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Chapter 2 - Fractions I

Chapter 2 - Fractions I. Math Skills – Week 2. Today’s Schedule. Turn in Homework Assignment #1 Quiz #1 Lecture on first half of Chapter 2 (Fractions) Stuff: Website Class listing and MyInfo Change your password Post Practice Final exam within the next week

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Chapter 2 - Fractions I

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  1. Chapter 2 - Fractions I Math Skills – Week 2

  2. Today’s Schedule • Turn in Homework Assignment #1 • Quiz #1 • Lecture on first half of Chapter 2 (Fractions) • Stuff: • Website • Class listing and MyInfo • Change your password • Post Practice Final exam within the next week • Office hour location and time • Virtual. Thursdays 6 – 6:45pm

  3. Week 2 - Fractions • Least Common Multiple (LCM) and Greatest Common Factor (GCF) • Section 2.1 • Introduction to Fractions • Section 2.2 • Writing Equivalent Fractions • Section 2.3 • Arithmetic with Fractions (Pt. 1) • Addition • Section 2.4 • Subtraction • Section 2.5

  4. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • The multiples of a number are the products of that number and the whole numbers 1, 2, 3, 4, 5, 6,… • Example: The multiples of 2 are: • 2 x 1 = 2 • 2 x 2 = 4 • 2 x 3 = 6 • 2 x 4 = 8 • … • Thus the multiples of 2 are • 2, 4, 6, 8, …

  5. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • A number that is a multiple of two or more numbers is called a common multiple of those numbers • For Example…8 is a common multiple of 2 and 4. • To find the Lowest Common Multiple (LCM) of a set of numbers use one of the following two methods

  6. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Method 1 (Listing multiples) • Steps • List the multiples of each number • Identify the common multiples • Identify which of those is the smallest number. • This is the LCM

  7. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Example: Find the LCM of 4 and 6 • Using Method 1 • Step 1: • The multiples of 4 are: • 4, 8, 12, 16, 20, 24, 28, 32, 36… • The multiples of 6 are: • 6, 12, 18, 24, 30, 36, 42,… • Step 2: • The common multiples of 4 and 6 are: • 12, 24, 36,… • Step 3: • By inspection, the LCM of 4 and 6 is: • 12

  8. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Method 2: Using Prime Factorizations • Steps: • Write the prime factorization of each number • Organize these prime factors into a “table of prime factors” (see pg. 65) • Circle the greatest product in each column • Multiply each of the circled quantities • This product is the LCM

  9. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Example: Find the LCM of 4 and 6 • Use Method 2: • Step 1: • The prime factorization of 4 is: • 2 x 2 • The prime factorization of 6 is: • 2 x 3 • Step 2: • Step 3: • Step 4: • LCM = 2 x 2 x 3 = 12 2 3 Organize 4 = 2 x 2 Circle greatest products 2 3 6 =

  10. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Which method is better? Which is easier? • Tougher example: Find the LCM of 24, 36, and 50 • Method 1 • Step 1 • Multiples of 24 are: • 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 456,…, 1800 • Multiples of 36 are: • 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540,…, 1800 • Multiples of 50 are: • 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850,….., 1800 • Step 2/3: LCM 1800

  11. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Same example: Find the LCM of 24, 36 and 50 • Method 2 • Step 1 • Prime factorization of 24: • 2 x 2 x 2 x 3 • Prime factorization of 36: • 2 x 2 x 3 x 3 • Prime factorization of 50: • 2 x 5 x 5 • Step 2: Prime Factors Table • Step 3: Circle largest products • Step 4: LCM is the product of circled quantities • 2 x 2 x 2 x 3 x 3 x 5 x 5 = 1800 2 3 5 3 24 = 2 x 2 x 2 2 x 2 3 x 3 36 = 50 = 2 5 x 5

  12. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Group Examples: Find the LCM of the following sets of numbers • 14, 21 • Ans = 42 • 12, 27, 50 • Ans = 2700 • Class Examples: • 2, 7, 14 • Ans = 14 • 5, 12, 15 • Ans = 60 Steps for finding LCM • Step 1: Find prime factorization of • each number • Step 2: Prime Factors Table • Step 3: Circle largest products • Step 4: LCM product of circled quantities

  13. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Recall that the factors of a number are the numbers (1, 2, 3, 4, 5, …) that divide the number evenly • Common factors of a set of numbers are the factors that those numbers have in common. • The Greatest Common Factor of a set of numbers is the largest number in the set of common factors.

