Pythagorean theorem

Pythagorean Theorem Report

By: Jessieca

8 Arctic

A.     Background

The Pythagorean theorem is important; because it will be apply on the daily life and also we will need this in jobs, works and other until were old (maybe). This is really important to us. It will be useful for jobs and other. There are some of examples where we need to learn Pythagorean, start from architecture, playing football and other. We can find Pythagorean everywhere. I will explain more about it in this report.

B.     Content

a.     What is Pythagorean theorem?

Before we learn more about Pythagorean theorem, we need to know what is Pythagorean theorem. Pythagorean theorem is the statement of right angle triangles, the Pythagorean theorem state that:

"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."

Resource: http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.html

Picture from: http://www.mathwarehouse.com/geometry/triangles/images/right-triangles/the-pythagorean-theorem/pythagorean-theorem-picture.png

b.     Proving Pythagorean theorem

To use the Pythagorean theorem, we need to prove if it works or not. There are many ways to prove the Pythagorean theorem. So, one of the way is to make what it shown in the picture:

Picture from: http://www.mathsisfun.com/geometry/images/pythagorean-theorem- proof.png
In that picture shows a square in side a square, and form 4 right angles. This is how to prove it:

            (²= 2ab+ c²

We also need to consider about this calculation (a+b)(a+b), this calculation come from the area of the big circle.

2ab+c²=(a+b)(a+b)

2ab+c²= a²+ab+ba+b²

2ab +c²=a²+2ab+b²

c²=a²+b²

This is how does the Pythagorean theorem proved. They are actually more ways but this is the obvious way to prove it.

c.      Pythagorean theorem in real life.

Let’s start from the right angle triangles. We can find them everywhere. Start from things in our home to the environment. Like example, TV, Pillows, AC, field, or even everything that have square, rectangle or straight up right angled triangles. These are some example of Pythagorean, and the world problem.

1.     Road trips

There are 2 Friend, 1 of them is in the market and he asks him to come to the market also. He wants to find the shortest way. There are 2 paths he could go to. The first path is to heading west 1mile and heading south 4 miles, the total distance is 5 miles. The other way is just pass the other road. What is the difference of both of the roads?

To find the difference, we need to know the distance of the other road

We will use the Pythagorean theorem

1²+4²=c²

1+16=c²

= c

4,5=c

5 -4,5=0,5 miles

The differences is 0,5 miles

2.     TV size

One day Mari wanted to buy a TV. She goes to the shop and saw a big TV, but they do not mention how many inches is that TV. It only says have a height of 15 inches height and 20 inches wide. How many inches is the TV?

To find how many inches is the TV is to use the Pythagorean theorem.

15²+20²=c²

225+400=c²

=c

25=c

The TV is 25 inches big

Those are some examples of Pythagorean theorem. There are many of other examples, but these are only some of them.

Resources: http://www.brighthubeducation.com/homework-math-help/36639-applications-of-pythagoras-theorem-in-real-life/

C.     Conclusion

My conclusion about Pythagorean theorem is that Pythagorean theorem is really important. It is useful for jobs, it is also useful in real life. The Pythagorean can be applied really easily and it has an easily to remembered formula.

Unit 6: Probability (chapter 15)

In this unit I learn about probability, start from the First

What is probability?

Probability is a type of ratio, were we compare the possible outcome and how often will it come out or maybe impossible to come out.

Language of probability:
-  Certain: will always happen, example, package of macaron you will always get a macron in it.
- Imposible: never happen or impossible to get that outcome, example, in a regular dice you never can get a number above 6.
- Very likely: usually happen, example, in a box there are 6 red ball and 1 blue ball, you will very likely get the red ball
- Very unlikely: usually won't happen, example, in a box there are 6 red ball and 1 blue ball, you will very unlikely get the blue ball
- even chance: 50% chance, example, in  a coin tossing.

If We put it in diagram, it will look like this:

I also learn about the complementary and compound event

Compound event

is a event went the outcome comes from 2 things, such as 2 dice.

Formula:

Example:

There are 2 dice, and we wanted to find the probability to get 6 and 6 and also odd number and 6.

Probability (6 and 6)






<---Those are the probability to get the number of each dice









Next step, is to times them together


Probability (odd and 6)

Just do the same thing, but what makes it different is the outcome that we wanted to get.

<---Those are the probability to get the number of each dice






Next step, is to times them together



Complementary event

is always the total of 1 or 100%

example:
We wanted to get a sum of 8 from 2 dice

first we need to find the list of the number that can sum a 8 from 2 dice:
2 + 6
3 + 5
4 + 4
5 + 3
6 + 2

There are 5 possible outcomes.

So the probability to get the sum of 8 from 2 dice is






and not to get the sum of 8 is






From this unit, I also learn to make games, not only making it but we can see some probability from games to, from me and my partner game that we make:

So, there are some differences from the regular snack and leader and our game

<--- from this one the differences is the snake that we change to a broken treasure chest,  and have 15 paper in it so there will be 15 outcomes from this chest

<----from this one the differences is the leader that we change to a golden treasure chest, and have 15 paper in it so there will be 15 outcomes from this chest

<---- we also change the dice, rather that 6 or 12 outcomes, how about 14 outcome. Using a cuboctahedron as a dice.



There are also other probability in our games:

• dice that is used in snake and ladder, we use a cuboctahedron that have 14 outcomes)
• Possibility to landed of getting  number we wanted to landed on.
• Possibility to take one of the paper from the 15 outcomes in each of the treasure box.
• Probability of getting good luck is 10 out of 100 blocks as well as the possibility of getting bad lucks.
• There’s 8 out of 14 possibilities on getting good and big value number when you roll the dice.
• There’s 50% chance to be winner as well as ending up being the loser.

Probability when rolling the dice:

• Getting the number that is larger than 1 but less than 6: 4/14
• Getting the number that is larger than 5 but less than 10: 4/14
• Getting the number that has a value of zero: Impossible
• Getting the number that is larger than number 1: Very likely
• Getting the number that is less that number 15: Certain

Unit 5: Statistic (Chapter 9 and 14)

In unit 5, I learn about statistic or specifically mean, median, mode and range. Also some types of table and how to make it, how to find mean, median, mode and range in the easier way in different tables or graphs.

So, for the first exercise is about mean, median, mode and range. 

Mean:

Mean is the average of all the number, by adding all the number than divided by how many numbers are there. Example the exercise below:

Find the mean of the number below:

63, 27, 19, 5, 47, 21

First just add all the number, or just follow the formula below;

So, if you follow the formula above, this is what will it look like if we put number in it

Than just solve it, first sum all the number in the numerator, than divided with the denominator. 

How does it work:

http://www.mathsisfun.com/mean.html 

Because when we sum everything together, it is equal to 3 part with 8 in each.

Median;

Median is the middle number. But before that, we need to arrange it from the smallest to he largest number. 

http://www.mathsisfun.com/definitions/median.html

To find a median, Example: 

Find the median of the number below:

4, 3, 7, 5, 1, 8, 3, 3, 7, 8, 3, and 6

First, arrange the number from the smallest to the