# Integrated Algebra I Answer Keys by gzn20404

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```									        Answers for Enrichment Activity Exercises
Enrichment Activity 1-1: Guessing a                                To find a word in the dictionary, choose a middle
Number by Bisection                                                word alphabetically and ask if the chosen word
a. 8                b. 6                                          comes before or after the word for which you are
c. A maximum of 11 guesses would be needed to                     searching. Continue the process. To find a name
locate a number between 1 and 2,000. The first                 on a list, choose a middle name and continue the
guess would be the average of 0 and 2,000, or                  process.
1,000. Assuming that the number itself was not
1,000, this would divide the possible numbers            Enrichment Activity 1-2: Repeating
into two equal sets: numbers between 0 and 999           Decimals
1
and numbers between 1,001 and 2,000. Sets of              1. 15                    2. 41
99
this size require a maximum of 10 guesses to              3.   7
4. 10
12                     33
find the number.                                               47
5.   111                 6. 322 or 325
99      99
Use a simpler related problem to find a general
solution.                                                  7. 1                     8. 155
198
9. 15                   10. 120
Range   Max. Number
of Numbers of Guesses                                        Enrichment Activity 1-3: A Piece of Pi
From 1 to 3         2        First guess would be 2.          1. 0 11        1 5         2 12       3 11       4 10
If 2 is not the number,             5 8         6 9         7 8        8 12       9 14
the next guess would             2. No. The digit 9 appears most often while the digit
be correct.                         1 appears least often.
From 1 to 7         3        First guess would be 4.          3. 0 12        1 11        2 12       3 8        4 12
If 4 is not the number,             5 12        6 7         7 4        8 13       9 9
the number must be               4. Answers will vary. Example: The digits 0, 1, 4, 5,
one of a set of three               and 8 appear more often than in the first group,
possible numbers, and               the digits 3, 6, 7, and 9 appear less often than in
at most two more                    the first group, and the digit 8 appears with same
guesses are needed.                 frequency as in the first group.
From 1 to 15        4        First guess would be 8.          5. 0 23        1 16        2 24       3 19       4 22
If 8 is not the number,             5 20        6 16        7 12       8 25       9 23
the number must be               6. Answers will vary. Examples: The digit 7 appears
one of a set of seven               significantly less frequently than the other digits.
possible numbers, and               Appearances of the digits 0, 2, 3, 4, 5, 8, and 9 are
at most three more                  evening out. As more digits in the expansion are
guesses are needed.                 checked, the digits may seem to appear the same
Range   Max. Number                                           number of times.
of Numbers of Guesses             Powers of 2
From 1 to 3         2               22   4                   Enrichment Activity 1-4: Making 100
2,148                    1,752
From 1 to 7         3               23   8                    1. 96 537                2. 96 438
From 1 to 15        4               24   16                        1,428                    1,578
3. 96 357                4. 94 263
From 1 to 31        5               25   32                           7,524                    5,823
From 1 to 63        6               26   64                   5. 91 836                6. 91 647
3,546                    7,524
From 1 to 127       7               27   128                  7. 82 197                8. 81 396
From 1 to 255       8               28   256                          5,643
9. 81 297        10. 3 714
69,258

If 2n 1 the number of integers in the set                                                         5,472
11. Answers will vary. Examples: 13 5 9 1,368,
from which the chosen number is selected,                                  3,576            5,148
n is the maximum number of guesses needed to                    16 5 12 894 , 40 5 27 396
identify the number.

264
Enrichment Activity 2-1: Special Ops                                             3.     z      0         2       4    6      8
5                             0      0         0       0    0      0
1. 7              2. 5                              3.   8
5                             2      0         4       8    2      6
4. 6.2            5. 20                             6.   8
7. 0.4            8. 75                             9. 23                              4      0         8       6    4      2
10. 8             11. 14                                                                6      0         2       4    6      8
12. a. a  b a 2b
8      0         6       2    8      4
b. 4
13. Answers will vary.                                                           4. a. Yes        b. Yes
c. Yes        d. 6
Enrichment Activity 2-2A: Clock Arithmetic                                          e. Number z Inverse Identity
2   z    8            6
a.          1   2       3       4   5   6   7   8    9 10 11 12                           4   z    6            6
1   2   3       4       5   6   7   8   9 10 11 12 1                              6   z    4            6
2   3   4       5       6   7   8   9 10 11 12 1             2                    8   z    2            6
5. Yes
3   4   5       6       7   8   9 10 11 12 1             2   3
Conclusion: This is a field; all 11 properties hold.
4   5   6       7       8   9 10 11 12 1             2   3   4
5   6   7       8       9 10 11 12 1         2       3   4   5         Enrichment Activity 2-5: Paying the Toll
6   7   8       9 10 11 12 1            2    3       4   5   6         1(7)    2( 3)        1       1(11)    11                    1(11)       1(7)    1(3)      21
1(11)    3( 3)           2   4(3)    12                     2(11)       22
7   8   9 10 11 12 1                2   3    4       5   6   7
1(3)    3                    1(7)    2(3)    13             2(7)    3(3)        23
8   9 10 11 12 1                2   3   4    5       6   7   8         1(7)    1( 3)        4       1(11)    1(3)     14           8(3)    24
9 10 11 12 1                2   3   4   5    6       7   8   9         1(11)       2( 3)        5   5(3)    15                     2(11)       1(3)     25
10 11 12 1                 2   3   4   5   6    7       8   9 10          2(3)    6                    1(7)    3(3)    16             3(11)       1( 7)        26
1(7)    7                    1(11)    2(3)     17           9(3)    27
11 12 1            2       3   4   5   6   7    8       9 10 11
1(11)       1( 3)        8   1(11)    1(7)     18           4(7)    28
12 1       2       3       4   5   6   7   8    9 10 11 12                3(3)    9                    1(7)    4(3)    19             2(11)       1(7)     29
b. 1.   Yes                                                                    1(7)    1(3)        10       2(7)    2(3)    20             3(7)    3(3)        30
2.   Yes                                                                    All purchases can be made since any whole
3.   Yes                                                                    number can be formed using multiples of the
4.   12                                                                     numbers above.
5.   Number 1 2 3 4 5 6 7 8 9 10 11 12
Inverse 11 10 9 8 7 6 5 4 3 2 1 12                                     Enrichment Activity 2-8: Graphing
Ordered Pairs of Numbers
Enrichment Activity 2-2B: Digital Addition;                                      a.
State Street

