Lesson 2 Rational Decisions Lesson 2 Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

LEARNING GOALS • Determine under which operations (addition, subtraction, multiplication, and division) number sets are closed. • Recognize that all numbers can be written as decimals and that rational numbers can be written as terminating or repeating decimals. • Write repeating decimals as fractions. • Identify numbers that are not rational as irrational numbers. Lesson 2: Rational Decisions

KEY TERMS • natural numbers • whole numbers • integers • closed • rational numbers • • irrational numbers terminating decimal repeating decimal bar notation You have learned about rational numbers. How are they different from other number sets? Lesson 2: Rational Decisions

The first set of numbers that you learned when you were very young was the set of counting numbers, or natural numbers. Natural numbers consists of the numbers that you use to count objects: {1, 2, 3, …}. Lesson 2: Rational Decisions

1. Consider the set of natural numbers. a. Why do you think this set of numbers is sometimes referred to as the set of counting numbers? Lesson 2: Rational Decisions

1. Consider the set of natural numbers. b. How many natural numbers are there? Lesson 2: Rational Decisions

1. Consider the set of natural numbers. c. Does it make sense to ask which natural number is the greatest? Explain why or why not. Lesson 2: Rational Decisions

You have also used the set of whole numbers. Whole numbers are made up of the set of natural numbers and the number 0, the additive identity. 2. Why is zero the additive identity? Lesson 2: Rational Decisions

You have also used the set of whole numbers. Whole numbers are made up of the set of natural numbers and the number 0, the additive identity. 3. Explain why having zero makes the set of whole numbers more useful than the set of natural numbers. Lesson 2: Rational Decisions

Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses. 4. What is the additive inverse of a number? Lesson 2: Rational Decisions

Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses. 5. Represent the set of integers. Use set notation and remember to use three dots to show that the numbers go on without end in both directions. Lesson 2: Rational Decisions

When you perform an operation such as addition or multiplication on the numbers in a set, the operation could produce a defined value that is also in the set. When this happens, the set is said to be closed under the operation. The set of integers is said to be closed under the operation of addition. This means that for every two integers a and b, the sum a + b is also an integer. Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. a. Are the natural numbers closed under addition? Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. b. Are the whole numbers closed under addition? Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. c. Are the natural numbers closed under subtraction? Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. d. Are the whole numbers closed under subtraction? Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. e. Are the integers closed under subtraction? Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. f. Are any of these sets closed under multiplication? Lesson 2: Rational Decisions

7. Determine if each set of numbers is closed under the given operation. Provide an example to support your response. g. Are any of these sets closed under division? Lesson 2: Rational Decisions

In previous courses, you have learned about the additive inverse, the multiplicative inverse, the additive identity, and the multiplicative identity. 8. Consider each set of numbers and determine if the set has an additive identity, additive inverse, multiplicative identity, or a multiplicative inverse. Explain your reasoning for each. a. the set of natural numbers Lesson 2: Rational Decisions

In previous courses, you have learned about the additive inverse, the multiplicative inverse, the additive identity, and the multiplicative identity. 8. Consider each set of numbers and determine if the set has an additive identity, additive inverse, multiplicative identity, or a multiplicative inverse. Explain your reasoning for each. b. the set of whole numbers Lesson 2: Rational Decisions

In previous courses, you have learned about the additive inverse, the multiplicative inverse, the additive identity, and the multiplicative identity. 8. Consider each set of numbers and determine if the set has an additive identity, additive inverse, multiplicative identity, or a multiplicative inverse. Explain your reasoning for each. c. the set of integers. Lesson 2: Rational Decisions

New number systems arise out of a need to create new types of numbers. If you divide two integers, what type of number have you created? You’ve created a rational number. Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

2. Order the fractions from least to greatest. Then, convert each fraction to a decimal by dividing the numerator by the denominator. Continue to divide until you see a pattern. Lesson 2: Rational Decisions

3. Explain why these decimal representations are called repeating decimals. Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

5. Identify the fraction represented by the repeating decimal 0. 44. . Lesson 2: Rational Decisions

6. Use this method to write the fraction that represents each repeating decimal. a. 0. 55… Lesson 2: Rational Decisions

6. Use this method to write the fraction that represents each repeating decimal. b. 0. 0505… Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Lesson 2: Rational Decisions

Complete the table to summarize the number sets you have learned about and reviewed in this lesson. Provide examples for each number set to address the four operations of addition, subtraction, multiplication, and division. Lesson 2: Rational Decisions

Lesson 2: Rational Decisions