ࡱ> k#` IFbjbj\.\. ?>D>D_=2222$Z "Pn.Zv"444W> $h8iW22445%ٸٸٸR244ٸٸٸ.MH64 pŒ@*0Z ٸZ...S... dv V 222222  Lesson Title: Properties of Parallelograms Lesson Summary: Students will engage in discovering properties of parallelograms by plotting points on a coordinate plane, calculating slopes, measuring angles and calculating distances of side lengths. Additionally, students will make conjectures about the relationship between diagonals in a parallelogram and how they bisect one another. Key Words: parallelogram parallel diagonals midpoint slope distance formula Background Knowledge: plot ordered pairs determine slope given two points calculate distance of a segment using the distance formula determine midpoint of a segment given the endpoints relationships between the slopes of parallel and perpendicular segments measure angles with a protractor using Cabri Jr. Apps in the TI-84 if using the extensions angle relationships supplementary, congruent, consecutive, opposite NCTM Standards Addressed: Geometric and Spatial Sense Indicator, grade 9, #3: Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. Learning Objectives: SWBAT discover the following properties of a parallelogram Opposite sides parallel Opposite sides equal Diagonals bisect each other Opposite angles congruent Consecutive angles supplementary Materials: pencil handout Geoboard with rubberbands protractor student lesson sheets TI-84 with Cabri Jr. Apps if doing the extensions Cabri Jr. Guidebook located at  HYPERLINK "http://education.ti.com/guidebooks/apps/83cabri_jr/ti83pcabrijr_eng.pdf" http://education.ti.com/guidebooks/apps/83cabri_jr/ti83pcabrijr_eng.pdf Suggested Procedures: The following activities may be modified to fit a jigsaw or activity stations format. Introduction: Find Parallelograms in the Room Look around the room and try to find at least 5 examples of parallelograms. Write and draw these items in the table on your paper. Activity 1: Lengths of Side Students will plot given points for a parallelogram on the coordinate plane, then calculate distance of the segments using the distance formula. They should then be able to determine that the opposite sides of a parallelogram are equal. Activity 2: Slopes Students will plot given points for a parallelogram on a coordinate plane, then determine the slopes of opposite sides. They should then conclude that the opposite sides have the same slope. Activity 3: Diagonal Relationship Students will plot given points for a parallelogram on a coordinate plane, then use the midpoint formula and the distance formula to determine the point of intersection. They should then conclude that the diagonals bisect each other. Activity 4: Angle Relationship Students will build parallelograms given coordinates and a Geoboard. Students will measure the angles in the constructed parallelogram and discover similarities and differences. They should see that consecutive angles are supplementary and opposite angles are congruent. Closure: Re-examine the Parallelograms around the Room Take out your list from the introduction of this lesson. Are there any objects you would change? How do you know that the objects on the list are parallelograms? Write a sentence next to each describing your thoughts. Extension 1: Properties of Parallelograms Extension 2: Parallelograms and TI Cabri Jr. Instructions Assessment: Student lesson sheets Questioning techniques during inquiry activity The Closure Activity Quiz Properties of Parallelograms Worksheet  Lesson Goals: To engage in discovering properties of parallelograms by plotting points on a coordinate plane, calculating slopes, measuring angles and calculating distances of side lengths. To conjecture about the relationship between diagonals in a parallelogram and how they bisect one another. Directions: Look around the room and try to find at least 5 examples of parallelograms. Write and draw these items in the first and second columns of the table below, titled INTRODUCTION. Please do not write in the third and fourth columns, titled CLOSURE. INTRODUCTIONCLOSUREItemObject namePictureIs it a parallelogram? (yes / no)Why or why not? 1  2  3  4  5  Activity One On a coordinate grid, plot and label the points A(6,2), B(17,5), C(13,12), and D(2,9).   What general shape will be formed if you connect A to B, B to C, C to D, and D to A? Connect each pair of points to verify your answer. Use the distance formula,  EMBED Equation.DSMT4 , to determine the length of each of the segments. Length of  EMBED Equation.DSMT4  = Length of  EMBED Equation.DSMT4  = Length of  EMBED Equation.DSMT4  = Length of  EMBED Equation.DSMT4  = What can you conclude about the lengths of the segments in this figure? Activity Two On a coordinate grid, plot and label the points A(-7,5), B(-2,-4), C(8,-8), and D(3,1).   