PLEASE NOTE:

This document may contain a number of charts and graphics that could be problematic for your computer configuration. It is recommended that you use the “pdf” version for printing this document and this file for working with or adapting the Course Profile to meet your instructional needs.

Course Profile (for a locally developed course)

Essential Mathematics, Grade 9

Unit 5: Exploring Geometric Relationships

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 | Activity 6

Time: 10 hours

Unit Description

In this unit, students use concrete materials, diagrams, drawings, and dynamic geometric software to develop their spatial sense and an operational understanding of geometric properties. Students explore geometric patterns and use appropriate mathematical language in identifying and applying geometric concepts so as to gain a better understanding of geometric relationships. Investigations include those related to angle properties of parallel lines and two-dimensional figures. Through the use of the tools of dynamic geometry software students construct geometric designs, explore, and solve simple problems and clearly explain the use of geometric properties in their constructions.

Strand(s) and Expectations

Measurement and Geometry Specific Expectations: MG3.01, 02, 03, 04, 05, 06, 07.

Activity Titles

What follows is a suggested sequence, with timing, for teaching Unit 5. Many skills are developed within the activities themselves. However, the need for remediation and further development of skills will arise from the activities. Seventy-five minutes have been allotted for this.

Activity 1	Triangle Power	150 minutes
Activity 2	Regular Polygons	75 minutes
Activity 3	On the Straight and Narrow Path	75 minutes
Activity 4	Investigating Transformations	75 minutes
Activity 5	A Kaleidoscope of Colours	75 minutes
Activity 6	Summative Activity: Is it Fair?	75 minutes

Unit Planning Notes

· This unit has been written primarily using a dynamic geometry software package to augment much of the current pencil and paper transformational work that is readily available. For this reason the worksheets are primarily written for use with such a program. Teachers may choose to modify the handouts so as not to require the use of technology.

· This unit blends instruction that uses dynamic geometric software and concrete materials. The Geometer's Sketchpad™ program is used within this unit. However, other dynamic software programs may be substituted.

· The teacher may wish to modify the handouts included, depending upon the familiarity of students with this program and/or their comfort level with computers,

· If the teacher decides to use a dynamic software program, working through the activities to become familiar with the different aspects of this software package is recommended before using it in their classroom. Furthermore, it would be beneficial if another person, perhaps a senior student, is available in the classroom to provide extra assistance to students.

· To further help students through the computer based activities it would be helpful if the steps were demonstrated using an overhead projection device.

Teaching/Learning Strategies

· Although many of the activities are designed to be guided discovery it would be advisable if the teacher read through the instructions with the students to ensure there are no misunderstandings and that the teacher periodically stopped students in the middle of an exercise to ensure that the investigations are proceeding properly.

· It is important that students record observations on their worksheets. The use of diagrams and words to illustrate their findings is important. The worksheets are designed to facilitate this.

· Since many of theses activities take advantage of dynamic geometry software, a computer resource lab may be required. Two students can share a computer but one student per computer is preferred.

· The initial worksheets included in this activity are quite prescriptive to allow students to develop a basic understanding of the functions of the software program. As students move through the different activities, the amount of direct instruction diminishes since it is assumed that students developing a fundamental understanding of the software package.

· The computer worksheets are written using different styles of instruction. Some instructions incorporate graphics, some are very prescriptive, and others assume some student familiarity with the software package. The teacher should try to determine which format students find the most comfortable and modify the other sheets accordingly.

· The software's ability to draw in different colours can be used effectively to highlight important concepts and can help students to check and organize their constructions.

Assessment/Evaluation

· The degree to which technology is employed within this unit has a direct influence on the types of assessment used. Assessment may in part reflect the use of concrete materials and dynamic software.

· Facts and basic skills can be assessed through the use of paper and pencil types of evaluation and technology can be used to assess students’ ability to perform investigative work.

· Since a majority of the concepts within this unit have been developed through the use of dynamic software, the summative assessment incorporates the use of The Geometer's Sketchpad™.

