Fractions!!.

Fractions!!

1. Relate and apply the concepts of prime numbers and factorization. 2
1. Relate and apply the concepts of prime numbers and factorization Perform calculations involving Greatest Common Factor Perform calculations involving Least Common Multiple Perform calculations involving estimation of fractions Identify whether a number is rational or irrational Perform calculations involving equivalence and simplification of fractions & mixed numbers Perform calculations involving add/sub of fractions with common denominators Perform calculations involving addition and subtraction with unlike denominators Perform calculations involving addition and subtraction of mixed numbers Perform calculations involving multiplication of fractions Perform calculations involving division of fractions Perform calculations involving multiplication and division with mixed numbers.

Prime Numbers and Factorization

How can we tell if a number is composite or prime?

Divisibility tests Divisible by 2? If a number is divisible by 2…

Divisibility tests Divisible by 2? If a number is divisible by 2… It ends with 2, 4, 6, 8, or 0 (in other words it’s an EVEN number.)

Divisibility tests Divisible by 3? If a number is divisible by 3… It’s digit sum is 3, 6, or 9. The digit sum is when you add the digits in the number repeating until you get to 1 digit.

Divisibility tests Divisible by 3? So the number 1467 has a digit sum of 9 = 18 then = 9 So 1467 is divisible by 3.

Divisibility tests Divisible by 4? If a number is divisible by 4… If we half the number, and the result is even, then our number is divisible by 4.

Divisibility tests Divisible by 4? If a number is divisible by 4… If the last 2 digits of a number are divisible by 4, then our number is as well. is divisible by 4 because 28 is divisible by 4.

Divisibility tests Divisible by 5? If a number is divisible by 5… It ends with a 5 or a 0. 5, 10, 15, 20, 25, 30, …

Divisibility tests Divisible by 6? If a number is divisible by 6… It passes the tests for both divisibility by 2 and divisibility by 3. In other words – an EVEN number with digit sum 3, 6 or 9

Divisibility tests Divisible by 7? If a number is divisible by 7… It appears in the 7 times table! Sorry! You’ll just have to learn it because there is no trick for 7s!

Divisibility tests Divisible by 8? If a number is divisible by 8… If you halve it, and halve it again, the result is an even number. If the last 3 digits are divisible by 8, the whole number will be.

Divisibility tests Divisible by 8? If a number is divisible by 8… So 2, 560, 104 is divisible by 8 because 104 is divisible by 8. ( 104 ÷ 2 = ÷ 2 = is EVEN. )

Divisibility tests Divisible by 9? If a number is divisible by 9… It’s digit sum is 9

Divisibility tests Divisible by 9? If a number is divisible by 9… It’s digit sum is 9 This means it is also divisible by 3!

Divisibility tests Divisible by 10? If a number is divisible by 10… It ends in a 0

Divisibility tests Divisible by 10? If a number is divisible by 10… It ends in a 0 So these numbers are also divisible by 2 and 5

Divisibility tests Divisible by 11? If a number is divisible by 11… If you sum every second digit and then subtract all other digits and the answer is: 0, or divisible by 11 1364 ((3+4) - (1+6) = 0) Yes 3729 ((7+9) - (3+2) = 11) Yes 25176 ((5+7) - (2+1+6) = 3) No

Divisibility tests Divisible by 1000? If a number is divisible by 1000… It ends in a 000 And so on…

4: halve last 2 digits and the result is even 5: ends with 5 or 0
Divisibility tests Divisible by …? 2: ends with 2, 4, 6, 8, or 0 3: digit sum is 3, 6, or 9 4: halve last 2 digits and the result is even 5: ends with 5 or 0 6: ends with 2, 4, 6, 8, or 0 AND digit sum is 3, 6, or 9 7: - 8: Halve the last 3 digits twice, and the result is even. 9: digit sum is 9 10: ends with 0

Competition Problems

How many positive prime numbers are less than 100?

Answer: 25

What is the sum of the prime numbers between π and 10π?

