Geometry - Yola



Geometry

Due January 4, 2010

The Pythagorean Spiral Project

 

 

In this project you, as students, will have the opportunity to explore the historical development of the concept of irrational numbers and the Pythagorean Theorem. You will research the life and times of the Greek mathematician Theodorus and will create a poster of the Pythagorean spiral.

 

Materials: + poster board + ruler + pencil, colored pencils or markers

 

TASK

               

You will complete the following tasks:

RESEARCH – SUMMARIZE – CONSTRUCT - ANALYZE

1.  Research the life and times of Theodorus of Crete and the secret society of Pythagoreans to which he belonged by using the links that are provided.

Information on Theodorus: 





 

Information on the Pythagoreans: 

 



 

Information on irrational numbers:



Other cool spirals:



2.  Use the information that you have gathered to summarize and complete the “Who was Theodorus?” worksheet. Answers must be in complete sentences!

3.  Construct the Pythagorean Spiral by following the instructions provided.     

4.  Use your wheel to explore some interesting mathematical ideas on the number system by analyzing and completing the “Analyzing the Wheel” worksheet.

TO TURN IN:

1) Your poster with light pencil lines shown for constructions and color used to decorate the pattern. Your triangles need to be numbered.

2) Your work for each hypotenuse length using the Pythagorean Theorem on the sheet of paper attached. Answers should be in reduced radical form.

3) Your “Who was Theodorus?” and “Analyzing the Spiral” worksheet (attached).

4) Your grading rubric that is attached.

You will be graded based on the rubric attached. This will count as a TEST grade.

TAKE THIS VERY SERIOUSLY!

PROCESS

Step 1: Place the poster board in landscape orientation. Start measuring from the top left hand corner and go 22 cm right then 14 cm down. This will be the starting point for your diagram. It will assure that your diagram stays on the page.

 

Step 2: Using your ruler create a segment that is 4 cm across starting from the starting point and heading towards the right of the poster. Make this segment perpendicular to the side of the poster. Use you ruler to construct a congruent segment that is perpendicular to the original. Connect the endpoints of the two segments to create a right isosceles triangle.

[pic]

Step 3: On a separate piece of paper, use the Pythagorean Theorem to calculate the length of the hypotenuse. Show all work and write your answer in reduced radical form.

 

Step 4: Using the hypotenuse of the first triangle, create another right triangle on top of the previous hypotenuse. The old hypotenuse will be the new base and construct a perpendicular segment to this, with a length of 4. Then connect the two segments to form a new hypotenuse.

[pic]

 

Step 5: On your same separate piece of paper, show the calculations to find the length of the new hypotenuse.

 

Step 6: Continue to repeat this process of connecting and constructing new triangles with a side length of 4, using the previous hypotenuse as the other side. Continue to show your calculations on your separate piece of paper. Construct triangles until you have formed a full spiral, 17 total triangles.

 

Step 7: Detail your Pythagorean Spiral with a design. Use color and a pattern to make a creative picture. Be creative! See me for examples.

Name______________________________________________ Period ______________

Grading Rubric for Pythagorean Spiral Project

|Number of Points |“Who was Theodorus?” & |Calculations for each |Poster Result |Creativity |Score |

| |“Ananlysis of the Wheel” |hypotenuse on separate | | | |

| |worksheets |sheet of paper | | | |

|4 |All the questions and the|All work is shown for |The result shows 17 right|The poster is | |

| |table were attempted and |each triangle using the |triangles that rotate |creatively colored | |

| |answered correctly in |Pythagorean Theorem and |around to the left and |and decorated. | |

| |complete sentences with |each answer is simplified|the last triangle | | |

| |thoughtful analysis. |neatly in the space |overlaps the original. | | |

| | |provided. | | | |

|3 |Most of the questions |All work is shown for |The result shows an error|The poster is | |

| |were attempted and |each triangle using the |in construction resulting|colored but the | |

| |answered correctly, with |Pythagorean Theorem but |in one fewer or one more |results are not | |

| |some answers not in |some answers are not |triangle. |neat. | |

| |complete sentences. The |properly simplified. | | | |

| |table was mostly |Work is not neatly | | | |

| |completed with only a few|represented in the space | | | |

| |errors and some attempt |provided. | | | |

| |at analysis. | | | | |

|2 |A few questions were |All work is shown but |The result goes the wrong|The poster is | |

| |attempted and answered |with errors in |direction and/or is off |partially colored or| |

| |correctly, with most |calculation and/or |by more than one |incomplete. | |

| |answers not in complete |simplification. Work is |triangle. | | |

| |sentences. Table |not in the space provided| | | |

| |partially completed, with|and is hard to follow. | | | |

| |many errors, and a | | | | |

| |limited attempt at | | | | |

| |analysis. | | | | |

|1 |Nothing submitted. |Only partial work is |The result does not |The poster is not | |

| | |shown and/or no evidence |appear to have followed |colored or | |

| | |of the Pythagorean |the proper requirements. |decorated. | |

| | |Theorem. Not done on |  | | |

| | |sheet. | | | |

 A: 15 - 16    B+: 13 -14    B: 11 -12    C+: 9 - 10   C: 7 -8   D: 5 - 6    F: 0 - 4   

 

[pic] [pic]

EXAMPLES

Who was Theodorus? Name: ____________________

1. When and where was Theodorus born?

2. Where did he spend most of his working life?

3. Name one of his most influential teachers.

4. Name one of his most influential pupils.

6. Theodorus belonged to a brotherhood of scholars who believed that numbers were pleasing to the Gods. Describe another one of their beliefs.

7. What is an irrational number?

8. According to legend how was the Pythagorean dissenter put to death?

9. Tell me something about another interesting spiral.

Analyzing the Wheel Name: _______________________

Complete the table by measuring the length of each hypotenuse with your ruler. Next, determine the length of the hypotenuse using the Pythagorean Theorem on a separate sheet of paper (answer should be in reduced radical form). Then use your calculator to find the value of each square root in decimal form.

| |(Ruler) |(Pythagorean theorem) |(Calculator) |

|Triangle |Measurement |Hypotenuse |Decimal |

| | | | |

|1 | | | |

| | | | |

|2 | | | |

| | | | |

|3 | | | |

| | | | |

|4 | | | |

| | | | |

|5 | | | |

| | | | |

|6 | | | |

| | | | |

|7 | | | |

| | | | |

|8 | | | |

| | | | |

|9 | | | |

| | | | |

|10 | | | |

| | | | |

|11 | | | |

| 12 | | | |

| 13 | | | |

| 14 | | | |

| 15 | | | |

| 16 | | | |

| 17 | | | |

|[pic] |[pic] |

|Kyle created a turkey while incorporating the assigned Wheel of Theodorus. |Mollie made a beautiful Wheel that reminded her of a |

| |shell on the beach. |

|[pic] |[pic] |

|John made his wheel into a French horn. |Dan created a hair style when he drew his Wheel. This|

| |is his Princess Leah. |

|[pic] |[pic] |

|Will made a scary monster-snail-thing for his Wheel. |Kristina's Wheel involved a tongue and a lolly pop. |

|[pic] |[pic] |

|Luke's was interesting |Melanie drew a beach ball flying high. |

|[pic] |[pic] |

|Jessica's Wheel is a sort of a sun. |Ben created a snail on the beach. |

Pythagorean Theorem work space (for 16 of the 17 triangles)

Name _____________________

|1) |2) |

|3) |4) |

|5) |6) |

|7) |8) |

|9) |10) |

|11) |12) |

|13) |14) |

|15) |16) |

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