Notes 7 th Grade Math Mc Dowell Order

Notes 7 th Grade Math Mc. Dowell

Order of Operations 9/11 PEMDASLR Please Excuse My Dear Aunt Sally’s Last Request

Parenthesis – Not just parenthesis • Any grouping symbol – Brackets – Fraction bars – Absolute Values Example 2+4 2 6 2 3 Simplify the top of the fraction 1 st Then divide

Exponents Simplify all possible exponents

Multiplication And Division Do multiplication and division in order from left to right Don’t do all multiplication and then all division Remember division is not commutative

Addition And Subtraction Do addition and subtraction in order from left to right Don’t do all addition and then all subtraction Remember subtraction is not commutative

You Try 1. 3 + 15 – 5 2 2. 48 8 – 1 3. 3[ 9 – (6 – 3)] – 10 4. 16 + 24 30 - 22

Exponents 9/11 Show repeated multiplication baseexponent Base The number being multiplied Exponent The number of times to multiply the base

Example 2³ 2 x 2 x 2 4 x 2 8

Expanded Notation When a repeated multiplication problem is written out long 3 x 3 x 3 x 3 Exponential Notation When a repeated multiplication problem is written out using powers 34

Example (-2)² -2 x – 2 4 -2² -1 x 2 x 2 -1 x 4 -4

Examples (12 – 3)² (2² - 1²) (-a)³ for a = -3 5(2(3)² – 4)³

Scientific Notation Powers Of Ten 9/14 Factors 10 10 x 10 x 10 x 10 Product 10 100 1, 000 10, 000 Power 101 102 103 104 # of 0 s 1 2 3 4

You Try Fill in the chart Factors 10 x 10 x 10 x 10 Product 10, 000 Power 107 # of 0 s 100, 000, 000 1010 10 14

Scientific Notation A short way to write really big or really small numbers using factors Looks like: 2. 4 x 104

One factor will always be a power of ten: 10 n The other factor will be less than 10 but greater than one 1 < factor < 10 And will usually have a decimal

The first factor tells us what the number looks like The exponent on the ten tells us how many places to move the decimal point

Example Convert between scientific notation and expanded notation 4. 6 x 106 Move the decimal 6 hops to the right 4. 600000 Rewrite 4600000

You Try Write in expanded notation 1. 2. 3 x 103 2. 5. 76 x 107 Answers 1. 2, 300 2. 57, 600, 000

Example Convert between expanded notation and scientific notation 13, 700, 000 Figure out how many hops left it takes to get a factor between 1 and 10 1. 3, 700, 000 1. 3 x 107 Rewrite: the number of hops is your exponent

You Try Write in scientific notation 1. 340, 000 2. 98, 200 Answers 1. 3. 4 x 108 2. 9. 82 x 104

Factor Trees and GCF Prime Numbers 9/15 Integers greater than one with two positive factors 1 and the original number 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

Composite Numbers Integers greater than one with more than two positive factors 4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, . . .

Factor Trees Steps A way to factor a number into its prime factors Is the number prime or composite? If prime: you’re done If Composite: Is the number even or odd? If even: divide by 2 If odd: divide by 3, 5, 7, 11, 13 or another prime number Write down the prime factor and the new number Is the new number prime or composite?

Example Find the prime factors of 99 prime or composite even or odd divide by 3 3 33 prime or composite even or odd divide by 3 3 11 prime or composite The prime factors of 99: 3, 3, 11

Example Find the prime factors of 12 prime or composite even or odd divide by 2 2 6 prime or composite even or odd divide by 2 2 3 prime or composite The prime factors of 12: 2, 2, 3

You Try Find the prime factors of 1. 8 2. 15 3. 82 4. 124 5. 26

GCF 9/15 Greatest Common Factor the largest factor two or more numbers have in common.

Steps to Finding GCF 1. Find the prime factors of each number or expression 2. Compare the factors 3. Pick out the prime factors that match 4. Multiply them together

Example Find the GCF of 126 and 130 126 2 130 63 3 75 2 21 15 5 5 3 7 The common factors are 2, 3 2 x 3 The GCF of 126 and 130 is 6 3

You Try Work Book p 47 # 1 -9 p 48 # 3 -33 3 rd

LCM 9/16 Least common multiple The smallest number that is a multiple of both numbers

Steps To Find LCM 1. Make a multiplication table for each number 2. Compare the multiplication tables 3. Pick the smallest number that both (all) tables have

Example Find the LCM of 8 and 3 1 2 3 4 5 6 7 8 8 16 24 32 40 48 56 64 1. Make a mult table 2. Compare 3 6 9 12 15 18 21 24 3. Find the smallest match The LCM of 8 and 3 is 24

