Fractions Pages 8 – 59

Fraction = Part of a Whole Page 11 What is a Fraction? Fraction = Part of a Whole Numerator = tells how many parts you have Top Number? 3 4 Denominator = tells how many parts are in the whole Bottom Number? Note: the fraction bar means to divide the numerator by the denominator

One Way To Remember 3 4 Page 11 Numerator = North # you have # of parts in whole Divided by Denominator = Down

What Fraction is Shaded? Page 11 What Fraction is Shaded? ¾ ⅝ 7/16

Identifying Forms of Fractions Page 13 Identifying Forms of Fractions There are three forms of fractions: Proper fraction: The numerator (top number) is always less than the denominator. The value of a proper fraction is less than 1 whole. Improper fraction: The numerator is equal to or greater than the denominator. When the numerator is equal to the denominator, an improper fraction is equal to 1 whole. Mixed Number: A whole number & a proper fraction are written next to each other. A mixed number always has a value of more than 1 whole.

Page 13 Example Tell whether each of the following is a proper fraction (P), an improper fraction (I) or a mixed number (M). 1 2 3 2 7 110 75 10 9

Page 13 Group Work Tell whether each of the following is a proper fraction (P), an improper fraction (I) or a mixed number (M). 1 2 3 1 2 17 9 6 4

Thinking About the Size of Fractions Page 14 Thinking About the Size of Fractions The size of the numerator compared to the size of the denominator tells you: A fraction is equal to ½ when the numerator is exactly half of the denominator A fraction is less then ½ when the numerator is less than half of the denominator A fraction is greater than ½ when the numerator is more than half of the denominator The symbol = means “is equal to” The symbol < means “is less than” The symbol > means “is greater than”

Page 14 Example In the box between each pair of fractions, write a symbol that makes the statement true. 3 5 1 2 8 16 1 2

Page 14 Group Work In the box between each pair of fractions, write a symbol that makes the statement true. 7 20 1 2 9 15 1 2

Reducing Fractions Pages 15 – 17 Reducing a fraction means dividing both the numerator & the denominator (top & bottom) by a number that goes into each evenly. Reducing changes the numbers in a fraction, but reducing does not change the value of a fraction. When both the numerator & the denominator of a fraction end with zeros, you can cancel the zeros one-for-one. This is a shortcut for reducing ten. Always check to see if you can continue to reduce. Sometimes a fraction can be reduced more than once to reach the lowest terms

Questions When Reducing Pages 15 – 17 Questions When Reducing Are the numerator & denominator both even? Divide by 2 Add the digits of the numerator separate from the digits of the denominator. Do they add up to a number that is divisible by 3? Divide by 3 Do the numerator & the denominator end in a 0 or 5? Divide by 5 If no to all previous questions: You just have to try 7, 11, 13 & so on

Pages 15 – 17 Example Reduce each fraction to lowest terms 1 6 12 = 3 33 77 = 2 25 30 =

Pages 15 – 17 Example Reduce each fraction to lowest terms 1 75 80 = 3 35 49 = 2 25 50 =

Raising Fractions to Higher Terms Page 18 Raising Fractions to Higher Terms An important skill in the addition & subtraction of fractions is raising a fraction to higher terms. Raising to higher terms is the opposite of reducing. To reduce a fraction, you must divide both the numerator & denominator by the same number. To raise a fraction to higher terms, multiply both the numerator & the denominator by the same number.

Page 18 Example Raise each fraction to higher terms by filling in the missing numerator. 1 4 5 30 = 2 4 7 35 =

Page 18 Group Work Raise each fraction to higher terms by filling in the missing numerator. 1 1 3 45 = 2 5 6 42 =

Changing Improper Fractions to Whole or Mixed Numbers Pages 19 – 20 Changing Improper Fractions to Whole or Mixed Numbers The answers to many fraction problems are improper fractions. These answers are easier to read if you change them to whole numbers or mixed numbers. To change an improper fraction, divide the denominator into the numerator.