  14. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • To find the GCF of a set of numbers use one of the following two methods • Method 1 (Listing factors) • Steps • List the factors of each number • Identify the common factors . • Identify which of those is the largest number. That number is the GCF • Example: Find the GCF of 30 and 105 • The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 • The factors of 105 are: 1, 3, 5, 7, 15, 21, 35, 105 • The Common Factors are 1, 3, 5, 15 • The GCF is: • 15

  15. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Method 2 (Using Prime Factorization) • Steps • Find the Prime Factorization of each number • Write out Prime Factorization table. • Circle the smallest product in each column that is not blank • Importante!If column has a blank for one of the numbers, don’t circle anything for that column • The product of the circled quantities is the GCF

  16. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Example using Method 2: • Find the GCF of 90, 168, 420 (Using method 2) • Step 1 • The Prime factorization of 90 is: • 2 x 3 x 3 x 5 • The Prime Factorization of 168 is: • 2 x 2 x 2 x 3 x 7 • The Prime Factorization of 420 is: • 2 x 2 x 3 x 5 x 7 • Step 2: Prime Factors table • Step 3: Circle Smallest Product (No Blanks) • Step 4: Product of circled numbers is GCF = 2 x 3 = 6 2 3 5 7 2 3 x 3 90 = 2 x 2 x 2 158 = 3 7 420 = 2 x 2 3 5

  17. Least Common Multiple (LCM) and Greatest Common Factor (GCF) – Section 2.1 • Group Examples: Find the GCF of the following sets of numbers • 12, 18 • Ans = 6 • 21, 27, 33 • Ans = 3 • Class Examples: • 24, 64 • Ans = 8 • 41, 67 • Ans = 1 Steps to find GCF • Step 1: Find Prime factorization of each • number • Step 2: Prime Factors table • Step 3: Circle Smallest Product in • each column (ignore columns with • blanks) • Step 4: Product of circled numbers is GCF

  18. Introduction to Fractions – Section 2.2 • A fraction is the representation of a specified portion of a whole number. 3 2 1 4 Numerator Fraction Bar 4 4 4 4 Denominator

  19. Introduction to Fractions – Section 2.2 • Definitions • Proper Fraction is a fraction that is less than 1 • Numerator is smaller than the denominator • Mixed number is a number greater than 1 • Whole number part and a fractional part. • Improper Fraction is a fraction greater than or equal to 1 • Numerator is greater than the denominator 3 4 3 1 4 7 4

  20. Introduction to Fractions – Section 2.2 • Convert Improper fractions  Mixed numbers • Steps • Divide the Numerator into the Denominator • Fractional Part: Write any remainder as a fraction by placing it over the original denominator • Example: Write 13/5 as a mixed number • Convert Mixed numbers  Improper Fractions • Steps • Multiply the denominator of the fractional part by the whole number part • Add this product to the numerator • Write the sum from step 2 over the denominator of the fractional part • Example: Write 7 3/8 as an improper fraction

  21. Introduction to Fractions – Section 2.2 • Class Examples: • Write 22/5 as a mixed number • 4 2/5 • Write 28/7 as a whole number • 4 • Write 14 5/8 as an improper fraction • 117/8 • Write 10/3 as a mixed number • 3 1/3

  22. Writing Equivalent Fractions – Section 2.3 • Equivalent fractions are equal fractions that look different • Example 4/6 is equivalent to 2/3 • Remember the ones property in multiplication? • 1 x Number = Number • Agree? 2/3 x 1 = 2/3 • 2/3 x 1/1 = 2/3 • 2/3 x 4/4 = 2/3 = 8/12 • 2/3 x 5000/5000 = 2/3 = 10000/15000

  23. Writing Equivalent Fractions – Section 2.3 • Example: (Finding equivalent fractions) What is an equivalent fraction to 5/8 that has a denominator of 32? • Ask yourself…self…what do I have to multiply the denominator of 5/8 by to get 32? • Or you could just divide 32 by 8 • 4 • 5/8 x 1 = 5/8 x 4/4 = 20/32 • 20/32 is a fraction with 32 in the denominator that is equivalent to 5/8 • Another example • Write 2/3 as an equivalent fraction that has a denominator of 42 • Divide 42 by 3 = 14 • 2/3 x 14/14 = 28/42 is equivalent to 2/3 • Example write 4 as a fraction with 12 in denominator