Digital Multiplication
1.   {      0       2       4       6   8
H                             S
0   0       2       4       6   8
2   2       4       6       8   0                                                                       A    L
V   C                                  Main Street
4   4       6       8       0   2                                                                              O
G        M   P                   F
6   6       8       0       2   4
8   8       0       2       4   6                                                                       D
2. a. Yes        b. Yes
c. Yes        d. 0                                                           b. M is two blocks west and one block south of
e. Number { Inverse Identity                                                    O( 2, 1).
0  {     0      0                                                     c. The first number gives the number of blocks east
2  {     8      0                                                        or west of O, the center of town. If the number is
4  {     6      0                                                        positive, the position is east of O; if the number
6  {     4      0                                                        is negative, the position is west of O. The second
8  {     2      0                                                        number gives the number of blocks north or

265
south of O. If the number is positive, the position         4.   16x 13,600 5. x 150; x 300; x 450
is north of O; if the number is negative, the               6.   16x 20,400 7. 33x 34,000
position is south of O.                                     8.   33x 34,000 65,000
9.   \$3,000     10. \$3,500    11. \$45,200
Enrichment Activity 3-1: Number Play
Columns 2 and 3 will vary. The answers for columns 1              Enrichment Activity 4-3: Consecutive
and 4 are shown below.                                            Integers
Col. 1           Col. 4                                 1.    Number        First                     Even
of Integers   Integer    Sum            or Odd
1.          15                 n
2.           3                2n                                           3          1            6            even
3.          38              2n 8                                           3          4            15           odd
4.        3,800          200n 800                                          3          7            24           even
5.        3,600          200n 600
6.         360            20n 60                                           3          x       3x        3    even or odd
7.         400            20n 100                                          4          1            10           even
8.          20              n 5                                            4          4            22           even
9.          15                 n
4          7            34           even
Conclusion: The result on line 9 will always equal the
4          x       4x        6       even
starting number on line 1.
5          1            15           odd
Enrichment Activity 3-2: Variable Codes                                     5          4            30           even
1. a. PAOIK                   b. ZUSUXXUC                                  5          7            45           odd
c. SGZNKSGZOIY             d. KDZXGUXJOTGXE                             5          x       5x        10   even or odd
3. I MUST STUDY FOR AN EXAM                                                6          1            21           odd
4. a. XIVRK                   b. GLQQCV                                    6          4            39           odd
c. RJJZXEDVEK              d. VOGCFIRKZFE                               6          7            57           odd
5. x 9
6          x       6x        15      odd
6. NZIXS, MLEVNYVI; the correspondence is
symmetric. If a line is drawn between M and N in                        7          1            28           even
the alphabet list, the corresponding letters are                        7          4            49           odd
equal distances from that line. A corresponds to                        7          7            70           even
Z, B corresponds to Y, and so on.
7          x       7x        21   even or odd
7. a. MLBNCFH
b. TUESDAY; the result is the original word.                  2. Let x be the first consecutive integer.
7 x is the encoding and decoding expression.                  a. Odd; 2x 1 is the sum of an even number and
8. a. YKBWTR                  b. RDUPMK; no                            an odd number.
b. Even when x is odd; 3x 3 is the sum of two
Enrichment Activity 3-7: Formulas for                                   odd numbers.
Health                                                                  Odd when x is even; 3x 3 is the sum of an
1.   22            2.   24            3.   17                        even number and an odd number.
4.   31             5.   26             6.   22                    c. Even; 4x 6 is the sum of two even numbers.
d. Even when x is even; 5x 10 is the sum of
7.   24            8.   34             9.   BMI 5 703W
2
H                  two even numbers.
10.   114 bpm       11.   19 bpm        12.   23 bpm                    Odd when x is odd; 5x 10 is the sum of an
13.   19 bpm        14.   84%                                           odd number and an even number.
e. Odd; 6x 15 is the sum of an even number
Enrichment Activity 4-2: Book Value                                     and an odd number.
1. Let x represent the middle book because the                      f. Even when x is odd; 7x 21 is the sum of two
values of the other books can be expressed in                       odd numbers.
relation to it.                                                     Odd when x is even; 7x 21 is the sum of an
2. 16 books        3. x 100; x 200; x 300                              even number and an odd number.

