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REAL RATIONAL NUMBERS

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:

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REAL RATIONAL NUMBERS

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

  2. 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

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

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

  5. Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers

  6. 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

  7. 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

  8. Real Numbers Rational numbers Irrational numbers Integers Whole numbers

  9. Rational Numbers Natural counting numbers. Natural 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. -0.76, -6/13, 0.08, 2/3 Ex:

  10. 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

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

  12. 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 Reminder

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

  14. Four Properties • Distributive • Commutative • Associative • Identity properties of one and zero

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

  16. Commutative Propertyof addition and multiplication Order doesn’t matter A x B = B x A A + B = B + A

  17. 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)

  18. 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

  19. 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 The factors are grouped differently. Associative Property of Addition

  20. Example 2: Using the Commutative and Associate Properties Simplify each expression. Justify each step. 29 + 37 + 1 Commutative Property of Addition 29 + 37 + 1 = 29 + 1 + 37 Associative Property of Addition = (29 + 1) + 37 = 30 + 37 Add. = 67

  21. 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

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