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Study Guide 2: Fractions Category 1: Numerical Representations and Relationships

Vocabulary Whole the entire amount of the figure or set of objects Fraction a number that stands for a part of a whole or a part of a group, equal parts of a whole (also called a fractional number) Equal parts the parts of a fraction must be divided equally Numerator top number in a fraction that names how many parts of a whole or group that are included in the fraction Denominator bottom number in a fraction that names the number of equal parts a whole or group is divided into Equivalent fraction two fractions that are equal or have the same amount or value Proper fraction a fraction with a numerator smaller than the denominator Improper fraction a fraction with a numerator larger than or equal to its denominator Mixed number a value that combines a whole number with a fraction Multiple the product (answer) of a whole number and any other whole

number Least common the smallest number that is a multiple of all the numbers in a set multiple Common a number that is a multiple of all denominators in a problem denominator Factor a number that is multiplied by another to get a product (answer) Greatest common the greatest number that is a factor of two or more numbers factor Simplify when the numerator and denominator have no common factor other

than 1 (also called lowest terms) Benchmark fractions such as 0, ¼, ½, ¾, or 1 that refer to the same whole, used fractions to evaluate the reasonableness of sums and differences of fractions

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Concepts Identifying Fractions

A whole refers to the entire parts of a figure or set of objects. The parts of a fraction must be divided into equal parts.

2 Numerator 4 Denominator

A fraction is a number that names a part of a set. The denominator (bottom number) tells how many equal parts are in the set and the numerator (top number) names how many of those parts are included in the fraction.

There are three types of fraction models: set models, region or area models, and length or measurement models.

In this fraction, the denominator (bottom number) is the number of equal parts in the set of stars, which is 8. The numerator (top number) is the number of shaded

stars, which is 4. This fraction is read as four eighths, or four parts out of 8 equal parts. It is written with a line between the numbers.

A fraction can also name the area or parts of a whole figure. This shape is divided into four equal parts and two of the parts are shaded. It is read as two fourths or 2 parts out of 4 equal parts. It is written with a line between the numbers.

You can also show fractions on a number line. The denominator (bottom number) shows the number of equal parts on a number line from 0 to 1. This number line is divided into six equal parts between 0 and 1. The first line after the 0 represents one sixth. The last line at one can also be named six sixths.

4 8

2 4

Unequal Parts Equal Parts


Whenever the numerator and denominator are the same number, such as 6/6, that is the same as one whole, or the number 1.

A unit fraction represents the quantity formed by one part of a whole. This whole has been divided into three equal parts and the unit fraction is 1/3.

+ + = or 1 whole + + = or 1 whole

You can count fractions just like you can count whole numbers. You count the numerators and the denominators stay the same. 1/4,

2/4, 3/4,

4/4, 5/4,

6/4, 7/4,

8/4, 9/4

1/5, 2/5,

3/5, 4/5,

5/5, 6/5,

7/5, 8/5,

9/5 Change an Improper Fraction to a Mixed Number

A proper fraction has a numerator smaller than the denominator.

An improper fraction has a numerator that is larger than the denominator. The proper fractions on this number line are 1/4, 2/4, and 3/4. The improper fractions on this number line are 5/4,

6/4, 7/4, and 8/4.

The number line has the improper fraction 6/4 marked. This could be recorded several other ways. However, they all are equal or the same as 6/4. 6/4 = 4/4 + 2/4 = 1 and 2/4 = 1 whole and 2/4 = 1 + 2/4

1 3

1 3

1 3

3 3

1 3

1 3

1 3

3 3

= 2 3

6 6 5

6

1 6

or 1 3 6

2 6

4 6

0

= 1 0 4

4 2 4

= 2 6 4

1 4

3 4

5 4

7 4

8 4


The fraction 1 + 2/4 is usually written as 1 2/4 with the plus sign being left out but understood. This is called a mixed number (1 2/4). A mixed number has a whole number and a fraction.

This is an example of the improper fraction 12/8. It also models 1 4/8.