266
g. Even when the number of integers divided by               2. x   4
2 is even.
Odd when the number of integers divided by
2 is odd.
h. Even when the first integer is even and the
number of integers after the first, divided by 2,
is even.
Odd when the first integer is odd and the
number of integers after the first, divided by 2,
is even.
Even when the first integer is odd and the
3. x       3
number of integers after the first, divided by 2,
is odd.
Odd when the first integer is even and the
number of integers after the first, divided by 2,
is odd.
3.   a. 3            b. 5                c. 7
d. The sum always has the number of consecutive
integers as a factor. The sum is always the
number of integers times the middle number.
4.   a. No
b. The sum is 6 more than 4 times the first integer.         4.             All real numbers
The sum is twice the sum of the second and                               between 4 and
third integers.                                                          4; [ 4, 4]
The sum is 6 less than 4 times the last integer.
5.   a. Student tables
b. The sum of even integers is always even.
6.   a. Student tables
b. The sum of odd integers is even if there is an
even number of integers and odd if there is an
odd number of integers.                                   5.             All real numbers
between 1 and
Enrichment Activity 4-4: Law of the Lever                                        1; ( 1, 1)
1.   2.4 ft
2.   15 lb, 25 lb
3.   Heavier carton, 9 ft; lighter carton, 12 ft
4.   Kelly, 49 lb; Laurie, 70 lb
5.   a. The side with the 32-lb weight
b. 3 ft
6.             All real numbers
Enrichment Activity 4-7: Graphing an                                             greater than 1;
Inequality                                                                       (1, ). Since there
1. x     4                                                                       is no real number
that is the square
root of a nega-
tive number, the
inequality is mean-
ingless for x 0.

267
7.                                                   All real numbers;                    Enrichment Activity 5-2: An Odd Triangle
(    , ).                             1. Row 6: 31, 33, 35, 37, 39, 41
Row 7: 43, 45, 47, 49, 51, 53, 55
2. 1, 9, 25, 49; the middle number in odd-numbered
Row n is n2.
3. 4, 16, 36; the average of the middle numbers in
even-numbered Row n is n2.
4. a. 169            b. 6
c. 157, 159, 161, 163, 165, 167, 169, 171, 173, 175,
8.                                                   All positive real                            179, 181, 183
numbers; (0, ).                       5. Row Number               Sum of Numbers in Row
1                       1
2                       8
3                       27
4                       64
5                      125
6                      216
9.                                                   Since x is always
positive, there is                                    7                      343
no real number                        6. Sum of the numbers in Row n n3
that makes this                       7. a. 1,728      b. Row 19
inequality true; .                    8. a. 3 22 1, 7 32 2, 13 42 3,
21 52 4, 31 62 5, 43 72 6
b. n2 (n 1) n2 n 1
9. a. 5 22 1, 11 32 2, 19 42 3,
29 52 4, 41 62 5, 55 72 6
b. n2 (n 1) n2 n 1
Enrichment Activity 5-1: Sums and Squares                                                 10. a.        Row         Sum of All     Sum Written
1.                   42           4        5        25         52                                      Number       Numbers         as a Square
1           1                 12
2           9                 32
3          36                 62
4          100               102
2.                                                                                                          5          225               152
n     + 1
6          441               212
2                                    2
n           n        (n    1)        n       2n       1                           7          784               282
n   n2     n
C           D
2                   ?                2
(n           2n    1) 5 (n           1)                              n(n 1 1) 2
?                                  b.          2
n2       2n        1 5 (n        1)(n 1)
+                                n2       2n        1 n2          2n 1
1   n+1                          n2       2n        1 (n          1)2
Enrichment Activity 5-4: Products, Sums,
and Cubes
3. 202 20 21    400               20           21        441                               1. 3(4)(5) 4       60 4      64 43
4. 102 100                                                                                    4(5)(6) 5 120 5 125 53
5. (n 1)2                                                                                     5(6)(7) 6 210 6 216 63
2. The product of three consecutive integers plus
the middle integer is equal to the cube of the
middle integer.

268
3. 6(7)(8) 7 336               7     343 73                                  Enrichment Activity 6-1B: Ratio: Estimates
4. 7(8)(9) 8 504               8     512 83                                  and Comparisons
5. (n 1)(n)(n 1)               n     n(n 1)(n 1) n                           Columns 1 and 2 will vary according to student
(n2 n)(n 1) n                           estimates.
n3 n2 n2 n n                                              Column 3
n3
?                                                     Actual Ratio     Source for Teacher:
6. (n)(n     1)(n      2)    (n     1) 5 (n     1)3                                            of Size           Area (sq mi)
?
(n)(n2     3n       2)    (n     1) 5 (n        1)(n    1)(n   1)         Australia            0.80              2,967,908
?
3        2                                    2
(n       3n       2n)    (n     1) 5 (n        1)(n     2n 1)            Brazil               0.88              3,286,487
n3      3n2    3n      1 n3          3n2     3n 1
Enrichment Activity 6-1A: Fibonacci                                           China                1.00              3,705,405
Sequence and the Golden Ratio                                                 India                0.34              1,269,345
Part 1. For clarity, the first 26 terms of the Fibonacci                      Japan                0.04               145,883
sequence are written below from smallest to
Mexico               0.20               761,606
largest in three-column format.
Russia               1.77              6,592,769
1      55           4,181
Spain                0.05               194,897
1      89           6,765
United States        1.00              3,718,709
2     144       10,946
3     233       17,711
Bonus:          Actual Ratio
5     377       28,657
of Size
8     610       46,368
Australia            3.90
13     987       75,025
Brazil               4.32
21    1,597     121,393
34    2,584
China                4.87
13                                 21                           India                1.67
Part 2.        8   5 1.625                       13   5 1.615384
34                                 55                           Japan                0.19
21   5 1.619047                    34   < 1.617647059
89                                144                           Mexico               1.00
55   5 1.618                       89   < 1.617977528
233                                377                           Russia               8.66
144   5 1.61805                    233   < 1.618025751
610                                987                           Spain                0.26
377   < 1.618037135                610   < 1.618032787
1,597                              2,584                           United States        4.88
987    < 1.618034448              1,597   < 1.618033813
4,181                              6,765
2,584   < 1.618034056              4,181   < 1.618033963           In a and b, student responses will vary.
10,946                             17,711                            a. With Mexico’s area used as the base of compari-
6,765    < 1.618033999             10,946   < 1.618033985
son, its ratio changes from 0.20 to 1.00. In turn,
28,657                             46,368
17,711   < 1.61803399              28,657   < 1.618033988               the ratio for every nation becomes about 5 times
75,025                            121,393                               as large as the ratio using the United States as
46,368   < 1.618033989            75,025    < 1.618033989
the base of comparison.

