Iulia Demeter
A five days unit
CIRCLES
LESSON 1: Circles, Arcs and Chords
LESSON 2: Tangents
LESSON 3: Central Angles and Inscribed Angles
LESSON 4: Other Angles
LESSON 5: Circles and Lengths of Segments
Lesson 1: Circles, Arcs and Chords
Objectives:
By the end of this lesson, the students should be able to:
* define the circle and terms related to them;
* apply the theorems about chords of a circle.
Materials:
* paper and pencils, pair of compasses
* computers with Geometer's Sketchpad
Standards (NCTM)*:
* Geometry_ analyze characteristics and properties of two dimensional shapes and
develop mathematical arguments about geometric relationships
* Reasoning and Proof_ make and investigate mathematical conjectures
* Measurement_ apply appropriate techniques, tools, and formulas to determine
measurements
* Communication_ communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
Lesson outline:
The teacher:
* lets the students know what the objectives of this lesson are;
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* introduces the definitions, drawing the picture of a circle on the blackboard:
circle: the set of all points in a plan at a given distance from a given point
radius: any segment that joining the center of the circle to a point of the circle
chord: a segment whose endpoints lie on the circle
* ask students to do the following:
Activity 1:
o Investigation - Dragging a line in a parallel manner** (using GSP):
+ In a new sketch draw a line AB and a circle with center C and radius CD
+ Drag the line in a parallel manner
+ Observe how the line meets the circle as you move the line
+ Observe in how many points a line can intersect a circle
+ Repeat the experiment by dragging the line with point A fixed
* introduces more definitions:
secant: a line that intersects the circle in two points
tangent: a line that intersects the circle in one point
congruent circles: circles that have the same radius
concentric circles: circles that have the same center
* In the following picture, the students are asked to name:
1. the center
2. the diameter
3. two radii
4. a tangent
5. two chords
6. a point of tangency
* gives the students a new assignment:
Activity 2 (using GSP):
o draw a circle and two chords that have the same lengths
o measure the arcs of the two chords
o what did you find?
o in a new sketch draw a circle, a chord and a diameter that is perpendicular on that
chord
o what do you observe?
o draw another circle with center O, and the chords AB and CD with the same
lengths
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o take S on AB and R on CD such that OS perpendicular on AB and OR
perpendicular on CD
o measure the segments OS and OR
o what do you observe?
o in a word document, state carefully all the observations you made during this
activity. Are you able to prove what you found?
Assessment:
The teacher should:
* constantly check the students' progress during the period class and offer help
whenever requested
* check students' GSP files/ word documents
Extensions:

Exploration 2.2- Midpoints of chords from Geometry Through the Circle with the
Geometer's Sketchpad
Lesson 2: Tangents
Objectives:
Upon completion of this lesson students will
* have discovered the theorems that relate tangents and radii
* have been proved the theorems discovered
Materials:
* paper, pencils, pair of compasses, protractor
* computers with Geometer's Sketchpad (GSP) and browser.
Standards (NCTM*):
* Geometry_ analyze characteristics and properties of two dimensional shapes and
develop mathematical arguments about geometric relationships
* Reasoning and Proof_ make and investigate mathematical conjectures
* Measurement _ apply appropriate techniques, tools, and formulas to determine
measurements
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Lesson outline:
The teacher:
* reminds the students what they have learned in the previous class;
* lets the students know what the objectives of this lesson are;
* explain the students how to do the assignments;
* after completing the assignments, brings the class together, asks the students to share
what they found and summarize their findings
* introduces the definitions of circumscribed polygons, inscribed circles and common
tangents of two circles.
The students perform the following activities:
Students' instructions
Activity 1 (using paper, pencil, protractor and pair of compasses):
* Draw a circle and a tangent to it
* Draw the radius to the point of tangency
* Measure the angle between the tangent and the radius
* Repeat the procedures two more times with different circles
* Make a conjecture about your findings (what is the relationship between the tangent
and the radius?) and prove it.
Activity 2 (using GSP)*:
In this activity you will learn to construct a line tangent to a circle at a point of the
circle:
* In a new sketch, draw a circle with center A through B
* Construct the point C on the circle
* Construct the line through C tangent to the circle (hint: first construct the segment
AC, then use the relationship that you discovered in the previous activity)
* Check your work: the line should remain tangent at C whether you drag C, A, or B.
Activity 3 (using GSP):
* Draw a circle and a point C outside the circle and construct two tangents lines (can
you construct more than two?) to the circle through C (ask for help if you need)
* Denote with A and B the points the tangency
* Measure the segments AC and BC
* Drag the point C and measure again the segments AC and BC
* Make a conjecture about your findings and prove it
State all your findings in a word document.
