Kyra Kopinski
Kyra Kopinski
Kyra Kopinski
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Course 2/Geometry - Grade 10<br />
Approximately 5 days<br />
Geometer’s Sketchpad<br />
By <strong>Kyra</strong> <strong>Kopinski</strong><br />
Niagara Falls High School<br />
2001
Table of Contents<br />
OVERALL OBJECTIVES........................................................................................2<br />
MATERIALS USED .................................................................................................4<br />
OVERVIEW OF UNIT ..............................................................................................5<br />
DAY 1 – DISCOVERING INTERIOR ANGLES ......................................................6<br />
DAY 2 – DISCOVERING EXTERIOR ANGLES.....................................................9<br />
DAY 3 – DISCOVERING VERTICAL ANGLES AND CORRESPONDING<br />
ANGLES.................................................................................................................11<br />
DAY 4 – REVIEW OF ANGLES FORMED WHEN TWO PARALLEL LINES ARE<br />
CUT BY A TRANSVERSAL..................................................................................13<br />
DAY 5 – REVIEW...................................................................................................15<br />
Project I2T2 – 2001 1
Overall Objectives<br />
♦ Students will practice using Geometer’s Sketchpad (GSP)<br />
♦ Students will discover the relationships among the angles formed when<br />
two parallel lines are cut by a transversal<br />
♦ Students will be able to define the following:<br />
1. Interior angles<br />
2. Alternate interior angles<br />
3. Same side interior angles<br />
4. Exterior angles<br />
5. Alternate exterior angles<br />
6. Vertical angles<br />
7. Corresponding angles<br />
New York State Learning Standards addressed:<br />
Standard 3 - Students are solving problems (ex. finding angle<br />
measures when parallel lines are cut by a transversal) using geometry<br />
and algebra.<br />
Standard 6 – Students can make the connection between mathematics<br />
and technology. They are able to discover certain mathematical<br />
concepts through the use of technology (ex. by creating the parallel<br />
lines cut by a transversal, and by measuring the angles using GSP, they<br />
are able to discover alternate interior angles and their properties).<br />
NCTM Standards addressed:<br />
Geometry Standard<br />
– After constructing parallel lines cut by a transversal, students can<br />
explore the relationship among the angles formed. They can “make<br />
conjectures” about the angles that are formed and test their<br />
conjectures through the use of GSP.<br />
Project I2T2 – 2001 2
Number and Operations Standard<br />
– When parallel lines are cut by a transversal, the students must find<br />
the angles formed. By doing this, students become “fluent” in<br />
operating with real numbers either by using “pencil and paper or by<br />
mental computation.”<br />
- Students may also test the “reasonableness” of their findings for<br />
the angle measures.<br />
Connections Standard<br />
- Students can make the connection between parallel lines and the<br />
“real world”. For example – telephone wires are parallel, the steps on a<br />
ladder are parallel, the lanes on the highway are parallel, piano keys<br />
are parallel and so forth.<br />
Project I2T2 – 2001 3
Materials Used<br />
• Textbook: Integrated Mathematics Course II<br />
Amsco School Publications, Inc.<br />
Second Edition<br />
Copyright 1982<br />
Chapter 5<br />
Pages 207- 221<br />
• Textbook: Sequential Mathematics Course 2<br />
West Sea Publishing Company<br />
Copyright 1990<br />
Chapter 4<br />
Pages 27 – 32<br />
• Textbook: Informal Geometry<br />
Merill Publishing Company<br />
Copyright 1988<br />
Chapter 7<br />
Pages 177 – 199<br />
• Textbook: Exploring Geometry with the Geometer’s Sketchpad<br />
Key Curriculum Press<br />
Copyright 1999<br />
Chapter 1<br />
Pages 17-18<br />
• Software: Geometer’s Sketchpad<br />
Version 3<br />
Key Press Curriculum<br />
Project I2T2 – 2001 4
Overview of Unit<br />
Day 1: Students will begin by defining parallel lines and transversals. The<br />
students will then use Geometer’s Sketchpad to construct two parallel lines<br />