Answer: 155

List the positive prime numbers less than 100 that have the units digit equal to 3.

Answer: 3, 13, 23, 43, 53, 73, 83

Prime Factorization

Factoring is like taking a number apart
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 x 4 = 12 3 and 4 are both factors of is a multiple of both 3 and 4.

READY? 3,2,1…GO!

Write the prime factorization of:
35

Answer: 5 ∙ 7

46

Answer: 2 ∙ 23

66

Answer: 2∙3∙11

48

Answer: 2∙2∙2∙2∙3 or 2⁴ ∙ 3 (Prime-Power Factorization)

40

Answer: 2∙2∙2∙5 or 2³ ∙ 5 (Prime-Power Factorization)

100

Answer: 2∙2∙5∙5 or 2² ∙ 5² (Prime-Power Factorization)

972

Answer: 2∙2∙3∙3∙3∙3∙3 or 2² ∙ 3⁵ (Prime-Power Factorization)

List the prime factorization of the following terms (includes variables):

Answer: 2·2·2·3·h

Answer: 5·5·n·n

92xy

Answer: 2·2·23·x·y

36x³

Answer: 2·2·3·3·x·x·x

Factors?

Factors Factors are the numbers you multiply together to get a product. For example, the product 24 has several factors. 24 = 1 x 24 24 = 2 x 12 24 = 3 x 8 24 = 4 x 6 SO, the factors are 1, 2, 3, 4, 6, 8, 12, 24

Finding Factors Start with 1 x the number. Try 2, 3, 4, etc.
When you repeat your factors, cross out the repeat - you’re done at this point. If you get doubles (such as 4 x 4), then you’re done. Repeats or doubles let you know you’re done.

What are the factors of 16? doubles = done
1 x 16 2 x 8 3 x ?? 3 is not a factor, so cross it out 4 x 4 doubles = done The factors of 16 are 1,2,4,8,16

What are the factors of 18? Repeat! Cross it out! We’re done! 1 x 18
The factors are 1,2,3,6,9,18 2 x 9 3 x 6 4 x ?? 5 x ?? 6 x 3 Repeat! Cross it out! We’re done!

What are the factors of 7? The only factors of 7 are 1,7 1 x 7 2 x ?? 3 x ?? 4 x ?? 5 x ?? 6 x ?? 7 x 1 This works, but it is a repeat. We are done.

READY? 3,2,1…GO!

List all the positive factors of
30

Answer: 1, 2, 3, 5, 6, 10, 15, 30

22

Answer: 1, 2, 11, 22

99

Answer: 1, 3, 9, 11, 33, 99

87

Answer: 1, 3, 29, 87

Greatest Common Factor (GCF)

Greatest Common Factor The 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.

Find the greatest common factor (GCF) of
39 & 6

Answer: 3

Find the GCF of 24 & 28

Answer: 4

Answer: 3

Find the GCF of 39v & 30uv

Answer: 3v

Answer: 7

Find the GCF of 35n²m & 21m²n

Answer: 7nm

Find the GCF of 36xy³ & 24y²

Answer: 12y²

Find the GCF of 105x, 30yx & 75x

Answer: 15x

Least Common Multiple (LCM)

Least Common Multiple The 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 so the Least Common Multiple of 4 and 5 is 20.

Finding the Least Common Multiple It is a really easy thing to do
Finding the Least Common Multiple It 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: 15 More than 2 Numbers You can also find the least common multiple of 3 (or more) numbers. Example: Find the least common multiple for 4, 6, and 8 Multiples 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 !)

using prime factorization
Least Common Multiple using prime factorization 1) Write the prime factorization of each number. Select all common factors ONCE. 3) Then select the remaining factors and multiply.

Least Common Multiple 18 20 2 9 2 10 3 3 2 5 Select all common factors once. Then select the remaining factors.

Least Common Multiple 45 72 5 9 8 9 3 3 2 4 3 3 2 2 Select all common factors once. Then select the remaining factors.

