Analysis in Geometry By Danielle Long Grade Level: 8 th Time: 5-50 minute periods Technology used: Geometer s sketchpad Geoboards NLVM website 1
NCTM Standards Addressed Problem Solving Geometry Algebra Representation NYS Standards Addressed Standard 1 Mathematical Analysis Standard 2 Information Systems Standard 3 Mathematics 2
Sources Used South Western Math Matters2 an Integrated Approach, Lynch and Olmstead. Chapter 3, pgs 91-124, Copyright 1995. http://www.matti.usu.edu/nlvm/nav/vlibrary.html Materials Needed Geometer s Sketchpad Access to NLVM website Straws (3in, 5in, 8in, 12in) Geoboard Math Matters textbook, Chapter 3 pg 92-124 3
Overall Objective students will discover important concepts of geometry pertaining to triangles, parallel lines and polygons students will be able to identify angles formed by parallel lines and transversals students will use technology to discover triangle postulates Students will determine the properties of quadrilaterals. 4
Overview of the Unit Day 1 Exploring Parallel Lines and Transversals Discussion parallel lines Discovery of angle relationships Assignment in book Day 2 Classifying Triangles Creating triangles on the Geoboard Straw Activity Assignment in book Day 3 Classifying Quadrilaterals Drawing 4 sided figures Geoboard Activity/Worksheet What s My Name Worksheet? Day 4 Angle Sums Creating triangles in Geometer s Sketchpad Measuring angles in any triangle Sum of angles in any polygon Assignment in book. Day 5 Triangle Congruency Definition of Congruency Proving postulates using NLVM website Congruency Problems Assignment in the book 5
Day 1 Objective Students will identify angles formed by parallel lines and transversals. Opening activity. Students will be asked to give examples of where they see parallel lines in their everyday lives. Some possible answers: telephone poles, railroad tracks, and street maps. Classroom Activity. Students will be given a worksheet. The worksheet provided requires students to create a set of parallel lines and a transversal on Geometer s Sketchpad. Students may require step by step directions to create this sketch. Please refer to Appendix A for directions on GSP. The activity allows the student to use the technology to make connections about the angles formed in this diagram. The teacher should walk around the room and assist any students that are having difficulty with the technology or assignment. Closing Activity Allow 5-10 minutes at the end of class for a full class discussion. Students should share their discoveries with the class. A student might say all the obtuse angles in their diagram were equal. To make this connection stronger the teacher might ask some students to share their measurements for the obtuse angles. Then ask the students if all their obtuse angles were also equal. Since all the measurements are different and each student had the same conclusion the class should get a better understanding that this equivalency is not a fluke. The teacher should summarize the main idea of this activity by restating Every angle in this diagram has a relationship, either the angles are equal to each other or they add up to be 180 degrees. Homework pg 95 1-23 All 6
Parallel Lines Sheet 1. Using GSP sketch a diagram of 2 parallel lines with a transversal through it. In the example below line A and B are parallel. 2. Label the diagram, and measure all angles. DIAGRAM 1 g Line A Line B h f b c a e d Draw and label the sketch you created in the space provided. 1.Line A and B are parallel to each other. Write out your own definition of what it means to be parallel. 2. What do you notice about all the obtuse angles? 3. What do you notice about all of the acute angle? 4. What is the sum of any obtuse angle and any acute angle on your diagram? 5. The interior angles are on the inside of the diagram they are: 7
6. The exterior angles are on the outside of the diagram they are: Alternate angles are non adjacent angles and are on opposite sides of the transversal. For example, in Diagram 1 Angle hfc and Angle fcd are examples of alternate interior angles. Write a statement that shows a relationship between alternate interior angles and alternate exterior angles 6. Write a statement about all the angles in this diagram. 8
Parallel Lines Sheet Answer Key 1. Using GSP sketch a diagram of 2 parallel lines with a transversal through it. In the example below line A and B are parallel. 2. Label the diagram, and measure all angles. Diagram 1 g Line A Line B h f b c a e d Draw and label the sketch you created in the space provided. f c e b d a g h 1.Line A and B are parallel to each other. Write out your own definition of what it means to be parallel. Parallel means that the lines will never cross, and they will always be the same distance apart. 2. What do you notice about all the obtuse angles? They are all equal to each other 3. What do you notice about all of the acute angle? 9