and a transversal that cuts through them (this should be done with the<br />
teacher’s assistance). They will then be asked to find the interior angles,<br />
and to discover the properties of the interior angles. This will lead to the<br />
students defining alternate interior angles and same side interior angles.<br />
Day 2: On their own, students will repeat the procedure for Day 1<br />
(Students will construct two parallel lines cut by a transversal), and will then<br />
determine where the exterior angles are located. Next, students will define<br />
exterior angles and alternate exterior angles. Students should save a copy<br />
of their work for the Day 3 activities.<br />
Day 3: Students will retrieve their saved copy of their work from Day 2.<br />
Today, students will explore the properties of vertical angles and<br />
corresponding angles, and define both.<br />
Day 4: Students will demonstrate their understanding of the angles formed<br />
by parallel lines. They will complete two activities to test their knowledge.<br />
Day 5: Students will demonstrate an understanding of the material covered<br />
in this chapter. They will work on a crossword puzzle that tests their<br />
knowledge of the definitions. Then they will work on a “chapter test” from<br />
the Informal Geometry textbook to test their ability to apply the concepts<br />
learned throughout the week.<br />
Project I2T2 – 2001 5
Day 1 – Discovering Interior Angles<br />
Objectives:<br />
♦ Students will determine where the interior angles lie when two parallel<br />
lines are cut by a transversal<br />
♦ Students will observe the location of interior angles. They will then<br />
discover the properties of same side interior angles and of alternate<br />
interior angles.<br />
Definitions needed:<br />
Parallel lines – lines that never intersect and have no points in common<br />
Transversal – a line that intersects two other lines in two different points<br />
Supplementary angles-two angles whose degree measure sums to 180°<br />
Interior angles-angles located within the parallel lines; there are four<br />
Alternate Interior Angles-a pair of interior angles on opposite sides of the<br />
transversal, not sharing a common vertex; they are congruent<br />
Same side interior angles – angles that are on the same side of the<br />
transversal; they are supplementary<br />
Once students are able to prove that two lines are parallel (they should have<br />
done this prior to this lesson), they are able to explore two parallel lines<br />
that are cut by a transversal. There are several types of angles that are<br />
formed when this occurs. Using GSP, we will discover these angles and the<br />
relationships among them.<br />
Activity<br />
• Using GSP, have students construct a line by choosing the line tool<br />
• Label the two existing points (A and B) on the line by using the text tool<br />
• Create a point not on line AB using the point tool and label it C<br />
• Select the line AB, and also select point C at the same time by holding<br />
down the shift key<br />
• Go to the construct menu and select “parallel line”<br />
• Select either point A or point B and select point C, then go to the<br />
construct menu and choose “line”<br />
Project I2T2 – 2001 6
• Create five more points and label them D, E, F, G and H. You can do this<br />
by selecting the line you wish to create the points on and choosing from<br />
the construct menu, “point on object.” Then drag the random point you<br />
created to the desired location on the line. See diagram below.<br />
G<br />
D<br />
A<br />
B<br />
E<br />
C<br />
F<br />
H<br />
• Determine where the interior angles lie and name them<br />
• Measure all four interior angles<br />
Note: You must select three points in order to measure an angle<br />
G<br />
D<br />
A<br />
m DAC =136°<br />
m BAC =44°<br />
B<br />
m ECA =44° m FCA =136°<br />
E<br />
C<br />
F<br />
H<br />
Project I2T2 – 2001 7
Discovery<br />
1. What relationship is there among the interior angles?<br />
-The two interior angles that lie on the same straight line sum to 180°<br />
(they are supplementary) Ex. m
Day 2 – Discovering Exterior Angles<br />
Objectives:<br />
♦ Students will use their information gathered from Day 1 to locate the<br />
exterior angles and alternate exterior angles, and then to find their<br />