  24. Writing Equivalent Fractions – Section 2.3 • Class Examples: • Write 3/5 as an equivalent fraction with a denominator of 45 • Fill in the blank • ½ = __ /32 • 2/3 = __ / 12 • 6 = __ / 11

  25. Writing Equivalent Fractions – Section 2.3 • A fraction is in simplest form when the numerator and denominator have no common factors (other than 1) • Example: 4/6 written in simplest form is 2/3 • To write a fraction in simplest form • Steps • Write prime factorization of the numerator and denominator • Cancel (divide) out all common factors. • Remaining products are the new Numerator and Denominator

  26. Writing Equivalent Fractions – Section 2.3 • Examples • Write 15/40 in simplest form • = 3 x 5 / 2 x 2 x 2 x 5 = 3/8 • Write 6/42 in simplest form • = 2 x 3 / 2 x 3 x 7 = 1/7 • Write 30/12 in simplest form • 2 x 3 x 5 / 2 x 2 x 3 = 5/2 = 2 1/2

  27. Writing Equivalent Fractions – Section 2.3 • Class Examples: • Write the following in simplest form • 16/24 • = 2 x 2 x 2 x 2 / 2 x 2 x 2 x 3 = 2/3 • 8/56 • 2 x 2 x 2 / 2 x 2 x 2 x 7 = 1/7 • 15/32 • = 3 x 5 / 2 x 2 x 2 x 2 x 2 = 15/32 • 48/36 • = 2 x 2 x 2 x 2 x 3 / 2 x 2 x 3 x 3 = 4/3 = 1 1/3

  28. Addition of Fractions and Mixed Numbers 2.4 • The key is the denominator. To add fractions together, each fraction must have the same denominator. • If the denominators are the same • Steps • Add the Numerators • Place the sum of the Numerators over the common denominator • Write the sum in simplest form 5 11 16 = + 12 12 12 4 = 3

  29. Addition of Fractions and Mixed Numbers 2.4 • If denominators are not the same: • Steps • Find the Lowest Common Denominator (LCD) of the two fractions • Note: this quantity is the LCM of the denominators • Rewrite each fraction as an equivalent fraction with the LCD as the denominator. • Add the numerators • Place this sum over the common denominator • Example: 1/2 + 1/3 = ? • LCM = 6, then 3/6 + 2/6 = 5/6

  30. Addition of Fractions and Mixed Numbers 2.4 • More Examples: • Find 7/12 more than 3/8 • Lowest Common Denominator (LCD) = 24 • 14/24 + 9/24 = 23/24 • Add 5/8 + 7/9 • LCD = 72 • 45/72 + 56/72 = 101/72 = 1 29/72 • Add 2/3 + 3/5 + 5/6 • LCD = 30 • 20/30 + 18/30 + 25/30 = 63/30 = 2 3/30 = 2 1/10

  31. Addition of Fractions and Mixed Numbers 2.4 • Class Examples: • Find the sum of 5/12 and 9/16 • LCM = 48 • 20/48 + 27/48 = 47/48 • Add 7/8 + 11/15 • LCM = 120 • 105/120 + 88/120 = 193/120 = 1 73/120 • Add 3/4 + 4/5 + 5/8 • LCM = 40 • 30/40 + 32/40 + 25/40 = 87/40 = 2 11/40

  32. Addition of Fractions and Mixed Numbers 2.4 • Addition of mixed numbers • Steps • Find the Lowest Common Denominator (LCD) of the two fractions • Note: this quantity is exactly the (LCM) of the denominators • Add the fractional parts • Add the whole number parts • Put fractional part in simplest form • Example: what is 6 14/15 added to 5 4/9 ? • LCM = 45, then5 20/45 + 6 42/45 = 11 62/45 = 11 + 1 17/45 = 12 17/45

  33. Addition of Fractions and Mixed Numbers 2.4 • More Examples: • Find 5 more than 3/8 • LCD = Don’t need this • 5 3/8 • Add 17 + 3 3/8 • LCD = Don’t need this • 20 3/8 • Add 5 2/3 + 11 5/6 + 12 7/9 • LCD = 18 • 5 12/18 + 11 15/18 + 12 14/18 = 28 41/18 = 30 5/18

  34. Addition of Fractions and Mixed Numbers 2.4 • Class Examples: • Find the sum of 29 and 7 5/12 • LCD = Don’t need this • 46 5/12 • Add 7 4/5 + 6 7/10 + 13 11/15 • LCD = 30 • 7 24/30 + 6 21/30 + 13 22/30 = 26 67/30 = 28 7/30 • Add 9 3/8 + 17 7/12 + 10 14/15 • LCD = 120 • 9 45/120 + 17 70/120 + 10 112/120 = 36 227/120 = 37 107/120

  35. Addition of Fractions and Mixed Numbers 2.4 • Word problems discussion • Pg 80, You Try It 9 • Add all time spent together • Pg 80, You Try It 10 • Add all time spent working overtime • Multiply total time spent working overtime by the overtime hourly rate.

