1 / 44

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

carter
Download Presentation

Chapter 2 - Fractions I

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related