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Fractions: Understanding Parts and Their Relationships, Schemes and Mind Maps of Elementary Mathematics

An introduction to fractions, explaining what they are, how they are represented, and how they can be used to represent parts of a whole. It covers topics such as complementary fractions, reducing fractions, and comparing fractions. exercises for practice.

What you will learn

  • How do you write a fraction that represents two fifths?
  • How do you find the complement of a fraction?
  • How do you reduce a fraction to its simplest form?
  • How do you compare two fractions to determine which is larger?
  • What is the meaning of a fraction?

Typology: Schemes and Mind Maps

2021/2022

Uploaded on 09/12/2022

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Intro to Fractions
Revised @ 2009 MLC page 1 of 12
Intro to Fractions
Reading Fractions
Fractions are parts. We use them to write and work with amounts that are
less than a whole number (one) but more than zero. The form of a fraction is
one number over another, separated by a fraction (divide) line.
i.e.
9
5
and,
4
3
,
2
1
These are fractions. Each of the two numbers tells certain information
about the fraction (partial number). The bottom number (denominator) tells
how many parts the whole (one) was divided into. The top number
(numerator) tells how many of the parts to count.
2
1
says, “Count one of two equal ports.”
4
3
says, “Count three of four equal parts.”
9
5
says, “Count five of nine equal parts.”
Fractions can be used to stand for information about wholes and their parts:
EX. A class of 20 students had 6 people absent one day. 6 absentees
are part of a whole class of 20 people.
20
6
represents the fraction of
people absent.
EX. A “Goodbar” candy breaks up into 16 small sections. If someone ate
5 of those sections, that person ate
16
5
of the “Goodbar”.
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Intro to Fractions

Reading Fractions

Fractions are parts. We use them to write and work with amounts that are

less than a whole number (one) but more than zero. The form of a fraction is

one number over another, separated by a fraction (divide) line.

i.e.

, and^5 4

,^3

These are fractions. Each of the two numbers tells certain information

about the fraction (partial number). The bottom number (denominator) tells

how many parts the whole (one) was divided into. The top number

(numerator) tells how many of the parts to count.

1 says, “Count one of two equal ports.”

3 says, “Count three of four equal parts.”

5 says, “Count five of nine equal parts.”

Fractions can be used to stand for information about wholes and their parts:

EX. A class of 20 students had 6 people absent one day. 6 absentees

are part of a whole class of 20 people.

6 represents the fraction of

people absent.

EX. A “Goodbar” candy breaks up into 16 small sections. If someone ate

5 of those sections, that person ate

5 of the “Goodbar”.

Exercise 1 Write fractions that tell the following information:

1. Count two of five equal parts

2. Count one of four equal parts

3. Count eleven of twelve equal parts

4. Count three of five equal parts

5. Count twenty of fifty equal parts

6. It’s 25 miles to Gramma’s. We have already driven 11 miles. What

fraction of the way have we driven?

7. A pizza was cut into twelve slices. Seven were eaten. What fraction

of the pizza was eaten?

8. There are 24 students in a class. 8 have passed the fractions test.

What fraction of the students have passed fractions?

The Fraction Form of One

Because fractions show how many parts the whole has been divided into and

how many of the parts to count, the form also hints at the number of parts

needed to make up the whole thing. If the bottom number (denominator) is

five, we need 5 parts to make a whole: 1

5. If the denominator is 18, we

need 18 parts to make a whole of 18 parts: 1

18. Any fraction whose top

and bottom numbers are the same is equal to 1.

Example: 1

1 ,^6

1,^11

1,^100

1 ,^4

Exercise 2

Write the complements to answer the following questions:

1. A cake had 16 slices. 5 were eaten. What fraction of the cake was

left?

2. There are 20 people in our class. 11 are women. What part of the

class are men?

3. It is 25 miles to grandma’s house. We have driven 11 miles already.

What fraction of the way do we have left to go?

4. There are 36 cookies in the jar. 10 are Oreos. What fraction of the

cookies are not Oreos?

Reducing Fractions

If I had 20 dollars and spent 10 dollars on a CD, it’s easy to see I’ve spent

half of my money. It must be that

10. Whenever the number of the part

(top) and the number of the whole (bottom) have the same relationship

between them that a pair of smaller numbers have, you should always give

the smaller pair answer. 2 is half of 4. 5 is half of 10.

1 is the reduced

form of

5 and

2 and

10 and many other fractions.

A fraction should be reduced any time both the top and bottom number can

be divided by the same smaller number. This way you can be sure the

fraction is as simple as it can be.

5 both 5 and 10 can be divided by 5

1 describes the same number relationship that

5 did, but with smaller

numbers.

1 is the reduced form of

6 both 6 and 8 can be divided by 2.

3 is the reduced form of

When you divide both the top and bottom numbers of a fraction by the same

number, you are dividing by a form of one so the value of the fraction

doesn’t change, only the size of the numbers used to express it.

12 These numbers are smaller but they can go lower

because both 6 and 8 can be divided by 2 again.

or^27 7

Exercise 3. Try these. Keep dividing until you can’t divide anymore.

Good knowledge of times tables will help you see the dividers you need to

reduce fractions.

Here are some hints you can use that will help, too.

Hint 1

If the top and bottom numbers are both even, use

Hint 2

If the sum of the digits is divisible by 3 then use

Exercise 4

Directions: Reduce these fractions to lowest terms

18

25

12

Higher Equivalents

There are good reasons for knowing how to build fractions up to a larger

form. It is exactly the opposite of what we do in reducing. If reducing is

done by division, it makes sense that building up should be done by

multiplication.

A fraction can be built up to an equivalent form by multiplying by any form

of one, any number over itself.

9

6 33

22 12

8 18

12 3

(^2) All are forms of 3

2 ; all will reduce to 3

Comparing Fractions

Sometimes it is necessary to compare the size of fractions to see which is

larger or smaller, or if the two are equal. Sometimes several fractions must

be placed in order of size. Unless fractions have the same bottom number

(denominator) and thus parts of the same size, you can’t know for certain

which is larger or if they are equal.

Which is larger

2 or 6

5? Who knows? A ruler might help, but rulers aren’t

usually graduated in thirds or sixths. Did you notice that if 3 were doubled,

it would be 6?

So build up

2 by 2

Then it’s easy to see that

5 is larger because it counts more sixth parts

than

4 , so

5 means

Which is larger

or^3 16

Build up

so^15 16

.^15

.^3

by^4 4

Exercise 5. Use <, >, or = to compare these fractions

1

6

27

(^312)

The answer is the whole number 3. The remainder 1 is the top number

of the fraction and the divider 2 is the denominator (bottom fraction

number).

Exercise 7. Reduce these top heavy fractions to mixed numbers.

Top heavy fractions may contain common factors as well. They will need to

be divided out either before or after the top heavy fraction is changed to a

mixed number.

26 but

2 can be divided by

  1. Then 8

If you had noticed that both 26 and 8 are even, you could divide out

2 right

away and then go for the mixed number. Either way, the mixed number is the

same.

Exercise 8

10

Answer Key

Exercise 1 Exercise 2 Exercise 3 Exercise 4

5

4

5

12

5

50

25

12

24

4

Exercise 5 Exercise 6 Exercise 7 10.

115 Exercise 8

4

5

9