Chapter 6 - Fractions. 6.1 The Set of Fractions
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1 Chpter 6 - Frctions 6.1 The Set of Frctions
2 The Concept of Frction Descrie the following:
3 Frction 1. Frctions re used s numerls to indicte the numer of prts of whole the frction / is clled the numertor is clled the denomintor
4 Frctions A frction is numer tht cn e represented y n ordered pir of whole numers where 0 In set nottion, the set of frctions is F = { nd re whole numers, 0}
5 Frction Attriutes - 1 When considering frction s numer, the focus is on the reltive mount ¼ descries the reltive mount without regrd to size, shpe, orienttion, etc Focus on numerousness
6 Frction Attriutes - 2 When considering frction s numerl representing prt-to-whole reltionship, mny different numerls cn e used
7 Frction Attriutes Equivlent frctions two frctions tht represent the sme reltive mount
8 Definition: Frction Equlity c Let nd e ny frctions. d c Then = if nd only if d d = c
9 Frction Equlity Two frctions re equl frctions if nd only if their cross-products, tht is, products d nd c otined y cross-multipliction, re equl. Is 3/12 = 4/16?
10 Theorem Let e ny frction nd n nonzero whole numer. Then: n = = n n n
11 Frctions with numertors greter thn or equl to their denomintors 1. Frctions with denomintors nd numertors equl e.g. 2/2, 3/3, etc represent the whole numer 1 2. Frctions where the numertors re greter thn the denomintors re clled improper frctions e.g. 7/2, 8/5, etc
12 Ordering Frctions 1. Use numer strip
13 Ordering frctions 2. Use frction numer line
14 Definition Less Thn for Frctions Let nd e ny frctions. c Then if nd only if c c c < <
15 Theorem Cross-Multipliction of Frction Inequlity Let nd e ny frctions. c Then if nd only if d c d < d < c
16 whole-numer line there re gps etween the whole numers on the whole numer line
17 whole-numer line There is frction etween ny two frctions There re infinitely mny frctions etween ny two different frctions
18 Theorem Let nd e ny frctions c d < c d Where Then c < + + d < c d This is known s the density property of frctions
19 Theorem Cross-Multipliction of Frction Inequlity Let nd e ny frctions. c Then if nd only if d c d < d < c
20 Section 6.2 Frctions: Addition nd Sutrction
21 Addition Properties Addition of frctions is n extension of whole-numer ddition nd cn e motivted using models Find the sum of 1/5 nd 3/5
22 Definition Addition of Frctions with Common Denomintors Let nd e ny frctions. Then c c c + = +
23 Addition of Frctions with Unlike Denomintors c Let nd e ny frctions. d Then c + = d d + c d
24 Alterntive method 17/15 + 5/12 =? Use the LCM (lest common Multiple of the denomintors) This is known s the LCD (lest common denomintor)
25 Properties to simplify frction ddition: Closure Property for Frction Addition: The sum of two frctions is frction Commuttive Property for Frction Addition c Let nd e ny Frctions. Then c c + = +
26 Properties to simplify frction ddition: Associtive Property for Frction Addition Let,, nd e ny fctions. Then c e c e c e + + = + +
27 Properties to simplify frction ddition: Additive Identity Property for Frction Addition Let e ny frction. There is 0 unique frction,, such tht = +
28 Definition: Sutrction of Frctions with Common Denomintors: c Let nd e ny frctions with Then c = c c
29 Sutrction of Frctions with Unlike Denomintors Sutrction is done y first finding common denomintors nd then sutrcting c d Let nd e ny frctions where Then c = d d d c c d
30 Section 6.3 Frctions: Multipliction nd Division
31 Multipliction nd Its Properties Cse 1: A Whole Numer Times Frction 3 x ¼=? Cse 2: A frction Times Whole Numer ½x 6 =?
32 Multipliction nd Its Properties Cse 3: A Frction of Frction 1/3 x 5/7 =?
33 Multipliction of Frctions Let nd e ny frctions. Then c d c = d c d
34 Multipliction of Frctions Exmples: 2/3* 5/13 =? 3/4 * 28/15 =? 2 1/3 * 7 8 2/5 =?
35 Multipliction of Frctions Compute nd simplify: 3/4 * 28/15 =? 18/13 * 39/72 =?
36 Properties of Frction Multipliction c Let, e ny frctions. d nd e f Closure Property for Frction Multipliction: The product of two frctions is frction. Commuttive Property for Frction Multipliction c c = d d
37 Properties of Frction Multipliction Associtive Property for Frction Multipliction c e c e = d f d f Multiplictive Identity Property for Frction Multipliction 1= = 1
38 Properties of Frction Multipliction Multiplictive Inverse Property for Frction Multipliction: For every nonzero frction, there is unique frction such tht =1
39 Distriutive Property of Frction Multipliction over Addition c Let, e ny frctions. d nd e f Then c d + e f c = + d e f
40 Division of Frctions We need to view division of frctions s n extension of whole-numer division This pproch provides meningful wy of lerning frction division This is n ttempt to deprt from the procedure of invert nd multiply
41 Division of Frctions with Common Denomintors c Let nd e ny frctions with d c 0 Then c = c
42 Division of Frctions with common denomintors Find the following quotients 12/13 4/13 =? 6/17 3/17 =? 16/19 2/19 =?
43 Division of Frctions with different denomintors To divide frctions with different denomintors, we cn rewrite the frctions so tt they hve the sme denomintor: c d = d d c d = d c
44 To divide frctions with different denomintors, we cn rewrite the frctions so tt they hve the sme denomintor: = = 27 35
45 Division of Frctions with different denomintors Another interprettion uses the missingfctor pproch nd refers directly to the multipliction of frctions Find: Then = 40 8 = = 3 5
46 Division of Frctions with Unlike Denomintors Invert the Divisor nd Multiply c d Let nd e ny frctions with Then c d = d c c 0
47 Visul wy to understnd invert nd multiply: 3 1/2 =?
48 Visul wy to understnd invert nd multiply: 4 2/3 =?
49 Approches to Division of frctions 1. The common-denomintor pproch 17/11 4/11 =? 17/11 4/11 = 17/4
50 Approches to Division of frctions 2. The divide-the-numertors-nddenomintors pproch 5/25 2/5 =? = = 3 5
51 Approches to Division of frctions 3. The invert-the-divisor-nd-multiply pproch 3/4 5/7 =? = = 21 20
52 For ll whole numers nd, = Find 17 6 using frctions = =
Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
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