What general shape will be formed if you connect A to B, B to C, C to D, and D to A? Connect each pair of points to verify your answer. Use the Slope formula,  EMBED Equation.DSMT4 , to determine the slopes of each of the segments. Slope of  EMBED Equation.DSMT4  = Slope of  EMBED Equation.DSMT4  = Slope of  EMBED Equation.DSMT4  = Slope of  EMBED Equation.DSMT4  = What can you conclude about the slopes of the segments in this figure? Activity Three On a coordinate grid, plot and label the points A(-3,0), B(-1,5), C(5,4), and D(3,-1).   What general shape will be formed if you connect A to B, B to C, C to D, and D to A? Connect each pair of points to verify your answer. Remember that diagonals are segments that connect two non-adjacent vertices of a polygon. How many diagonals will the figure above have? Use a straightedge to draw the diagonals and verify your answer. Using the Midpoint formula,  EMBED Equation.DSMT4 , find the midpoint of diagonal  EMBED Equation.DSMT4 . Plot and label this midpoint on the above coordinate plane as point F. Use the distance formula,  EMBED Equation.DSMT4 , to determine the length of each of these segments. Length of  EMBED Equation.DSMT4  = Length of  EMBED Equation.DSMT4  = From these lengths, what can you verify? Use the distance formula,  EMBED Equation.DSMT4 , to determine the length of each of these segments. Length of  EMBED Equation.DSMT4  = Length of  EMBED Equation.DSMT4  = From these lengths, what can you conclude? What can you conclude about the diagonals in this figure? Activity Four Consecutive angles of a quadrilateral are two angles that share a common ray. Opposite angles of a quadrilateral are two angles that are located on either end of a diagonal, they are directly across from one another. Consider your Geoboard to be the first quadrant of the coordinate plane, as the picture illustrates below.      Steps: Assume that the bottom outside edge of the Geoboard is the x-axis and the left outside edge is the y-axis. Stretch the rubber band around these 4 coordinates to create a parallelogram. A(1,1) B(3,1) C(3,5) D(5,5) Use a protractor to measure the angles at each of the vertices and record them below. m( A = EMBED Equation.DSMT4  m( B = EMBED Equation.DSMT4  m( C = EMBED Equation.DSMT4  m( D = EMBED Equation.DSMT4  Do you notice any similarities or differences among the angle measurements? Explain. Now stretch the rubber band around these 4 coordinates to create a different parallelogram. E(2,1) F(2,4) G(4,2) H(4,5) Use a protractor to measure the angles at each of the vertices and record them below m( E = EMBED Equation.DSMT4  m( F = EMBED Equation.DSMT4  m( G = EMBED Equation.DSMT4  m( H = EMBED Equation.DSMT4  Do you notice any similarities or differences among the angle measurements in the second parallelogram? Explain. Based on your responses to the #1-6, what general statement(s) can you make about a property or properties found in all parallelograms regarding angle measurements? Using your protractor draw a parallelogram below using only the angle measurements to guide you. Label the angle measurements in your parallelogram. QUIZ On a coordinate grid, plot and label the points A(-3,3), B(8,5), C(2,-1).   Using the information that you discovered in the previous activities, where could you place point D in order to make a parallelogram? Please verify your solution by showing the slopes of the sides, the lengths of the sides, the midpoint of the diagonals as well as the angle measures. After you have found one point that makes a parallelogram, is it possible to find another point that will make a different parallelogram with the three given points? How many different points exist that will form a parallelogram with the three given points? List as many of these points as you can. Extension 1 I. Constructing a parallelogram A. What is the definition of a parallelogram? B. Open the Cabri Jr. APP on your calculator. Construct a parallelogram using the instructions that follow below or use the instructions for Method I or Method II that follow this extension.  1. In F1 open a New window. 2. Using the Segment tool under F2 draw a segment. 3. Label it  EMBED Equation.DSMT4  with the Alp-Num tool in F5.  