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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 1 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. Project I2T2 – 2001 2 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. Project I2T2 – 2001 3 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 Project I2T2 – 2001 4 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. Project I2T2 – 2001 5 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” Project I2T2 – 2001 6 • 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 Project I2T2 – 2001 7 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 Project I2T2 – 2001 8 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 Project I2T2 – 2001 9 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 Project I2T2 – 2001 10 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 Project I2T2 – 2001 11 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 Project I2T2 – 2001 12 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 Project I2T2 – 2001 13 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 Project I2T2 – 2001 14 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 Project I2T2 – 2001 15