.
Include all parameters of the setting including grouping, space, time
and props. Include some critical directions and questions for the classroom
dialogue or attach a worksheet of activity directions.
These
activities are written for use in a setting with one computer for each
pair or group of students. Each activity is designed to be completed in
a 4050 ndnute class period.
Day 1:
Introduction to Geometer's Sketchpad  Have students work in pairs. Introduce
Geometer's Sketchpad (GSP). Distribute the Quick Reference Guide by Key
Curriculum Press
Have
students boot up GSP and guide them through the construction of congruent
line segments. This will enable the students become familiar with the
construction tools and the techniques of constructing a segment, defining
and selecting a point, constructing a circle by center + radius, and
hiding a figure. Why are the segments congruent?

Have
students measure each segment and use GSP to record the lengths. What
happens when you change the length of the segments?

Have
students draw a third segment and try to make it the same length as
the other segments by measuring. What happens when you change the
lengths of the segments now? What is the difference between constructing
and drawing?

Be
sure the students know how to LABEL. Have them use the TEXT tool to
create a caption for their sketch including the names of the students
in the group. SAVE and PRINT the sketch. If there is time introduce
GSP TABLES for collecting and organizing data.
Day 2:
Investigating theorems with GSP  Diameter Perpendicular to a Chord 
This lesson will guide students through the process of using GSP to formulate,
investigate and verify theorems through construction and measuring.

Construct
a circle with a diameter. Draw a chord. Measure and record the length
of the chord, the segments of the chord and the measure of the major
arcs and minor arcs and the angles formed by the chord and the diameter.
Drag the endpoints of the chord to find how the relationships change.
When are the chord segments congruent? What happens to the arc measures?
What happens to the angle measures?

Have
each group write their conjectures. Discuss the conjectures with the
class. How can we create a construction to test the conjectures?

Have
students start a new sketch constructing a circle and a chord. Construct
a diameter which is perpendicular to the chord. Measure the segments
of the chord. Measure the arcs. Move the chord and observe what happens.
How is it different from what happened before?

Create
a table and record at least 3 different measures for each segment
and arc. Discuss the findings with your partner. Write a statement
summarizing all the relationships found.
Day 3:
Constructing and Investigating chords in a circle  Students will construct
congruent chords and determine properties. Students will discover how
to use chords to find the center of a circle. Students will construct
parallel chords and verify conjectures about thenl Students will be able
to find the slope of a line segment.

Have
students start a new sketch and construct a circle and a chord. Construct
a congruent chord by choosing a third point on the circle then constructing
another circle with a center at the third point and a radius equal
to the length of the first chord. This circle will define a chord
congruent to the first. MDE the second circle.

Have
the students measure to verify that the chords are congruent. Drag
the circle to detennine what happens when one chords length changes.
Why must these chords be congruent?

Measure
the arcs of the chords. What do you expect about the arc measure?

Have
the students construct the perpendicular bisector of each chord. Where
do the perpendicular bisectors intersect?

Construct
another chord and its perpendicular bisector. Write a conjecture based
on this experiment.

Delete
all of the chords but one. Construct a chord parallel to the remaining
chord. Measure the arcs. Drag the chord to move it. What happens to
the relationship of the two chords? How can we be sure they remain
parallel? Show how to measure slope. What happens to the arc measures?
Write a conjecture.
Day 4
and 5: Discovering properties of tangents  Students WM create sketches
to determine that the tangent to a circle is perpendicular to the radius
at the point of tangency and investigate how to use this property in constructions.
Students will discover that tangent segments are congruent.

Have
the students starts a new sketch constructing a circle and a radius.
Construct a secant at the endpoint of the radius. Measure the angle
formed by the radius and the secant.

Drag
the secant until it becomes a tangent. What happens to the angle measure?

Construct
another radius in the same circle. Construct a line perpendicular
to the radius at its endpoint. What is this line? What conjectures
can you make? How can you use this property to construct a tangent
to any circle?

Drag
the tangent until the lines intersect. Measure the distance from the
point of intersection to the point of tangency. What conjecture can
you make?

Verify
your conjecture with a new sketch. Construct a circle and 2 tangents
which intersect. Measure the lengths of the tangent segments. Write
a caption which explains the relationships and print out your sketch.

How
can we construct a circle that is tangent to a given line? Start a
new sketch. Construct a line. Choose a point on the line and construct
a perpendicular fine.
Choose
a point on the perpendicular line as the center, and construct a circle
by point and radius.

Hide
the perpendicular line to only leave the circle and the tangent. Construct
another circle tangent to the same line. Define common tangent.

