Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Ethnomathematics wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Surreal number wikipedia , lookup
Location arithmetic wikipedia , lookup
Positional notation wikipedia , lookup
Large numbers wikipedia , lookup
Real number wikipedia , lookup
PRE-ALGEBRA Lesson 5-1 Warm-Up PRE-ALGEBRA “Comparing and Ordering Rational Numbers” (5-1) What is the a “multiple”? Multiple: The multiple of a number is the product of that number and any nonzero number (when you count by a number, you are finding its multiples) Example: Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36,…. Example: Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54,…. What is the the “least common multiple” or LCM? Least Common Multiple (LCM): the smallest multiple shared by all of the numbers Example: Common Multiples of 4 are 6 are 12, 24, and 36. The smallest multiple of both numbers , or Least Common Multiple (LCM) is 12. How do you find the LCM? To find the LCM: 1. list the multiples of both numbers until you find the first one that they share in common, or 2. multiply the greatest power of the factors the numbers. Example: Find the LCM of 18, 27, and 36. Method 1: List the multiples of each number until you find a common one. Multiples of 18 are 18, 36, 54, 72, 90, 108,…. Find the multiple of each number. Stop when you find a multiple the Multiples of 27 are 27, 54, 81, 108,… numbers share in common. Multiples of 36 are 36, 72, 108,…. The LCM of 18, 27, and 36 is 108. PRE-ALGEBRA “Comparing and Ordering Rational Numbers” (5-1) Method 2: Multiply the greatest power of all factors together. Example: Find the LCM of 18, 27, and 36. Create a factor tree number to find the prime factors of each number. 18 = 3 • 3 • 2 = 32 • 2 27 = 3 • 3 • 3 = 33 36 = 3 • 3 • 2 • 2 = 32 • 22 33 • 22 = 27 • 4 = 108 Write each number in prime factorization. form Multiply the greatest powers of all factors together. Example: Find the LCM of 6a2 and 18a3. LCM of 6a2 and 18a3 is 18a3. PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Additional Examples Today, the school’s baseball and soccer teams had games. The baseball team plays every 7 days. The soccer team plays every 3 days. When will the teams have games on the same day again? 7, 14, 21, 28, 35, 42, . . . List the multiples of 7. 3, 6, 9, 12, 15, 18, 21, . . . List the multiples of 3. The LCM is 21. In 21 days both teams will have games again. PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Additional Examples Find the LCM of 16 and 36. 16 = 24 36 = 22 • 32 LCM = 24 • 32 = 144 Write the prime factorizations. Use the greatest power of each factor. Multiply. The LCM of 16 and 36 is 144. PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Additional Examples Find the LCM of 5a4 and 15a. 5a4 = 5 • a4 15a = 3 • 5 • a LCM = 3 • 5 • a4 = 15a4 Write the prime factorizations. Use the greatest power of each factor. Multiply. The LCM of 5a4 and 15a is 15a4. PRE-ALGEBRA “Comparing and Ordering Rational Numbers” (5-1) How do you compare fractions? To compare fractions, you can: 1. use a number line (numbers to the right are greater than numbers to the left), or 2. compare the numerators (number of parts) if the denominator (size of the parts) are equal. So, if the denominators aren’t the same, you need to change one or more of the fractions into equivalent fractions with a common denominator. Method 1: Use a number line. -1 -1 Example: Compare and . 2 10 -1 -1 -1 -1 is on the left of , so . 2 10 10 2 Method 1: Compare the numerators. Example: Compare 2 and 3 . 3 4 3 • 4 = 12 Multiply the denominators together to find a common denominator 2 • 4 = 8_ Write equivalent fractions with a denominator of 3 • 4 = 12 12 and compare the numerators (Hint: Notice 3 • 3 = 9_ that you multiply the each fraction by the 4 • 3 = 12 other fractions denominator) Since 8 9 , then 3 2 . 12 12 4 3 PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Additional Examples Graph and compare the fractions in each pair. a. 7 , 3 8 8 3 8 7 8 3 3 7 is on the left, so < . 8 8 8 1 1 b. – , – 3 6 –1 –1 3 6 1 1 – 1 is on the right, so – 6 > – 3 . 6 PRE-ALGEBRA “Comparing and Ordering Rational Numbers” (5-1) What is the “least Least Common Denominator (LCD): the LCM of two or more denominators (in other words, the smallest common denominator) common denominator” (LCD)? Example: List the multiples of each denominator until you find a multiple that is shared by both numbers (LCM). LCM = 36 Rewrite the fractions into equivalent fractions with a denominator of 36 (The LCD is 36). Then, compare the numerators. Since 16 15 , then 4 5 . 36 36 9 12 PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Additional Examples 6 The softball team won of its games and the 7 7 hockey team won 9 of its games. Which team won the greater fraction of its games? Step 1 Step 2 Find the LCM of 7 and 9. 7 = 7 and 9 = 32 LCM = 7 • 32 = 63 Write equivalent fractions with a denominator of 63. 6 • 9 54 = 7 • 9 63 7 • 7 49 = 9 • 7 63 Step 3 Compare the fractions. 54 49 6 7 > , so > 63 63 7 9 The softball team won the greater fraction of its games. PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Additional Examples Order 3 , 1 , and 2 from least to greatest. 7 4 3 3 3 • 12 36 = = 7 7 • 12 84 1 1 • 21 21 = = 4 4 • 21 84 The LCM of 7, 4, and 3 is 84. Use 84 as the common denominator. 2 2 • 28 56 = = 3 3 • 28 84 21 < 36 < 56 , so 1 < 3 < 2 . 84 84 84 4 7 3 PRE-ALGEBRA Comparing and Ordering Rational Numbers LESSON 5-1 Lesson Quiz Find the LCM of each pair of numbers. 1. 8, 6 2. 12, 16 24 48 3. Compare and order 3 , – 8 , and – 3 from least to greatest. 16 10 16 – 8 < – 3 < 3 10 16 16 PRE-ALGEBRA