Pages 19 – 20 Example Change each fraction to a whole number or a mixed number. Reducing any remaining fractions. 1 14 8 = 2 30 9 =

Pages 19 – 20 Group Work Change each fraction to a whole number or a mixed number. Reducing any remaining fractions. 1 13 12 = 2 36 12 =

Changing Mixed Numbers to Improper Fractions Page 21 Changing Mixed Numbers to Improper Fractions When you multiply & divide fractions, you will have to change mixed numbers to improper fractions. To change a mixed number to an improper fraction, follow these steps: Multiply the denominator (bottom number) by the whole number. Add that product to the numerator (top number) Write the sum over the denominator.

Page 21 Example Change each mixed number to an improper fraction 1 3 4 = 2 2 1 2 9 =

Page 21 Group Work Change each mixed number to an improper fraction 1 1 3 = 10 2 4 5 3 =

Addition of Fractions with the Same Denominators Pages 22 – 24 Addition of Fractions with the Same Denominators To add fractions with the same denominators (bottom numbers), first add the numerators. Then write the total (or sum) over the denominator. Don’t forget to check to see if you can reduce your answer.

Pages 22 – 24 Example 3 8 Add 12 2 1 7 13 1 8 8 9 4 13 2 8 6 10 + +

Addition of Fractions with Different Denominators Page 25 Addition of Fractions with Different Denominators If the fractions in an addition problem do not have the same denominators, you must find a common denominator. common denominator = a number that can be divided evenly by every denominator in the problem. lowest common denominator or LCD = The lowest denominator that can be divided evenly by every denominator in the problem.

Finding a Common Denominator Pages 26 – 28 Finding a Common Denominator Method 1: Multiply the denominators together. Brute force method: List the multiples of the larger number until you find a multiple of the smaller number Prime factorization method: find prime factors of both numbers. Circle the numbers they have in common. Write those once then write in the rest of the numbers and multiply to find the LCM

Pages 26 – 28 Example 11 16 Add 2 1 7 10 1 3 3 4 7 8 + +

Pages 26 – 28 Group Work 5 7 2 1 5 12 4 9 5 9 2 3 + +

Subtracting Fractions with the Same Denominators Page 31 Subtracting Fractions with the Same Denominators To subtract fractions, subtract the numerators & put the difference (the answer) over the denominator.

Page 31 Example 2 1 5 6 5 9 23 1 6 2 9 – 7 –

Subtracting Fractions with Different Denominators Pages 32 – 33 Subtracting Fractions with Different Denominators When fractions do not have the same denominators, first find a common denominator. Change each fraction to a new fraction with the common denominator. Then subtract.

Pages 32 – 33 Example Subtract & Reduce 2 1 5 8 11 18 25 13 2 5 1 2 – 22 – 8

Pages 32 – 33 Group Work Subtract & Reduce 2 1 3 5 5 6 18 16 3 10 7 10 – 9 – 9

Borrowing & Subtracting Fractions Pages 34 – 36 Borrowing & Subtracting Fractions Sometimes there is no top fraction to subtract the bottom fraction from. Other times the top fraction is not big enough to subtract the bottom fraction from. To get something in the position of the top fraction, you must borrow. To borrow means to write the whole number on top as a whole number & an improper fraction. For example, 12 = 11 . The numerator & denominator of the improper fraction should be the same as the denominator of the other fraction in the problem. 8

Pages 34 – 36 Example Subtract & Reduce 2 1 2 9 3 16 13 24 5 6 2 3 – 7 – 9

Pages 34 – 36 Group Work Subtract & Reduce 2 1 1 6 1 3 12 30 7 12 8 11 – 10 – 16

Multiplication of Fractions Page 39 Multiplication of Fractions When you multiply whole numbers (except 1 & 0), the answer is bigger than the two numbers you multiply. When you multiply two proper fractions, the answer is smaller than either of the two fractions. When you multiply two fractions, you find a fraction of a fraction or a part of a part. To multiply fractions, multiply the numerators together & the denominators. Then reduce.

