Lesson Designed By:

TOPIC:

DISCOVERY LESSON: Pythagorean Theorem

LEVEL:

GEOMETER'S SKETCHPAD PROFICIENCY:

CLASS TIME:

Considering time constraints and/or student ability level the teacher may choose to allow students to use script tools which have been previously created. However, worksheets were designed to allow students to construct their own right triangle, square and polygon "script tools." It is fascinating to watch students as they "talk math" and "discover" how to construct regular polygons using the various tools provided within the sketchpad environment.

GEOMETER'S SKETCHPAD SKILLS NEEDED:

NOTES TO TEACHER:

*** You need version 3 of The Geometer's Sketchpad for this activity. ***

In an attempt to help students move sequentially through the geometric cognitive levels as defined by the van Hiele paradigm, this project incorporates a computerized investigation of the Pythagorean Theorem and it's converse into an existing geometry curriculum. The lesson was designed to facilitate the use of higher order thinking skills as students employed the concepts of similarity, measurement and transformation geometry in conjunction with traditional proofs to explore, understand and justifiy the theorem and it's converse. Use of The Geometer's Sketchpad as an investigative mathematical tool plays an integral part in the activities by providing informal, concrete experiences for students as they explore abstract geometric concepts. The project addresses two of the topics which the NCTM Standards suggest should receive increased attention in geometry classrooms: 1) Transformation approaches to geometry and 2) Computer-based explorations of figures.

The unit is an adaptation and extension of several activities found in Dan Bennett's Pythagoras Plugged In (Key Curriculum Press). His activity book includes a brief introduction to and history of The Pythagorean Theorem; detailed activity sheets for the construction, labeling and manipulation of a right triangle, and the construction of a "script tool" for a square. The book also contains a series of proofs for the Pythagorean Theorem.

*** See the v3 Review for more information on Sketchpad activity books. ***

The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equivalent to the square of the hypotenuse. It is expressed algebraically as: a2 + b2 = c2. From a geometry point of view, one can show that the sum of the areas of squares built on the legs is equivalent to the area of a square built on the hypotenuse. i.e., it can be expressed geometrically as: a(square)+ b(square)= c(square). Later stages of this activity encourage an investigation into the possibility that it might also be expressed as a(triangle) + b(triangle) = c(triangle), and/or a(pentagon) + b(pentagon) = c(pentagon), etc.?

• Construct, measure and manipulate two-dimensional objects using The Geometer's Sketchpad.
• Use induction/deduction to: a) make and test conjectures and b) construct, and judge the validity of, an argument when classifying and constructing figures and investigating the Pythagorean Theorem and it's converse.
• Construct a right triangle; classify and construct regular polygons based upon information regarding angle and/or side measure.
• Use transformations, recognizing their effects on geometric figures and to develop alternative methods of proof and construction.
• Identify similar figures and use their properties.
• State and apply the Pythagorean Theorem and it's converse.

INVESTIGATION: Pythagorean Theorem

STATEMENT OF THE PROBLEM:

The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equivalent to the square of the hypotenuse. It is expressed algebraically as: a2 + b2 = c2.

From a geometry point of view, does this mean that the sum of the areas of squares built on the legs is equivalent to the area of a square built on the hypotenuse? In other words, can this be expressed geometrically as a(square)+ b(square)= c(square)?

PROCEDURE:

INVESTIGATION 1:

SET UP THE SKETCH

1. Construct a right triangle. Move the vertices and sides to be certain it remains a right triangle.
   
2. Construct a square and shade it in. Manipulate the figure and be sure it is always a square.
   
3. Make a script for the square.
   
4. Save the script in the Script Tool folder as square, then delete the existing square.
   
5. Use your new script tool to construct a square on each side of the right triangle. Match points carefully!
6. Measure the areas of the three squares.
7. Relabel the areas as a2, b2 and c2. [c2 should represent the area of the hypotenuse.]
8. Use the calculator to determine the sum: a2 + b2.
9. Use the Measure Menu - Tabulate option to place the four measurements into a chart.

ACTIVITIES:

1. Move the vertices and sides of the right triangle in your sketch. Notice that the area measurements change.
2. Double-click on the chart to add your new measurements to the chart after each of several alterations.
3. Write a summary of your findings. Use complete sentences.

INVESTIGATION 2:

i.e. Is it true that a(triangle) + b(triangle) = c(triangle), and/or a(pentagon) + b(pentagon) = c(pentagon), etc.?

SET UP THE SKETCH:

1. Construct a script tool for one of the following regular polygons: equiangular triangle, pentagon, hexagon and octagon. Check with the groups next to you. Do NOT use the same polygon that they select.
2. Follow the procedure outlined in the original investigation using your new script tool.

ACTIVITIES:

1. Summarize your findings. Use complete sentences!
2. Compare your results with the groups next to you.
3. Do your conclusions match the other groups? If not, can you explain why?
4. Do the Similar Triangle Proof activity on the Pythagoras Plugged In handout.
5. What would happen if the three triangle interiors were reflected across their respective hypotenuses in this exercise?
6. Does this activity support or contradict your earlier findings? In what way?

INVESTIGATION 3:

ACTIVITIES:

1. Do the Unsquare Pythagoras activity on the Pythagoras Plugged In handout.
2. What transformation is being used as the objects are changed proportionally? Is this an isometric transformation? Why/why not?
3. What transformations are being used as the objects are moved? Are they isometric?
4. Does this activity support or contradict your earlier findings? In what way?
5. Draw any triangle that does NOT have a 90 degree angle.
6. Construct a square on each side of the new triangle and measure the areas.
7. Does the Pythagorean Theorem still work? Can you alter this triangle so that a2 + b2 = c2?
8. State the converse of the Pythagorean Theorem.
9. How would you explain what the Pythagorean Theorem meant to a 5th grader who did not understand algebra?

EVALUATION:

This time also provides me with an opportunity to observe their:

• ability to follow directions
• ability to work in a group setting
• spatial reasoning abilities
• problem-solving and data analysis strategies
• it gives me some insight as to which students have a workable understanding of the concept of a construction vs a drawing.