Part 3. 1 12"5 < 1.618033989
b. The ratios in both lists show the sizes of the
countries in relation to one another: Japan has
As the terms of the Fibonacci sequence                                    the smallest area, followed by Spain, up to Russia
increase, the ratio comparing the greater                                 with the largest area.
of two consecutive Fibonacci numbers to
the smaller approaches the value of the
golden ratio.

269
Enrichment Activity 6-2: Population Density                          9. a. False      b. Leads to the contradiction (y   0)
a   a1c     c
Part 1.                                                             10. b , b 1 d , d
State   Land Area Population
Population (sq mi)   Density                      Enrichment Activity 7-4: Hero’s Formula
Alaska               648,818         571,951             1           1. a. 24 sq in.    b. 24 sq in.
2. a. 431.6 m2     b. 11,161.36 sq ft
California        35,484,453         155,959           228           3. a. 388.8 cm2
Florida           17,019,068          53,927           316              b. Since the area and base are known, substitute
Montana              917,621         145,552             6                 the values into the formula A 5 1bh; b 18 cm
2
New Jersey         8,638,396             7,414        1,165             c. Right triangle
New York          19,190,115          47,214           406
Enrichment Activity 7-5: Rectangle
North Carolina     8,407,248          48,711           173          Cover-Up
Texas             22,118,509         261,797            84           1. 5 ways
Part 2. a. Student estimates will vary.
b. Canada, 9; China, 361; India, 943;
Japan, 880; Mexico, 141; Russia, 22

Enrichment Activity 6-3: Catching Up
1. a. 35 steps      b. 5 steps          c. 12.5 or 25               2. 8 ways
35     70
2. a. 43 steps      b. 5 steps; 10 steps
25
c. 43
d. 43 . 25 ; Evan is gaining on Marisa.
25
70
37.5    75
3. 51 or 102, 50, 62.5 or 125, 75
59 67      134 75
4. Evan continues to gain on Marisa until he
catches her after he has taken a total of 30 steps
and she has taken a total of 75 steps.

Enrichment Activity 6-4: Ratios and
Inequalities
1. Approach 1 proof:            a       c
b   ,   d
a              c
b (b)(d)   ,   d (b)(d)                    3.              Rectangles of Width 2
Number of Ways to
a        c                                Length of Rectangle     Cover with Dominoes
Approach 2 proof:              ,
, AbBd
b        d
a d     c                                              0                        1
b
, bd
bd
2                        2
c. d b         d. d bc                                                          4                        5
1     1
3. b . b 1 1 must be true because 1(b           1)    b(1)                         5                        8
or b 1 b.                                                                       6                       13
a
4. b . 1 is true because a2     b2, so "a . "b and                                 7                       21
a b.                                                                          8                       34
5.   a b is true because a(b c) b(a c), so
ab ac ba bc and ac bc.                                                        9                       55
7.   a. True       b. Consistent with x 0, y 0                      4. Each number is the sum of the two previous
8.   a. False      b. Leads to contradiction (y 0)                     numbers. This is the Fibonacci sequence.

270
Enrichment Activity 7-6: Area of Polygons                                         6. 69, 260, 269 or 69, 92, 115 (Other answers are possible.)
1. a. 60 cm2        2. a. 480 cm2              3. a.   88 sq in.                  7. a. Yes; 6, 8, 10
b. 1r(ED)                                           5                             b. No. If a number is odd, its square is odd. The
2                b. 4r(ED)                  b.   2 r(YZ)
sum of the squares of two odd numbers is even.
4. 1nsr
2                                                                                     Therefore, the third number must be even.
5. a. 9"3 sq in.       b. 96"3 cm2                c. 8 m2                            c. Yes; 3, 4, 5
d. 28,500 cm2       e. 800 mm2                                                    d. No. If either a or b is odd and the other even,
then c must be odd. If a and b are both even,
Enrichment Activity 8-1: Pythagorean Triples                                             then c must be even. Therefore, it is not
1. a.   U      V     U2           V2      2UV          U2           V2                   possible to have exactly one odd number in a
Pythagorean triple.
2      1             3             4                   5
3      1             8             6                   10                 Enrichment Activity 8-3: Polytans
4      1             15            8                   17                 1.
5      1             24           10                   26
6      1             35           12                   37
4 + √2    2 + 3√2        2 + 3√2       2 + 3√2
2. 4 1 "2, 2 1 3"2
b. The difference between U 2 V 2 and U 2 V 2
is 2. The sum of U 2 V 2 and U 2 V 2 is U times
3.
2U [5 3 2(4)]. The sum of the three num-
bers is twice the product of U and the next con-
secutive integer [3 4 5 2(2 3)]. Other
relationships are possible.                                                        4√2        6     4 + 2√2        4 + 2√2    4 + 2√2
2            2                    2            2
2. a.   U      V     U            V       2UV          U            V
3      2             5            12                   13
4      3             7            24                   25
4 + 2√2      4 + 2√2     4 + 2√2            4 + 2√2
5      4             9            40                   41
6      5             11           60                   61
7      6             13           84                   85
b. The difference between U2 V2 and 2UV is 1.
The sum of U2 V2 and 2UV is the square of                                        4 + 4√2      4 + 4√2     4 + 4√2            4 + 4√2        4 + 4√2
4. 4"2, 6, 4 1 2"2, 2 1 4"2
U2 V2. The sum of the three numbers is the
product of U2 V2 and U2 V2 1 [5 12
13 5 6]. The smallest number, U2 V2, is
Enrichment Activity 8-6: Trigonometric
the sum of U and V. Other relationships are
Identities
possible.
1. a. 1            b. 1                 c. 1
3. a.   U      V     U2           V2      2UV          U2           V2               d. 1            e. 1                 f. 1
4      2             12           16                   20                 2. tan A tan (90° A) 1, where 0° A                               90°
a
5      3             16           30                   34                 3. tan A 5 b
b
6      4             20           48                   52                     tan B 5 a
a
tan A 3 tan B 5 b 3 b 5 1
a
7      5             24           70                   74
B 90° A
8      6             28           96               100                       tan A tan (90° A) 1
b. The difference between U2 V2 and 2UV is 4.                                  4. a. 1            b. 1                 c. 1
The sum of U2 V2 and U2 V2 is twice U2                                        d. 1            e. 1                 f. 1
[20 12 2(42)]. The sum of U2 V2 and                                        5. (sin A)2 (cos A)2 1
6. sin A 5 a, (sin A) 2 5 A a B 2 5 a2
2
2UV is the square of half the smallest number,                                         c               c       c
cos A 5 b, (cos A) 2 5 A b B 2 5 b2
U2 V2 [20 16 (12 2)2]. Other relation-                                                                             2

