2. REAL RATIONAL NUMBERS (as opposed to fake numbers?) and Properties Part 1 (introduction)

3. 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

4. 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

5. Essential Questions: How do you know if a number is a rational number? What are the properties used to evaluate rational numbers?

6. Two Kinds of Real Numbers Rational Numbers Irrational Numbers

7. 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

8. 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

9. Irrational numbersRational numbers Real Numbers Integers Whole numbers

10. 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

11. 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

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

13. 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

14. Properties A property is something that is true for all situations.

15. Four Properties 1.Distributive 2.Commutative 3.Associative 4.Identity properties of one and zero

17. Distributive Property A(B + C) = AB + BC 4(3 + 5) = 4x3 + 4x5

18. Commutative Property of addition and multiplication Order doesn’t matter A x B = B x A A + B = B + A

19. 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)

20. 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

21. 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.

22. 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 = 30 + 37 Associative Property of Addition = 67 Add.

23. 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 22 392 224