Vocabulary for "Looking for Pythagoras"
Square - A two-dimensional figure with 4, congruent sides and 4 right angles. A type of rectangle.
Perfect square numbers - numbers (of tiles) that can be form a square. The square root of a perfect square number is a whole number (or integer if considering the negative square root). A whole number squared is a perfect square number. The first ten perfect square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Area - the number of squares (square units) within the perimeter of a figure. The surface area of a 3 dimensional figure is the number of squares on its surface. Area is always measured in square units. A square cm is 1 cm by 1 cm, a square inch is 1 inch by 1 inch, etc.
Click the link below to see a short video on estimating square roots. It also includes how to simplify square roots.
Click the link below to get view a video showing a strategy to convert repeating decimals to fractions
Click the link below to see a video with a different strategy to convert repeating decimals to fractions
Natural numbers - counting numbers (positive numbers with no fractions or decimals). 1,2,3,4,...
Whole numbers - the natural numbers and 0. 0,1,2,3,....
Integers - the whole numbers and their opposites. ...,-3,-2,-1,0,1,2,3,...
Rational numbers - any number that can be written as a fraction (integer/integer). This includes all integers, fractions, decimals that end, and decimals that repeat, but never end.
Irrational numbers - numbers that cannot be written as fractions - decimals that go on forever with no repeating patterns. Includes pi, and many square roots.
Real numbers - the set of all rational and irrational numbers
PROOFS OF THE PYTHAGOREAN THEOREM (these prove that the Pythagorean Theorem works for any right triangle. This is not the same as the main idea behind the Pythagorean Theorem or how it works. The main idea is that the sum of the areas of the squares placed on the legs of right triangle is equal to the area of the square placed on the hypotenuse).
**note** To meet the standard, you need to understand and be able to explain one proof of the Pythagorean Theorem. To exceed the standard, you need to understand and be able to explain more than one proof. You do not need to understand them all. Below, you will find links for some of the different proofs. There are hundreds more out there if you wish to further explore.
Pythagoras' geometrical proof (the one that we examined in class based on the puzzles we cut out):
An algebraic look at Pythagoras' proof:
Bhaskara's First Proof (similar to the algebraic version of Pythagoras' proof):
Bhaskara's Second Proof (this requires an understanding of similar triangles):
Garfield's Proof (this requires an understanding of how to find the area of a trapezoid):