ships are possible.                                                                     c                c       c
2      2       2    2    2
4. 10 and 7                                                                            (sin A) 2 1 (cos A) 2 5 a2 1 b2 5 a 1 b 5 c2 5 1
c    c      c2    c
5. 33, 56, 65 or 42, 56, 70 (Other answers are possible.)                              (Note the use of the Pythagorean Theorem.)

271
sin
7. a. cos 208 5 tan 208
208
sin
b. cos 428 5 tan 428
428
11.                            z
sin
c. cos 788 5 tan 788
788
sin
d. cos 348 5 tan 348
348
sin
e. cos 158 5 tan 158
158
sin
f. cos 78 5 tan 78
78                                                           (0, 0, 4)
sin
8. cos A 5 tan A, where 0° A 90°
A
9. sin A 5 ac
cos A 5 bc                                                                         (0, –4, 0)         1
a
tan A 5 b                                                                                                                        y
sin A  a   b   a    c  a                                                                                   1
cos A 5 c 4 c 5 c 3 b 5 b 5 tan A                                                        (2, 0, 0)

1
Enrichment Activity 9-3: Graphing with
Three Variables
1–4.                                     z

x
(2, –5, 7)

(3, 2, 6)
Enrichment Activity 9-7: Iterating Linear
Functions
1                                     1.                       y    0.4x       6
y                Starting Value 5               Starting Value 15
1
8                                  12
1

(5, –4, 1)                                                                 9.2                                10.8
9.68                               10.32
9.872                              10.128
(4, 2, –3)
x

9.9488                             10.0512
9.97952                            10.02048
2. The iterates are approaching 10; for the starting
5. Octant 6              6. Octant 5       7. Octant 4
value 5, the iterates approach 10 from below;
8. Octant 8              9. Answers will vary; (0, y, 0)
for the starting value 15, the iterates approach
10.                                   z                                        10 from above.
3. All the y-values (outputs) are exactly 10.
4. a. 3             b. 4
c. 0             d. 2
(0, 0, 4)                                      5. y x
b
6. a. x 5 1 2 m
b. Undefined for m 1 since the denominator
1                                           would be 0. Functions of the form y x b
y                (b 0) cannot have a fixed point because the
1      (0, 3, 0)
y-value can never be the same as the x-value
1

if a nonzero number is being added to the
x-value.
7. a. Starting point 0: 2, 6, 14, 30, 62, 126
(6, 0, 0)                                                    Starting point 4: 6, 10, 18, 34, 66, 130
b. No
x

c. Here, m 0; in question 1, 0 m 1.

272
In 8–11, answers will vary.                                             c.                          y
40
8. No fixed point;
Starting point 0: 3, 6, 9, 12, 15, 18                                                      36
Each iterate is b more than the previous.                                                  32

Cost of parking
9. Fixed point 100;                                                                           28
Starting point 101: 100.8, 100.64, 100.512,                                                24
100.4096, 100.32768, 100.262144                                                            20
Starting point 99: 99.2, 99.36, 99.488, 99.5904,
16
99.67232, 99.737856
Iterates approach the fixed point.                                                         12
10. Fixed point 3.5;                                                                            8
Starting point 4.5: 6.5, 12.5, 30.5, 84.5, 246.5,                                           4
732.5                                                                                                                                     x
Starting point 2.5: 0.5, 5.5, 23.5, 77.5,                                        2 4 6 8 10 12 14 16 18 20 22 24
0
239.5, 725.5                                                                        Number of hours
Iterates move away from the fixed point                             d. From 1 to 19 hours, the graph would be a
11. Fixed point 0;                                                         straight line.
Starting point 1: 6, 36, 216, 1,296, 7,776, 46,656                             y
Starting point 1: 6; 36; 216; 1,296;                                      40
7,776; 46,656                                                           36
Iterates move away from the fixed point.                                  32
Cost of parking
28
Enrichment Activity 9-10: Graphing Step
Functions                                                                     24
a.                                                                           20
16
Hours      Cost        Hours    Cost       Hours       Cost
12
1
4    \$4.00         2       \$6.00       173
4       \$38.00
8
2
3    \$4.00         21
4      \$8.00       18        \$38.00                  4
x
1     \$4.00         6      \$14.00       19        \$40.00
0        2 4 6 8 10 12 14 16 18 20 22 24
11
2    \$6.00        121
2     \$28.00       201
3       \$40.00                                               Number of hours
b.
Time       Time                 Time       Time                       Enrichment Activity 9-11: Holes, Holes, and
In        Out         Cost      In        Out         Cost           More Holes: An Exponential Investigation
Part I
9:15 A.M. 9:50 A.M.    \$4.00   10:00 A.M. 2:30 P.M.    \$12.00
9:30 A.M. 10:29 A.M.   \$4.00   10:10 A.M. 10:00 P.M.   \$26.00
# of folds                               0   1    2    3    4    5
9:30 A.M. 10:35 A.M.   \$6.00   12:15 A.M. 8:00 A.M. \$40.00
# of holes                               1   2    4    8    16   32
10:00 A.M. 12:45 A.M.   \$8.00   12:40 A.M. 11:00 A.M. \$40.00            # of holes expressed 20                      21   22   23   24   25
as a power of 2
a. The total number of holes doubles with each
fold, or the total number of holes is a power of 2,
the power being the number of folds.
b. 2n               c. H 2n
# of folds                               0   1    2    3    4    5
# of holes                               2   4    8    16   32   64
# of holes expressed 21                      22   23   24   25   26
as a power of 2

