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Fractions!!. Prime Numbers and Factorization. How many positive prime numbers are less than 100?. Answer: 25. What is the sum of the prime numbers between π and 10 π ?. Answer: 155. List the positive prime numbers less than 100 that have the units digit equal to 3.

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slide11

Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).The number 12 is a multiple of 3, because it can be divided evenly by 3.3 · 4 = 123 and 4 are both factors of 123 · 2 · 2 (prime factorization of 12)12 is a multiple of both 3 and 4.

slide26

Greatest Common FactorThe highest number that divides exactly into two or more numbers. If you find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.Abbreviated "GCF". Also called "Highest Common Factor"Example: the GCF of 12 and 30 is 6, because 1, 2, 3 and 6 are factors of both 12 and 30, and 6 is the greatest.

slide29

Find the GCF of

39v & 30uv

slide31

Find the GCF of

35n²m & 21m²n

slide33

Find the GCF of

36xy³ & 24y²

slide35

Find the GCF of

105x, 30yx & 75x

slide38

Least Common MultipleThe smallest (non-zero) number that is a multiple of two or more numbers.Least Common Multiple is made up of the words Least, Common and Multiple:What is a "Multiple" ?The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table. Here are some examples: The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ... The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc... What is a "Common Multiple" ?When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of those numbers. For example, when you write down the multiples of 4 and 5, the common multiples are those that are found in both lists:The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...Notice that 20 and 40 appear in both lists? So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too) What is the "Least Common Multiple" ?It is simply the smallest of the common multiples. In our previous example, the smallest of the common multiples is 20 ...... so the Least Common Multiple of 4 and 5 is 20.

slide39

Finding the Least Common MultipleIt is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.Example: Find the least common multiple for 3 and 5:The multiples of 3 are 3, 6, 9, 12, 15, ..., and the multiples of 5 are 5, 10, 15, 20, ..., like this:As you can see on this number line, the first time the multiples match up is 15. Answer: 15More than 2 NumbersYou can also find the least common multiple of 3 (or more) numbers.Example: Find the least common multiple for 4, 6, and 8Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...Multiples of 8 are: 8, 16, 24, 32, 40, ....So, 24 is the least common multiple (I can't find a smaller one !)

slide42

Find the LCM of

30, 25 & 10

slide48

Find the LCM of

20y & 14y²

slide50

Find the LCM of

16x²y & 32x

slide52

Find the LCM of

8y², 16xy & 16y

slide54

What is the negative difference of the least common multiple (LCM) and the greatest common factor (GCF) of:

80, 140, and 200

types of fractional numbers
Types of Fractional Numbers
  • A proper fraction is a fraction whose value is less than 1 (numerator < denominator)
  • An improper fraction is a fraction whose value is greater than or equal to 1 (numerator > denominator)
  • A mixed number is a number whose value is greater than 1 made up of a whole part and a fraction part
improper fraction mixed number
Improper FractionMixed Number
  • Denominator: tells us how many parts make up a whole
  • Numerator: tells us how many parts we have
  • How many wholes can we make out of the parts we have?
  • Divide the numerator by the denominator  the quotient is the whole part
  • How many parts do we have remaining?
  • The remainder (over the denominator) makes up the fraction part
simplest form of a fraction
Simplest Form of a Fraction
  • A fraction is in simplest form when there are no common factors in the numerator and the denominator.
ex simplest form
Ex: Simplest Form

Ex: 6/8 and 3/4 are equivalent

The fraction 6/8 is written in simplest form as 3/4

=

=

=

1 x

Magic one

ex write 12 42 in simplest form
Ex: Write 12/42 in simplest form
  • First prime factor the numerator and the denominator:
  • 12 = 2 x 2 x 3 and 42 = 2 x 3 x 7
  • Look for Magic Ones
  • Simplify

=

=

=

1 x 1 x

=

Notice: 2 x 3 = 6 = GCF(12, 42)

 factoring (dividing) out the GCF will simplify the fraction

ex write 7 28 in simplest form
Ex: Write 7/28 in simplest form
  • What is the GCF(7, 28)?
    • Hint: prime factor 7 = 7
    • prime factor 28 = 2 x 2 x 7

= 7

=

=

=

1 x

=

Dividing out the GCF from the numerator and denominator simplifies the fraction.

ex write 27 56 in simplest form
Ex: Write 27/56 in simplest form
  • What is the GCF(27, 56)?
    • Hint: prime factor 27 = 3 x 3 x 3
    • prime factor 56 = 2 x 2 x 2 x 7

= 1

There is no common factor to the numerator and denominator (other than 1)

Therefore, 27/56 is in simplest form.

slide64

Rational and Irrational Numbers Essential Question

How do I distinguish between rational and irrational numbers?

slide65

Real Numbers

Rational numbers

Irrational numbers

Integers

Whole

numbers

The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.

slide66

Rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.

⅔, ⅕, ¼ or

144 = 12

slide67

Caution!

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Irrational numberscan be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, the square root of 2 is not a perfect square, so the square root of 2 is irrational. Also, π is irrational.

slide68

Make a Venn Diagram that displays the following sets of numbers:

Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.

Reals

Rationals

-2.65

Integers

-3

-19

Wholes

0

Irrationals

Naturals

1, 2, 3...

slide69

Classifying Real Numbers

Write all classifications that apply to each number.

A.

5 is a whole number that is not a perfect square.

5

irrational, real

B.

–12.75

–12.75 is a terminating decimal.

rational, real

16

C.

whole, integer, rational, real

slide70

9 = 3

Write all classifications that apply to each number.

9

A.

whole, integer, rational, real

–35.9

–35.9 is a terminating decimal.

B.

rational, real

81

C.

whole, integer, rational, real

slide71

A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

slide125

Simplify completely:

16x⁴y¯¹

4x²y¯²

slide127

Simplify completely:

36x³y⁶z¹²

4x¯¹y³z¹⁰

slide129

Simplify completely:

21x³y⁷z¹⁴30x³z¯⁵

18x⁴y⁶ y¹²z¯⁶

·

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