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Mathematics lesson: Sec: Fractional arithmetic
What is a break
Amelie, Ben, Clara and Dean just had bio and have math straight away. But now they have a break first.
- D: I have gummy bears with me again!
- B: Cool.
- A: 1, 2, 3 ...
- D: You don't need to count! My mother always gives me exactly 25.
- B: That's a stupid number ... how do we divide it up?
- C: Well, each of us gets 6. Then there's one left.
- B: I think Dean gets it ...
- D: Hmmm ... but I want them to be divided fairly.
- A: Well we could cut the gummy bear into four pieces and give each one a piece.
- B: Yes. That is fair.
Without knowing it, the four of them have just taught each other what a break is. This is useful because fractions will be the topic of the next math lessons. Even so, they will not be able to rest because there is a lot to learn here.
- Such as or is called .
- The line in a fraction is called. The fraction line has the same meaning like a split sign.
So you can also as write.
- The number above the fraction line is called.
- In the break is 3 so the counter.
- The number under the fraction line is called. In the break is 5 so the denominator.
- In general, a fraction is the following: or Numerator divided by denominator.
- One becomes the numerator and denominator of a fraction exchanged. So the numerator becomes the denominator and the denominator becomes the numerator.
- Generally: When we break is the downside of it .
- Example: Of is the turning point .
- Is in a fraction of the Counter = 1, we call the fraction. So .
- The majority of fracture is Fractions. So we say: "There are two breaks".
- If two breaks denote the same denominator we call the fractions.
- Example of fractions with the same name:
- and are two fractions of the same name.
- If two breaks denote the same counter we call the fractions.
- Example of fractions of the same number:
- and are two equal fractions.
Types of fractions
There are essentially three types of fraction. These are:
a) real fraction:
- the counter is smaller than the denominator
- z. B. is a proper fraction because 3 (the numerator) is less than 5 (the denominator).
- A real fraction is smaller than 1. It denotes a (fraction) part of a whole (an integer)
b) improper fraction
- the counter is greater than the denominator
- e.g. is an improper fraction because 7 (the numerator) is greater than 5 (the denominator).
- An improper fraction is greater than 1. It denotes one or more wholes and a fraction of that whole
c) mixed fraction
- consists of a whole number and a fraction.
- z. B.
- This is a simplified form of writing
If you remember the story at the beginning of the chapter, a gummy bear was divided into four parts:
So it means everyone gets Gummy bear.
In the fraction, 1 is the numerator and 4 is the denominator. is also automatically a trunk fraction, since the counter is 1. The break is also a real break because describes part of the whole. The piece is a fraction of the whole gummy bear.
Shortening and expanding fractions
The math lesson is over. Now Amelie, Ben, Clara and Dean have German with Mr. Schnipsel. As always, he does very boring lessons. And of our four friends, only Amelie seems to be interested in the class:
- B: Man, what do I care about grammar? I can speak properly.
- D: Exactly!
- A: Pssst! I can not concentrate!
- D: Why do you want to concentrate?
- A: Because I care.
- B: Well, I don't care.
- D: You Clara, where do you look all the time?
- C: On the clock.
- D: So if you hope that the time will go faster I wouldn't look at the clock! Because then time goes slower.
- B: Oh nonsense! This is total nonsense.
- C: guys! I don't look at the clock to make it run faster, but because I'm wondering why we say half past twelve when the hand is on 30.
- B: Yes, because the pointer is halfway around then.
- C: Yes, that's clear, but how can it be that half of them are 30? I'm wondering that right now.
- D: That's probably like 50 being half.
- B: But here the 50 is much more than half! Even more than three quarters.
- C: Actually, Dean!
- A: Now be quiet at last.
- C: If 50 is half, 100 is everything. And at the clock everything, or all of it, is 60. And half of it is 30.
- A: If you keep talking, I will condemn you to Mr. Schnipsel.
- D: Don't act like that. Clara is solving a math puzzle!
- C: guys! It all makes total sense! 50 out of 100 is half. 30 out of 60 is half. 15 out of 60 is a quarter and 45 out of 60 is three quarters!
- B: Well, I don't quite understand that yet ...
- D: Neither do I! Do you understand Amelie?
