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學好基礎數學的第一步,就是要有正確扎實的基本觀念,良好觀念的建立原則─先理解再記憶。 在理解的過程中,因為數學的本質是抽象的,所以會造成學習上的困難。為了讓同學容易理解抽象的數學觀念,葛倫老師借助數位工具的力量,將原本抽象難懂的數學概念,轉化為具體有趣的多媒體,透過筆跡、圖片、聲音等動畫影音呈現,搭配iPad直覺易用的操作介面,讓同學以最有效率、最具創意的未來學習方式來學好基礎數學! ※本書為試閱版,若要下載完整版請前往 ibooks.Liveism.com

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  • PREFACE

    m

    Preface

    7{is@O9Ea`Sq~q("%ULOq}XX

    fSG..*&("%9(--)0:?7E|(3&(3&(0+$-+

  • 1-1

    1.

    2.

    3.

    4.

    5.

    1-1

    2

  • 1-1

    1. x + y2x 3y + 45 6a + 7b + 8a 9b

    3

    1.1

  • 2. (1)

    (4x 5y + 6) (x 2y + 3)= 4x 5y + 6 x + 2y 3= (4x x)+ (5y + 2y)+ (6 3)= 3x + (3y)+ 3= 3x 3y + 3

    (2)

    4x 5y + 6

    ) x 2y + 33x 3y + 3

    4

    1-1

  • 3. (1) x = 5y = 4

    3x 2y +1= 35 2 4+1= 158+1= 8

    (2) x = 3y = 2

    3x 2y +1= 3 3 2 (2)+1= 9 (4)+1= 9+ 4+1= 14

    5

    1-1

  • 4. 2x 3y = 45a = 6b + 7

    6

    1-1

    Check Answer

    Question 1 of 2

    A. 2x 3x + 4

    B. 2x 3y = 4

    C. 2x 3y + 4

    D. x2 3y + 4

  • 5. xy xy x = 4y = 6 3x 2y = 0 x = 4y = 6

    = 3x 2y = 3 4 2 6 = 12 12 = 0 = x = 4y = 6 3x 2y = 0

    x = 4y = 6 3x 2y = 0

    (4,6) {x = 4y = 6

    7

    1-1

  • 1.

    2.

    3.

    4.

    5.

    1-2

    81-2

  • 1.

    {2x 3y = 45x + 6y = 7

    9

    1-2

  • 2. xy xy

    x = 2y = 1 {3x + 2y = 44x y = 9 x = 2y = 1 3x + 2y = 3 2 + 2 ( 1) = 6 + ( 2) = 4

    x = 2y = 1 3x + 2y = 4

    x = 2y = 1 4x y = 4 2 ( 1) = 8 + 1 = 9

    x = 2y = 1 4x y = 9

    {x = 2y = 1 {3x + 2y = 44x y = 9

    10

    1-2

  • 3.

    {x = 3y2x y = 5 x = 3y 2 (3y) y = 5

    6y y = 5

    5y = 5

    y = 1

    y = 1 x = 3 1 = 3

    x = 3 y = 1

    11

    1-2

  • 4.

    {3x + 2y = 45x 2y = 12 (3x + 2y) + (5x 2y) = 4 + 12

    3x + 2y + 5x 2y = 16

    8x = 16 x = 2

    x = 2 3 2 + 2y = 4

    6 + 2y = 4

    2y = 4 6 = 2 y = 1

    x = 2 y = 1

    12

    1-2

  • 5.

    13

    1-2

    {a1x + b1y = c1a2x + b2y = c2

    a1a2 b1

    b2a1a2

    = b1b2

    = c1c2

    a1a2

    = b1b2 c1

    c2

    {3x 2y = 45x 4y = 6{3x 2y = 46x 4y = 8{3x 2y = 46x 4y = 5

  • 1.

    2.

    1-3

    141-3

  • 1. (1) x y (2) (3) (4) (5) 510201405 x 10 y

    {x + y = 205x + 10y = 140 {x + y = 20x + 2y = 28 y = 8 y = 8 x + 8 = 20 x = 12

    512108

    15

    1-3

  • 2. 35 x y

    {x = 3yx y = 5

    3y y = 5 2y = 5 y = 52

    y = 52 x = 3 52 =

    152

    x = 152 y =52

    16

    1-3

  • 1.

    2.

    3.