Accommodations

· To assist students who have difficulty constructing some of the more complicated diagrams, construct the complete or partial diagram for the students and save it to a file. Later students can open these files and start from that stage.

· Students who have fine motor skill difficulties using a mouse can be paired with another student for the investigations using The Geometer's Sketchpad™.

Resources

Bennett, D. Exploring Geometry with the Geometer's Sketchpad. Key Curriculum Press, 1999.

Battista, M.T. Shape Makers: Developing Geometric Reasoning with the Geometer's Sketchpad. Key Curriculum Press, 1998.

Geddes, D. Geometry in the Middle Grades. NCTM, 1992.

Margaret, J.K. and J.B. Stanley. Informal Geometry Explorations. Dale Seymour Publications, 1992.

Serra, M. Patty. Paper Geometry. Key Curriculum Press, 1994.

Wyatt, K.W., A. Lawrence, and G.M. Foletta. Geometry Activities for the Middle School Students with the Geometer's Sketchpad. Key Curriculum Press, 1998.

Woodward, E. and T. Hamel. Visualized Geometry: A Van Hiele Level Approach. J. Weston Watch, 1990.

Activity 1: Triangle Power

Time: 150 minutes

Description

Students investigate the properties of triangles with and without the aid of The Geometer's Sketchpad™ software package. Students also begin to develop a glossary of important terms and definitions.

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Specific Expectations: MG3.01, .02, .04, .06.

Planning Notes

· A computer resource lab is required for this exercise.

· Students require the worksheets in this package. Depending upon the students’ comfort level with computers, the worksheet may require greater detail than currently exists in this activity.

· The exercises in this activity can be performed without the use of technology. This choice requires the creation of other worksheets.

· Teachers are encouraged to work through the handouts prior to students using the program and modify the sheets accordingly to meet the needs of the in students.

Teaching/Learning Strategies

Student Activity

· Students work through the handouts to investigate different mathematical properties of triangles. The teacher facilitates as required.

· They are initially led through the first activity and allowed to continue on their own.

· It is important that students not only perform the necessary work on the computer but that they complete the written portion of the handouts as they work through the exercises.

Teacher Facilitation

· Prior to the use of the software programs the teacher can demonstrate or involve their students in a quick activity that demonstrates that the sum of the angles in a triangle is 180^o. A piece of paper can be cut in the form of a triangle and the three angles can be torn off and re-oriented to form a straight angle. This can then be a starting point for making a transition from a concrete proof to a software proof.

· Terms such as vertex and line segments and how to identify angles (e.g., ÊABC) should be reviewed before students begin the worksheets.

· Students should have an opportunity to explore the software package. It is important that students are aware of the basic functions of the program (e.g., construction of lines, rays, line segments, circles, measuring angles, setting preferences, set the level of accuracy of measurement to units, tenths, hundredths and thousands, and changing colours). This can be facilitated with the sample files included with the software package.

· It is important for teachers and students to use clear and precise language whenever possible.

· Lead students through each step in the first activity to ensure that students develop specific software related skills.

· Once students have begun the second activity, the teacher may choose to circulate to help students as required.

· Once students have completed the exercises there should be a teacher-led discussion to summarize what students have determined. With the teacher's assistance students can create a glossary of terms and definitions that may not have been touched upon in the investigations such as the different types of angles and triangles (e.g., acute, obtuse, reflex). A chart similar to the one below may be developed and kept at the front of the students' unit in their binders. Students can use texts or other resource books to complete their charts.

· Students should begin with a new screen after each exercise.

Term	Diagram	Definition/Properties
Triangle
etc.

Assessment/Evaluation

Observations of student's individual learning skills could be recorded at this time using a modified version of the Rubric for Observing Students in Appendix 2. The worksheet can be assessed for accuracy of calculations, quality of communication and completeness. The component of the exercise that requires the computer can be assessed through the use of The Geometer's Sketchpad™ Investigation Rubric at the end of unit.

EXERCISE A

INVESTIGATING TRIANGLES

In this activity, use the preferences option under the Display menu so that only points will be automatically labelled and the lengths of the sides and measures of the angles will be accurate to the nearest tenths.