They are all equal to each other 4. What is the sum of any obtuse angle and any acute angle on this diagram? The sum equals 180 5. The interior angles are on the inside of the diagram they are: (answers vary depending on how student labels their diagram) Angle bde, Angle deg, Angle ced, and Angle edh. The exterior angles are on the outside of the diagram they are: Angle feg, Angle fec, Angle adh, and Angle bda. Alternate angles are non adjacent angles and are on opposite sides of the transversal. For example, in Diagram 1 Angle hfc and Angle fcd are examples of alternate interior angles. Write a statement that shows a relationship between alternate interior angles and alternate exterior angles. Alternate interior angles are equal to each other. Alternate Exterior angles are also equal to each other. 6. Write a statement about all the angles in this diagram. Either 2 angles are equal to each other or they add up to be 180 degrees. 10
Day 2 Triangle Classification Objectives: Using Geoboards and straws students will be able to classify triangles according to their sides. Opening activity: Have students define the following terms isosceles triangles, scalene triangles, equilateral triangles, acute and obtuse angles. Have students share their responses with their classmates. Classroom Activity Students will be paired off and be given geoboards and rubber bands. Together with their partners students are asked to complete the geoboard worksheet. The time allowed will be 10 minutes. After this time is completed, the will discuss their findings. Important part of the discussion includes: What is the relationship between the smallest angle and the smallest side? How many obtuse angles can one triangle have? Students will be given 12 straws cut into 3in, 5 in, 8 in and 12 pieces. Students are to try and create triangles using the 4 different straws recording their results. Teacher should walk around the room making sure the results are being recorded. Students will be asked if they can find connection with the lengths of sides and triangles. (i.e. Why can t we make a triangle with 3, 5, 8 and we can make a triangle with 8, 12, and 3.) With some prompting students should recognize that the sum of any 2 sides of a triangle is greater than the third side. Allow 10-15 minutes for this activity. Closing Students will do an exit pass explaining one new thing they learned about triangles and one thing they had already knew. Homework PG 96-97 Try these 3-5 Exercise 1-4; 18-22 11
Geoboard Activity Use your geoboard to create the following triangles. Trace your solution on the dot pattern provided. Important questions to look at Where is the largest angle always located? The smallest? Is it possible to make a triangle that has more than one obtuse angle? Acute Isosceles Obtuse Scalene Right Isosceles Obtuse Isosceles Right Scalene Acute Scalene 12
Geoboard Activity Answer key Use your geoboard to create the following triangles. Trace your solution on the dot pattern provided. Important questions to look at Where is the largest angle always located? Across from the largest side The smallest? The smallest angle is across from the smallest side Acute Isosceles Obtuse Scalene Right Isosceles G H B Obtuse Isosceles Right Scalene Acute Scalene J I K 13
Lesson 3: Classifying Quadrilaterals Lesson Objectives: Students will be able to recognize a trapezoid, parallelogram, rhombus, rectangle, and square Students will be able to describe the properties of the figures above. Opening Activity: Have students draw a four sided figure. Classroom Activity 1. Tell the students that all the figures drawn in the class have one thing in common. That they are quadrilaterals. Ask the students if they can guess what it means to be a quadrilateral. Have them write their guess down, walk around the room to look at student s responses. Ask a student what their response was. It should be It has four sides 2. Show a picture of a parallelogram. Ask the students how many sets of parallel lines does the parallelogram have? Ask the students to write down the pairs of parallel lines. Ask the students if there are any sides that are equal, and what types of angles are there in this figure. 3. Students will be given out Quadrilateral Classification worksheets. Geoboards and dot paper. Students may work with their partners to complete the work. Go around the room and ask prompting questions about the figures. Such as, What do you notice the difference is between a rhombus and a square? What do they have in common? Once the students have completed the worksheets, review the answers on the overhead. Ask the students to go back through the worksheets and find out what each of the five special case quadrilaterals have in common and what are there differences. Students should be thinking about what is the difference between a square and a rhombus. Go around the room and ask prompting questions. Some questions to pose: Can a square also be a rectangle? Can a rectangle also be described as a parallelogram? Why or why not? Summarize and review the main point of the activity on the overhead. Closure: Ask the students to draw/or name all the four sided figures that have one set of parallel lines and have them hand it in before leaving. HMWK Have students complete what s my name worksheet. ` 14