4. Draw  EMBED Equation.DSMT4  using the Segment tool.  5. With the Parallel tool in F3, 6. Then draw a line parallel to draw a line parallel to  EMBED Equation.DSMT4   EMBED Equation.DSMT4  through pt. B through pt. D.  7. Construct the point of 8. With the Quad tool in F2 intersection of the two lines draw quadrilateral ABCD. and label it C using Intersection under Point in F2.  9. Hide  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  using the Hide/Show tool in F5.  C. Grab point A. To do this select point A, press clear (the cursor will switch to a hollow arrow), then press the Alpha key (the cursor will change to a hand), and then move point A around using the arrow keys. Does ABCD appear to stay a parallelogram? Try grabbing points B, C, and D. What happens? Which point can you not grab? Why do you think this is so? II. The Properties A. Sides Use D & Length under the Measure tool in F5 to measure the four sides. Then, grab pt. A and change the shape of ABCD. Record the new values. SideLength in 1st shapeLength in 2nd shape EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Change the shape several more times. What do you notice is true about the lengths of the sides? B. Angles Using Angle in the Measure tool, find the number of degrees in each angle. Then, grab a point and change the shape of ABCD. Record the values. Angle1st shape2nd shape A B C D Find two relationships that appear to be true about the angles of a parallelogram. 1. 2. C. Diagonals Construct diagonals  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Construct their point of intersection and label it E. Measure the segments listed below. Then, change the shape and record the values. Length1st shape2nd shape EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  What property about parallelograms can you state? What is not necessarily true about the diagonals? D. Angles formed by the diagonals 1. Measure the angles formed by the intersection of the diagonals. Change shapes. Can you make any conjecture about the angles? 2. Measure ( BAE and ( DAE. Change shapes. Is there any property that seems to be true? E. List the properties of parallelograms that you have found. F. Constructions using the properties. Each person in your group should construct a parallelogram using a different property. Constructing a Parallelogram Using Cabri Jr. - Method I 1. In the F1 menu highlight New 2. Open the F2 window, scroll to and press ENTER. Segment, and press ENTER.  3. Draw a line segment. 4. Highlight Alp-Num in the F5 menu.  5. Label the segment  EMBED Equation.DSMT4 . 6. Draw line segment  EMBED Equation.DSMT4 .  7. In F3 select Parallel. 8. Highlight  EMBED Equation.DSMT4 and press ENTER. 9. With the arrow keys move the 10. Construct a line parallel to  EMBED Equation.DSMT4 . parallel line up to pt. D so that D is blinking and press ENTER.  11. There are three choices under the 12. Move the cursor until both lines Point tool. Select Intersection. are blinking. Press ENTER.  Label the point C.  13. Under F2 select the Quad tool. 14. Highlight the Hide/Show tool in Select A, then B, C, and D. F5 and press ENTER.  15. Select  EMBED Equation.DSMT4  and press ENTER. 16. Hide  EMBED Equation.DSMT4 .   A figure will not disappear until the cursor is moved away from it. Constructing a Parallelogram Using Cabri Jr. Method II 1. In the F1 menu highlight New 2. Open the F2 window, scroll to and press ENTER. Segment, and press ENTER.  3. Draw a line segment. 4. Highlight Alp-Num in the F5 menu.  5. Label the segment  EMBED Equation.DSMT4 . 6. Draw line segment  EMBED Equation.DSMT4 .  7. In F3 select Parallel. 8. Highlight  EMBED Equation.DSMT4 and press ENTER. 9. With the arrow keys move the 10. We need to copy the length of  EMBED Equation.DSMT4  parallel line up to pt. D so that onto the line through D. For this D is blinking and press ENTER. use the Compass tool under F3.  11. Highlight pt. B and press ENTER. 12. Using the arrow keys move the Then highlight pt. A and press center of the circle from A to D. ENTER. Be sure D is blinking. ENTER.  Note: When you use the Compass tool, the first point that you highlight will be on the circle and the second point will be the center. It does not really matter which one you highlight first since you can move the circle. 13. There are three choices under the 14. Move the cursor until both the Point tool. Select Intersection. line and the circle are blinking.  Press ENTER. Label pt. as C.  15. Under F2 select the Quad tool. 16. 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