  36. Subtraction of Fractions and Mixed Numbers 2.5 • Again…the key is the denominator. To subtract fractions, each fraction must have the same denominator. • If the denominators are the same • Steps • Subtract the Numerators • Place the difference of the new numerators over the common denominator • Write the difference in simplest form 11 5 7 = - 12 12 12

  37. Subtraction of Fractions and Mixed Numbers 2.5 • If denominators are not the same: • Steps • Find the Lowest Common Denominator (LCD) of the two fractions • Note: this quantity is exactly the Least Common Multiple (LCM) of the denominators • Rewrite each fraction as an equivalent fraction with the LCD as the denominator. • Subtract the numerators • Place this difference over the common denominator • Example: 5/6 – 1/4 = ? • LCM = 12, thus 10/12 - 3/12 = 7/12

  38. Subtraction of Fractions and Mixed Numbers 2.5 • More Examples: • Subtract 3/4 - 2/5 • LCD = 20 • 15/20 – 8/20 = 7/20 • Subtract 53/60 - 7/12 • LCD = 60 • 53/60 – 35/60 = 18/60 = 3/10 • 11/16 – 5/12 = ? • LCD = 48 • 33/48 - 20/48 = 13/48

  39. Subtraction of Fractions and Mixed Numbers 2.5 • Class Examples: • Subtract 5/6 – 4/15 • LCD = 30 • 25/30 – 8/30 = 17/30 • Subtract 13/18 – 7/24 • LCD = 72 • 52/72 – 21/72 = 31/72

  40. Subtraction of Fractions and Mixed Numbers 2.5 • Subtraction of mixed numbers • Steps • Find the Lowest Common Denominator (LCD) of the two fractions • Note: this quantity is the LCM of the denominators • Subtract the fractional parts • Borrow if necessary • Borrow 1 from the whole number part and rewrite it as an equivalent fraction to 1 using with the same LCD • Subtract the whole numbers

  41. Subtraction of Fractions and Mixed Numbers 2.5 • Subtraction of mixed numbers • Example: (No Borrowing) what is 5 5/6 subtracted from 2 3/4? • LCD = 12 • 5 10/12 – 2 9/12 = 3 1/12 • Example: (With Borrowing): Subtract 5 – 2 5/8 • LCD = Don’t need it • 4 8/8 – 2 5/8 = 2 3/8 • Example: (With Borrowing): Subtract 7 1/6 – 2 5/8 • LCD = 24 • 7 4/24 - 2 15/24 = 6 28/24 – 2 15/24 = 4 13/24

  42. Subtraction of Fractions and Mixed Numbers 2.5 • More Examples: • Subtract 15 7/8 – 12 2/3 • LCD = 24 • 15 21/24 – 12 16/24 = 3 5/24 • Subtract 9 – 4 3/11 • LCD = Don’t need this • 8 11/11 – 4 3/11 = 4 8/11 • Find 11 5/12 decreased by 2 11/16 • LCD = 48 • 11 20/48 – 2 33/48 = 10 68/48 – 2 33/48 = 8 35/48

  43. Subtraction of Fractions and Mixed Numbers 2.5 • Class Examples: • Subtract 17 5/9 – 11 5/12 • LCD = 36 • 17 20/36 – 11 15/36 = 6 5/36 • Subtract 8 – 2 4/13 • LCD = Don’t need this • 7 13/13 – 2 4/13 = 5 9/13 • Find 21 7/9 minus 7 11/12 • LCD = 36 • 21 28/36 – 7 33/36 = 21 64/36 – 7 33/36 = 14 31/36

  44. Subtraction of Fractions and Mixed Numbers 2.5 • Word problems discussion • Pg 88, • 6 • Add all time spent together • You Try It 7 • How would you approach this problem? • Add all the weight lost over the first two months • 13 ¼ pounds lost in the first two months • Subtract 13 ¼ from the total of 24. (10 ¾ pounds left)

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