Quadrilateral Classification On your geoboards create the following quadrilaterals and record your answers on the dot grids. Then describe the figures. Things you might want to consider are: Are any sides equal. If so which ones? What types of interior angles are there? Which sides are parallel, if any? Are there any interior angles that are equal to each other? (Use a corner of a piece of paper to measure angles) Parallelogram Description Square Description: 15
Rectangle Description Trapezoid Description Rhombus 16
Description 17
What s my name? I am a quadrilateral with exactly one pair of parallel lines. I am a quadrilateral with all sides equal, but no right angles I am a rhombus with all 4 angles congruent My diagonals are bisect each other and are perpendicular to each other. (Hint you want to draw the diagonals first then connect the points) ` 18
Quadrilateral Classification On your geoboards create the following quadrilaterals and record your answers on the dot grids. Then describe the figures. Things you might want to consider are: Are any sides equal. If so which ones? What types of interior angles are there? Which sides are parallel, if any? Are there any interior angles that are equal to each other? (Use a corner of a piece of paper to measure angles) Parallelogram Description There are 2 sets of sides that are parallel. Opposite angles are equal to each other. Square Description: All four sides and four angles are equal. There are 2 sides of lines that are parallel to each other. There are 4 right angles 19
Rectangle Description There are 4 right angles. Opposite sides are parallel and equal to each other. Trapezoid Description There is one set of parallel lines. Rhombus 20
Description All sides are equal. Opposite angles are equal. Opposite sides are parallel to each other. 21
What s my name? I am a quadrilateral with exactly one pair of parallel lines. Trapezoid I am a quadrilateral with all sides equal, but right no angles Rhombus I am a rhombus with all 4 angles congruent Square My diagonals are bisect each other and are perpendicular to each other. (Hint you want to draw the diagonals first then connect the points) Rhombus 22
Day 4 Triangle and Polygon Sums Objectives: Using GSP students will discover the sum of the angles of a triangle and other polygons Opening Activity Students will be asked to open up a new sketch in GSP and create a line. Classroom Activity Students will be using GSP to work on the Angle worksheet. Please refer to Appendix A for directions on creating triangles/ measuring angles on GSP. During the activity the teacher should be walking around the room to ensure the students are on task and asking any probing questions. Closing Activity Chart will be displayed on the overhead. Students will be asked if they found a pattern with the number of triangles that can be formed and the number of sides in any polygon. Students should respond that the number of triangles is always 2 less than the number of sides of the polygon. Homework Pg 99: 5-14 Pg 108 7-12 23
TRIANGLE SUMS Using Geometer's Sketchpad create a triangle and label it ABC. Use the measuring tool and measure the triangle s angles. Calculate the sum of all three angle measurements. Copy your findings in the space provided. a b.angle ABC= Angle BCA= Angle CAB= c Angle ABC+ Angle BCA +Angle CAB= Highlight point A and drag the point to a new position. (This should create a new triangle) What happened to the angle measurements? What happened to the sum? Can you create a triangle whose sum does not equal 180 degrees? Draw and label a 4 sided figure on Geometer s sketchpad. 24
Highlight 2 points that are diagonal from each other. Construct a segment. Recreate your sketch here What do you notice about the interior of this 4 sided figure? Using your knowledge of triangle, how many degrees are in any 4 sided figures? Use the measurement tool to prove your answer. Using Geometers sketchpad complete the chart below Name of Polygon Number of sides Number of Triangles Triangle 3 1 Quadrilateral 4 2 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Degrees Do you notice a pattern between the number of triangles and the number of sides? How many degrees are in a polygon with 20 sides? 25
TRIANGLE SUMS Answer Key Using Geometer's Sketchpad create a triangle and label it ABC. Use the measuring tool and measure the triangle s angles. Calculate the sum of all three angle measurements. Copy your findings in the space provided. a b c m! a m! a m! c m! c Highlight point A and drag the point to a new position. (This should create a new triangle) What happened to the angle measurements? They Changed What happened to the sum? It is still 180. Can you create a triangle whose sum does not equal 180 degrees? NO Draw and label a 4 sided figure on Geometer s sketchpad. 26
a c b Highlight 2 points that are diagonal from each other. Construct a segment. Recreate your sketch here a P c b What do you notice about the interior of this 4 sided figure? It creates 2 triangles. Using your knowledge of triangle, how many degrees are in any 4 sided figures? 360 degrees. Use the measurement tool to prove your answer. m! c m! c m! a m! a m! a 27
Using Geometers sketchpad complete the chart below Name of Polygon Number of sides Number of Triangles Degrees Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 Hexagon 6 4 720 Heptagon 7 5 900 Octagon 8 6 1080 Nonagon 9 7 1260 Decagon 10 8 1800 Do you notice a pattern between the number of triangles and the number of sides? Yes. The number of triangles is always 2 less than the number of sides How many degrees are in a polygon with 20 sides? There would be 18 triangles so 18*180=3240 degrees. 28