You change the improper fraction 12/8 into a mixed number by dividing the numerator by the denominator. Twelve divided by 8 equals 1 with 4 left over. The 1 is the whole number for the mixed number. The 4 becomes the numerator and the 8 is the denominator of the fraction

for the mixed number 1 4/8. Change a Mixed Number to an Improper Fraction

You change the mixed number into an improper fraction by doing the opposite of division. To change a mixed number you multiply the whole number times the

denominator (1 x 8 = 8). Then you add the 8 + 4 = 12 to get the numerator. The denominator is 8 (12/8).

Comparing Fractions with the Same Denominator and Different Numerators

The two fractions below have the same denominator (bottom number) and different numerators (top numbers). Each model is divided into equal parts of four.

When the denominators or bottom numbers are the same, you can compare these two fractions by looking at the shaded portions in each model, which is the numerator.

1/4 < (is less than) 2/4

1 4 8

12 8

1 4 8


This same example can also be compared using a number line.

Comparing Fractions with the Same Numerators but Different Denominators

These two fractions have the same numerators (top numbers) but different denominators (bottom numbers).

Each model is divided into equal parts, but one model is divided into fourths and the other model is divided into thirds.

In both models, only one of the equal parts is shaded. The numerators in both models are 1.

When the numerators or top numbers are the same, you can compare these two fractions by looking at the size of the pieces or the denominators (bottom numbers).

> (is greater than)

The two fractions can also be compared using number lines. > (is greater than)

1 4

1 2 4

1 4

0

1 3

2 3

1 1 3

0

1 2 4

1 4

0 3 4

1 3

1 4


Comparing Fractions with Different Numerators and Denominators

These two fractions have different numerators and denominators. This example compares 1/2 to 2/3.

< (is less than)

This example compares 6/8 to 3/6. > (is greater than)

3 6

6 8

2 3

1 1 3

0

1 1 2

1 2

0

2 3

or 1 3 6

2 6

4 6

0

0 4 8

2 8 or 1

1

6 8

8 8

6 6 5

6

1 6


1 3

Equivalent Fractions with Models

Fractions that name the same amount are called equivalent fractions. Here are some examples of equivalent fractions using models.

= (is equal to) = (is equal to)

= (is equal to)

Equivalent fractions can also be shown on a number line.

2 6

1 2

2 4

3 6

4 8

1 2 4

1 4

3 4

0

0 4 8

2 8

1 6 8

1 8

3 8

5 8

7 8


Equivalent Fractions Using the Multiplication Identity Property

This is a model of the fraction 3/4.

This model doubles (multiplies by 2) the fraction 3/4.

This models the equivalent fraction of 6/8. When we double the number of squares, we multiply by 2/2. Two halves are the same as 1 whole.

The multiplication identity property states that any number multiplied by 1 remains unchanged (1 x 2 = 2, 1 x 3 = 3). This also means that any fraction with the same numerator and

denominator such as 1/1, 2/2,

3/3, 4/4,

5/5, 6/6,

7/7, etc. can be used as the identity element for an equivalent fraction.

Therefore when you multiply by 2/2, you have the equivalent fraction 6/8 (3/4 x 2/2 = 6/8).

Multiplying 3/4 by 3/3, 4/4,

5/5, 6/6,

7/7, etc. will also give equivalent fractions for 3/4.

Least Common Multiple

A multiple is the product (answer) of a whole number and any other whole number (2 x 4 = 8). Eight is a multiple of both 2 and 4. When you skip count you are saying multiples of a number.

The least common multiple is the smallest number that is a multiple of all the numbers in a set.

Use the chart to find the least common multiple for 5 and 7.

Multiples of 5 and 7

Have them look at the multiples of the larger number to see if it is in the list of the smaller number. They will draw a line through the 7, 14, 21, and 28. They will underline the 35 in both columns. This is the least common multiple of 5 and 7.

Number of Rows

Multiples of Five

Multiples of Seven

1 2 3 4 5 6 7 8 9

5 10 15 20 25 30 35 40 45

7 14 21 28 35 42 49 56 63

3/4 x 2/2 = 6/8


Comparing Fractions with the Least Common Multiple

This example compares 2/3 and 3/5. It is very difficult to look at these two fractions and determine which one is larger. They are both larger than one half, but less than 1. You cannot use benchmarks to compare them.

