Learning Math: Geometry
The Pythagorean Theorem Part A: The Pythagorean Theorem (20 minutes)
Session 6, Part A
In this part
- Calculating Area
- Squares Around a Right Triangle
- The Theorem
Here is a square drawn on dot paper:
Come up with a method to find the exact area of the square in square units. You can either use calculations or count the square units.
Problem A1 adapted from IMPACT Mathematics, Course 1, developed by Educational Development Center, Inc. p. 536. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
Squares Around a Right Triangle
Each of the three figures in Problem A2 shows a right triangle with squares built on the sides. Determine the exact area of all three squares for each figure.
Area of Square on Side a
Area of Square on Side b
Area of Square on Side c
Look at your table, and come up with a relationship between the three squares that holds for all of the pictures.
In this video segment, the participants calculate the areas of the squares built on the right triangles. As they do the activity, a pattern begins to emerge, and they are able to formulate the relationship they discovered.
Was your method of calculating areas similar to or different from the one shown here? Can you come up with yet another way of doing this?
You can find this segment on the session video approximately 4 minutes and 58 seconds after the Annenberg Media logo.
Problem A2 and the Video Segment problems adapted from IMPACT Mathematics, Course 1, developed by Educational Development Center, Inc. p. 537. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
In a right triangle, the side opposite the right angle (side c in all of the pictures in Problem A2) is called the hypotenuse.
The problems you just solved illustrate the Pythagorean theorem: In a right triangle, the square built on the hypotenuse is equal in area to the sum of the squares built on the other two sides.
Today, most people think of the theorem as stating a relationship among three numbers, a, b, and c, which represent the lengths of the sides of a right triangle.
The Pythagorean theorem is named for Pythagoras, a Greek mathematician who lived from about 569-500 B.C.E., around the same time as Lao-Tse, Buddha, and Confucius. Pythagoras was the leader of a society that would likely be considered a cult by modern standards. They studied mathematics and numerology, were very superstitious about what they ate and how they lived, and were sworn to secrecy.
Part A: The Theorem adapted from Connected Geometry, developed by Educational Development Center, Inc. p. 197. © 2000 Glencoe/McGraw Hill. Used with permission. www.glencoe.com/sec/math
Draw a square with side lengths of 5 units around the given square so that the vertices of the given square are on the sides of the new square. The area of the new square is 25 square units (since its side length is 5), and its area is larger than the area of the original square by the area of the four right triangles whose legs have lengths of 2 and 3 units. So the area of the original square is 25 – 4 • (3 • 2 / 2) = 13 square units.
|It looks like this:
||Another method is the following:
Again, the area of the square is 4 • (2 • 3 / 2) + 1 • 1 = 13 square units.
Using the same method from A1, here are the areas:
The area of the square on side c is always the sum of the areas of the squares on the other two sides.
Session 1 What Is Geometry?
Explore the basics of geometric thinking using rich visualization problems and mathematical language. Use your intuition and visual tools for geometric construction. Reflect on the basic objects of geometry and their representation.
Session 2 Triangles and Quadrilaterals
Learn about the classifications of triangles, their different properties, and relationships between them. Examine concepts such as triangle inequality, triangle rigidity, and side–side–side congruence, and look at the conditions that cause them. Compare how these concepts apply to quadrilaterals. Explore properties of triangles and quadrilaterals through practical applications such as building structures.
Session 3 Polygons
Explore the properties of polygons through puzzles and games, then proceed into a more formal classification of polygons. Look at mathematical definitions more formally, and explore how terms can have different but equivalent definitions.
Session 4 Parallel Lines and Circles
Use dynamic geometry software to construct figures with given characteristics, such as segments that are perpendicular, parallel, or of equal length, and to examine the properties of parallel lines and circles. Look past formal definitions and discover the properties and relationships among geometric figures for yourself.
Session 5 Dissections and Proof
Review and explore transformations such as translation, reflection, and rotation. Apply these ideas to solve more complex geometric problems. Use your knowledge of properties of figures to reason through, solve, and justify your solutions to problems. Analyze and prove the midline theorem.
Session 6 The Pythagorean Theorem
Continue to examine the idea of mathematical proof. Look at several geometric or algebraic proofs of one of the most famous theorems in mathematics: the Pythagorean theorem. Explore different applications of the Pythagorean theorem, such as the distance formula.
Session 7 Symmetry
Investigate symmetry, one of the most important ideas in mathematics. Explore geometric notions of symmetry by creating designs and examining their properties. Investigate line symmetry and rotation symmetry; then learn about frieze patterns.
Session 8 Similarity
Examine your intuitive notions of what makes a "good copy" and then progress toward a more formal definition of similarity. Explore similar triangles and look into some applications of similar triangles, including trigonometry.
Session 9 Solids
Explore various aspects of solid geometry. Examine platonic solids and why there are a finite number of them. Investigate nets and cross-sections for solids as a way of establishing the relationships between two–dimensional and three–dimensional geometry.