In this activity we create the following diagram.

1. Create a triangle using the Line Segment tool, , by clicking and dragging your mouse to create a line segment.

1. Select the Selection Arrow tool, . Hold down the Shift key, and click on each of the three points in order A, B, C and release the Shift key. Each point should now be highlighted.

2. Go to the Measure menu and choose Angle. You notice that the measure of the angle (ÊABC) is in the top left corner of the screen. Now find the measure of the other two angles. Write down each angle measure and label accordingly.

3. Go to the Measure menu and select Calculate. A calculator appears. You can move the calculator over to the right hand side of the screen by clicking and holding on the blue bar of the calculator and moving it.

Now, click on the first angle measurements on the top left hand side (you know you have done so when it also appears on the calculator). Then click on the addition sign (+) on the calculator, click on the next angle, then hit the addition sign (+) again, then the next angle and then click on the OK button on the calculator.

4. Now, using your Selection Arrow tool, click on any point and move your mouse without letting go of the mouse button.

5. What is changing? What is not changing? What does this tell you?

EXERCISE B

TYPES OF TRIANGLES

Construct a triangle and measure all of the angles and the lengths of all the sides

Describe what you observe about the measures of the angles and the name of each triangle (isosceles, scalene, or equilateral).

INVESTIGATION		OBSERVATIONS	SKETCH
1	Drag a vertex so two sides of the triangles are the same length.
2	Drag the vertices so that all three sides are the same length.
3	Drag the vertices so that all three sides are unequal.

EXERCISE C

CONSTRUCTING SPECIAL TRIANGLES

PART 1

CONSTRUCTION				DIAGRAM
1	Use the Circle tool to construct a circle.
2	Use the Point tool to place two points on the circumference of the circle
3	Connect the points together
4	Select the circumference of the circle and then go to the Display Menu and select Hide Circle.
		INVESTIGATION	OBSERVATIONS
Drag one of the vertices (not the centre of the circle). How does the triangle change? What type of triangle do you think this is?

PART 2

CONSTRUCTION			DIAGRAM
1		Use the Circle tool to construct a circle.
2		Construct a second circle so that the centres of the circles lie on the others circumference.
3		Select the Point Tool and mark another point at the point of intersection of the two circles.
4		Connect the points
5		Select the circumference of the circles and then go to the Display Menu and select Hide Circle.
	INVESTIGATION			OBSERVATIONS
Drag one of the vertices. How does the triangle change? What type of triangle do you think this is?

EXERCISE D

PERPENDICULAR LINES OF A TRIANGLE

CONSTRUCTION		DIAGRAM
1. 2. 3.	Draw a triangle Select all three sides Go to the Display Menu and select Line Style then Thick
4	Select one vertex of the triangle and the opposite side (remember to hold down the shift key)
5	Go to the Construct menu and select Perpendicular Line (this line runs through the altitude of the triangle).
	INVESTIGATION	OBSERVATIONS
Drag the points on the triangle. Does the perpendicular line always cross inside the triangle?
6	Construct perpendicular lines to the other sides of the triangle
	INVESTIGATION	OBSERVATIONS
Drag the points on the triangle. What do you notice? Are the altitudes always inside the triangle? Do the altitudes ever cross?

EXERCISE E

MEDIANS

CONSTRUCTION		DIAGRAM
1	Draw a triangle with thick blue lines.
2	Select all three sides
3	Go to the Construct menu and select Point at Midpoint.
4	Connect each vertex to the midpoint of the opposite side of the triangle (this is called the median of the triangle)
	INVESTIGATION	OBSERVATIONS
Drag each vertex on the triangle. Are the medians always inside of the triangle? What do you notice that remains the same?

Activity 2: Regular Polygons

Time: 75 minutes

Description

In this lesson students construct and investigate the properties of regular polygons. Students also determine the interior angle measures through the use of patterns. The Geometer's Sketchpad™ is used to help develop the properties of regular polygons and the use of rotations in the construction of geometric figures.