- A: Now be quiet Dean !!! Bloody hell!
Mr. Schnipsel: What's the matter with Amelie?
- A: They talk all the time ...
- D: Sneak!
Mr. Schnibsel: The corner is quiet now, otherwise you'll have to copy a poem for tomorrow!
- C: Okay. I quickly write the explanation down on a piece of paper.
After a short time, Clara gives Dean the following note:
But of course there are more fractions that add value have e.g. B. or
If you write down a few fractions that add up to 5 it looks like this:
- As you can see, the denominator and the numerator are taken with 2 times in the first row.
- In the second row, the denominator and the numerator are always taken with 3 times.
- In the third row, the denominator and the numerator are always taken with 4 times.
- In all of the three rows, the value of the fractions always remains 5.
The value of the fraction remains the same if the denominator and the numerator are multiplied by the same number. This is called one Expand fracture.
- If you go backwards in the row, we see that in the first row, the denominator and the numerator are divided by 2.
- If we go backwards the second row, the denominator and numerator in the row are always divided by 3.
- If we go backwards the third row, the denominator and numerator in the row are always divided by 4.
- In all of the three rows, the value of the fractions always remains 5.
So the value of the fraction stays the same if the denominator and numerator are divided by the same number. This is called one Shorten the fraction.
Expand: Denominator and numerator are used with the same number multiplied. The value of the fraction remains the same.
Shorten: Denominator and numerator are given by the same number divided. The value of the fraction remains the same.
If you remember the story at the beginning of the section: There Clara asked herself how a 30 can represent a half. 30 minutes out of 60 minutes have passed. This explains why 30 past minutes can represent half of the past.
Below you will find a few exercises, it can happen that you find large numbers difficult to calculate. Then use a trick:
If said cross-sum is included in the numerator as well as in the denominator in the 3 series, the entire fraction can be reduced by 3.
Adding and subtracting fractions
Amelie, Ben, Clara and Dean have a break now. You go to the canteen.
B: Oh cool! Today we have pizza.
C: man! I forgot my money for food.
D: It doesn't matter. You can get something from my pizza!
A: You can also get something from me.
Ben, Clara and Dean each bought a pizza. She and Amelie sit down at their regulars' table.
D: Here you have 3 of my 6 pieces of pizza, Clara.
C: Thank you!
A: You will get 2 pieces from me.
C: Thank you! Your pieces are bigger than Dean's ...
A: I only cut my pizza into 4 pieces.
B: I love this pizza ... I can eat a hundred slices of it.
D: Do you want another slice of my pizza?
B: If you still get full ...
B: Thank you.
How much pizza do Amelie, Ben, Clara and Dean each have?Amelie divided her pizza into 4 pieces. So she has or simply written, . She gives 2 pieces to Clara: or simply written . So the calculation is: . Ben has a pizza: . He gets one of his 6 pieces from Dean: or it is written more simply . So the calculation is: . Clara has . But she gets from Amelie or simply written . So the calculation is: . Additionally she gets from Dean . So the whole bill is: . Dean split his pizza into 6 pieces. With that he has or simply written, . He gives 3 pieces to Clara: or simply written . So the calculation is: . In addition, he gives 1 piece of 6 pieces to Ben, so or simply written . So the whole bill is: .
Write down all invoices again:
But how do you calculate that?A) When Amelie surrenders 2 of her 4 pieces, she only has 2 of 4 pieces left. Then she wrote in the fraction . So we know that we have to subtract the number of pieces from each other to get the result. Because actually we did the math: B) Like us we have already learned above: C) Clara has the invoice . Here we cannot simply count the pieces together because they are of different sizes. But we could make the pieces so much smaller that they are the same size. So we can shorten the fractions. We learned how to do this above. If Amelie and Dean had split their pieces into 12 pieces, the pieces of the two would be the same size. So Amelie can divide all of her pieces into three parts again. Dean can divide all of his pieces into two parts again. We can now do the math: D) The bill for Dean looks like this: Now we know we just need to peel pieces off each other:
GenerallyTo be able to add two fractions, the denominator the breaks be equal. This often requires fractions expand and or shorten. The Counters added man The Denominator remains equal.
With different denominators: . Example:
If the denominators are the same:
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