273
a. The pattern is similar but begins with 21 rather          5. The function is maximized at (6, 2). The
than 20.                                                     maximum profit of \$18 is obtained when 6
b. 21 2 20, 22 2 21, 23 2 22, 24 2 23,                          bracelets and 2 necklaces are produced.
25 2 24, 26 2 25                                         Exercises
c. H 2 2n                                                    1. Max 20 at (4, 4), min 0 at (0, 0)
Part II                                                       2. Max 68 at (12, 4), min 14 at (2, 2)
a.     H                                                     3. a. S 30c 40t
84                                                           b. 2c 4t 800, c t 300, c 0, t 0
78                                                           c.        y
72
66                                                                  400
60
54
48                                                                  300
H = 2n
42
36                                                                               x + y = 300
(0, 200)
30
24                                                                                    (200, 100)
18                                                                  100                        2x + 4y = 800
12
6                                            n
0         1     2      3         4   5                           (0, 0)     100 200 (300, 0) 400 500 x
d. (0, 0), (0, 200), (200, 100), (300, 0)
Number of folds                            e. At (0, 0), S 0; at (0, 200), S 8,000;
at (200, 100), S 10,000; at (300, 0), S 9,000
Part III
f. The function is maximized at (200, 100). The
Answers will vary. Example: Yes. It is appropriate                 maximum sales of \$10,000 is obtained when
because as n, the number of fold, increases, H,                    200 chairs and 100 tables are produced.
the number of holes punched, increases rapidly
(or exponentially).                                          Enrichment Activity 11-1: Finding
Primes—The Sieve of Eratosthenes
Enrichment Activity 10-5: Solving Systems
1. 17
Using Matrices
2. Composite numbers can be written as pairs
1.   x 2, y 1                                                   of factors, such that when the number divided
2.   x     1, y 6                                               by a factor is less than the factor, all positive
3.   x 4, y       1                                             integral factors have been found. In this case,
4.   x     3, y 5                                               the greatest number, 200, divided by 15 yields
5.   a. Error message results.                                  131 . Therefore, all composite numbers between
3
b. There is no solution to the system.                     15 and 200 that have a factor greater than 15
must also have a factor less than 15 and have
Enrichment Activity 10-6: Systems with                           already been crossed out.
Three Variables                                               3. 19, the largest prime less than 20
1. (3, 5, 2)          2. (1, 2, 2)                           4. a. x 1, 12 1 41 41
3. (2, 1, 2)          4. (4, 3, 0.5)                                x 2, 22 2 41 43
5. (10, 3, 7)         6. ( 6, 4, 2)                                 x 3, 32 3 41 47
x 4, 42 4 41 53
Enrichment Activity 10-8: Linear                                     x 5, 52 5 41 61
Programming                                                          x 6, 62 6 41 71
Example                                                              x 7, 72 7 41 83
3. B (0, 5), C (6, 2), D (7, 0)                                     x 8, 82 8 41 97
4. At B (0, 5), 2x 3y 15                                            x 9, 92 9 41 113
At C (6, 2), 2x 3y 18                                            x 10, 102 10 41 131
At D (7, 0), 2x 3y 14                                        b. x 41, 412 41 41 1,681 41 41

274
Enrichment Activity 11-5: Differences of                           2.             Square            Equation Relating Sides
Squares                                                                                   a                       —
1. a. 92 82         b. 72 52          c. 302 292
d. 16 2
14 2
e. 36 2
35 2
f. 262 242                                         b                   "8 5 2"2
g. 512 502 h. 732 722
2       2       2     2
i. 492 472                                         c                  "18 5 3"2
j. 152     151 k. 97      95       l. 5032 5022
2n 2 1         1
2. 2 5 n 2 2 , which is between n 1 and n,                                               d                  "32 5 4"2
so using the rule, 2n 1 can be written as                                             e                  "50 5 5"2
n2 (n 1)2. To check, n2 (n 1)2
3.
n2 (n2 2n 1) 2n 1.
3. 4n 4 n, which is between n 1 and n 1,                                        Side
(square                                   Equation
so using the rule, 4n can be written as
root of Side (decimal                     Relating
(n 1)2 (n 1)2. To check, (n 1)2 (n 1)2
Square Area area) approximation)                       Sides
n2 2n 1 (n2 2n 1) 4n.
f             5        "5          2.236067977      —

