Download Kyra Kopinski

Course 2/Geometry - Grade 10
Approximately 5 days
Geometer’s Sketchpad
By Kyra Kopinski
Niagara Falls High School
2001
Table of Contents
OVERALL OBJECTIVES ........................................................................................2
MATERIALS USED .................................................................................................4
OVERVIEW OF UNIT ..............................................................................................5
DAY 1 – DISCOVERING INTERIOR ANGLES ......................................................6
DAY 2 – DISCOVERING EXTERIOR ANGLES .....................................................9
DAY 3 – DISCOVERING VERTICAL ANGLES AND CORRESPONDING
ANGLES.................................................................................................................11
DAY 4 – REVIEW OF ANGLES FORMED WHEN TWO PARALLEL LINES ARE
CUT BY A TRANSVERSAL ..................................................................................13
DAY 5 – REVIEW...................................................................................................15
Project I2T2 – 2001
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Overall Objectives
♦ Students will practice using Geometer’s Sketchpad (GSP)
♦ Students will discover the relationships among the angles formed when
two parallel lines are cut by a transversal
♦ Students will be able to define the following:
1.
2.
3.
4.
5.
6.
7.
Interior angles
Alternate interior angles
Same side interior angles
Exterior angles
Alternate exterior angles
Vertical angles
Corresponding angles
New York State Learning Standards addressed:
Standard 3 - Students are solving problems (ex. finding angle
measures when parallel lines are cut by a transversal) using geometry
and algebra.
Standard 6 – Students can make the connection between mathematics
and technology. They are able to discover certain mathematical
concepts through the use of technology (ex. by creating the parallel
lines cut by a transversal, and by measuring the angles using GSP, they
are able to discover alternate interior angles and their properties).
NCTM Standards addressed:
Geometry Standard
– After constructing parallel lines cut by a transversal, students can
explore the relationship among the angles formed. They can “make
conjectures” about the angles that are formed and test their
conjectures through the use of GSP.
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Number and Operations Standard
– When parallel lines are cut by a transversal, the students must find
the angles formed. By doing this, students become “fluent” in
operating with real numbers either by using “pencil and paper or by
mental computation.”
- Students may also test the “reasonableness” of their findings for
the angle measures.
Connections Standard
- Students can make the connection between parallel lines and the
“real world”. For example – telephone wires are parallel, the steps on a
ladder are parallel, the lanes on the highway are parallel, piano keys
are parallel and so forth.
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Materials Used
•
Textbook: Integrated Mathematics Course II
Amsco School Publications, Inc.
Second Edition
Copyright 1982
Chapter 5
Pages 207- 221
•
Textbook: Sequential Mathematics Course 2
West Sea Publishing Company
Copyright 1990
Chapter 4
Pages 27 – 32
•
Textbook: Informal Geometry
Merill Publishing Company
Copyright 1988
Chapter 7
Pages 177 – 199
•
Textbook: Exploring Geometry with the Geometer’s Sketchpad
Key Curriculum Press
Copyright 1999
Chapter 1
Pages 17-18
•
Software: Geometer’s Sketchpad
Version 3
Key Press Curriculum
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Overview of Unit
Day 1: Students will begin by defining parallel lines and transversals. The
students will then use Geometer’s Sketchpad to construct two parallel lines
and a transversal that cuts through them (this should be done with the
teacher’s assistance). They will then be asked to find the interior angles,
and to discover the properties of the interior angles. This will lead to the
students defining alternate interior angles and same side interior angles.
Day 2: On their own, students will repeat the procedure for Day 1
(Students will construct two parallel lines cut by a transversal), and will then
determine where the exterior angles are located. Next, students will define
exterior angles and alternate exterior angles. Students should save a copy
of their work for the Day 3 activities.
Day 3: Students will retrieve their saved copy of their work from Day 2.
Today, students will explore the properties of vertical angles and
corresponding angles, and define both.
Day 4: Students will demonstrate their understanding of the angles formed
by parallel lines. They will complete two activities to test their knowledge.
Day 5: Students will demonstrate an understanding of the material covered
in this chapter. They will work on a crossword puzzle that tests their
knowledge of the definitions. Then they will work on a “chapter test” from
the Informal Geometry textbook to test their ability to apply the concepts
learned throughout the week.
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Day 1 – Discovering Interior Angles
Objectives:
♦ Students will determine where the interior angles lie when two parallel
lines are cut by a transversal
♦ Students will observe the location of interior angles. They will then
discover the properties of same side interior angles and of alternate
interior angles.
Definitions needed:
Parallel lines – lines that never intersect and have no points in common
Transversal – a line that intersects two other lines in two different points
Supplementary angles-two angles whose degree measure sums to 180°
Interior angles-angles located within the parallel lines; there are four
Alternate Interior Angles-a pair of interior angles on opposite sides of the
transversal, not sharing a common vertex; they are congruent
Same side interior angles – angles that are on the same side of the
transversal; they are supplementary
Once students are able to prove that two lines are parallel (they should have
done this prior to this lesson), they are able to explore two parallel lines
that are cut by a transversal. There are several types of angles that are
formed when this occurs. Using GSP, we will discover these angles and the
relationships among them.
Activity
• Using GSP, have students construct a line by choosing the line tool
• Label the two existing points (A and B) on the line by using the text tool
• Create a point not on line AB using the point tool and label it C
• Select the line AB, and also select point C at the same time by holding
down the shift key
• Go to the construct menu and select “parallel line”
• Select either point A or point B and select point C, then go to the
construct menu and choose “line”
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•
Create five more points and label them D, E, F, G and H. You can do this
by selecting the line you wish to create the points on and choosing from
the construct menu, “point on object.” Then drag the random point you
created to the desired location on the line. See diagram below.