Least Common Multiple Find the LCM of 48 and 80. 2 48 80 Common
Common Factors Once 2 2 2 Remaining Factors

SO I GUESS YOU ARE WONDERING
Least Common Multiple SO I GUESS YOU ARE WONDERING When Are We Ever Gonna Have To Use This?

Least Common Multiple What is the LCM used for?
The LCM is used to find common denominators so that fractions may be easily compared, added, or subtracted.

Least Common Multiple The GCF and LCM are used so regularly that
most people find them mentally. GCF = 1 GCF = 5 1) 2) LCM = 72 LCM = 60

Least Common Multiple The GCF and LCM are used so regularly that
most people find them mentally. GCF = 4 GCF = 4 1) 2) LCM = 60 LCM = 24

READY? 3,2,1…GO!

Find the least common multiple (LCM) of
10 & 3

Answer: 30

Find the LCM of 14 & 6

Answer: 42

Find the LCM of 35 & 25

Answer: 175

Find the LCM of 28, 14 & 21

Answer: 84

Find the LCM of 30, 25 & 10

Answer: 150

Find the LCM of 18 & 6v

Answer: 18v

Find the LCM of 3x² & 10

Answer: 30x²

Find the LCM of 20y & 14y²

Answer: 140y²

Find the LCM of 25x² & 25y

Answer: 25x²y

Find the LCM of 16x²y & 32x

Answer: 32x²y

Find the LCM of 18xy² & 15y³

Answer: 90xy³

Find the LCM of 20x³ & 16x⁴

Answer: 80x⁴

Find the LCM of 8y², 16xy & 16y

Answer: 16y²x

Competition Problem

What is the negative difference of the least common multiple (LCM) and the greatest common factor (GCF) of: 80, 140, and 200

Answer:

INTRODUCTION TO FRACTIONS!!

Introduction to Fractions
A fraction represents the number of equal parts of a whole Fraction = numerator (up North) denominator (Down south) = numerator/denominator Numerator = # of equal parts Denominator = # of equal parts that make up a whole

Example: My friend and I ordered a large Papa John’s pizza
Example: My friend and I ordered a large Papa John’s pizza. The large pizza is cut into 8 (equal) slices. If my friend ate 3 slices, then he ate 3/8 of the pizza

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

Converting Between Fraction Types
Any integer can be written as an improper fraction Any improper fraction can be written as a mixed number Any mixed number can be written as an improper fraction

Integer  Improper Fraction
The fraction bar also represents division The denominator is the divisor The numerator is the dividend The original integer (number) is the quotient To write an integer as a division problem, what do we divide a number by to get the number? One n = n/1

Ex: Write 17 as an improper fraction
17 = 17 / ? 17 divided by what is 17? 1 Therefore, 17 = 17 / 1

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

Write 11/8 as a mixed number.
How many parts make up a whole? 8 Draw a whole with 8 parts: How many parts do we have? 11 To represent 11/8 we must shade 11 parts . . . But we only have 8 parts. Therefore, draw another whole with 8 parts . . . Keep shading . . . 9 10 11 This is what 11/8 looks like.

Given the representation of 11/8, how many wholes are there?
Dividing 11 parts by 8 will tell us how many wholes we can make: 11/8 = 1 R ? The remainder tells us how much of another whole we have left: 1 R 3 Since 8 parts make a whole, we have 3/8 left. Therefore, 11/8 = 1 3/8.

Mixed Number  Improper Fraction
Denominator: tells us how many parts make up a whole. Chop each whole into that many parts. How many parts do we get? Multiply the whole number by the denominator. Numerator: tells us how many parts we already have. How many parts do we now have in total? Add the number of parts we get from chopping the wholes to the number of parts we already have Form the improper fraction: # of parts # of parts that make a whole

Write 2 5/8 as an improper fraction.
Draw the mixed number Looking at the fraction, how many parts make up a whole? 8 Chop each whole into 8 pieces. How many parts do we now have? 8 + 8 + 5 = 8 * = 21 = parts from whole + original parts

Therefore 2 5/8 = 21/8

Finding Equivalent Fractions
Equal fractions with different denominators are called equivalent fractions. Ex: 6/8 and 3/4 are equivalent.

