<p> P r o p o r t i o n a l R e a s o n i n g</p><p>Proportional reasoning is one of the four types of mathematical reasoning that we explored in Chapter 1 of the O’Daffer text (p. 29-30). O’Daffer states, “Proportional reasoning involves drawing conclusions or solving problems with either the formal or informal use of proportions” (2002, p. 29). When two rational expressions are equal to one another, they form a proportion. So the problem with solving and working with proportions is two fold: 1) rational numbers and 2) rational number equivalencies. Rational Numbers</p><p>Rational numbers are discussed in chapter 6 of the O’Daffer text (p. 276-343).  Modeling Q numbers  Rational Numbers a o ; a  , b   , b  0 b o Why can’t b be zero? o Dividing by zero is undefined  Expanded notation: 100 . 10-1 10-2 10-3 …  Decimals o Terminating: 5, 2.5, 3.1254689, … o Repeating: 2.3 ; 0.56 ; 23.18568; … o Irrational numbers: , e, 2 ; 3 ; 5 ; 6 ; non - square number , 1.2121121112…, …  Decimal  Rational  Rational  Decimal Rational Number Equivalents  Proportions</p><p>Proportional reasoning is discussed in chapter 7 of the O’Daffer text (p. 344-389).  Equivalent fractions a c o  iff ad = bc ; a  , b  , c  , d  , b  0 , d  0 b d 4 ? 20 o  example 7 35 5 ? 30 o  non-example 9 45  Fundamental Law of Fractions a a ac o , a  , b  , c   , b  0 , c  0, exists, then  b b bc</p><p>See Also  DAR – Q – Add, Subt, Mult, Divide  Power Points on Equivalents and Integer Rod Operations  Quizzes “Money makes the world go ‘round” is a common saying. Elementary students come to school with a wide variety of information and miss-information regarding how to count and do problems involving money. Today we are going to take a look at the links between money, decimal fractions, and rational numbers. We will explore some of the historical ideas surrounding money and some of the mathematical ideas surrounding how we keep money safe in banks and on the internet.</p><p> Equivalencies – money, decimal fractions, and rational numbers o Why is a penny called one cent? o Why is five cents called a nickel? o Why is ten cents called a dime? o Why is twenty-five cents called a quarter? o Why is fifty cents called a half? o Coin money . Use whole numbers if you say “cents” . Use decimal equivalents if you say “dollars” . Can you use both at the same time? Why or why not? . Practice some equivalents  $0.23  $0.593  27¢  125¢ o Paper money . Use whole numbers if you say “dollars” . Multiply dollar amount by 100 if say “cents” . Practice some equivalents  $1  $5  $23.899  234¢  20589¢ o Making change . How many different ways can you make 1¢? . How many different ways can you make 5¢? . How many different ways can you make 10¢? . How many different ways can you make 25¢? . How many different ways can you make 50¢? . How many different ways can you make 37¢? . How many different ways can you make 48¢? . How many different ways can you make $1 using paper money? . How many different ways can you make $5 using paper money? . How many different ways can you make $10 using paper money? . How many different ways can you make $20 using paper money? . How many different ways can you make $18 using paper money? Once fundamental ideas have been explored then more complicated schemes can be addressed. Remember – we teach concepts when errors occur in procedures, the underlying problem is a misunderstanding of the concepts. Straighten out the misconception about the concept and the student owns the material – re-teach the process until the student finally memorizes it and they are only borrowing the knowledge.</p><p>Take a look at “Math, Money, and You” and work through the exercises. There is some encryption and use of modulo to see how money is protected on the internet. Discuss and do exercises one through five. You may try exercises six and seven if you have caught the coding theory bug! </p><p> Be able to encrypt a message using modulo  Be able to decrypt a message using modulo </p>