Fractions Progressive Mathematics Initiative Presentation Part 1 - - PDF document

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Fractions Presentation Part 1

www.njctl.org 2011-11-29

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Fractions

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· Meaning of Fractions · Equivalent Fractions · Lowest Term Fractions · Improper Fractions and Mixed Numbers · Using Fractions in Measurement · Adding Fractions with Common Denominators · Adding Mixed Numbers with Common Denominators · Subtracting Fractions with Common Denominators · Subtracting Mixed Numbers with Common Denominators · Finding Common Denominators · Comparing Fractional Numbers

Table of Contents

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· Adding Fractions with Unlike Denominators · Subtracting Fractions with Unlike Denominators · Adding Mixed Numbers with Unlike Denominators · Subtracting Mixed Numbers with Unlike Denominators · Multiplying Fractions · Multiplying Fractions and Whole Numbers · Multiplying with Mixed Numbers · Dividing Fractions · Dividing with Whole Numbers and Mixed Numbers

Table of Contents for Presentation 2

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Meaning of Fractions

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Fraction - An expression that indicates the quotient ( ) of two quantities. Numerator - The number above the fraction bar. The numerator answers the question "How many parts?" Denominator - The number below the fraction

• bar. The denominator answers the question

"How many total?"

Key Terms

Numerator 3 Denominator 7 Fraction = =

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Proper Fraction - A fraction where the numerator (top number) is less than the denominator (bottom number) 5 7 11 9 13 17 Improper Fraction - A fraction where the numerator (top number) is greater than or equal to the denominator (bottom number) 8 12 9 3 7 9

Key Terms Slide 9 / 114

Equivalent Fractions - Fractions that represent the same number or are equal to each other. 2 4 6 8 10 12 14 3 6 9 12 15 18 21 Mixed Number - A fraction with a whole number and a proper fraction. 5 1 2 2

= = = = = =

2

Improper Fraction Mixed Number

Key Terms Slide 10 / 114 Slide 11 / 114

What fractions are represented below?

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Fractions are used to measure ingredients for baking and cooking.

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Halves

Pull

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Thirds

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Fourths

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Fifths

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Sixths

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Sevenths

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Eighths

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1 What fraction of the whole is shaded?

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2 What fraction of the whole is white?

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Internet links for more practice Fractions Naming Model link Fractions Parts of a Whole Model link

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Equivalent Fractions

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1 9 1 9 1 9

What do you notice about the denominators in each set of equivalent fractions?

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Click below to use this interactive number line.

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2 5 = ? ? Use the multiplication table to make equivalent fractions.

Pull

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To create equivalent fractions, multiply (or divide) the numerator and denominator by the same number. 2 x 3 6 x 2 12 x 3 36 7 x 3 21 x 2 42 x 3 126 36 / 6 6 / 3 2 126 / 6 21 / 3 7 = = = = =

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3 Which set of fractions is equivalent?

A B C D

1 2 2 2 = 4 1 8 4 1 3 3 6 3 9 7 21 = = =

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4 Write a fraction equivalent to 4 7

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5 Which fraction is equivalent to ?

A B C D 3 8 16 6 12 24 12 32 9 16

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6 Which set of fractions is equivalent?

A B C D

3 9 4 16 = 9 2 12 3 6 1 18 3 3 9 7 14 = = =

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7 Write a fraction equivalent to 5 9

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Lowest Term Fractions

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Reducing Fractions to Lowest Terms Once you complete operations with fractions, you must write your answer in lowest terms. The easiest way to determine if your answer is in simplest form is to check:

• 1. Do the numerator and denominator have any

common factors?

• 2. If they do not, the answer is in simplest form.
• 3. If they do, divide both the numerator and

denominator by common factors until there are none left. 40 10 4 2 2 220 10 22 2 11 = =

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Remember: Whatever you do to the numerator, you must do to the denominator. 18 24 In this problem, 18 and 24 are divisible by 2, 3 and 6. If you divide by 6 (the GCF), you will reach the simplest form in one step. 18 6 3 24 6 4 =

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If you divide by 2 and 3 (common factors), then you will reach the simplest form in two steps. Either way, the simplest form is the same. 18 3 6 2 3 24 3 8 2 4

= =

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8 Simplify to lowest terms. 21 35