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Specific Expectations: MG3.01, .02, .03, .04, .05, .06.

Planning Notes

· A computer resource lab is required for this exercise.

· Students require the worksheets in this package. Depending upon the comfort level of the students, the worksheet may require greater detail than currently exists in this activity.

· The exercises in this activity can be performed without the use of technology. This may require the creation of other worksheets.

Teaching/Learning Strategies

Student Activity

· Students work with the teacher to complete the investigation of the interior angles of a polygon.

· Students use protractors and The Geometer's Sketchpad™ to help them construct polygons.

Teacher Facilitation

· Teachers work with the students through the investigation of the interior angles of a polygon worksheet. Students may require assistance in determining the pattern.

· The teacher should draw attention to the fact that a construction using rotations ensures that the diagram retains it shape despite how it is dragged whereas a free hand drawn triangle changes as you drag a vertex. Thus some of the properties do not remain constant.

· It is important for teachers and students to use clear and precise language whenever possible.

Assessment/Evaluation

Observations of student's individual learning skills could be recorded at this time using a modified version of the Rubric for Observing Students in Appendix 2. Assess student worksheets for accuracy of calculations, quality of communication, and completeness. Assess students through a quiz in which students can use a protractor and straight edge to construct and measure basic figures. Ask students to construct a pentagon using the same tools. Assess the component of the exercise that requires the computer through the use of The Geometer's Sketchpad™ Investigation Rubric at the end of the unit.

HOT TUB FUN

Designers of the Great Canadian Hot Tub Company have decided to expand their line of hot tubs to include some new designs. To manufacture the new tubs they must determine the measure of each interior angle in their design. If each hot tub is in the shape of a regular polygon determine the measure of each interior angle.

Sample Chart

Regular Polygon Name	Diagram	Number of Triangles	Sum of interior angles	Measure of each interior angle
Triangle		1	1 x 180 = 180	180 ) 3 = 60
Square		2	2 x 180 = 360	360 ) 4 = 90
Pentagon		3	3 x 180 = 540	540 ) 5 = 120
$ $ $
Octagon

What would the measure of each angle be of a Loonie (11 sided)? A one hundred sided regular polygon?

This would be an opportunity for students to review how to draw and measure angles using a protractor and straight edge to classify angles by measure.

REGULAR POLYGONS ON THE GEOMETER'S SKETCHPAD™

Constructing an equilateral triangle

CONSTRUCTION		DIAGRAM
1	Draw a line segment AB approximately 4 cm in length.
2. 3.	Select point B. Go to the Transform menu and select Mark Centre "B"
4	Select points A and B and line segment AB (remember to hold down the shift key).
5. 6.	Go to the Transform menu and select Rotate. A pop-up menu appears, similar to the one to the right. Change the degree measure to read 60^O then hit OK
7	Select point A' and go to the Transform menu and select Mark Centre"A' "
8	Select line segment A'B, points A'
9	Go to the Transform menu and select Rotate
	INVESTIGATION	OBSERVATIONS
What type of triangle have you created? How do you know?

Use the steps above to help you construct a square and a regular pentagon.

Activity 3: On the Straight and Narrow Path

Time: 75 minutes

Description

Students investigate the properties of parallel lines with and without the aid of The Geometer's Sketchpad™.

Strand(s) and Expectations

Strands: Measurement and Geometry

Specific Expectations: MG3.01, .02, .03, .04, .05, .06.

Planning Notes

· This activity requires the use of The Geometer's Sketchpad™. A computer resource lab is required.

· Students require the worksheets in this package.

· The exercises in this activity can be performed without the use of technology. A teacher-directed lesson may be substituted.

Teaching/Learning Strategies

Student Activity

· Students work through the handouts to determine the angle relationships that result when parallel lines are cut by a transversal.

· To help introduce the topic of parallel lines and angle relationships the teacher can first discuss a situation in which a road is built to run past two towns. Each town is constructing half of the road to eventually meet the other town's section. Ask students what the relationship between the two indicated angles would be to ensure that the two roads meet.