The Magic One We can find equivalent fractions by using the Multiplication Property of 1: for any number a, a * 1 = 1 * a = a (magic one) We will just disguise the form of the magic one Do you agree that 2/2 = 1? How about 3/3 = 1? 4/4 = 1? 25/25 = 1? /17643 = 1? 1 has many different forms . . . 1 = n/n for any n not 0

Find another fraction equivalent to 1/3
1/3 = 1/3 * 1 We can write 1/3 many ways just be using the Magic One = 1/3 * 2/2 = 2/6 or 1/3 = 1/3 * 1 = 1/3 * 3/3 = 3/9

Find a fraction equivalent to ½ but with a denominator of 8
1/2 = 1/2 * 1 We can write 1/2 many ways just be using the Magic One. We want a particular denominator – 8. What can we multiply 2 by to get 8? = 1/2 * 4/4 = 4/8 Notice: 4 so choose the form of the Magic One

Ex: Find a fraction equivalent to 2/3 but with a denominator 12
2/3 = 2/3 * 1 We can write 2/3 many ways just be using the Magic One. We want a particular denominator – 12. What can we multiply 3 by to get 12? = 2/3 * 4/4 = 8/12 4 so choose the form of the Magic One

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: 6/8 and 3/4 are equivalent
The fraction 6/8 is written in simplest form as 3/4 = = = 1 x Magic one

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

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.

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.

Estimation of Fractions

You can estimate fractions by rounding to 0,½, or 1.
2 __ 8 4 • 1 3 1 1 1 3 5 7 3 4 __ 1 2 __ The fraction is halfway between and 1, but we usually round up. So the fraction rounds to 1. 3 4 __

You can round fractions by comparing the numerator and denominator.
1 2 __ 1 • • closer to Each numerator is about half the denominator, so the fractions are close to . 1 2 __ closer to 1 Each numerator is about the same as the denominator, so the fractions are close to 1. closer to 0 Each numerator is much less than half the denominator, so the fractions are close to 0. 1 5 __ 2 11 __ 2 15 __ 5 11 __ 4 7 __ 9 20 __ 9 10 __ 16 19 __ 6 7 __ 1 2 __

Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1. + 1 2 __ 6 7 __ 3 8 __ 6 7 __ 3 8 __ 6 7 __ 3 8 __ 1 2 __ + Think: rounds to 1 and rounds to 1 2 __ 1 2 __ = 1 6 7 __ 3 8 __ 1 2 __ is about

Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1. – 1 2 __ 9 10 __ 7 8 __ 9 10 __ 7 8 __ 9 10 __ 7 8 __ – Think: rounds to 1 and rounds to 1. 1 – 1 = 0 9 10 __ 7 8 __ – is about 0.

Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1. + 1 2 __ 5 6 __ 3 7 __ 5 6 __ 3 7 __ 5 6 __ 3 7 __ 1 2 __ + Think: rounds to 1 and rounds to . 1 2 __ 1 2 __ = 1 5 6 __ 3 7 __ 1 2 __ is about

Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1. – 1 2 __ 13 19 __ 2 11 __ 13 19 __ 2 11 __ – 2 11 __ 13 19 Think: rounds to 1 and rounds to 0. 1 – 0 = 1 13 19 __ 2 11 __ – is about 1.

Rational and Irrational Numbers

Rational and Irrational Numbers Essential Question
How do I distinguish between rational and irrational 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. Irrational numbers Rational numbers Real Numbers Integers Whole numbers

Rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. ⅔, ⅕, ¼ or = 12

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

Reals 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 Irrationals Naturals 1, 2, 3...