You Try Find the LCM between 1. 2 and 5 2. 9 and 7

Simplifying Fractions Simplest form 9/16 When the numerator and denominator have no common factors

Simplifying fractions 1. Find the GCF between the numerator and denominator 2. Divide both the numerator and denominator of the fraction by that GCF

Example Simplify 28 52 28 s Prime factors: 2, 2, 7 52 s Prime factors: 2, 2, 13 28 4 = 7 52 4 13 Use a factor tree to find the prime factors of both numbers and then the GCF: 2 x 2 4

You Try Write each fraction in simplest form 1. 27/30 2. 12/16

Equivalent fractions Fractions that represent the same amount ½ and 2/4 are equivalent fractions

Making Equivalent Fractions 1. Pick a number 2. Multiply the numerator and denominator by that same number 5 x 3 = 15 8 x 3 24

You Try Find 3 equivalent fractions to 6 11

Are the Fractions equivalent? 1. Simplify each fraction 2. Compare the simplified fraction 3. If they are the same then they are equivalent

You try Work Book p 49 #1 -17 odd

Least common Denominator 9/17 Common When fractions have the same Denominator denominator

Steps to Making Common Denominators 1. Find the LCM of all the denominators 2. Turn the denominator of each fraction into that LCM using multiplication Remember: what ever you multiply by on the bottom, you have to multiply by on the top!

Example Make each fraction have a common denominator 5/6, 4/9 Find the LCM of 6 and 9 6 12 18 24 30 36 42 48 9 18 27 36 45 64 73 82 Multiply to change 5 x 3 = 15 each denominator 6 x 3 18 to 18 4 x 2=8 9 x 2 18

You try What are the least common denominators? 1. ¼ and 1/3 2. 5/7 and 13/12

Comparing And Ordering fractions Manipulate the fractions so each has the same denominator Compare/order the fractions using the numerators (the denominators are the same)

Project Group work! 1. Get into groups of 3 or 4 2. Pick 3 or 4 different fractions 3. Each person make a picture of their fraction 4. Get together as a group and put the fractions/pictures in order from least to greatest

You try Workbook p 51 #1 -17 odd p 52 #3 -36 3 rd

Mixed Numbers and Improper Fractions 9/18 Improper fractions When the numerator is bigger than the denominator 7 4 Represents more than 1

You try http: //www. youtube. com/user/M ath. Raps#play/uploads/3/VZQ Dvb 5 Yjvw

Mixed Numbers The sum of a whole number and a fraction 1+¾ 1¾ 7 1¾ = 4

Converting A Mixed Number To an Improper MAD face Multiply, Add, keep the Denominator Multiply the denominator by the whole number then add the product to the numerator That is the new numerator— keep the old denominator

6 x 5+3 6 30 + 3 6 33 6

You try Convert to an improper fraction 1. 2.

Converting An Improper To a Mixed # Divide the numerator by the dominator The quotient is the whole number The remainder is the new numerator Keep the same denominator

Example Convert 26 to a mixed number 3 8 R 2 3 26 -24 2

You try Convert each improper fraction to a mixed number 1. 14 3 2. 25 5

Fractions and Decimals 9/21 Terminating Decimal a decimal that ends Repeating Decimal When the same group of numbers continues to repeat forever 1. 25 4. 333333 4. 3

Converting Fractions To decimals Divide the numerator by the denominator 5 16 Insert the decimal and some place holders to divide

You try Convert each fraction to a decimal. Determine if the decimal is a terminating decimal or a repeating decimal 1. 3 5 2. 1 6

Converting decimals to fractions Remember place values

Converting decimals to fractions Find the place value of the terminating decimal Place the numbers after the decimal over the place value Keep whole numbers as whole numbers Simplify the fraction to lowest terms

Example 0. 925 The 5 is in the thousandths place so 1000 is the denominator 925 1000 925 25 1000 25 37 40 Simplify

You try Convert each decimal to a fraction 1. 0. 05 2. 4. 7 3. 0. 84

Number Sets Whole Numbers Natural Numbers 9/22 0, 1, 2, 3, . . . for short Also known as the counting numbers 1, 2, 3, 4, . . .

Integers Positive and negative whole numbers for short. . . – 2, -1, 0, 1, 2, . . . Rational Numbers that can be written as Numbers fractions for short ½, ¾, -¼, 1. 6, 8, -5. 92

You Try Copy and fill in the Venn Diagram that compares Whole Numbers, Natural Numbers, Integers, and Rational Numbers Whole #s

Ordering Rational Numbers Two Options 1. Change each number to a decimal and compare 2. Write each number as a fraction with a common denominator and compare

You Try Order from least to greatest 1. 2. 7, -0. 3, -4/11 2. -5/6, 2. 2, -0. 5 3. 2. 56, -2. 5, 24/10