Chapter 6: Rational Number Operations and Properties
6.1Rational Number Ideas and Symbols
6.1.1.Modeling Rational Numbers
6.1.1.1.Used to describe a quantity between 0 and 1
6.1.1.1.1.identify the whole representing the numeral 1
6.1.1.1.2.separate the whole into equal parts
6.1.1.1.3.use an ordered pair of numbers to describe the portion of the whole under consideration
6.1.1.2.Identifying the whole and separating it into equal parts
6.1.1.2.1.Egg carton fractions
6.1.1.2.1.1.
6.1.1.2.1.2.
6.1.1.2.2.Integer Rods
6.1.1.2.2.1.
6.1.1.2.2.2.
6.1.1.2.3.Make your own fraction kit
6.1.1.2.3.1.
6.1.1.2.4.NCIS Mountain demonstration
6.1.1.3.Using two integers to describe part of a whole
6.1.1.3.1.Need more language to describe part-whole relationship
6.1.1.3.2.number of pieces of interest vs. number of pieces found in the original whole
6.1.1.3.3.Your turn p. 280
6.1.1.3.4.Do the practice and the reflect
6.1.2.Defining Rational Numbers
6.1.2.1.Description of a rational number: A rational number is the relationship represented by an infinite set of ordered pairs, each of which describes the same quantity – (1,2); (2,4); (3,6); (6,12); etc.
6.1.3.Using Fractions to Represent Rational Numbers
6.1.3.1.Fractions and Equivalent Fractions
6.1.3.1.1.Definition of a fraction: A fraction is a symbol, , where a and b are numbers and b 0. Here, a is the numerator of the fraction and b is the denominator of the fraction
6.1.3.1.2.Proper fraction: when the numerator of the fraction is less than the denominator of the fraction and both the numerator and the denominator are integers
6.1.3.1.3.Improper fraction: when the numerator of the fraction is greater than the denominator of the fraction (fractions with non-integers in the numerator or denominators are also improper)
6.1.3.1.4.Definition of equivalent fractions : Two fractions, and , are equivalent fractions if and only if ad = bc
6.1.3.2.Using fractions to represent rational numbers
6.1.3.2.1.every rational number can be represented by an integer in the numerator and the denominator
6.1.3.2.2.sometimes rational numbers are represented by non-integers
6.1.4.Properties of Fractions
6.1.4.1.The Fundamental Law of Fractions: Given a fraction and a number c 0,
6.1.4.2.Fractions in simplest form
6.1.4.2.1.Description of the simplest form of a fraction: a fraction representing a rational number is in simplest form when the numerator and the denominator are both integers that are relatively prime and the denominator is greater than zero.
6.1.4.2.1.1.Finding equivalent fractions on the number line
6.1.4.2.1.2.Folding paper
6.1.4.2.1.3.Using a calculator
6.1.4.2.1.4.Using Integer rods
6.1.4.3.Your turn p. 286
6.1.4.4.Do the practice and the reflect
6.1.5.Using Decimals to Represent Rational Numbers
6.1.5.1.Decimals
6.1.5.1.1.Description of a decimal: A decimal is a symbol that uses a base-ten place-value system with tenths and multiples of tenths to represent rational numbers
6.1.5.1.2.decimal point divides the decimal portion of the number from the whole number portion of the number
6.1.5.1.3.Using base ten blocks to explore decimals – see p. 287-8
6.1.5.2.Expanded notation
6.1.5.2.1.
6.1.5.3.Writing a decimal for a fraction
6.1.5.3.1. - divide 3 by 4 to get the decimal equivalent
6.1.5.3.2. - use the Fundamental Law of Fractions
6.1.5.4.Your turn p. 288
6.1.5.5.Do the practice and the reflect
6.1.5.5.1.terminating decimals – rational numbers that have a finite number of decimal places when written as decimals:
6.1.5.5.2.repeating decimals – rational numbers that have an infinite number of decimal places filled by the same number or a fixed number of digits repeated over an infinite number of decimal places:
6.1.5.5.2.1.
6.1.5.5.2.2.
6.1.5.5.2.3.Generalization about decimals for rational numbers: Every rational number can be expressed as a terminating or a repeating decimal
6.1.5.6.Scientific notation
6.1.5.6.1.Description of scientific notation: A rational number is expressed in scientific notation when it is written as a product where one factor is a decimal grater than or equal to 1 and less than 10 and the other factor is a power of 10
6.1.5.6.1.1.0.1234 = 1.234 x 10-1
6.1.5.6.1.2.0.0000001234 = 1.234 x 10-7
6.1.5.6.2.Your turn p. 286
6.1.5.6.3.Do the practice and the reflect
6.1.6.Connecting Rational Numbers to Whole Numbers, Integers, and Other Numbers
6.1.6.1.a = a 1 =
6.1.6.2.The set of rational numbers is denoted by Q
6.1.6.3.The set of real numbers is denoted by R
6.1.6.4.R(Q(Z(W(N)))) or N W Z Q R – All of the natural numbers are contained within the whole numbers which are contained within the integers which are contained within the rational numbers which are contained within the real numbers.
6.1.6.5.The real numbers are composed of the rational numbers and the irrational numbers (, 1.23223222322223…, , etc.)
6.1.6.6.The number line is dense – there are no holes in the number line
6.1.6.6.1.Between any pair of rational numbers is an irrational number
6.1.6.6.2.Between any pair of irrational numbers there is a rational number
6.1.7.Problems and Exercises p. 292
6.1.7.1.Home work: 1-3, 7, 8ac, 9abc, 10c, 11, 12, 14, 19, 22