The only way to compare these two fractions is to use the multiplication identity property to make equivalent fractions that have the same denominator.

2/3 x 5/5 = 10/15 2/3 = 10/15 > 3/5 = 9/15 3/5 x 3/3 = 9/15

Multiples of 5: 5, 10, 15

List the multiples of 5 below that fraction until you reach a number that is a multiple of 3. Underline the 15 because it is also a multiple of 3 (3 x 5 = 15).

Change both fractions to equivalent fractions with a common denominator of 15. Multiply the 3 in 2/3 by 5 to get 15. If you multiply the bottom number by 5, then you must also multiply the top number by 5. Remember, whatever you do the bottom number you must do the same thing to the top number to keep the fractions equivalent. Multiply both the numerator and denominator of 2/3 by 5/5 to get 10/15.

Multiply 3/5 by 3/3 to get 9/15.

Now you can compare 10/15 to 9/15. Since the denominators are now the same (15), you look at the numerator to see which fraction is larger. The greater sign goes in the circle because two thirds is greater than three fifths.

Greatest Common Factors

A factor is a number that is multiplied by another number to get a product (2 x 4 = 8). The 8 is a multiple of 2 and 4. The 2 and the 4 are factors of 8.

The two numbers you multiply to get another number is called a factor pair (2 and 4). Another factor pair of 8 is 1 and 8. You list the factors of 8 in order (1, 2, 4, 8).

To list all the factors of a number you always start with 1 and the number. The example lists all the factors of 36. You put the 1 on the left and the 36 on the right with space between for the other factors. Then try 2, 3, 4, 5, etc. to see if they are factors. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

List all the factors of 18. Factors of 18: 1, 2, 3, 6, 9, 18

Factors that are the same for two or more numbers are common factors. The common factors for 18 and 36 are 1, 2, 3, 6, 9, and 18.

The greatest number that is a factor of two or more numbers is called the greatest common factor. The greatest common factor of 18 and 36 is 18.


Simplifying Fractions

A fraction is in simplest form when the numerator and denominator have no common factor other than 1 (also called lowest terms).

Factors of 8: 1, 2, 4, 8 Simplify: 8

12 ÷

4

4 =

2

3

To simplify the fraction 8/12 find the factors of the smallest number which is 8. The factors of 8 are 1, 2, 4, and 8.

Start with the largest factor, which is 8.

There are two ways to check if 8 is a factor of 12. Divide 12 by 8 to see if it divides evenly with nothing left over. Or you can think what do you multiply by 8 to get a product of 12?

Neither of these processes can be done, so 8 is not the greatest common factor.

Next try 4. You can multiply 4 x 3 = 12, so 4 is the greatest common factor of 8 and 12.

To simplify 8/12 divide both the numerator and denominator by 4/4. Remember, whatever you do to the top number you must do the same

thing to the bottom number.

You will get the simplified fraction of 2/3. The fraction 2/3 is equivalent to the fraction 8/12.

Adding Fractions

Adding fractions is just like adding 3 marbles and 9 marbles to get 12 marbles. You are putting things together. Then 3 thirds and 9 thirds would be 12 thirds. Three halves and 9 halves would be 12 halves. When you add the same thing (the same denominator), you add the two

numerators together to get the sum.

This example adds three fourths and one fourth. The equation is 3/4 + 1/4 = 4/4.

This example adds three fourths and two fourths. Because the denominators are the same you would add the numerators.

The equation is 3/4 + 2/4 = 5/4 or the mixed number 1 1/4.

3 4

3 4

5 4

+

+

1 4

or 1 whole 4 4 =

2 4 =

or 1 1 4


Addition of fractions can be modeled on the number line.

You can add 3/4 + 3/4 = 6/4. Start at three fourths. You are going to count four fourths, five fourths, and six fourths to get the

sum. Adding Fractions with the Associative Property

The associative property of addition states that the sum of two numbers is the same no matter how the numbers are grouped. This property also works with adding fractions.