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Assessment:
The teacher should:
* constantly check the students' progress during the period class and offer help
whenever requested
* check students' GSP files/ word documents
Extension:
Finding the number of common tangents of two circles, using Common Tangents
applet.
Lesson 3: Central Angles and Inscribed Angles
Objectives:
Upon completion of this lesson, the students will:
* have understood the definitions of central and inscribed angles
* have discovered properties of central and inscribed angles
* have discovered the relationship between the central and inscribed angles that
intercept the same arc
Materials:
* Computers with Geometer's Sketchpad (GSP) and browser.
Standards (NCTM):
* Geometry _ analyzes characteristics and properties of two dimensional shapes and
develop mathematical arguments about geometric relationships
* Measurement_ apply appropriate techniques, tools, and formulas to determine
measurements
* Communication_ communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
Lesson outline:
The teacher:
* reminds the students what they have learned in the previous class;
* lets the students know what the objectives of this lesson are;
* introduces the definitions of central and inscribed angles:
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A central angle is an angle whose vertex is the center of the circle.
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords
of the circle
* explain the students how to do the assignments;
The students perform the following activities:
Students' instructions:
Activity 1* (using GSP):
* Construct a circle with center O
* Construct a central angle AOB
* Label 3 other points P, Q and R that are on the circle, but not on arc AB
* Construct and measure the angles APB, AQB and ARB (what kind of angles are
these?)
* Place point C on the circumference of the circle between A and B
o select the points B, C and A (in this order) and under “construct " choose “arc
through 3 points"
o select the arc you just created and then under "measure" choose "arc angle"
* Drag the construction and make careful observations based on the following
questions:
1. What is the relationship between the measures of the angles APB, AQB and
ARB?
2. What is the relationship between the measures of the angles AOB and APB?
3. What is the relationship between the measure of a central angle and the measure
of its intercepted arc?
4. Make a conjecture about the measure of an inscribed angle and the measure of its
intercepted arc.
Activity 2* (using GSP):
* Draw a large circle and create angle ABC inside the circle
* Measure angle ABC (ABC is a inscribed angle because it is formed by 2 chords of a
circle)
* Create point D on arc AC
* Select points A, D and C in this order and click on "create an arc through 3 points"
* Select the arc and measure the arc in the measure menu
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Find a relation between the measure of the angle and the measure of its arc and state
it in your word document.
* Move the point B on the circle and measure the angle ABC
* Repeat the previous step two more time.
What can you say about the measure of the angle ABC?
* Move the point C on the circle until CB becomes the diameter of the circle
* Measure the angle ABC. As above, measure the arc ADC
State the relationship between the measure of the angle ABC and the arc ADC. What
is the measure of the angle BAC? How do you explain?
* Move the point C on the circle until CB becomes tangent to the circle at point B
* Measure the angle ABC and the arc ADC
What kind of angle is angle ABC? State the relationship between the measure of the
angle ABC and the minor arc AC
Activity 3 (using GSP):
Make the following construction:
* Draw a large circle and inscribe a quadrilateral ABCD inside the circle
* Measure each angle of the quadrilateral.
State in a word document the relationship between opposite angles of an inscribed
quadrilateral and explain why is true.
Activity 4 (using GSP):
* Draw a large circle and inscribe a hexagon ABCDEF inside the circle (not need to be
a regular hexagon)
* Measure the angles of the hexagon
* Calculate the sums of alternating angles
State in a word document your findings and use your conjecture about inscribed
angles to try to explain why the sum of alternate angles of an inscribed hexagon is this
number.
Activity 5 (if time allows):
* ask the students to provide proofs for the conjectures made in Activity 2 from Lesson
1
* Investigate the following problem.
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*Note: As an alternative for Activity 1 it can be used Inscribed Angle & Central Angle
applet.
Assessment:
The teacher should:
* constantly check the students' progress during the period class and offer help
whenever requested
* check students' GSP files/ word documents
Extensions:
* More about the inscribed angles can be discovered using A Flashlight applet.
Closure:
* after completing the assignments, the teacher brings the class together, asks the
students to share what they found and summarize their findings.
Lesson 4: Other Angles
Objectives:
Upon completion of this lesson, the students will:
* have discovered the relationships among the arcs of a circle and the angles formed by
chords, secants, and tangents
* have solved problems using the theorems discovered
Materials:
* paper, pencils, pair of compasses, protractor
* computers with browser and Geometer's Sketchpad.