Page 39 Example Multiply & Reduce 1 2 3 4 5 x = 2 1 3 4 7 2 3 x x =

Page 39 Group Work Multiply & Reduce 1 5 7 2 9 x = 2 2 5 7 9 1 3 x x =

Canceling & Multiplying Fractions Pages 40 – 41 Canceling & Multiplying Fractions Canceling is a way of making multiplication of fractions problems easier. Canceling is similar to reducing. To cancel, divide a numerator & denominator by a number that goes evenly into both of them. You don’t have to cancel to get the right answer, but it makes the multiplication easier.

Pages 40 – 41 Example Multiply & Reduce 1 15 28 7 16 12 45 x x = 2 17 21 14 51 7 11 x x =

Pages 40 – 41 Group Work Multiply & Reduce 1 11 39 10 11 13 18 x x = 2 19 36 7 10 3 7 x x =

Multiplying Fractions & Whole Numbers Page 42 Multiplying Fractions & Whole Numbers To multiply a whole number & a fraction, first write the whole number as a fraction. Write the whole number as the numerator & 1 a the denominator.

Page 42 Example Multiply & Reduce 1 9 10 x 2 = 2 7 12 x = 36

Page 42 Group Work Multiply & Reduce 1 5 24 x 16 = 2 7 30 x = 35

Multiplying Mixed Numbers Page 43 Multiplying Mixed Numbers To multiply mixed numbers, first change the mixed numbers to improper fractions. Then multiply the improper fractions. Reduce the answer.

Page 43 Example Multiply & Reduce 1 2 15 1 4 1 2 2 x 5 x 7 = 2 2 5 3 8 7 9 2 x 3 x 2 =

Page 43 Group Work Multiply & Reduce 1 3 4 8 9 1 5 3 x x 1 = 2 1 3 5 14 16 x 2 =

Dividing Fractions by Fractions Pages 47 – 48 Dividing Fractions by Fractions To divide fractions, take the reciprocal (or inverse) of the divisor (the number at the right of the ÷ sign) & follow the rules for multiplying fractions. To make a reciprocal means to write the numerator on the bottom & the denominator on the top.

Pages 47 – 48 Example Divide & Reduce 1 16 21 3 4   = 2 12 19 18 38   =

Pages 47 – 48 Group Work Divide & Reduce 1 3 10 6 7   = 2 5 11 25 33   =

Pages 49 – 54 Dividing Whole Numbers by Fractions & Dividing Fractions by Whole numbers & Dividing with Mixed Numbers In fraction division problems, change whole numbers & mixed numbers to improper fractions. Then take the reciprocal of the fraction you are dividing by & follow the rules for multiplying fractions.

Pages 49 – 54 Example Divide & Reduce 1 24 25   56 = 3 2 5 8 1 2 10   4 = 7 24   35 =

Pages 49 – 54 Group Work Divide & Reduce 1 21 25   7 = 3 2 5 6 5 12 5   3 = 5 18   15 =

Finding a Number When a Fraction of It Is Given Page 55 Finding a Number When a Fraction of It Is Given There is a kind of division problem that is sometimes hard to recognize. Think about the question ½ of what number is 12? Without using pencil & paper, you can probably come up with the answer 24. You know that ½ of 24 is 12. To solve the problem, you find a solution to the statement ½ x ? = 12. The statement asks you to find the missing number in a multiplication problem. To find the missing number, divide 12 by ½.

Example 1 7 12 of what number is 35? 2 3 10 of what number is 45? Page 55 Example Solve 1 7 12 of what number is 35? 2 3 10 of what number is 45?

Group Work 1 8 9 of what number is 32? 2 4 5 of what number is 80? Page 55 Group Work Solve 1 8 9 of what number is 32? 2 4 5 of what number is 80?