· To investigate the relationship, have students construct two parallel lines with a line intersecting both. At this time discuss terms such as transversal and parallel lines. The sheet of paper can be torn in half and translated or rotated so as to compare angles.

· When completing the computer worksheets it is important that students not only perform the necessary work on the computer but that they complete the written portion of the handouts.

Assessment/Evaluation

Observations of students' individual learning skills could be recorded at this time using a modified version of the Rubric for Observing Students in Appendix 2. The worksheet can be assessed for accuracy of calculations, quality of communication, and completeness. A quiz can be used to assess basic skills in determining angle relationship between parallel lines and a transversal. The component of the exercise that requires the computer can be assessed through the use of The Geometer's Sketchpad Investigation Rubric at the end of unit.

ACTIVITY A

STRAIGHT ANGLES

CONSTRUCTION		DIAGRAM
1	Construct the following diagram
2. 3.	Measure ÊADC and ÊCDB Determine the sum of the two angles using the calculator tool.
	INVESTIGATION	OBSERVATIONS
Drag point C left and right. What do you notice about the sum of the angles?

ACTIVITY B

OPPOSITE ANGLES

CONSTRUCTION		DIAGRAM
1	Construct the following diagram
2	Measure ÊAEC, ÊDEB, ÊAED and, ÊCEB
	INVESTIGATION	OBSERVATIONS
Drag point A left and right. What do you observe about opposite angles?

ACTIVITY C

INVESTIGATING PARALLEL LINES

Construct the following diagram.

CONSTRUCTION		DIAGRAM
1	Select the Line Tool, , and create a horizontal line approximately 4 cm down from the top of the page.
2	Use the Point Tool, , to make a point approximately 5 cm below point A.
3. 4.	Select the Selection Arrow Tool, . Holding down the SHIFT key select line AB and point C. Go to the CONSTRUCT menu and select Parallel Line.
5	Use the Point Tool, , to make create point D as in the diagram.
6	Now draw a line that crosses through both line segments. This line is called a transversal.
7	By using the Point Tool, label the intersection points of the transversal and the parallel lines. To do this, select two lines AB and EF then go to the CONSTRUCT menu and select Point at Intersection. Do the same for the other point of intersection

Complete each investigation and draw a sketch to indicate the angles observed.

INVESTIGATION	OBSERVATIONS	SKETCH
Measure Ê BGH and Ê CHG. Now using the Selection Arrow Tool to move point E to the left and right. What do you observe about the angles? Are there any other angles equal to these angles?
Measure Ê AGH. Move point E. What is the relationship between this angle and Ê CHG ? (Hint: Use the calculator.)
Measure Ê CHF. Move point E. What is the relationship between this angle and Ê AGH? Are there any other angles with similar relationships?

· The teacher can review the exercise and high light the relationships. A lesson can follow that looks at different patterns (C, F, Z)

This exercise can be followed by opportunities for students to practise determining angles involving parallel lines.

Activity 4: Investigating Transformations

Time: 75 minutes

Description

In this activity students investigate reflections, translations, and rotations. This activity uses centimetre dot paper and tracing paper.

Strand(s) and Expectations

Strands: Measurement and Geometry

Specific Expectations: MG3.03, .04, .06.

Planning Notes

· Students use centimetre dot paper, tracing paper, and coloured pencils to complete this exercise.

· A copy of the worksheets on an overhead acetate with different coloured overhead markers would be helpful in demonstrating the transformations.

· The worksheets in this package are only an example of the types of questions students can work through. Greater development of the exercises is required (see resource list).

· Similar activities may be found in the optics units in the science program.

Teaching/Learning Strategies

Student Activity

· Students can work through worksheets similar to those provided in this activity.

Teacher Facilitation

· A teacher-directed lesson is required at the beginning of the exercise. The teacher can demonstrate the work with the use of an overhead.

· As student complete their worksheets, the teacher or students may periodically place answers on the overhead.

· The teacher can circulate around the classroom providing encouragement and assistance as required.