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9 Simplify to lowest terms. 3 9

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10 Simplify to lowest terms. 21 29

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11 Simplify to lowest terms. 17 51

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12 Simplify to lowest terms. 24 96

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Internet links for more practice Simplifying Fractions link (interactive at bottom of page) Rename in Lowest Terms interactive

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Improper Fractions and Mixed Numbers

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In a proper fraction, the numerator is always less than the denominator. The value of a proper fraction is always less than 1. 3 5 In an improper fraction, the numerator is equal to

• r greater than the denominator. The value of an

improper fraction is equal to or greater than 1. 5 7 5 5

Proper or Improper? Slide 46 / 114

An improper fraction can be expressed as a whole number and a fraction. This is called a mixed number. "Seven Halves"

3 1

2 7 2 = Seven halves can fill up 3 whole strips and of another whole strip. That's and we say "three and one half", meaning "three plus one half".

3

1 2 1 2

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Converting Improper Fractions to Mixed Numbers Remember, the fraction bar symbolizes division! Therefore: · Divide the numerator by the denominator to see how many "wholes" there are. · Write the remainder over the denominator Let's look at "seven halves" again. 7 2

3 1

2 7 2 = 3 2 7

• 6

1 2

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Let's look at another example. Now try this one. 17 8 13 5 2 5 13

• 10

3 5

2 3

5 13 5 =

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Converting Mixed Numbers into Improper Fractions To do this, complete a multiplication problem. · Multiply the whole number by the denominator to create an improper fraction (of the original whole number). · Add the new fraction to the original fraction (from the mixed number). Let's look at "seven halves" again.

3 1

2 7 2 = = 6 2 + 1 2

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Let's look at another example. Now try this one.

4 2

3 14 3 = = 12 3 + 2 3

6 3

7

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13 is a proper fraction.

True False

8 5

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14 Change to a mixed number.

1 7

8

A 15 8

2 7

8

B

1 3

8

C

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15 Change this mixed number to an improper fraction.

1 3

4

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16 Change this mixed number to an improper fraction.

3 9

10

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17 Change this improper fraction to a mixed number. 28 6

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Using Fractions in Measurement

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Each one inch segment on this ruler is like

• ne strip folded into four parts. For every

inch, there are , and inch markings.

1 4 1 2 3 4

1 4 1 4 1 4 1 4

Fraction Ruler Slide 58 / 114

Move the pink arrows to two locations on the ruler. Then hit the large blue arrow for the distance between the arrows to be measured.

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18 What is the length between the two arrows?

1 9

16 in.

A

1 5

8 in.

C

3 4 in.

D

1 3

4 in.

B

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19 What is the length between the two arrows? 13 16 in.

A

3 4 in.

C

7 8 in.

D

1 in.

B

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Adding Fractions with Common Denominators

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Adding Fractions with Common Denominators To add fractions with common denominators, add the numerators and leave the denominator the same. Make sure your answer is in simplest form. The denominator indicates the number of parts

• f the whole. If the fractions have a common

denominator, they are the same "size" so we can add the numerators (or number of parts). 2 6 3 6 5 6 +

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Try these! Move the boxes to see work and answers. Be sure to simplify all answers. 2 4 1 4 3 4 + 3 7 1 7 4 7 + 5 12 4 12 9 12 + 3 4 11 30 13 30 24 30 + 4 5

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20 3 10 2 10 +

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21 5 8 1 8 +

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22 7 14 3 14 +

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23 5 12 2 12 +

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24 8 20 6 20 +

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Adding Mixed Numbers with Common Denominators

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Adding Mixed Numbers with Common Denominators To add mixed numbers with common denominators, add the fractions then add the whole numbers. Make sure your answer is in simplest form.

2 1

6 + 1 4 6

3 5

6

5 1

9 + 2 2 9

7 3

9 = 7 1 3

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25 Is the equation below true or false? True False

1 1

4 + 3 2 4

4 3

4

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26 Is the equation below true or false? True False

4 1

4 + 4 1 4

8 2

4

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27 Find the sum.

2 5

12 + 3 2 12

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28 Find the sum.

5 3

10 + 7 5 10

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Adding Mixed Numbers with Common Denominators Sometimes after you add the mixed numbers, the fraction is improper. When this occurs, you must rename the improper fraction as a mixed number and add it to the whole number.