7/8 + 6/8 could be grouped as 7/8 + (1/8 + 5/8) = (7/8 + 1/8) + 5/8 = 8/8 + 5/8 = 1 5/8

To evaluate the reasonableness of this sum, use benchmark fractions to estimate. When estimating, one might estimate 7/8 as 1 and 6/8 as 1/2 since the

estimate of 1 is a bit larger than 7/8. The estimated sum would be 1 1/2. This estimate is very close to the

answer 1 5/8, so this is a reasonable sum. Subtracting Fractions with Models

Subtracting fractions is just like subtracting marbles. How many marbles are left from 6 marbles if you take away (minus) 4 marbles? There are 2 marbles left. Then how much is 6 fourths minus 4 fourths? (2 fourths) How much is 6 thirds minus 4 thirds? (2 thirds) What about 6 tenths minus 4 tenths? (2 tenths)

When you subtract the same things (the same denominators), you subtract the numerators to get the difference.

0 4 4

2 4

2 6 4

1 4

3 4

5 4

7 4

7 8

6 8 +


This example models the equation 5/6 - 2/6 = 3/6. -

Take away 2/6 from 5/6. What fraction is left? (3/6)

Evaluate the reasonableness of this difference by using benchmarks such as 0, 1/4, 1/2,

3/4, or 1. The fraction 5/6 is close to 1 and the fraction 2/6 is close to 1/2. The estimated difference would be 1/2.

Subtracting a Fraction from a Mixed Number

The next example models the equation 1 3/6 – 5/6 = 4/6.

The model shows the mixed number 1 3/6. Take away the 5/6 to get 4/6.

Subtracting Fractions on a Number Line

Subtraction of fractions can also be modeled on the number line.

In the equation 6/4 - 3/4 = 3/4, you begin at the dot on the 6/4 and jump back 3 times to get the answer 3/4.

Multiples and Fractions in Place Value

The chart below shows the numeral 111,111,111. Each place value on the chart is ten times the value of the place to its right and is a multiple of 10. The tens place is 10 x 1 and the hundreds place is 10 x 10. The one thousands place is 10 x 100, the ten thousands place is 10 x

1,000, and the hundred thousands place is 10 x 10,000.

0 4 4

2 4

2

5 6

2 6

3 6

=

_ 9

6

1 5

6

=

4 6

or

6 4

3 6

1 4

3 4

5 4

7 4


The one millions place is 10 x 100,000, the ten millions place is

10 x 1,000,000, and the hundred millions place is 10 x 10,000,000.

Each place value on the chart is 1/10 the value of the place to its left. The ones place is 1/10 of 10, the tens place is 1/10 of 100, and the hundreds

place is 1/10 of 1,000. The one thousands place is 1/10 of 10,000, the ten thousands place is 1/10

of 100,000, and the hundred thousands place is 1/10 of 1,000,000. The one millions place is 1/10 of 10,000,000, the ten millions place is

1/10 of 100,000,000, and the hundred millions place is 1/10 of 1,000,000,000.

The numbers below show the patterns for the relationship between place value positions.

Millions Period Thousands Period Ones Period

Hu

nd

red

s

Te

ns

On

es

Hu

nd

red

s

Te

ns

On

es

Hu

nd

red

s

Te

ns

On

es

1 1 1 , 1 1 1 , 1 1 1 100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

10 x 10.000.000

10 x 1,000,000

10 x 100,000

10 x 10,000

10 x 1,000

10 x 100 10 x 10 10 x 1

Millions Period Thousands Period Ones Period

Hu

nd

red

s

Te

ns

On

es

Hu

nd

red

s

Te

ns

On

es

Hu

nd

red

s

Te

ns

On

es

1 1 1 , 1 1 1 , 1 1 1 100,000,000 10,000,000 1,000,000 100,000 10.000 1,000 100 10 1

1/10 of

1,000,000,000

1/10 of

100,000,000

1/10 of

10,000,000

1/10 of

1,000,000

1/10 of

100,000

1/10 of

10,000

1/10 of

1,000

1/10 of 100 1/10 of 10

Number 10 Times as Much as the Number

1/10 of the Number

20 200 2

400 4,000 40

50,000 500,000 5,000

20,000,000 200,000,000 2,000,000

6,000 60,000 600

300,000 3,000,000 30,000

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