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 real-world 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, semi-circle.
• 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.

• 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 40-50 n-dnute 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 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 deten-nine 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 non-computer 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.
  

Sample Project - Circles with GSP

If the diameter is drawn perpendicular to the chord, it bisects the arcs
 
 

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 '