"20                        "20 5 2"5
Enrichment Activity 11-7: Factoring                                     g            20                    4.472135955
Trinomials
1. (1) 4x2( 10)            40x2                                        h            45       "45          6.708203932   "45 5 3"5
(2) 8 and 5
(3) 4x2 8x         5x 10                                       Enrichment Activity 12-7: Operations with
(4) 4x(x 2)         5(x 2)                                     Radicals
(5) (x 2)(4x         5)                                        1. a. 3"2                        b. 8"11
c. 3"2                      d. 6"3 1 3"5
2. a. (x 2)(2x         5)              b.   (x 3)(4x 3)

e. 2"10                     f. 5 2 2"3
c. (x 5)(3x         1)              d.   (x 4)(3x 2)
e. (2x 3)(3x         2)             f.   (3x 4)(4x 1)
g. (x 5)(8x         3)              h.   (x 5)(5x 2)                g. 0                        h. 2"5
i. (x 3)(7x         1)                                              i.    0                     j. 5 2 2"3
k. 10"6                     l. 4 1 2"3
m. 11"7                     n. 6"3 1 3"5
Enrichment Activity 12-2: Square Root:
Divide and Average
1.   7.07             2.   6.32              3.   2.45                 o. 11"7                     p. 3
4.   9.38             5.   10.39             6.   24.78                q. 10"6                     r. 2"5
7.   3.32             8.   23                9.   24.78                s. "5                       t. 4 1 2"3
u. 8"11                     v. 23"5
10.   0.77            11.   0.55             12.   50.25
13. "529 is rational because 232
x. 23"5
529.
w. 3
y. 5"2
14. 3.162         15. 10.100

Enrichment Activity 12-4: Equivalent                               2.                         Common                     Common
In 1 and 3, answers will vary depending on the number                         a, c              3"2               k, q     10"6
of decimal places displayed on the calculator.
b, u              8"11              l, t   4 1 2"3
1.               Side (square Side (decimal
Square Area root of area) approximation)                                d, n            6"3 1 3"5           m, o     11"7
a        2                "2         1.414213562                     f, j             5 2 2"3            p, w       3
b        8                "8         2.828427125                     g, i                  0             v, x    23"5
c   18            "18            4.242640687                     h, r              2"5
d       32            "32            5.6565854249         3. a. 100
e   50            "50            7.071067812               b. Yes

275
Enrichment Activity 13-2: Carpet Squares                                            5. The graphs for squares with fringe on two sides
1. 4 with fringe on two sides, 8 with fringe on one                                    and one side are linear. The graph for squares
side, 4 with no fringe.                                                             with no fringe is steepest.
6. a 4, b 4(x 2), c (x 2)2
7. 4 with fringe on two sides, 192 with fringe on one
side, 2,304 with no fringe.
8. The formulas are graphed in Exercise 4 for the
domain of 2 x 10, where x is an integer. In
general, the formulas hold true for the domain
x 2. The graph of c is quadratic.
2.                                                                                  9. No. For the number of squares to be equal,
4(x 2) (x 2)2. Simplifying gives x2 8x
Dimensions Fringe on Fringe on No     Total                                            12 0, which factors to (x 6)(x 2) 0. The
of Carpet Two Sides One Side Fringe Squares                                           only solutions for x are 2 and 6.
2 ft by 2 ft                               4         0       0          4         10. (n 1)2 n2 2n 3
3 ft by 3 ft                               4         4       1          9
4 ft by 4 ft                               4         8       4         16
Inequalities
5 ft by 5 ft                               4        12       9         25
1.                        y
6 ft by 6 ft                               4        16      16         36
7 ft by 7 ft                               4        20      25         49
8 ft by 8 ft                               4        24      36         64
9 ft by 9 ft                               4        28      49         81
10 ft by 10 ft                              4        32      64        100
1
3. The number of squares with fringe on two sides is 4                                                        O                   x
–1    1
for any size because all squares have 4 corners. The
number of squares with fringe on one side starts
at 0 and increases by 4 each time the length of the
side increases by 1 foot. The number of squares                                  2.   (0, 0) is not in the region, (0, 3) is in the region
with no fringe starts at 0 and is the sequence of                                3.   See graph
square numbers. The total number of squares is                                   4.   All points inside of the graph of y x2 6x 8
also the square numbers starting with 4.                                         5.   2 x 4
4.            y                                                                     6.                                  y
64
60
Number of each type of square

1
56                                                                                 O             x
52                                                                           –1
–1
1
48
44
40
36
32
28
24
20
16
12
8
4     * * * * * * * * *            x
0    1 2 3 4 5 6 7 8 9 10
Length of side of square

