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§6.1 Fractions 3/9/17 Today We’ll Discuss What are fractions, improper fractions, proper fractions, and mixed numbers? What are models for fractions? How do we recognize equivalent fractions? How do we use the real number line to graph and order fractions? Fractions Definition: A fraction is a number of the form numerator denominator where a is called the numerator and b (b ≠ 0) is called the denominator. Fractions Verbiage: Fractions are read as the numerator followed by the denominator with the appropriate “half”, “third”, “th” 1 2 one-half (not “one over two”) 2 3 two-thirds (not “two over three”) 2 4 three-fourths (not “three over four”) Area Models for Fractions In an area model, the entire geometric shape represents the whole (denominator), while certain highlighted pieces represent the fractional part (numerator). Examples Set Models for Fractions In a set model, the largest set represents the whole (denominator), while a subset represents the fractional part (numerator). Example Explain how you could use the set model to represent three different fractions. Linear Models for Fractions In a linear model, the entire length of a line/rod/strip represents the whole (denominator), while the smaller parts are the fractional parts (numerator). Fraction Strips Interlocking Fraction Bars Cuisenaire Rods upload.wikimedia.org/wikipedia/commons/8/85/Cuisenaire_staircase.JPG Example Design a quick linear model activity that shows students Proper and Improper Fractions Definition: A proper fraction is a fraction in which the numerator is less than the denominator. (They are between 0 and 1) Definition: An improper fraction is a fraction in which the numerator is greater than or equal to the denominator. (They are greater than 1) Examples 2 3 1 2 7 3 , , , , , 3 2 5 4 2 3 Mixed Numbers An improper fraction not equal to 1 can be rewritten as a mixed number. (Note: Division with remainder gives exactly the mixed number!) Examples Convert the following improper fractions to mixed numbers. a) b) Examples Represent the following improper fractions with area models. Then rewrite each as a mixed number. a) b) Examples Represent the following improper fraction using a set model. Then rewrite the fraction as a mixed number. Equivalent Fractions Two fractions are equivalent if they represent the same quantity. We use models first to help students understand the concept before talking about “mathematical” methods. Example On each school day, you start with a whole round of cheddar cheese and cut it into pieces of equal size. The amount you ate each day is highlighted below. On which days do you eat the same amount of cheese? Simplest Form A fraction is in simplest form when its numerator and denominator have no common factors other than 1. Generally, we simplify fractions by identifying and dividing the GCF of the numerator and denominator. Example Simplify the following fractions. a) b) c) Two Tests for Equivalent Fractions To see if two fractions equivalent we can… and are 1)Compare Simplified Fractions Write and in simplest form. If the simplest form for both is the same, they are equivalent. Two Tests for Equivalent Fractions To see if two fractions equivalent we can… and are 2)Compare Cross Products Take the “cross product” of and by multiplying opposite sides of the fractions. If ad = bc, the original fractions are equivalent. Example Determine whether or not the following fractions are equivalent. a) b) Ordering Fractions Using a Number Line To order any two fractions, graph them on a number line. The fraction that is to the left is less than the fraction that is to the right. Ordering Tips and Tricks Let students know that… 1) A fraction with the same denominator but lesser numerator than another fraction is less than the other fraction. 2) A fraction with the same numerator but lesser denominator than another fraction is greater than the other fraction. Ordering Tips and Tricks Benchmarks can be used to graph certain proper fractions. 1) Numerator closer to 0 → benchmark 0 2) Numerator close to half of denominator → benchmark ½ 3) Numerator closer to denominator → benchmark 1 Example Order the following fractions. a) b) Homework #13 - §6.1 Pages 204-207 #4,6,10,12,17,19,22,24,32,39