3 3

5 + 2 4 5

5 7

5 = 5 + 1 2 5 = 6 2 5

6 5

9 + 1 7 9

7 12

9 = 7 + 1 3 9 = 8 1 3

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29 Is the equation below true or false? True False

1 8

12 + 1 5 12

3 1

12

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30 Find the sum.

2 4

9 + 5 2 9

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31 Find the sum.

3 3

14 + 2 4 14

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32 Find the sum.

4 3

8 + 2 3 8

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Subtracting Fractions with Common Denominators

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Subtracting Fractions with Common Denominators To subtract fractions with common denominators, subtract the numerators and leave the denominator the same. Make sure your answer is in simplest form. The denominator indicates the number of parts

• f the whole. If the fractions have a common

denominator, they are the same "size" so we can subtract the numerators (or number of parts). 5 6 4 6 1 6

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Try these! Move the boxes to see work and answers. Be sure to simplify all answers. 2 4 1 4 1 4 3 7 1 7 2 7 11 12 3 12 8 12 2 3 19 30 13 30 6 30 1 5

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33 7 8 4 8

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34 7 10 3 10

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35 5 6 3 6

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36 9 14 5 14

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37 7 9 5 9

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Subtracting Mixed Numbers with Common Denominators

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Subtracting Mixed Numbers with Common Denominators To subtract mixed numbers with common denominators, subtract the fractions then subtract the whole numbers. Make sure your answer is in simplest form.

2 4

6

1 3

6

1 1

6

5 7

9

2 4

9

3 3

9 = 3 1 3

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38 Is the equation below true or false? True False

4 5

9 3 9

3 2

9

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39 Is the equation below true or false? True False

2 7

9 1 9

1 2

3

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40 Find the difference.

4 7

8

2 3

8

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41 Find the difference.

6 7

12

1 4

12

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42 Find the difference.

13 5

8

5 2

8

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Finding Common Denominators

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How many halves make a whole circle?

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How many fourths make half of this circle?

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How many sixths make 1/3 of this circle?

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How many eighths can fit in 1/4 of this circle?

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How many different combinations can you make to fill the circle? Keep track of what pieces you use. (You may need to rotate your pieces.)

1/8 1/4 1/2 1/3 1/6 1/7 1/5

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Fix the Sticks You can use the set of Skip Counting Sticks to find common denominators for two fractions with unlike denominators. If you don't have a set of sticks, you can create them by listing the multiples of the denominator. For the fractions and , line up the sticks this way for the denominator of each fraction: Find the smallest number in the "denominator" sticks that is common in both fractions. It's 12. The least common denominator of and is 12.

3 4

4 8 12 16 2024283236 6 1218 24 3036424854

... ...

1 6

3 4 1 6

3 4 1 6

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A quick way to find LCDs... List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator. Ex: and Multiples of 5: 5, 10, 15 Ex: and Multiples of 9: 9, 18, 27, 36 2 5 1 3 3 4 2 9

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43 Find the LCD of this pair of fractions. 2 4 1 6

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44 Find the LCD of this pair of fractions. 5 6 3 8

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Comparing Fractional Numbers

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Comparing Fractional Numbers Common Denominators When you have two fraction with common denominators, all you have to do is compare the numerators. Unlike Denominators To compare fractions with unlike denominators, you have to rewrite both fractions with a common denominator. Then compare the numerators.

>

8 9 7 9 2 3 7 10 2 3 20 30

=

7 10 21 30

= <

2 3 7 10

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Compare the fractions 1. 4 2 7 5 4 7 20 35

=

2 5 14 35

=

4 2 7 5

>

2. 11 13 17 17 11 13 17 17

<

3. 4 3 5 4 4 5 16 20

=

3 4 15 20

=

4 3 5 4

>

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45 True or false? 2 3 3 4

>

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46 True or false? 5 6 5 8

>

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47 Compare the two fractions. A > 8 11 3 4 B < C =

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48 Compare the two fractions. A > 3 12 1 4 B < C =

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49 Compare the two fractions. A > 4 9 5 8 B < C =

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Internet links for more practice Finding fractions on a number line link Comparing Fractions Model

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