measures.<br />
Definitions needed:<br />
Exterior angles-angles located outside the parallel lines; there are four<br />
Alternate Exterior angles-a pair of exterior angles on opposite sides of the<br />
transversal not sharing a common vertex; they are congruent<br />
*The first several steps are the same as Day 1, except now the students will<br />
work on their own to construct the lines, measure the angles, and discover<br />
the relationships among exterior angles.<br />
• Using GSP, have students construct a line by choosing the line tool<br />
• Label the two existing points (A and B) on the line by using the text tool<br />
• Create a point not on line AB using the point tool and label it C<br />
• Select the line AB, and also select point C at the same time by holding<br />
down the shift key<br />
• Go to the construct menu and select “parallel line”<br />
• Select either point A or point B and select point C, then go to the<br />
construct menu and choose “line”<br />
• Create five more points and label them D, E, F, G and H<br />
• Determine where the exterior angles lie<br />
• Measure all four exterior angles<br />
Project I2T2 – 2001 9
m GAD = 44¡<br />
D<br />
G<br />
A<br />
m GAB = 136¡<br />
B<br />
E<br />
C<br />
F<br />
m ECH = 136¡ m FCH = 44¡<br />
H<br />
Discovery<br />
1. What relationship is there among the exterior angles?<br />
-The two exterior angles that lie on the same straight line sum to 180°<br />
Ex. m
Day 3 – Discovering Vertical Angles and<br />
Corresponding Angles<br />
Objectives:<br />
♦ Students will be able to locate and identify vertical and corresponding<br />
angles and determine the relationship between their measures.<br />
Definitions needed:<br />
Vertical angles-angles formed by intersecting lines; vertical angles are<br />
congruent<br />
Corresponding Angles-a pair of angles on the same side of the transversal,<br />
not sharing a common vertex, one interior and one exterior; they are<br />
congruent<br />
• Refer to saved copy from Day 2<br />
• Determine where the vertical angles lie<br />
• Measure these angles<br />
G<br />
m GAD = 44¡<br />
m GAB = 136¡<br />
D<br />
A<br />
m DAC = 136¡ m BAC = 44¡<br />
B<br />
m ECA = 44¡ m FCA = 136¡<br />
E<br />
C<br />
F<br />
m ECH = 136¡ m FCH = 44¡<br />
H<br />
Project I2T2 – 2001 11
Discovery<br />
1. What relationship is there among these angles?<br />
-Vertical angles are congruent. Ex. m
Day 4 – Review of angles formed when two<br />
Parallel Lines are cut by a transversal<br />
Objectives:<br />
♦ Students will demonstrate their knowledge of pairs of angles formed<br />
when two parallel lines are cut by a transversal. The angles are listed<br />
below:<br />
1. Interior angles<br />
2. Alternate interior angles<br />
3. Exterior angles<br />
4. Alternate exterior angles<br />
5. Same side interior angles<br />
6. Corresponding angles<br />
7. Vertical angles<br />
Activity 1:<br />
Given: Lines l and m are parallel, and line n is a transversal.<br />
n<br />
l<br />
1 2<br />
3<br />
4<br />
m<br />
6<br />
5<br />
7 8<br />
Complete the following table:<br />
Angles Name of Angles Measures of<br />
Angles<br />
6 and 5<br />
6 and 7<br />
4 and 5<br />
3 and 5<br />
1 and 5<br />
1, 2, 7, and 8<br />
3, 4, 5, and 6<br />
Project I2T2 – 2001 13
Activity 2:<br />
• Have the students use GSP to create two parallel lines cut by a<br />
transversal<br />
• Next, have students label all eight angles formed<br />
• Have students give examples of each of the following:<br />
1. Interior angles<br />
2. Alternate interior angles<br />
3. Exterior angles<br />
4. Alternate exterior angles<br />
5. Same side interior angles<br />
6. Corresponding angles<br />
7. Vertical angles<br />
Students should print out their work when finished<br />
Homework: Integrated Mathematics – pages 220 - 221<br />
Project I2T2 – 2001 14
Day 5 – Review<br />
Objectives:<br />
♦ Students will demonstrate their knowledge of the definitions from<br />
the chapter<br />
♦ Students will demonstrate their ability to apply the concepts learned<br />
First, have the students complete the crossword puzzle on the following<br />
page, and then have them work on the “chapter test” from the Informal<br />
Geometry textbook on pages 185 – 187.<br />
Homework: Study for test on parallel lines<br />
Project I2T2 – 2001 15