Define
externally tangent and internally tangent circles. Do constructions
to illustrate these terms.
Day 6: Assessment Activity
 Students will use the skills and properties they have learned to do
an original construct with GSP.

Use
GSP to construct a circle inscribe in a triangle. Students should
be able to drag the circle to change the radius and the figure will
remain inscribed. Circulate around the room to verify this through
observations.

Have
the students hand in the sketch with a list of the steps they followed
to do this construction.

BONUS:
Construct a circle inscribed in a square. Write a summary of the steps
followed to do this construction.
Day 7
and 8: Verifying theorems about angle measures and circle  Students will
create sketches to illustrate and confirm angle measure theorems

Review
the angles that are associated with circles: central inscribed, formed
by 2 chords intersecting inside a circle, formed by a chord and a
tangent intersecting at the point of tangency, formed by 2 secants,
a tangent and a secant or 2 tangents intersecting outside the circle.

Have
students use GSP to construct each type of angle and use angle measure,
arc measure to determine the relationships.

Have
students use a caption on each sketch to report their conclusions
about the relationships. Print the sketches.

Have
students create a list of formulas to summarize the angle measure
properties they have found.
Day 9:
Working with theorems about angles in a circle  Students will create
sketches to illustrate some angle measure corollaries.

Start
a new sketch. Construct a circle and a diameter. Choose a point on
the circle. Construct 2 chords connecting the point and the endpoints
of the diameter.

Measure
the angle formed by the chords. Drag the vertex point. What happens
to the angle measure? Write your observations and an explanation of
the property you discovered. Is this related to a property we already
discovered?

Discuss
the conclusions with the class and define a corollary.

Construct
a circle and inscribe a quadrilateral. Can you predict something about
the opposite angles? Measure the angles of the quadrilateral. Drag
the quadrilateral to change its shape. Does the relationship remain
the same? Write another corollary.
Day 10
: Formal Assessment Activity  This can be done as a noncomputer exercise
with students working independently or in pairs.

Distribute
a sketch with a circle and multiple angles. Give some of the measures.
Using their summary sheets as a reference, have the students determine
the measures of all the arcs and angles shown. To document their work,
students should fill in a table identifing the arc or angle, stating
the measure and writing a reason or an explanation of how it was found.

Distribute
a sketch showing a diameter perpendicular to a chord. Have the students
write an explanation of the relationships and solve a number of algebra
problems based on the sketch.
Geometer's Sketchpad
Project
Sequential Math III  Circles

ATTEMPTED 
ADEQUATE 
PROFICIENT 
OUTSTANDING

SKETCHES

Some sketches
created printed and handed in

Multiple sketches
created, printed and handed in
Sketches relate
to problems
Most labels
are present

Almost all
sketches created, printed and handed in
Sketches correctly
represent the problems
Construction
techniques are evident
Correct labels
are present in most cases
Captions present

All sketches
created, printed and handed in
Sketches correctly
represent the problem and show insight into properties and relationships
Construction
techniques are use correctly
All labeling
is correct
Captions are
correct and demonstrate logical thinking

USE OF GSP

Needs help
using GSP
Only familiar
with a few functions and commands

Can use GSP
independently
Aware of most
functions and commands

Comfortable
using GSP independently
Can use most
functions and commands

Displays facility
using GSP
Knows functions
and commands and can apply for problem solving

TIME MANAGEMENT

Sometimes
unprepared for class
Sometimes
off task during class time
Assignments
late
Unfinished
work

Prepared for
class with notebooks and supplies
Works throughout
the class period
Assignments
on time with supervision

Productive
use of class time
Assignments
on time with no reminders

Productive
use of class time
Evidence of
work or preparation outside class
Assignments
on time and complete

Sample
Project  Circles with GSP
If the diameter is drawn perpendicular to the chord, it
bisects the arcs
arc DF 
53.558 
65.856 
31.740 
90.421 
arc CF 
53.556 
65.855 
31.740 
90.421 
arc CD 
107.113 
131.711 
63.479 
179.158 
major Arc CE 
123.107 
110.443 
154.138 
major Arc DE 
123.107 
110.443 
154.138 
major Arc CD 
246.214 
220.887 
308.276 
Conjecture:
The measure of a central angle is equal to the measure of its intercepted
arc.
Angle(DCE)
= 39.98 '
Angle(DCE)*2=79.97'
ArcAngle(D
to E) = 79.97'
Angle(DAE) 79.97'
Conjecture:
The measure of an inscribed angle is 1/2 the measure of its intercepted
arc.
Angle(DCE)
= 39.98 '
ArcAngle(D to E) 79.97 '
Angle(DCE)*2 79.97 '
