lesson title: properties of · web viewswbat discover the following properties of a...

Project AMP Dr. Antonio R. Quesada Director, Project AMP

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:

1. parallelogram2. parallel3. diagonals4. midpoint5. slope6. distance formula

Background Knowledge:1. plot ordered pairs2. determine slope given two points3. calculate distance of a segment using the distance formula4. determine midpoint of a segment given the endpoints5. relationships between the slopes of parallel and perpendicular segments6. measure angles with a protractor7. using Cabri Jr. Apps in the TI-84 if using the extensions8. 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:1. SWBAT discover the following properties of a parallelogram

a. Opposite sides parallelb. Opposite sides equalc. Diagonals bisect each otherd. Opposite angles congruente. Consecutive angles supplementary

Materials:1. pencil2. handout3. Geoboard with rubberbands4. protractor5. student lesson sheets6. TI-84 with Cabri Jr. Apps if doing the extensions7. Cabri Jr. Guidebook located at

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Project AMP Dr. Antonio R. Quesada Director, Project AMPhttp://education.ti.com/guidebooks/apps/83cabri_jr/ti83pcabrijr_eng.pdf

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http://education.ti.com/guidebooks/apps/83cabri_jr/ti83pcabrijr_eng.pdf

Project AMP Dr. Antonio R. Quesada Director, Project AMPSuggested Procedures: The following activities may be modified to fit a jigsaw or activity stations format.

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. Extension 1: Properties of Parallelograms

8. Extension 2: Parallelograms and TI Cabri Jr. Instructions

Assessment: 1. Student lesson sheets

2. Questioning techniques during inquiry activity

3. The Closure Activity

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Project AMP Dr. Antonio R. Quesada Director, Project AMP4. Quiz

Properties of ParallelogramsWorksheet

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.

INTRODUCTION CLOSURE

Item Object name PictureIs it a

parallelogram?(yes / no)

Why orwhy not?

1

2

3

4

5

4


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Activity One1. On a coordinate grid, plot and label the points A(6,2), B(17,5), C(13,12), and D(2,9).

2. 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.

3. Use the distance formula, , to determine the length of each of the segments.

Length of =

Length of =

Length of =

Length of =

4. What can you conclude about the lengths of the segments in this figure?

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Activity Two1. On a coordinate grid, plot and label the points A(-7,5), B(-2,-4), C(8,-8), and D(3,1).

2. 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.

3. Use the Slope formula, , to determine the slopes of each of the segments.

Slope of =

Slope of =

Slope of =

Slope of =

4. What can you conclude about the slopes of the segments in this figure?

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Activity Three

1. On a coordinate grid, plot and label the points A(-3,0), B(-1,5), C(5,4), and D(3,-1).

2. 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.

3. 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.

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4. Using the Midpoint formula, , find the midpoint of

diagonal . Plot and label this midpoint on the above coordinate plane as point F.

5. Use the distance formula, , to determine the length of each of these segments.

Length of =

Length of =

From these lengths, what can you verify?

6. Use the distance formula, , to determine the length of each of these segments.

Length of =

Length of =

From these lengths, what can you conclude?

7. What can you conclude about the diagonals in this figure?

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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:1. 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)

2. Use a protractor to measure the angles at each of the vertices and record them below.

m A =

m B =

m C =

m D =

3. Do you notice any similarities or differences among the angle measurements? Explain.

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x-axis

y-axis


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4. 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)

5. Use a protractor to measure the angles at each of the vertices and record them below

m E =

m F =

m G =

m H =

6. Do you notice any similarities or differences among the angle measurements in the second parallelogram? Explain.

7. 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?

8. Using your protractor draw a parallelogram below using only the angle measurements to guide you. Label the angle measurements in your parallelogram.

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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.

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Project AMP Dr. Antonio R. Quesada Director, Project AMPExtension 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 with the Alp-Num tool in F5.

4. Draw using the Segment tool.

5. With the Parallel tool in F3, 6. Then draw a line parallel to draw a line parallel to 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 and using the Hide/Show tool in F5.

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Project AMP Dr. Antonio R. Quesada Director, Project AMPC. 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.

Side

Length in 1st shape

Length in 2nd shape

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.

Find two relationships that appear to be true about the angles of a parallelogram. 1.

2.

Angle

1st shape 2nd shape

A B C D

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Project AMP Dr. Antonio R. Quesada Director, Project AMPC. Diagonals

Construct diagonals and . Construct their point of intersection and label it E.

Measure the segments listed below. Then, change the shape and record the values.

Length

1st shape 2nd shape

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.

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Name: ______________________________________ Date: __________ Period:_____

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 . 6. Draw line segment .

7. In F3 select Parallel. 8. Highlight and press ENTER.

9. With the arrow keys move the 10. Construct a line parallel to . parallel line up to pt. D so that D

is blinking and press ENTER.

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Name: ______________________________________ Date: __________ Period:_____

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 and press ENTER. 16. Hide .

A figure will not disappear until the cursor is moved away from it.

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Name: ______________________________________ Date: __________ Period:_____

Constructing a ParallelogramUsing 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 . 6. Draw line segment .

7. In F3 select Parallel. 8. Highlight and press ENTER.

9. With the arrow keys move the 10. We need to copy the length of

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.

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Name: ______________________________________ Date: __________ Period:_____

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. Highlight Hide/Show in F5 and Select A, then B, C, and D. press ENTER.

Note: The order in which you select the points does not matter.

17. Move the cursor until the circle is 18. Hide line . A figure will not

blinking. Press ENTER. disappear until the cursor is

moved away from it

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lesson title: properties of · web viewswbat discover the following properties of a...

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