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REAL RATIONAL NUMBERS (as opposed to fake numbers?) and Properties Part 1 (introduction) STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions: and justify each step in the process. Student Objective: • Students will apply order of operations to solve problems with rational numbers and apply their properties, by performing the correct operations, using math facts skills, writing reflective summaries, and scoring 80% proficiency Vocabulary Set Set Notation Natural numbers Whole Numbers Integers A collection of objects. { } Counting numbers {1,2,3, …} Natural numbers and 0. {0,1,2,3, …} Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …} A real number that can be expressed Rational as a ratio of integers (fraction) Number Any real number that is not rational. Irrational 2 , Number Real Numbers All numbers associated with the number line. Essential Questions: • How do you know if a number is a rational number? • What are the properties used to evaluate rational numbers? Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers Rational Numbers • A rational number is a real number that can be written as a ratio of two integers. • A rational number written in decimal form is terminating or repeating. EXAMPLES OF RATIONAL NUMBERS •16 •1/2 •3.56 •-8 •1.3333… •-3/4 To write a fraction as a decimal, divide the numerator by the denominator. You can use long division. numerator denominator denominator numerator Additional Example 3A: Writing Fractions as Decimals Write the fraction as a decimal. 11 9 The fraction 1 .2 9 11 .0 –9 20 –1 8 2 The pattern repeats. Writing Math A repeating decimal can be written with a bar over the digits_that repeat. So 1.2222… = 1.2. 11 is equivalent to the decimal 1.2. 9 Additional Example 3B: Writing Fractions as Decimals Write the fraction as a decimal. 7 20 0.3 5 This is a terminating decimal. 20 7.0 0 –0 70 –6 0 1 00 –1 0 0 0 The remainder is 0. The fraction 7 is equivalent to the decimal 0.35. 20 Irrational Numbers • An irrational • Square roots of number is a non-perfect number that “squares” cannot be written as a ratio of two 17 integers. • Irrational numbers written as • Pi- īī decimals are nonterminating and non-repeating. Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all. So it is not a real number, it is not rational or irrational. Real Numbers Rational numbers Integers Whole numbers Irrational numbers Rational Numbers Natural Numbers - Natural counting numbers. 1, 2, 3, 4 … Whole Numbers - Natural counting numbers and zero. 0, 1, 2, 3 … Integers - Whole numbers and their opposites. … -3, -2, -1, 0, 1, 2, 3 … Rational Numbers - Integers, fractions, and decimals. Ex: -0.76, -6/13, 0.08, 2/3 Rational Numbers on a Number Line Integers Whole Numbers Natural Numbers | – 4 | | | | | | | | – 3 – 2 – 1 0 1 2 3 4 Negative numbers Positive numbers Zero is neither negative nor positive Making Connections Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. Rational Numbers are classified this way as well! Animal Reptile Lizard Gecko Venn Diagram: Naturals, Wholes, Integers, Rational Real Numbers Rationals 6.7 5 9 0.8 Integers 11 Wholes Naturals 1, 2, 3... 5 0 3 2 7 Make a Venn Diagram that displays the following sets of numbers: Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Reals Rationals 2 3 -3 -2.65 Integers -19 Wholes 1 0 6 Naturals 1, 2, 3... 4 Irrationals 2 Reminder • Real numbers are all the positive, negative, fraction, and decimal numbers you have heard of. • They are also called Rational Numbers. • IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever. • Examples: π 2 3 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. A. 21 irrational B. 0 3 rational 0 =0 3 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. C. 4 0 not a real number Check It Out! Example 2 State if each number is rational, irrational, or not a real number. A. 23 23 is a whole number that is not a perfect square. irrational B. 9 0 undefined, so not a real number Rational Numbers on a Number Line To graph a set of numbers means to draw, or plot, the points named by those numbers on a number line. The number that corresponds to a point on a number line is called the coordinate of that point. Identify Coordinates on a Number Line Name the coordinates of the points graphed on the number line. The dots indicate each point on the graph. Answer: The coordinates are {–9, –7, –6, –3}. Identify Coordinates on a Number Line Name the coordinates of the points graphed on the number line. The bold arrow on the graph indicates that the graph continues infinitely in that direction. Answer: The coordinates are {11, 12, 13, 14, …}. Identify Coordinates on a Number Line Name the coordinates of the points graphed on each number line. a. Answer: {6, 9, 11, 12} b. Answer: {–0.5, 0, 0.5, 1, 1.5, …} Graph Numbers on a Number Line Graph Answer: . Graph Numbers on a Number Line Graph {–1.5, 0, 1.5, …}. Answer: Graph Numbers on a Number Line Graph {integers less than –6 or greater than or equal to 1}. Answer: Graph Numbers on a Number Line Graph each set of numbers. a. {–5, 2, 3, 5} Answer: b. Answer: Graph Numbers on a Number Line c. {integers less than or equal to –2 or greater than 4} Answer: Properties A property is something that is true for all situations. Four Properties 1. Distributive 2. Commutative 3. Associative 4. Identity properties of one and zero Distributive Property A(B + C) = AB + BC 4(3 + 5) = 4x3 + 4x5 Commutative Property of addition and multiplication Order doesn’t matter Ax B= B xA A+B = B +A Associative Property of multiplication and Addition Associative Property (a · b) · c = a · (b · c) Example: (6 · 4) · 3 = 6 · (4 · 3) Associative Property (a + b) + c = a + (b + c) Example: (6 + 4) + 3 = 6 + (4 + 3) Identity Properties If you add 0 to any number, the number stays the same. A + 0 = A or 5 + 0 = 5 If you multiply any number times 1, the number stays the same. A x 1 = A or 5 x 1 = 5 Example 1: Identifying Properties of Addition and Multiplication Name the property that is illustrated in each equation. A. (–4) 9 = 9 (–4) (–4) 9 = 9 (–4) The order of the numbers changed. Commutative Property of Multiplication B. The factors are grouped differently. Associative Property of Addition Example 2: Using the Commutative and Associate Properties Simplify each expression. Justify each step. 29 + 37 + 1 29 + 37 + 1 = 29 + 1 + 37 Commutative Property of Addition = (29 + 1) + 37 Associative Property of Addition = 30 + 37 Add. = 67 Exit Slip! Name the property that is illustrated in each equation. 1. (–3 + 1) + 2 = –3 + (1 + 2) 2. 6 y 7=6 ● 7 ● y Associative Property of Add. Commutative Property of Multiplication Simplify the expression. Justify each step. 3. 22 Write each product using the Distributive Property. Then simplify 4. 4(98) 392 5. 7(32) 224