# Multiplying and Dividing Fractions

 6.3 Multiplying and Dividing Fractions 6.3.1. Modeling multiplication of fractions 6.3.1.1. Repeated addition can be used when we have a whole number times a rational number: 6.3.1.2. Joining of equal-sized groups can be used when we have a mixed number times a rational number :see figure 6.12, p. 326 6.3.1.3. Area model can also be used for multiplying a mixed number times a rational number:see figure 6.13, p. 326 6.3.1.4. Additionally the area model can be used to show multiplication of a rational number times a rational number. 6.3.1.5. Your turn p. 327: Do the practice and reflect 6.3.2. Multiplying fractions 6.3.2.1. Fraction with a numerator of one is called a unit fraction 6.3.2.2. Generalization about multiplying rational numbers represented by unit fractions: For rational numbers 6.3.2.3. Procedure for multiplying rational numbers in fraction form: For rational numbers 6.3.2.4. Your turn p. 329: Do the practice and reflect 6.3.2.5. Class demonstration using paper folding to show multiplication of rational numbers: 6.3.2.5.1.half OF a third 6.3.2.5.2.two-thirds OF three-fourths 6.3.2.5.3.half OF two-thirds 6.3.2.5.4.three-fourths OF four-fifths 6.3.2.6. Integer rod steps (always use least number of rods possible) 6.3.2.7. Class demonstration using integer rods to show multiplication of rational numbers: 6.3.2.7.1. 6.3.2.7.2. 6.3.2.7.3. 6.3.2.7.4. 6.3.3. Properties of rational number multiplication 6.3.3.1. Basic properties of rational numbers 6.3.3.1.1. Multiplicative inverse ( reciprocal ) analogous to additive inverse property 6.3.3.2. Your turn p. 330: Do the practice and reflect 6.3.3.3. Basic properties for multiplication of rational numbers • Closure property: For rational numbers is a unique rational number • Identity property: A unique rational number, 1, exists such that ; 1 is the multiplicative identity element • Zero property : For each rational number • Commutative property: For rational numbers • Associative property: For rational numbers • Distributive property: For rational numbers • Multiplicative inverse: For every nonzero rational number b/a , a unique rational number, a/b , exists such that 6.3.3.4. Property for multiplying an integer by a unit fraction: For any integer a and any unit fraction 6.3.3.5. Using the properties to verify (prove) the procedure for multiplication of rational numbers: see p. 331 6.3.4. Modeling Division of fractions 6.3.4.1. used to separate a quantity into groups of the same size 6.3.4.2. no remainders in division of rational numbers 6.3.4.3. Partition model – fig. 6.16 p. 332 6.3.4.4. Measurement model – fig.6.17 p. 332 6.3.4.5. Integer rod steps (always use least number of rods possible) 6.3.4.6. Class demonstration using integer rods to show division of rational numbers: 6.3.4.6.1. 6.3.4.6.2. 6.3.4.6.3. 6.3.4.6.4. 3/2 6.3.4.6.5. 6.3.4.6.6. 6.3.4.6.7. 3/10 6.3.5. Definition and properties of rational number division 6.3.5.1. Definition of rational number division in terms of multiplication : for rational numbersif and only if e/f is a unique rational number such that 6.3.5.2. Closure property of division for nonzero rational numbers: For nonzero rational numbers is a unique nonzero rational number 6.3.6. Dividing fractions 6.3.6.1. Procedure for dividing fractions – multiplying by the reciprocal method : for rational numbers a/b and c/d , where c, b, and d ≠ 0, 6.3.6.2. Procedure for dividing fractions – common denominator method : for rational numbers a/b and c/d , where c ≠ 0, 6.3.6.3. Procedure for dividing fractions – complex fraction method: for rational numbers a/b and c/d , where c ≠ 0, 6.3.6.4. Procedure for dividing fractions – missing factor method : for rational numbers a/b and c/d , where c, b, and d ≠ 0,, where To find f, 6.3.7. Estimation strategies 6.3.7.1. rounding 6.3.7.2. front-end estimation 6.3.7.3. substituting compatible numbers 6.3.7.4. Where does the decimal point go? 6.3.7.4.1. 6.25 x 0.89 = 55625 6.3.7.4.2. 4.3 x 0.49 = 2107 6.3.7.4.3. 5.75 x 1.39 = 79925 6.3.8. Problems and Exercises p. 340 6.3.8.1. Home work: 1, 6, 7, 8, 9ac, 10, 14, 15, 16, 17, 18
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