Day 5 Triangle Congruency Objective: students will be able to identify congruent triangles. Opening Activity Students will be asked to define congruent. Classroom Activity Students will log onto the internet and NLVM website. Students will be asked to go to Geometry, grades 6-8 and click on congruent triangles. Using NLVM website and the worksheet students will work with SSS, SAS and ASA Postulates to gain a better understanding of why they prove triangle congruencies. Allow 15-20 minutes for this activity. Students should discuss with the class the answers to the worksheet. The teacher will then put up examples from the book, pg 104 Try these 1-4 and ask the students to apply their knowledge to answer which triangles are congruent and why. Homework pg 104 1-14 ALL 29
Triangle Congruency In SSS, each blue segment was equal to each red segment. What happened when you formed the blue and red triangle? Was it possible to have the red triangle not congruent to the blue triangle? Why or Why Not? In SAS each blue segment was equal to the red segments and the blue angle was equal to the red angle. After fitting both the red segments onto the red angle, why do you think it automatically formed a triangle? How does this show that if 2 triangles have 2 sides and the include angle equal they are congruent to each other? ASA means that if two triangles have 2 equal angles and the included side is also equal then the triangles must be congruent. How were you able to show this using the technology? 30
Triangle Congruency In SSS, each blue segment was equal to each red segment. What happened when you formed the blue and red triangle? Was it possible to have the red triangle not congruent to the blue triangle? Why or Why Not? Both of the triangles were equal. No it is not possible. All three segments would only fit together one way to form a triangle. In SAS each blue segment was equal to the red segments and the blue angle was equal to the red angle. After fitting both the red segments onto the red angle, why do you think it automatically formed a triangle? Because the there is only one line segment that would connect the two endpoints of the 2 segments How does this show that if 2 triangles have 2 sides and the include angle equal they are congruent to each other? There is only one triangle that can be formed when you have 2 sides and the included angle already given. Therefore if another triangle has the same 2 sides and included angle that new triangle must be congruent to the first. ASA means that if two triangles have 2 equal angles and the included side is also equal then the triangles must be congruent. How were you able to show this using the technology? When building the triangle the line segment had to fit onto the angle ray of each of the 2 angles given. The rays not used are going opposite directions and they will meet at one point only. Meaning the lengths of the other 2 sides is determined by this one point. Therefore if 2 triangles have 2 equal angles and the included side is also equal the triangles must be congruent. 31
Appendix A GUIDE TO GSP ACTIVITIES Directions for Geometer s Sketchpad Remember when working with Sketchpad, often an error occurs when incorrect information is highlighted. Parallel Lines Using the line tool construct a line. Use the pointer tool to highlight the line. Go up to Transform, Choose Translate. Parallel Lines with a Transversal Follow the procedure for parallel lines above. Using the point tool place a point on each of the lines. Highlight the points. Go up to Construct, choose line. Naming the Angles: an angle is made up of 3 points. Using the case above, additional points must be added to the line in order for an angle to be named. Using the point tool place points on the lines. Then go up to display, choose label points. This will automatically label all points on the lines. 32
L E F K I J G H Directions for Measuring Angles In order to measure angles you must click on three points. For instance, if measuring Angle LEF first highlight point L then point F then point E. Go up to Measure, choose angle. To Measure angle LFK repeat steps only first highlight L then F then K. In order to find the sum of the angles, go up to Measure, choose calculate and click on values and choose the measurements you would like to include in your sum. m! L m! L m! L L E F K I J G H Directions to create a triangle on Geometer s Sketchpad. Using the pointer tool, make three points. Use the arrow tool and highlight each point. Go up to the menu, Construct then choose Segments. This will create a triangle. To Label the triangle highlight the points choose Display and Label Points. l l 1 m n m n 2 3 33
Measure a triangle angles. If measuring Angle 123 in the triangle click on 1 then 2 then 3. Select Measure then Angle To measure angle 231 click on 2 then 3 then 1. To calculate the sum of all the angles in the triangle, Select Calculate, then click on the angle measures you would like to include. 34