MATH 113 Section 5.2: Fractions and Rational Numbers

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusA Brief History of Rational NumbersBecause rational numbers are a more complicated concept thatwhole numbers, they are a relatively recent innovation.Rational Number FactsEgyptians used fractions, but only “unit fractions” of the form1n. These were denoted by placing an oval over the symbolicrepresentation of the denominator.Greeks did not like non-whole numbers so they used ratiosinstead of fractions. For example, to denote the idea that25ths of a pie has been eaten, Greeks would say the ratio ofthe pie that was eaten to the pie that remained is 2 : 3.

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusSelecting an Appropriate ModelFractions are used in many different situations and some modelsare more appropriate than others depending on the situation.ExampleDetermine which model is most useful for each situations.Fractions and MeasureI went 4 5th of the total distance.Fractions as QuotientsI have 4 cookies which I want to split between 5 people.Fractions as an Operationths of your class of 20 people missed Friday’s class.45Fractions as Ratiosths of your ideas are good ones.45What properties or ideas are common to each situation?

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusFractions and UnitsIn many situations we model fractions by representing the unit(number 1) as a whole object and then dividing it into equalpieces. However, there are situations when it is more convenient touse some other number as the whole and divide it into pieces.ExampleUse the set model below to represent 2 3rd. Justify this in twodifferent ways.= 2

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusUsing Fraction ModelsFraction models are useful for more than just representing afraction using pictures. They can also be used to help formulatethe solution to a story problem.ExampleYou paid $6 for 1 1 2pound of potato salad. What is the cost of onepound of potato salad?While there are several ways to approach this problem, using anarea model to represent a whole of 1 1 2pounds we can find thesolution.If you do not wish to draw a picture, you can solve this problemusing ratios.

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusMore Story ProblemsExampleSally has some free time to split between exercising and reading.She spends 1 6of her time exercising. If she spends 2 hours morereading than she did exercising, how much free time did she have?ExampleA thief stole some apples from an orchard. On his way out, he met3 watchmen, one after the other. He gave each watchman 1 2of theapples that he had at the time and two more besides. He escapedwith one apple. How many apples did he originally steal?ExampleA business man spends 1 3of his travel budget on food. He splitsthis evenly between groceries and dining out. If he spends $20 oneach of the 6 meals he eats out, what is his travel budget.

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusDefining EquivalenceAt the beginning of the class, we commented that there are manydifferent ways to represent a single number using fractions.Equivalent FractionsFractions are called equivalent if they represent the same value.ExampleIn each example, use two different models to show that thefractions are equivalent.1 34 = 9122 32 = 1510ProcedureWhat is the procedure which we follow to determine if 3 4 and 912are equivalent?

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusSimplest FormSince there are so many different ways to write a single rationalnumber, it is convenient to have a single preferred form.Simplest FormA fraction is in simplest form (also called reduced) if thenumerator and denominator are relatively prime.Reducing to Simplest FormThere are several methods to reduce fractions to simplest form.ExampleGuess and CheckRecognizing FactorsPrime FactorizationReduce 5632to simplest form.

Introduction to Rational Numbers Modeling Rational Numbers Fraction Models and Story Problems Equivalent Fractions ConclusImportant ConceptsThings to Remember from Section 5.21 The definition of rational numbers and how they relate toother number sets2 Three models for fractions:Area modelLinear modelSet model3 Using fraction models to solve story problems4 Identifying equivalent fractions