    2-1

    182-1

  • 1. (1) x x (2) y y (3) x y O

    O(0,0)

    19

    2-1

    1

    1

    Ox

    y

    x

    y

    x y

  • 2. (m,n) P P (m,n) P(m,n)(1) m P x ()

    n P y ()(2) P x |n |

    P y |m |(3) x (a,0)

    y (0,b) (0,0)

    20

    2-1

  • 3. x y x y x y x y x y x y

    21

    2-1

  • 22

    2-1

    2.1

    Check Answer

    11

    2

    2

    3

    3

    4

    4

    5

    5

    11

    2

    2

    3

    3

    4

    4

    5

    5

    Ox

    y

    R(4,3)

    R(4,3)

    A(4,0)

    A(4,0)

    B(0,3)

    B(0,3)

    S(-4,-3)

    S(-4,-3)

    C(-4,0)

    C(-4,0)

    D(0,-3)

    D(0,-3)

  • 1.

    2.

    3.

    4. GeoGebra

    5.

    6. x

    7. y

    8.

    2-2

    232-2

  • 1. xy xy xy (x, y) y = 2x + 1

    x -3 -2 -1 0 1 2 y -5 -3 -1 1 3 5

    24

    2-2

  • 2. x 0 y 0 2x 3y = 6 (1) 2x 3y = 6

    x 3 0y 0 -2

    (2) (3,0) (0, 2)

    (3) (3,0) (0, 2) 2x 3y = 6

    25

    2-2

  • 3.

    26

    2-2

    y = ax + b

    (a 0)

    a > 0b > 0

    y = ax + b

    (a 0)a > 0b = 0

    y = ax + b

    (a 0)

    a > 0b < 0

    x

    y

    11O

    x

    y

    11

    O

    x

    y

    11O

  • 27

    2-2

    y = ax + b

    (a 0)

    a < 0b > 0

    y = ax + b

    (a 0)a < 0b = 0

    y = ax + b

    (a 0)

    a < 0b < 0

    x

    y

    11O

    x

    y

    11O

    x

    y

    11O

  • x

    28

    2-2

    x y = b

    b > 0

    x y = b b = 0 x y = b

    b < 0

    x

    y

    11O

    x

    y

    11O

    x

    y

    11O

  • y

    29

    2-2

    y x = k

    k > 0

    y x = k k = 0 y x = k

    k < 0

    x

    y

    11O

    x

    y

    11O

    x

    y

    11O

  • 4. GeoGebra

    30

    2-2

  • 31

    2-2

    2.1 GeoGebra y = ax+b

    a,b

  • x y

    32

    2-2

  • 33

    2-2

    2.2 GeoGebra y = b x = k

    b,k

  • 5. (p, q)(r, s) y = ax + b

    (p, q)(r, s) y = ax + b {q = ap + bs = ar + b ab ab y = ax + b (p, q)(r, s)

    (3,1)(2, 1) y = ax + b

    (3,1)(2, 1) y = ax + b

    {1 = 3a + b1 = 2a + b a = 2b = 5

    y = 2x 5

    34

    2-2

  • 6. x

    x (m, n) y = b (m, n) y = b b b y = b (m, n) x x (3,2) y = b

    (3,2) y = b b = 2

    y = 2

    35

    2-2

  • 7. y

    y (t, u) x = k (t, u) x = k k k x = k (t, u) y y (4,5) x = k

    (4,5) x = k k = 4

    x = 4

    36

    2-2

  • 8.

    {x + y = 2x 2y = 1

    x + y = 2

    x 2 0y 0 2

    x 2y = 1

    x -1 0y 0 1/2

    (1,1) x = 1y = 1 {x + y = 2x 2y = 1

    37

    x

    y

    ( , )1 1

    ( , )2 0

    ( , )0 2

    1

    1O

    ( , )0 12

    ( , )1 0

    x y+ =2

    x y =2 1

    2-2

  • {x + y = 22x + 2y = 4 x + y = 2

    x 2 0y 0 2

    2x + 2y = 4

    x 2 0y 0 2

    {x + y = 22x + 2y = 4

    38

    x

    y

    ( , )2 0

    ( , )0 2

    1

    1O

    x y+ =2

    2 2 4x y+ =

    2-2

  • {x + y = 22x + 2y = 6 x + y = 2

    x 2 0y 0 2

    2x + 2y = 6

    x 3 0y 0 3

    {x + y = 22x + 2y = 6

    39

    x

    y

    ( , )2 0( , )3 0

    ( , )0 2

    ( , )0 3

    11

    O

    x y+ =2

    2 2 6x y+ =

    2-2

  • 40

    2-2

  • 1233-1490(Galleria dell' Accademia)wikipedia.org

  • 1.

    2.

    3.

    4.

    5.

