Mathematics Local News Update
Did you know?
Breaking news!!! New mathematical way has been discovred
But what is the Pythagorean theorem about?
But what is a hypotenuse?
1st usage of the Pythagorean theorem
2nd usage of the Pythagorean theorem
3rd usage of the Pythagorean theorem
The Pythagorean Theorem tells us how to predict certain distance relationships from other distance relationships. If you see a pendulum swinging and you want to predict the coordinates of its tip over time, the Pythagorean Theorem is there. If you place a box on a ramp, and you want to know how quickly it will slide down, the Pythagorean Theorem is there. If you do any physics or engineering of any kind that involves the notions of "distance", "length", "angle", or "rotation"... well, the Pythagorean Theorem is there, and how much more real-life can you get?
Radicals have been introduced to math!!!
The History of Radicals
The mathematical concept of square roots has been in existence for many thousands of years. A square root of a number can be defined as a number that when multiplied by itself produces the first number. For example, 2 is the square root of 4, 3 is the square root of 9. Exactly how the concept was discovered is not clearly known, but several different methods of exacting square roots were used by early mathematicians. Recently discovered Babylonian clay tablets from 1900 to 1600 b.c. contain the squares and cubes of the integers 1 to 30 in Babylonian base 60 Akkadian notation. Whole number roots were specifically stated, while irrational roots were expressed in surprisingly accurate approximations.
Egyptian papyrus dating from about 1700 b.c. have revealed that the Egyptians were knowledgeable of square roots, using a hieroglyphic symbol similar to ⌈ to denote square roots. By the Greek Classical Period (600 to 300 b.c.), square root operations were improved upon by the use of better arithmetic methods. However, because Greek mathematicians were unsettled by inharmonious phenomenon such as irrational numbers (e.g., the square root of 2 is 1.4142135...), they regarded geometry more highly than algebra and arithmetic for its elegance and harmony.
When Hindu mathematics became significant about a.d. 628, mathematicians accepted irrational numbers and used square root operations freely in their equations. They used the term ka, from the word karana, to denote a square root. Thus, ka 9 would equal 3. Borrowing much from Hindu mathematics, Arabian mathematicians continued working with irrational number operations. It is the Middle Eastern mathematician al-Khwarizmi who developed our familiar term root to denote a solution to a problem. In the sixteenth century, the German mathematician Cristoff Rduolff was the first to use the square root symbol, , in his book Coss, written in 1525.
1st usage of radicals
In the real world people are interested in finding out what values are "normal" and what values are outside of normal, those values that are in the tails of the distribution. Students can't control how tall they grow so you don't want to call the shortest and tallest kids in your class abnormal! Student height was just an easy example to look at and understand.
But many factories use the normal distribution to make sure that the products that they are making are of good quality. The business people at the factory develop a normal distribution of the product and do not sell any items that measure in the tails of the distribution. There are lots of other uses for the normal distribution; factories are just one simple example.
What does the normal distribution have to do with squares and square roots? Plenty! The equations for finding the tails of the normal distribution use squares and square roots!
2nd usage of radicals
Here is one idea that showcases an important real-life application of square roots and at the same time lets students ponder where math is needed. This idea will work best after you have already taught the concept of square roots but have not yet touched on the Pythagorean theorem.
- Draw a square on board or paper, and draw one diagonal into it. Make the sides of the square to be, say, 5 units. Then make the picture to be a right triangle by wiping out the two sides of square. Then ask students how to find the length of the longest side of the triangle.
The students probably can't find the length if they haven't studied the Pythagorean theorem yet — but that is part of the "game". Have you ever seen an advertisement where you couldn't tell what they were advertising? Then, in a few weeks the ad would change and reveal what it was all about. It makes you curious.
So, let them think about it for a few minutes (don't tell them the answer at first). Hopefully it will pique their interest. Soon you will probably study the Pythagorean theorem anyway, since it often follows square roots in the curriculum.
- Then go on to the question: In what occupations or situations would you need to find the longest side of a right triangle if you know the two other sides? This can get them involved!
The answer is: in any kind of job that deals with triangles. For example, it is needful for carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers.
One time I observed construction people who were measuring and marking on the ground where a building would go. They had the sides marked, and they had a tape measure to measure the diagonals, and they asked ME what the measure should be, because they couldn't quite remember how to do it. This diagonal check is to ensure that the building is really going to be a rectangle and not a trapezoid or some other shape.
3rd usage of radicals
And for the last subject today, we have Quadrilaterals!!
Definition of every quadrilateral
Also opposite sides are parallel and of equal length.
-Rhombus:A rhombus is a four-sided shape where all sides have equal length.
Also opposite sides are parallel and opposite angles are equal. Another interesting thing is that the diagonals meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles. A rhombus is sometimes called a rhomb or a diamond.
-Square:A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).
-Parallelogram: A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same).
NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!
-Trapezoid:A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown. And a trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides
Who discovered them and when did they show up?
1st usage of quadrilaterals
2nd usage of quadrilaterals
3rd usage of quadrilaterals
There are a number of different types of quadrilaterals, some of which are harder to find in real life, such as a trapezoid. But, look around you – at buildings, at patterns on fabric, at jewelry – and you can find them!
We Hope that you are more informed about mat everyday
Math is always fun!!!
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