words are bolded and are Adding and Subtracting Fractions and Mixed … · Adding and Subtracting Fractions and Mixed Numbers ©2019 . When adding and subtracting fractions, the most

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SOL 5.6a Computation and Estimation TEACHER GUIDE Page 1

Adding and Subtracting Fractions and Mixed Numbers

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When adding and subtracting fractions, the most important part of the process is to make sure that the denominators are the same. Adding and subtracting fractions is easy if we have like or

common denominators. When we have like or common denominators, we just add or subtract the numerators. The denominators are just names of the parts we are working with.

Consider this problem: 710

+ 810

= 1510

That’s pretty easy. We add the numerators, 7 plus 8 equals 15. The denominator is tenths.

The solution is 1510

or fifteen-tenths.

Improper Fractions

However, we are not finished with the sum in this problem. The fraction 1510

is called an improper

fraction. An improper fraction is a fraction where the numerator is greater than the

denominator. 49 and

711 are also examples of improper fractions. We have to rewrite an

improper fraction as a mixed number.

Mixed Numbers

A mixed number is a number that has two parts: a whole number and a fraction.

The improper fraction 49 becomes 2

41 , a mixed number. The improper fraction

711 becomes 1

74 , a mixed number. The value of a mixed number is the sum of its two parts.

Math Vocabulary: Content words are bolded and are defined through context or example.

Questions, Cues, Advance Organizers/Activate Prior Knowledge! Open your brain folders on fractions and mixed numbers. How are fractions and mixed numbers related? Show your knowledge on your What I Know page. Add pictures and words. Then, share with a partner.

Show students how to define word using context clues.

1

Activate prior knowledge.

ASK QUESTIONS! Number and box each paragraph one at a time. For each paragraph, students ask questions that are important (go back to objective) and underline the answers in that paragraph.



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Writing an Improper Fraction as a Mixed Number

Let’s write the improper fraction 1510

as a mixed number.

We know that 1010

= 1. Therefore, 1510

represents more than one whole.

How much more? To find out we subtract:

We now know that the improper fraction 1510

can be written as 1 510

, a mixed number. This

mixed number could also be written as: 1 + 510

That’s because a mixed number is the sum of

its two parts: the whole number and the fraction.

Writing a Fraction in its Simplest Form

We are not finished with this sum yet. The fraction 510

is not in its simplest form.

A fraction is in its simplest form when the numerator and the denominator have no common

factors other than 1. Factors are numbers that are multiplied to get a product. First, let’s look at the factors of 5 and 10.

5 – 1, 5 10 – 1, 2, 5, 10

Now we are ready to put 510

in simplest form.

To write a fraction in simplest form, we need to divide both the numerator and denominator

by their greatest common factor, which we know is 5.

This works because 55

is the same as 1, and anything divided by 1 is the same number.

510

÷ 55

= 12

5 is the only common factor of 5 and 10 other than 1.

1510

- 1010

= 510



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We can see that 12

is in its simplest form by checking the factors of 1 and 2.

1 – 1 2 – 1, 2

Now, let’s put it all together!

Here is the solution to the original problem: 710

+ 810

1510

= 1 510

= 1 12

PRACTICE!

1. 82 +

83

2. 63 +

65

An easy way to change an improper fraction to a mixed number is to divide the numerator by the denominator and write the remainder as a fraction. Let’s try it!

11510 = 1

105 We still simplify! = 1

21

PRACTICE!

1. Change the improper fraction 8

12 into a mixed number by following the example above.

Think! In your opinion, which way works best?

The only common factor is 1; therefore, the fraction 1

2 is in simplest form.

- 10 5

The divisor 10 becomes the denominator again.

The remainder 5 becomes the numerator.



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Let’s see what the same problem would look like using a picture.

710

+ 810

= 1 510

or 12

+ =

We can see from the pie pieces that adding the two pies with 710

and 810

shaded gives us

a total of 1 510

. 1 510

= 1 1

2 They are the same.

PRACTICE! Use a picture to represent the following problem: Equivalent Fractions

The fractions 510

and 12

are called equivalent fractions because they name the same amount.

We find equivalent fractions by multiplying or dividing the numerator and the denominator of a fraction by the same non-zero number.

In our last problem, the numerator and denominator of the fraction 510

were both divided by 5 in

order to come up with 12

When adding or subtracting fractions that have different or unlike denominators, we have to rewrite the fractions to have a common or like denominator. The reason we do this is because

we can only add or subtract fractions that name the same kind of thing or amount.

510

÷ 55

= 12

1. 141 + 1

2



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Least Common Multiple (LCM) & Least Common Denominator (LCD)

In order to add two fractions with unlike denominators, we have to find a common denominator. To find it, we have to find the LCM, or the least common multiple of the two denominators.

When we find the LCM or least common multiple of the unlike denominators, we will have the LCD or least common denominator.

Let’s try the example 41 +

32 to find the LCD, or least common denominator.

First, we have to find the LCM, or least common multiple for the denominators 4 and 3. Remember, once we have found the least common multiple or LCM of the two denominators, we

can find the LCD or least common denominator. Multiples of 4: 4, 8, 12, 16, 20, 24, etc. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, etc.

We can see that 12 and 24 are common multiples, but 12 is the smallest or least common multiple. That makes 12 the LCD or least common denominator. Using paper and pencil, we rewrite the fractions with unlike denominators like this:

14

= 14 x

x33

=

123

32 =

32

xx

44 =

128

1211

+

We now have a common, or like denominator.