    6.

    7.

    8.

    3-1

    423-1

  • 1. a b a : b a b a b

    a : b a b = ab(b 0)

    1.5:2 34

    1.5:2 34 = 1.5 234 =

    32

    114 =

    32

    411 =

    611

    43

    3-1

  • 2.

    3:5 3 3 3:5 3:5 3 5 3:5

    1:2 1 1 1:20001:22000 2000 1:2000

    33301

    330:3 330 110 330:3 3 1110:1

    1110

    44

    3-1

    110:1

    1110

  • 3.

    a : b (b 0) c : d (d 0) ab

    cd

    a : b = c : d

    1 : 2 123 : 6 36 =

    12

    1 : 2 = 3 : 6

    b 0m 0 (1) a:b = (a m): (b m)(2) a:b = (a m): (b m) 2:4 = (2 2): (4 2) = 4:8 2:4 = (2 2): (4 2) = 1:2

    45

    3-1

  • 4. ab (b 0)

    1 ab

    a : b

    1:2 = 12( 7):11 =711

    12

    711

    1 : 2 (7) : 11

    3:6 = 36 =12( 4): ( 8) =

    48 =

    12

    36

    48

    3 : 6 (4) : (8)

    46

    3-1

  • 5. 0 0

    (1)ab ab

    23

    23 2 23 3

    23

    23

    (2) ab

    ab

    5

    5 5 5 1 51

    5147

    3-1

  • (3)ab

    aa + b

    b

    a + b

    2 : 3

    2 : 3

    22 + 3 =

    25

    3

    2 + 3 =35

    48

    3-1

  • 6.

    a : b (b 0) c : d (d 0) ab= c

    d a : b c : d

    a : b = c : d b c a d

    a:b = c:d ad = bc(b 0, d 0)

    a:b = c:d(b 0, d 0) ab= c

    d

    bd

    ab bd = c

    d bd ad = bc

    3 : 4 = 5 : x x 3 : 4 = 5 : x 3x = 5 4 3x = 20

    x = 203

    49

    3-1

  • 7. abcd ad = bc (1) a : b = c : d

    ad = bc 1bd

    ad 1bd

    = bc 1bd

    ab= c

    d

    a : b = c : d

    (2) a : c = b : d

    ad = bc 1cd

    ad 1cd

    = bc 1cd

    ac= b

    d

    a : c = b : d

    50

    3-1

    abcd ad = bc

  • xy 0 (1)3x = 2y x : y (2)2x : 5y = 3 : 7 x : y (3)x : 4 = y : 6 x : y(1)3x = 2y x:y = 2:3 (2)2x:5y = 3:7 14x = 15y x:y = 15:14 (3)x:4 = y:6 x:y = 4:6 = 2:3

    51

    3-1

  • 8. abcd a : b = c : d k (k 0) a = ck b = dk

    a : b = c : d a : c = b : d ac= b

    d

    ac= b

    d= k(k 0)

    ac = kbd = k

    {a = ckb = dk x : y = 2 : 3 (1)4x : 5y (2)(x + y) : y (3)x2 : y2

    x : y = 2 : 3 x = 2ky = 3k (k 0) (1)4x:5y = 4(2k):5(3k) = 8k:15k = 8:15 (2)(x + y):y = (2k + 3k):3k = 5k:3k = 5:3 (3)x2:y2 = (2k)2: (3k)2 = 4k2:9k2 = 4:9

    52

    3-1

  • 1.

    2.

    3.

    4.

    5.

    3-2

    533-2

  • 1. 3 0

    a : b : c abc abc 1 a : b : c

    x : y : z a : b : c x : y : z = a : b : c x : y : z = a : b : c x : y = a : by : z = b : cz : x = c : a x : y : z = 2 : 3 : 5 x : y = 2 : 3y : z = 3 : 5z : x = 5 : 2

    54

    3-2

  • 2. x : y = a : by : z = b : cz : x = c : a x : y : z = a : b : c x : y = 2 : 3y : z = 3 : 4 x : y : z

    x : y : z2 : 3

    3 : 42 : 3 : 4 x : y : z = 2 : 3 : 4

    x : y = 2 : 3y : z = 4 : (5) x : y : z

    x : y : z2 : 3

    4 : (5)(24) : (34)

    (43) : (53)8 : 12 : (15)

    x:y:z = 8:12: ( 15) x : y : z

    55

    3-2

  • 3. abcm 0(1) a:b:c = (a m): (b m): (c m)(2) a:b:c = (a m): (b m): (c m)

    56

    3-2

    Check Answer