Name: Tara Lauria

Marple Newtown High School

Title of Lesson: The Centroid : Balancing the History

1. Objective: Have students discover the centroid of a triangle based on Euclid’s beginning constructions. Have students discover certain features of the centroid; Locus of points of a median, properties of the segments that form the median, areas of the smaller sections(triangles) formed by the medians.

B: Historical Perspective ( Explanation of how history fits into the lesson):

Through lecture and activities, general and specific information about Euclid and his works will be highlighted. The construction of medians will highlight Euclid’s beginning constructions found in Book I of The Elements.

Emphasize at this point, that the construction of the medians cannot happen without Euclid’s beginning constructions. Also Emphasize the where and when of Euclid. Show a map of Greece, Explain that we do not have many facts about Euclid, but his time was approximately 300 B.C. , he was from Alexandria, he was considered the father of geometry, he wrote the book called The Elements, which is the basis of the geometry we are learning today. At this time, Alexandria was known as a center of knowledge. Once the medians are constructed, students recognize a point of concurrency. Students, with guidance will use Euclid’s Book IV, Propositions 4 and 5 as references to the locus of points of special lines in triangles to discover the importance of the locus of points of the medians just constructed. Proposition 4 references the incenter of a triangle, which inscribes a circle in a triangle. The incenter is constructed using angle bisectors. The locus of points of the angle bisectors are equidistant from the sides of the triangle. Proposition 5 references the circumcenter, which circumscribes a circle about a triangle. The circumcenter is constructed by constructing the perpendicular bisectors of a line. This is from Euclid’s First Book. The locus of points of the perpendicular bisectors are equidistant from the vertices of the triangle. Euclid’s Book I , proposition 38 will allow students to make the connection that equal triangles through equal bases between the same parallel lines will have equal areas. Therefore the locus of points of the medians creates equal areas. Therefore the point of concurrency of the medians divides the triangle into equal areas creating a balance point or center of gravity called the centroid. The connection made by using Euclidean constructions to discover this special point of concurrency and then to analyze this point gives the lesson depth and in turn gives the students a background to dig into.

3. Prerequisites: Euclidean constructions including; incenter, circumcenter.

Working knowledge of Geometry Sketchpad

4. Materials needed to teach lesson: Compass, straight edge, transparencies, overhead projector or chalkboard , markers, paper, cardboard, string, small weight or paper clips, pin/nail, corkboard, access to Geometry Sketchpad

E. Procedures: ( step by step guidelines for executing the lesson. Please

discuss any student handouts that you intend to use and include the handouts as appendices to the lesson plan.)

1. Hand out 3 page packet which includes an acute triangle, obtuse triangle and a right triangle.(Appendix A)
2. Divide class into thirds, each third will construct the medians on the assigned triangle, while demonstrated by the teacher using the overhead projector.

3. Connect this point to the opposite vertex of the triangle.
1. Have the students construct the second median and stop.

5. Have students construct the third median. Recognize point of concurrency as the centroid.
6. Demonstrate: what the centroid does. a.)Punch small holes at vertices on cardboard or wood triangle. b.) Construct the medians on it. C.) Hang the triangle on the wall or corkboard. d.) Using string with an attached small weight , pin the string to one of the vertices and watch what happens. e.) attach the string at a different vertex. The weighted string should fall directly on the path of the median previously constructed. Check at all three vertices and see if it is accurate to the medians and the centroid. ( Centroid is the balance point , or center of gravity of any shape)
7. Assign for Homework: 1.) What is important about the locus of points of a median and why does this make the centroid important? Jot some ideas down in packet 2.) Construct centroids of the other triangles left in packet. Make some observations in packet Guide/refer students back to Euclid for connections and say no more! (see historical perspective for importance of the locus of the points)

DAY 2:

1. Put students in groups let them chat about their ideas and have them draw some conclusions. Now let them see if they are correct!
2. Geometry Sketchpad (Key Curriculum Press) (Appendix B) Self - contained packet with instructions, and guidelines to discovering the special features of the medians and centroid of a triangle. Let students confirm, connect, and communicate their discoveries!

6. References:

Bunt. Jones. Bedient, Jack D. The Historical Roots of Elementary Mathematics. Dover 1976.

Heath, Sir Thomas L. Euclid, The Thirteen Books of The Elements, Vol.2

Dover Publications, NY 1956.

Heilbron, J.L. From Geometry Civilized, History, Culture, and Technique.

Calerendon Press, Oxford 1998.

Serra, Michael. Discovering Geometry. Key Curriculum Press, 1993

Professor Wolfson’s Lectures, West Chester , PA July 1999

Steiner point construction

Nine point circle construction

Euler line construction

Analyze centroids of other shapes

8. Careers in the field:

Engineers

Architects

Geographic information systems

Interior designer

Jeweler

Observations/ Conjectures :

worksheet for Geometry Sketchpad

1. Construct Triangle ABC.

2. Construct the midpoints of Two sides.

3. Construct two medians.(connect midpoints to

opposite vertex)

4.) Construct the point of intersection of the two medians.

( If you jumped ahead and constructed all three medians, select two of them and then construct point of intersection.)

5. Construct the third median

What can you observe from this third median? Drag any vertex to discover and prove that this is true for all triangles.

6. The point of Concurrency is the centroid.

Label it . Then double click on label to change it ,

Label it C .

7. Measure the distance from point A to C

and from C to the midpoint D. Recall: measuring

distance between two points requires that you select

two points, or select a segment and measure length.

8. Drag vertices of ABC and look for a relationship between AC and C F.

Select the two measures,

go to measure menu, select tabulate.

10. Change the triangle by dragging a vertex and double click on the table of

values to automatically add other pieces of data.

11. Continue changing the triangle and adding data to the table until a

relationship between the distances is confirmed.

12. Based on the relationship discovered from the data, use the calculator to

investigate the relationship further. Double click on a measurement to

bring up the calculator. Click once to put a measurement into a

calculation.

Write down the expression calculated and the result.

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13. Plot data from the table. Select the table. Go to the Graph menu and select

plot table data. In the dialog box click plot. Do not change the data.

What do the points form?_____________

14. Construct a line through any two points plotted from the data and

measure the slope.

15. What is the significance of the slope of this line?

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1. Construct a triangle and its medians using the previous steps, or use

triangle already constructed.

2. Make sure the vertices and centroid are labeled and the

midpoints are also labeled.

3. Select three vertices of any of the 6 interior

triangles. Construct polygon interior.

This shades and selects the region enclosed by the three points

chosen.

4. While the polygon interior is selected, measure the area.

5. Repeat the process for a different triangle formed by different points.

Continue until you see a pattern. It helps to change the color of each

polygon interior, select display and color.

6. Drag on a vertex. What Happens? Make a conjecture .

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Is the "balance point" or "center of gravity" a good description for the centroid? Why or why not? (Support your answer)

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