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+

+ =

– –

Here’s how that same problem can be represented using fraction strips:

41

32

123

128

1211

We can see that 14

is equal or equivalent to 312

We can see that 23

is equal or equivalent to 812

When we combine the numerators 3 and 8, we get a sum of 11 or 1112

PRACTICE!

1. Why is the addition problem below incorrect? Show the correct way to solve the problem.

14

+ 23

= 37

2. Solve the following example as both a paper and pencil model and an area model.

We started with 38

of a pie. Someone ate 14

of the pie. How much is left?

Paper & Pencil Area Model

83

41



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Adding and Subtracting Mixed Numbers

As we know, a mixed number has two parts: a whole number and a fraction. The value of a mixed number is the sum of these two parts.

578

Let’s look at the addition problem and follow the steps above: 3 14

+ 2 78

1. Add the fraction parts. Before we can do this, we must have like denominators. To find like denominators, we must find the LCM of the two denominators.

LCM or least common multiple: 4 – 4, 8, 12, 16 8 – 8, 16, 24

We see that 8 and 16 are common multiples, but 8 is the least common multiple.

Therefore . . . 14

x ?? =

8? 1

4 x

22 =

82

87 x

?? =

8?

87 x

11 =

87

2. We are now ready to add the fraction parts.

3 14

= 3 28

2 78

=+ 2 78

98

whole number fraction

1. When we add or subtract mixed numbers, we add and subtract the fraction part first.

We must make sure that we have common or like denominators when we add or subtract the fraction part. That means we have to find equivalent fractions with the least common denominator (LCD).

2. Then, we add and subtract the whole number part. Sometimes, we will have to regroup. 3. Finally, make sure the answer is in simplest form.



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3. We are now ready to add the whole numbers. 3 14

= 3 28

2 78

=+ 2 78

5 98

4. Finally, we write the fraction in its simplest form.

How do we change 5 98

to its simplest form?

Take the fraction 98

and divide the numerator into the denominator:

198

We aren’t done yet!

We take the whole number 1 in 1 18

and add it to the whole number 5 (5 98

).

1 + 5 = 6

Include the remainder 18

and put it all together!

Answer = 6 18

PRACTICE!

1. 552 + 1

32

2. 132 + 3

43

-8 1

The divisor 8 becomes the denominator.

The remainder 1 becomes the numerator.

81



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Changing a Mixed Number to an Improper Fraction

Sometimes we may need to change a mixed number into an improper fraction before we

subtract two numbers like 1 35

– 45

In this problem, our fraction parts have like denominators, but we cannot subtract a larger

number like 45

from 35

, a smaller number.

In order to subtract, we need to change the mixed number 1 35

into an improper fraction.

Follow these steps:

Change the number 1 into the fraction 55

. We will use 5 as the denominator because the fraction 35

has a 5 as the denominator. We know 55

is equal to 1 ( 55

= 1).

We then add 55

to the fraction part 35

and add: 55

+ 35

= 85

We see that the improper fraction 85

is also 1 35

, the mixed number. They are equivalent!

PRACTICE!

Change the mixed numbers into improper fractions.

1. 552

2. 132



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Using the same mixed number 1 35

, let’s use the algorithm or mathematical process for changing a

mixed number into an improper fraction:

85

- 45

= 45

Example:

Let’s try changing the mixed number 241 to an improper fraction following the algorithm or

mathematical process above. 1. Multiply the denominator 4 by the whole number 2. 4 x 2 = 8 2. Add the numerators. 8 + 1 = 9 3. Use the same denominator.

49

4. We turned the mixed number 241 into the improper fraction

49

.

1. Multiply the denominator by the whole number.

In this problem, we multiply 5, the denominator in the fraction 35 by the

whole number 1 in 1 35

, the mixed number. 5 x 1 = 5.

2. Add the numerator.

In this problem, we added 5 + 3 ( 35

) = 8.

3. Use the same denominator.

In this problem, the denominator is 5. Answer: 85

Result: We turned the mixed number 1 35

into 85

, the improper fraction.



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PRACTICE!

1. Change the mixed number 172

into an improper fraction. Be prepared to explain the process.

2. Change the improper fraction 58

into a mixed number. Be prepared to explain the process.

Adding & Subtracting Fractions in Word Problems

Example: A family orders a pepperoni pizza. The pepperoni pizza is cut into 12 slices.

The son eats 41 of the pizza. The daughter eats

21 of the pizza. Mom eats 2 pieces.

Is there any left for dad? If so, how many pieces?

Steps: 1. Set up an estimate:

21 (daughter’s share) +

21 (son’s

41 and mom’s share) 1

2. Set up and solve the problem:

21

= 126 1 pizza =

1212

41

= 123 –

1211

+ 122

= 122

121

1211

3. Check answer for reasonableness.

Compare our estimate (1 – whole pizza) and exact answer (1211 or 11 pieces out of 12).

The answers are definitely close.

Subtract the whole pizza from the part the family ate.

There is 1 piece left out of 12.

To add up how much of the pizza was eaten, change

unlike denominators to common or like



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PRACTICE!

Use the problem-solving steps above to solve the following word problems:

1. 3 pizzas were delivered for the pool party. 241 of the pizzas were eaten.

How much pizza is left?

2. Sam walked 221 miles the first day. He walked 6 2

3 miles the second day.

How many total miles did Sam walk? Remember! Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. Inverse operations are opposite operations. They undo each other.

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words are bolded and are Adding and Subtracting Fractions and Mixed … · Adding and Subtracting Fractions and Mixed Numbers ©2019 . When adding and subtracting fractions, the most