G
D
A
B
E
C
F
H
• Determine where the interior angles lie and name them
• Measure all four interior angles
Note: You must select three points in order to measure an angle
G
D
A
m DAC =136°
m ECA =44°
E
B
m BAC =44°
m FCA =136°
C
F
H
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Discovery
1. What relationship is there among the interior angles?
-The two interior angles that lie on the same straight line sum to 180°
(they are supplementary) Ex. m<DAC + m<BAC = 180°
-The interior angles that are on opposite sides of the transversal are the
same measure– they have just discovered alternate interior angles and
the fact they are congruent. Ex. m<BAC ≅ m<ECA
-Same side interior sum to 180° Ex. m<DAC ≅ m<ECA
2. Have the students define interior angles, alternate interior angles, and
same side interior angles.
*Teachers: Refer to movie01.gsp, movie01.gss, and follow the directions
below for a demonstration of constructing two parallel lines cut by a
transversal.
1. Open GSP File (movie01.gsp)
2. Select the points in alphabetical order
3. Open GSP Script file (movie01.gss)
4. Select play button in GSP script file window
Homework: West Sea – pages 27 and 29
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Day 2 – Discovering Exterior Angles
Objectives:
♦ Students will use their information gathered from Day 1 to locate the
exterior angles and alternate exterior angles, and then to find their
measures.
Definitions needed:
Exterior angles-angles located outside the parallel lines; there are four
Alternate Exterior angles-a pair of exterior angles on opposite sides of the
transversal not sharing a common vertex; they are congruent
*The first several steps are the same as Day 1, except now the students will
work on their own to construct the lines, measure the angles, and discover
the relationships among exterior angles.
•
•
•
•
•
•
•
•
•
Using GSP, have students construct a line by choosing the line tool
Label the two existing points (A and B) on the line by using the text tool
Create a point not on line AB using the point tool and label it C
Select the line AB, and also select point C at the same time by holding
down the shift key
Go to the construct menu and select “parallel line”
Select either point A or point B and select point C, then go to the
construct menu and choose “line”
Create five more points and label them D, E, F, G and H
Determine where the exterior angles lie
Measure all four exterior angles
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G
m GAD=
D
44¡
m GAB=
136¡
A
B
E
m ECH=
136¡
C
F
m FCH=
44¡
H
Discovery
1. What relationship is there among the exterior angles?
-The two exterior angles that lie on the same straight line sum to 180°
Ex. m<GAD + m<GAB = 180°
-The exterior angles that are on opposite sides of the transversal are
the same measure – they just discovered alternate exterior angles and
the fact they are congruent. Ex. m<GAB ≅ m<ECH
2. Have students define exterior angles and alternate exterior angles.
Students should save a copy of their work for the Day 3 activities.
Homework: None
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Day 3 – Discovering Vertical Angles and
Corresponding Angles
Objectives:
♦ Students will be able to locate and identify vertical and corresponding
angles and determine the relationship between their measures.
Definitions needed:
Vertical angles-angles formed by intersecting lines; vertical angles are
congruent
Corresponding Angles-a pair of angles on the same side of the transversal,
not sharing a common vertex, one interior and one exterior; they are
congruent
•
•
•
Refer to saved copy from Day 2
Determine where the vertical angles lie
Measure these angles
G
m GAD=
D
44¡
m DAC=
m GAB=
136¡
136¡
A
m BAC=
m ECA=
44¡
m FCA=
E
m ECH=
B
44¡
136¡
136¡
C
F
m FCH=
44¡
H
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Discovery
1. What relationship is there among these angles?
-Vertical angles are congruent. Ex. m<GAD ≅ m<BAC
•
•
•
Have students print out a copy of their work, take a sheet of tracing
paper, and trace the diagram
Take the copy on tracing paper, and cut down the middle of the two
parallel lines
Place the top part of the copy directly on the bottom part of the
copy
2. What relationship is there among these angles?
-They should notice that the angles match up exactly and therefore,
have the same angle measure
3. Have the students check their findings by measuring the angles to show
that corresponding angles are congruent
Homework: West Sea – pages 28 and 30
Informal Geometry – pages 182 - 183
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Day 4 – Review of angles formed when two
Parallel Lines are cut by a transversal
Objectives:
♦ Students will demonstrate their knowledge of pairs of angles formed
when two parallel lines are cut by a transversal. The angles are listed
below:
1.
2.
3.
4.
5.
6.
7.
Interior angles
Alternate interior angles
Exterior angles
Alternate exterior angles
Same side interior angles
Corresponding angles
Vertical angles
Activity 1:
Given: Lines l and m are parallel, and line n is a transversal.
n
l
1
2
3
4
6
m
5
7
8
Complete the following table:
Angles
Name of Angles
Measures of
Angles
6 and 5
6 and 7
4 and 5
3 and 5
1 and 5
1, 2, 7, and 8
3, 4, 5, and 6
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Activity 2:
•
•
•
Have the students use GSP to create two parallel lines cut by a
transversal
Next, have students label all eight angles formed
Have students give examples of each of the following:
1.
2.
3.
4.
5.
6.
7.
Interior angles
Alternate interior angles
Exterior angles
Alternate exterior angles
Same side interior angles
Corresponding angles
Vertical angles
Students should print out their work when finished
Homework: Integrated Mathematics – pages 220 - 221
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Day 5 – Review
Objectives:
♦ Students will demonstrate their knowledge of the definitions from
the chapter
♦ Students will demonstrate their ability to apply the concepts learned
First, have the students complete the crossword puzzle on the following
page, and then have them work on the “chapter test” from the Informal
Geometry textbook on pages 185 – 187.
Homework: Study for test on parallel lines
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