Math Operations: PEMDAS, Exponents, Scientific Notation, Factor Trees, GCF, LCM, Simplifying Fractions

1. Notes Chapter 3 7th Grade Math McDowell

2. PEMDASLR Order of Operations 9/11 Please Excuse My Dear Aunt Sally’s Last Request arenthesis xponents ultiplication ivision ddition ubtraction eft ight

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

4. Exponents Simplify all possible exponents

5. 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

6. 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

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

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

9. Example 2³ 2 x 2 x 2 • 4 x 2 8

10. 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

11. Example (-2)² • -2 x –2 4 • -2² • -1 x 2² • -1 x 2 x 2 • -1 x 4 • -4

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

13. Powers Of Ten Scientific Notation 9/14

14. You Try Fill in the chart

15. Scientific Notation • A short way to write really big or really small numbers using factors • Looks like: • 2.4 x 104

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

17. 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

18. Convert between scientific notation and expanded notation Example • Move the decimal 6 hops to the right • 4.6 x 106 • 4.600000 • Rewrite • 4600000

19. You Try • Write in expanded notation • 2.3 x 103 • 5.76 x 107 • Answers • 2,300 • 57,600,000

20. Convert between expanded notation and scientific notation Example • 13,700,000 • Figure out how many hops left it takes to get a factor between 1 and 10 • 1.3,700,000 • Rewrite: the number of hops is your exponent • 1.3 x 107

21. You Try • Write in scientific notation • 340,000,000 • 98,200 • Answers • 3.4 x 108 • 9.82 x 104

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

23. 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, . . .

24. Factor Trees A way to factor a number into its prime factors • Is the number prime or composite? Steps • 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?

25. 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

26. 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

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

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

29. 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

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

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

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

33. 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

34. Example Find the LCM of 8 and 3 1. Make a mult table 2. Compare 3. Find the smallest match • The LCM of 8 and 3 is 24

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

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

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

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

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

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

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

42. You Try Find 3 equivalent fractions to 6 11

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

44. You try Work Book p 49 #1-17 odd

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

46. 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!

48. 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 each denominator to 18 • 5 x 3 • 6 x 3 • = 15 • 18 • 4 x 2 • 9 x 2 • = 8 • 18

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

50. 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)