Assessment/Evaluation

Observations of student's individual learning skills could be recorded at this time using a modified version of the Rubric for Observing Students in Appendix 2. Assess the worksheet for accuracy of calculations, quality of communication, and completeness.

REFLECTION

Draw a reflection of each figure in the mirror line.

1. Choose any point of your original figure. How far away is it from the mirror line? How far away is its reflection?

2. Did any points not move? Which ones?

3. Did the reflection change the shape?

4. Did the reflection change the size?

TRANSLATIONS

1. Did the translation change the shape?

2. Did the translation change the size?

ROTATIONS

1. Did the rotation change the shape?

2. Did the rotation change the size?

COMBINATION TRANSFORMATIONS

Complete two separate transformations of the diagram above.

Transformation #1: Reflection, translation, rotation

Transformation #2: Translation, rotation, reflection

Do the transformations result in the same shape?

Is the location of the figures, after the transformations the same?

Activity 5: A Kaleidoscope of Colours

Time: 75 minutes

Description

In this summative activity students use translations, reflections, and rotations to construct a “kaleidoscope” with the aid of The Geometer's Sketchpad™.

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Specific Expectations: MG3.03, .04, .05, .06, .07.

Planning Notes

· This activity requires the use of The Geometer's Sketchpad™. A computer resource lab is required.

· Students use the worksheets in this package.

· A similar activity can be substituted that does not use technology. This alternate activity may involve using transformations to create geometric designs.

Teaching/Learning Strategies

Student Activity

· Students work through the worksheets provided.

· Once students have completed their first "kaleidoscope" they then can print out a copy of it and experiment in creating another.

Teacher Facilitation

· Circulate around the classroom providing encouragement and assistance as required.

· It is important that students complete the written portion of the exercise as well the constructions. Teachers may wish to periodically remind their students to complete the written component of the activity along with diagrams.

CONSTRUCTING A KALEIDOSCOPE

CONSTRUCTING A TRIANGLE

Turn Autoshow Labels for Points On by going to the Display menu and selecting preferences.

CONSTRUCTION		DIAGRAM
1. 2.	Construct a small triangle in the centre of your screen. Select all 3 vertices.
3. 4.	Go to the Construct menu and select Polygon Interior. Colour your triangle blue.

TRANSLATION

Turn Autoshow Labels for Points Off.

CONSTRUCTION		DIAGRAM
1.	Measure side BC.
2. 3.	Select your triangle by clicking on it. Go to the Transform menu and select Translate.
4	Make your translate menu look similar to the one on the right and click OK. Step 1: Click on By Polar Vector Step 2: Click on the length of BC above the menu.
5	Colour your new triangle red.
	INVESTIGATION	OBSERVATIONS
Drag each point, what do you notice about the two objects size, shape, and location?

REFLECTION

CONSTRUCTION		DIAGRAM
1. 2.	Select line segment BC. Go to Transform and choose Mark Mirror.
3. 4.	Go to Transform and choose Reflect. Colour your new triangle green.

ROTATION

CONSTRUCTION		DIAGRAM
1	Place a small circle to the left of your diagram.
2. 3.	Select the centre of the circle. Go to the Transform menu and select Mark Centre.
4	Now select all three triangles. (remember to hold down the shift key)
4. 5.	Go to the Transform menu and select Rotate. Rotate by 120^0.
6	Go to the Transform menu and select Rotate again and hit OK.

ANIMATION

6. Select point C and the circumference of the circle (remember to hold down the shift key)

7. Go to the Edit Menu select Action Button then Animation

8. Click on the Animate Button

9. An Animate button has now been created on your screen. Double click on it.

10. Select point B and the circumference of the circle and repeat steps 2 to 4.

Assessment/Evaluation

The component of the exercise that requires the computer can be assessed through the use of The Geometer's Sketchpad™ Investigation Rubric at the end of unit.

Activity 6: Summative Activity: Is It Fair?

Time: 75 minutes

Description

This summative activity allows the teacher to blend the use of technology with typical pencil and paper forms of assessment. Only the technology component has been fully developed here.

