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

Set A collection of objects. Set Notation { } Natural numbers Counting numbers {1,2,3, …} Whole Numbers Natural numbers and 0. {0,1,2,3, …} Rational Number Integers Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …} A real number that can be expressed as a ratio of integers (fraction) Irrational Number Any real number that is not rational. Real Numbers All numbers associated with the number line. Vocabulary

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/ … -3/4

denominator numerator To write a fraction as a decimal, divide the numerator by the denominator. You can use long division. numerator denominator

9 11 The pattern repeats. 1 – –1 8 Additional Example 3A: Writing Fractions as Decimals Write the fraction as a decimal. The fraction is equivalent to the decimal A repeating decimal can be written with a bar over the digits that repeat. So … = 1.2. Writing Math _

This is a terminating decimal The remainder is – –6 0 –1 0 0 Additional Example 3B: Writing Fractions as Decimals Write the fraction as a decimal. The fraction is equivalent to the decimal

Irrational Numbers An irrational number is a number that cannot be written as a ratio of two integers. Irrational numbers written as decimals are non- terminating and non-repeating. Square roots of non-perfect “ squares ” Pi- īī 17

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution! 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.

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.

Irrational numbersRational numbers Real Numbers Integers Whole numbers

Whole numbers and their opposites. Natural Numbers - Natural counting numbers. 1, 2, 3, 4 … Whole Numbers - Natural counting numbers and zero. 0, 1, 2, 3 … Integers - … -3, -2, -1, 0, 1, 2, 3 … Integers, fractions, and decimals. Rational Numbers - Ex: -0.76, -6/13, 0.08, 2/3 Rational Numbers

Rational Numbers on a Number Line – 4 – 3 – 2 – |||||||||||||||||| Negative numbers Positive numbers Zero is neither negative nor positive Natural Numbers Whole Numbers Integers

Animal Reptile 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! Lizard Gecko Making Connections

Venn Diagram: Naturals, Wholes, Integers, Rationals Naturals Wholes Integers Rationals Real Numbers

Make a Venn Diagram that displays the following sets of numbers: Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Naturals 1, 2, 3... Wholes 0 Integers Rationals Irrationals Reals

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 IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever. Examples: π

State if each number is rational, irrational, or not a real number. 21 irrational 0303 rational 0303 = 0 Additional Example 2: Determining the Classification of All Numbers A. B.

not a real number Additional Example 2: Determining the Classification of All Numbers 4040 C. State if each number is rational, irrational, or not a real number.

23 is a whole number that is not a perfect square. 23 irrational 9090 undefined, so not a real number Check It Out! Example 2 A. B. State if each number is rational, irrational, or 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.

Example 1-1a 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

Example 1-1b 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

Example 1-1c Name the coordinates of the points graphed on each number line. a. b. Answer: {6, 9, 11, 12} Answer: {–0.5, 0, 0.5, 1, 1.5, … } Identify Coordinates on a Number Line

Example 1-2a Graph. Answer: Graph Numbers on a Number Line

Example 1-2b Graph {–1.5, 0, 1.5, … }. Answer: Graph Numbers on a Number Line

Example 1-2c Graph { integers less than –6 or greater than or equal to 1}. Answer: Graph Numbers on a Number Line

Example 1-2d Graph each set of numbers. a. {–5, 2, 3, 5} b. Answer: Graph Numbers on a Number Line

Example 1-2e c. { integers less than or equal to –2 or greater than 4} Answer: Graph Numbers on a Number Line

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 A x B = B x A 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 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) B. (–4)  9 = 9  (–4)The order of the numbers changed. Commutative Property of Multiplication Associative Property of Addition The factors are grouped differently.

Example 2: Using the Commutative and Associate Properties Simplify each expression. Justify each step = Commutative Property of Addition = (29 + 1) + 37 = Associative Property of Addition = 67 Add.

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 Simplify the expression. Justify each step. 3. Write each product using the Distributive Property. Then simplify 4. 4(98) 5. 7(32) Associative Property of Add. Commutative Property of Multiplication