Standards (NCTM)*:
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* Geometry _ analyze characteristics and properties of two dimensional shapes and
develop mathematical arguments about geometric relationships
* Reasoning and Proof_ make and investigate mathematical conjectures
* Measurement_ apply appropriate techniques, tools, and formulas to determine
measurements
* Communication_ communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
Lesson outline:
The teacher:
* reminds the students what they have learned in the previous class;
* lets the students know what the objectives of this lesson are;
* organizes the students for the activity:
o forms groups of four students;
o each student in the groups is asked to prepare a worksheet with various examples
of one of the following:
1. a circle and two chords that intersect inside the circle
2. a circle and two secants that intersect outside the circle
3. a circle and two tangents that intersect
4. a circle, a tangent and secant that intersect outside the circle
o ask each student of the group to measure the various angles and arcs on their
worksheet and to find a relationship between the angle between the two lines and the
measures of the intercepted arc.
o each member of the group has to explain his (her) findings to the other members
of the groups
o the students in each group should work together to come up with a proof for
these theorems
* gives the students the following worksheet with problems to be solved independently
1. Let ABC be an isosceles triangle with measure of angle A equal to 100 degrees.
Denote by M, N, and P the points of intersection of the circle inscribed in the triangle
with the sides BC, AC and AB respectively. Find the measure of the angle MNP.
2. A secant and a tangent to a circle intersect in a 42 degree angle. The two arcs of the
circle intercepted by the secant and tangent have measures in a 7:3 ratio. Find the
measure of the third arc.
Assessment:
The teacher should:
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* constantly check the students' progress during the period class and offer help
whenever requested
* check students' GSP files/ word documents
Extensions:
* Investigation of the Quoits applet.
* Exploration 8.4 from Geometry Through the Circle with the Geometer's Sketchpad.
**
Lesson 5: Circles and Lengths of Segments
Objectives:
Upon completion of this lesson, the students will:
* have discovered the relationships involving products of parts of chords, parts of
secants and parts of tangents and secants
* have solved problems using the theorems discovered
Materials:
* paper, pencils, pair of compasses, protractor
* computers with browser and Geometer's Sketchpad.
Standards (NCTM)*:
* Geometry_ analyze characteristics and properties of two dimensional shapes and
develop mathematical arguments about geometric relationships
* Reasoning and Proof _ make and investigate mathematical conjectures
* Measurement_ apply appropriate techniques, tools, and formulas to determine
measurements
* Communication _ communicate their mathematical thinking coherently and clearly
to peers, teachers, and others
Lesson outline:
The teacher:
* reminds the students what they have learned in the previous class;
* lets the students know what the objectives of this lesson are;
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* explains to the students how to do the assignments:
Activity 1(using GSP):
* Draw a circle
* Choose the points A, B, C, and D on the circle and draw two intersecting chords AC
and BD using the points as endpoints
* Denote with O the point of intersection of the two chords
* Measure the lengths of the pieces of chords and compute the products AO · OC and
BO · OD.
* In a word document state what you discovered. Prove your finding. (Hint: What can
you say about the triangles AOD and BOC?)
Activity 1 (using GSP) - A dynamic Relation between Products of Lengths**:
* Construction:
o Construct a circle with center A through B and a point O outside the circle
o Construct a point D in the plane, then construct lines OD and OA
o intersect these lines with the circle and to form the chords PQ and RS
o Measure the distances OP, OQ, OR, and OS. Calculate the products OP· OQ and
OR· OS
o Make a conjecture using your findings and prove it.
* Experiment:
o Drag the point O or D, although one of the products will disappear when line OD
does not intersect the circle
o With O located outside the circle, construct a point T on the circle so that OT is a
tangent segment
o Measure OT and calculate OT²
+ Observe that this quantity is equal to the two products. Explain why this
seems natural as a limiting condition when line OD approaches OT , and the intersections
R and S both approach T
+ Instead of dragging O or D, drag the circle (the circle itself , not points A or
B) and observe the quantities as the line OD becomes tangent
o Measure the distance OA and the distance AP, which is the radius
o Calculate OA²-AP² and OP· OQ.
* Conclusions:
State your conclusions in a word document, using as guidelines the following
questions:
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1. If you denote OA by d and the radius of the circle AB by r, then what
relationship do you observe between the products OR· OS and OP· OQ and the quantity
d²- r²?
2. Write the distances OP and OQ in terms of d and r. Use this expression to write
the product OP· OQ as an expression in d and r.
3. Let t be the distance OT. What is the relationship among d, r ant t is given by the
Pythagorean Theorem? Does this fit with the product relations explored in this
investigation?
Assessment:
The teacher should :
* constantly check the students' progress during the period class and offer help
whenever requested
* check students' GSP files/ word documents.
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