Strand(s) and Expectations

Strand(s): Measurement and Geometry

Specific Expectations: MG3.01, .02, .04, .05, .06.

Planning Notes

· This activity requires the use of The Geometer's Sketchpad™ and a computer resource lab.

· Students require the worksheets in this package.

· A similar activity can be substituted that does not use technology; however, this would require a re-write of the worksheets.

· This assessment activity can be organized so that while one half of the students are working at the computers the other half can be writing a pencil and paper assessment. After one half of the period students switch activities.

· A written assessment is required that would take approximately one half period to complete. This assessment may test basic knowledge but students may also be asked to apply their knowledge through geometric constructions. The degree to which dynamic software was utilized in this unit determines the amount of assessment using software and a straight edge and protractor. This task is left to the teacher to complete.

Teaching/Learning Strategies

Student Activity

· Students work through the assessment exercise worksheets provided.

Teacher Facilitation

· The teacher circulates around the classroom providing encouragement and appropriate assistance as required.

Assessment/Evaluation

The component of the exercise that requires the computer can be assessed through the use of The Geometer's Sketchpad™ Investigation Rubric at the end of this unit. The written component can be marked on completeness and quality of response.

INVESTIGATION

IS IT FAIR?

The tradition of a certain family of farmers has been always to give part of their land to their children. The parents have decided to divide a piece of their land among their four children. They would do this by keeping one half of their farm for themselves and divide the remaining half equally among the four children. Use the diagram below and the instructions to help you determine a fair way to divide the land.

11. Construct a triangle to represent the land given to the children

12. Measure the area and perimeter by selecting the triangle using the Measure menu.

13. Determine the midpoint of any two sides by selecting the side and using the Construct menu.

14. Connect the midpoints.

15. Drag any of the vertices on the new smaller triangle.

INVESTIGATION	OBSERVATION
What do you notice about the line joining the midpoints?
How can you use what you learned about parallel lines and a transversal to show that these two lines are parallel?
Use The Geometer's Sketchpad™ to prove that these two lines are parallel.

16. Construct the third midpoint of the side of your original triangle

17. Connect this midpoint to each of the other midpoints to form a triangle.

18. Colour the middle triangle blue.

19. Measure the area and perimeter of the blue triangle.

INVESTIGATION	OBSERVATION
Compare the areas of the blue and large triangles? What do you notice?
Compare the perimeters of the blue and large triangles? What do you notice?

INVESTIGATION	OBSERVATION
Are all four of the triangles of equal area? Is this a fair way to divide the land?

20. Print a copy of your diagram.

The Geometer's Sketchpad™ Investigation Rubric

	Level 1	Level 2	Level 3	Level 4
Construction of graphical model	- diagram contains minor errors	- diagram is constructed through manual estimation (**correct only in static form)	- diagram is accurate and constructed using appropriate tools and relationships (dynamic)	- diagram is constructed using appropriate tools and relationships (dynamic) with appropriate information shown using correct measures of accuracy
Uses appropriate methodology to construct model	- with assistance interprets and constructs model with technology	- constructs model with the use of technology but may contain minor errors due to lack of understanding of appropriate relationships	- constructs proper model with the use of technology	- construct model with the use of technology and recognizes and measures features of the model.
Describes relationships observed in diagram	- makes correct inference with frequent teacher support	- makes correct inference with some prompting	- makes correct inference independently	- makes correct inference independently and identifies other geometric properties and relationships
Communication	- communicates with limited clarity - uses little or no justification	- communicates results with inappropriate forms - uses faulty logic to justify conclusions	- communicates clearly using appropriate forms - justifies conclusions	- communicates clearly and concisely - justifies and generalizes relationships

** It should be noted that students may be able to construct diagrams through visual estimation. For example students may be able to estimate the midpoint of a side of a triangle instead of using the software program to determine its location. When a triangle vertice is moved (dragged) the estimated midpoint will not move dynamically. This hinders appropriate investigations of the diagrams properties. The diagram may only be correct when it remains static (unchanged; no points dragged).

Continue to Unit 6 | Back to Unit 4 | Back to Course Profiles main menu