Classifying Real Numbers
Write all classifications that apply to each number. A. 5 5 is a whole number that is not a perfect square. irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 C. whole, integer, rational, real

Write all classifications that apply to each number.
9 9 = 3 whole, integer, rational, real B. –35.9 –35.9 is a terminating decimal. rational, real 81 C. whole, integer, rational, real

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

Classification of All Numbers
State if each number is rational, irrational, or not a real number. A. 21 irrational 3 3 = 0 B. rational

Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number. 4 0 C. not a real number

State if each number is rational, irrational, or not a real number.
23 23 is a whole number that is not a perfect square. irrational 9 B. undefined, so not a real number

State if each number is rational, irrational, or not a real number.
8 9 = 64 81 64 81 C. rational

Adding and Subtracting Fractions

Evaluate the expression: ⅖ + ⅘

Answer: 6/5

Evaluate the expression: ⅕ + ⅕

Answer: 2/5

Evaluate the expression: ⅓ + ⅔

Answer: 1

Evaluate the expression: ¾ + ¼

Answer: 1

Evaluate the expression: ⁵⁄₄ - ¾

Answer: ½

Evaluate the expression: ³⁄₂ - ½

Answer: 1

Evaluate the expression: ½ - ½

Answer: 0

Evaluate the expression: 6 - ¹⁄₆

Answer: ³⁵/₆

Evaluate the expression: ¼ + ½

Answer: 3/4

Evaluate the expression: ⅕ + ½

Answer: ⁷/₁₀

Evaluate the expression: ⅕ + ¾

Answer: ¹⁹/₂₀

Evaluate the expression: 1/3 – (-5/3)

Answer: 2

Evaluate the expression: (-4/5) – 7/8

Answer: - ⁶⁷/₄₀

Evaluate the expression: (-10/7) + 1/6

Answer: - ⁵³/₄₂

Evaluate the expression: 2 – ¹³/₈

Answer: 3/8

Evaluate the expression: (-4/3) – (-3/2)

Answer: 1/6

Evaluate the expression: - 3 ⅗ - 4 ⅖

Answer: -8

Evaluate the expression: 1²/₇ – 3 ⁴/₇

Answer: -2 ²/₇

Evaluate the expression: (-2⁷/₈) + (-1 ¹/₂)

Answer: -4 ³/₈

Evaluate the expression: (-2⁵/₆) - (-1 ¹/₄)

Answer: -1 ⁷/₁₂

Evaluate the expression: 2⁴/₅ - ⁵/₈

Answer: 2 ⁷/₄₀

Multiplying Fractions

Evaluate the expression: -⁵/₄ · ¹/₃

Answer: -⁵/₁₂

Evaluate the expression: ⁸/₇ · ⁷/₁₀

Answer: ⁴/₅

Evaluate the expression: -²/₃ · ⁵/₄

Answer: -⁵/₆

Evaluate the expression: -2 · ³/₇

Answer: -⁶/₇

Dividing Fractions

Evaluate the expression: - ¹/₅ ÷ ⁷/₄

Answer: - ⁴/₃₅

Evaluate the expression: - ¹/₂ ÷ ⁵/₄

Answer: - ²/₅

Evaluate the expression: - ³/₂ ÷ ⁻¹⁰/₇

Answer: ²¹/₂₀

Evaluate the expression: - ⁹/₅ ÷ 2

Answer: -⁹/₁₀

Multiplying and Dividing Mixed Numbers

Evaluate the expression: -2²/₃ · 4 ¹/₁₀

Answer: ¹⁴/₁₅

Evaluate the expression: -2 ¹/₅ · (-1 ³/₄)

Answer: 3 ¹⁷/₂₀

Evaluate the expression: -2 ÷ (-3 ⁴/₅)

Answer: + ¹⁰/₁₉

Evaluate the expression: - 3 ⁷/₁₀ ÷ 2 ¹/₄

Answer: -1 ²⁹/₄₅

Challenge Problem

2 + 3_________ ___ /2 2 +

Answer: 3 ¹/₂₀

Fractions!!.

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