276
7. All points outside the graph of y                      x2    x       12         2. Student-generated examples.
a1c
8. x     3 or x 4                                                                  3. No. Cite any one example to show that b 1 d is
a      c
not the average of b and d .
Enrichment Activity 14-2: Is It Magic or Is                                        4. a. 3 , 3 1 9 , 9, 3 , 12 , 9, 0.75 , 1.5 , 2.25
4    414     4 4     8     4
It Math?                                                                                                      6
b. 1 , 1 1 5 , 5, 1 , 12 , 5, 0.16 , 0.5 , 0.83
6    616      6 6          6
1. a. 3 S 5 S 13 S 13 S 21 S 34 ; yes
5   8
8
21   34   55                                                      c. 2 , 9 1 10 , 10, 9 , 19 , 10, 4.5 , 4.75 , 5
9
212       2 2      4     2
13
b. 13 S 15 S 15 S 28 S 43 S 114 ; yes
2
28   43   71
71
5. a. If b , b, then b , a 2b c , b (a 0, b 0, c 0)
a   c       a     1       c
2. Student-generated results. Each result should                                                               a      c
b. Yes. The average of b and b is equal to
equal 0.6 when rounded to the nearest tenth.                                          a1c           a1c
x       y          x1y           x 1 2y      2x 1 3y       3x 1 5y                    b 4 2 5 2b .
3. y S x 1 y S x 1 2y S 2x 1 3y S 3x 1 5y S 5x 1 8y
3x   3x 1 5y   5y                  3   3x 1 5y   5                              Enrichment Activity 14-8: Estimating
4. 5x , 5x 1 8y , 8y is equivalent to 5 , 5x 1 8y , 8
3x 1 5y                                                       Solutions to Quadratic Equations
or 0.6 , 5x 1 8y , 0.625
1. a. x 1.32 or 1.33          b. x 1.316 or 1.317
3x 1 5y
Therefore, 5x 1 8y 5 0.6 to the nearest tenth.                                  2. y     2; x      5.32 or 5.33 (to the nearest tenth),
5. a. 0.62                                                                            x     5.316 or 5.317 (to the nearest hundredth)
b. The seventh term in the sequence beginning                                   3. Calculator check.          4. x 1.45, x        3.45
x                             5x 1 8y                             5. x 6.54, x 0.46             6. x 1.09, x        10.09
with y (x . 0, y . 0) is 8x 1 13y . Apply this
7. x 2.27, x 5.73
algebraic fraction to the rule illustrated                                8. The product of x and x 4 equals 7, not each
in Exercise 4, placing terms in a correct                                    factor. If each factor were to equal 7, then their
8y     8y 1 5x        5x
product would be 49.
order, to discover: 13y , 13y 1 8x , 8x ,
8     8y 1 5x     5                               Enrichment Activity 15-3: Probability and
which is equivalent to 13 , 13y 1 8x , 8 or
8y 1 5x                                         a Digital Clock
0.615384615               13y 1 8x     0.625. Therefore,
1. Answers will vary; student guesses
8y 1 5x                                                                               450      5
13y 1 8x        0.62 to the nearest hundredth.                            2. a. P(0) 1,440 5 16 .3125
720
b. P(1) 1,440 5 12     .5
Enrichment Activity 14-5: Operations with
540
Fractions                                                                             c. P(2)    1,440   53
8       .375
450         5
A       B             A     B          A B             A        B            d. P(3)    1,440   5   16   .3125
x1w                   x2w               xw                 x                             450         5
1.        y                     y                                   w                 e. P(4)    1,440   5   16   .3125
y2
450         5
x 1 wy                x 2 wy            xw                 x                 f. P(5)    1,440   5   16   .3125
2.          y                     y               y                 wy                            252         7
g. P(6)    1,440   5   40   .175
3. (2)             4. (4)            5. (4)
252         7
6. (2)             7. 1                                                               h. P(7)    1,440   5   40   .175
8. If the sum and the difference in Exercise 1 are                                                252         7
i. P(8)    1,440   5   40   .175
equal, then w 0. This would contradict the                                                     252         7
j. P(9)    1,440  5    40 .175
given statement, w 0.
3. The digit appearing most often on a digital clock
9. No. If x 1 and y 1, then x 1 w 5 xw becomes
2
y         y                          is 1, and the digit appearing the second most
1 w w (an impossible statement),                                                  often is 2. The digits 0, 3, 4, and 5 appear the same
or 1 0 (also impossible).                                                         number of times over a 24-hour period. The digits
10. w 2          11. x 5 5
4     12. y 5 2
3                                              6, 7, 8, and 9 also appear the same number of
times over a 24-hour period, and these digits
Enrichment Activity 14-7: Fractions                                                   appear less than all the others.
Between Fractions                                                                  4. The answers will be exactly the same for any
1. a. 2 , 2 1 4 , 4, 2 , 6 , 4, 0.6 , 0.75 , 0.8
3   315     5 3     8    5
12-hour period as for the 24-hour period discussed
earlier because, in a 24-hour period, a 12-hour
9
b. 14 , 9 1 14 , 14, 9 , 23 , 14, 2.25 , 2.5 , 2.8
415      5 4     9   55                                                   cycle is repeated twice. Doubling (or halving) the
c. 7 , 7 1 19 , 19, 7 , 26 , 19,
8   8 1 20   20 8    28   20                                                  numerator and the denominator of the probability
0.875 0.9285714… 0.95                                                         fraction does not change the value of the fraction.

277
Enrichment Activity 15-6: Probability and                     Enrichment Activity 15-7: Probability on
Area: And, Or, Not                                            a Dartboard
7
1. a. P(D) 25 .28                                             1. a.
18                                           b.   3  (from 4         )
b. P(not D) 25 .72
7
2. a. P(D) 25 .28                                                c.   5  (from 9        4 )
18                                           d.   7  (from 16        9 )
b. P(not D) 25 .72
7                                                e.   9  (from 25        16 )
3. a. P(D) 25 .28                                                     1                      3
18
2. a.   25   .04           b. 25      .12
b. P(not D) 25 .72                                               5                       7
c. 25 5 1 .2
5               d. 25      .28
4. Answers will vary. Example: The areas of the                      9
three shaded regions in Exercises 1–3 are equal;              e. 25 .36
10
each is 7 square units. The probability that a             3. a. 25 5 2
5         .4       14
b. 25      .56
point on the 5-by-5 grid lies in the shaded region               15
c. 25 5 3         .6    d.   24
.96
7                                                                  5                    25
is 25 for each region.                                           9
e. 25       .36
5. a. A and B intersect in two squares. Example:                    4                        9
4. a. 25       .16         b. 25      .36
A
c.    1
625     .0016      d. 576
625     .9216
B                            e.    81
.1296         256
f. 625     .4096
625
49                     81
g.   625     .0784      h.   625   .1296

Enrichment Activity 15-11: Expectation
1. a. Responses will vary.
b. P(A or B) 10 5 2 .4
25     5                                       b. Most should say “left” or “negative.”
2. E(game) 5 A 1 B 3 (13) 1 A 5 B 3 (21)
6. a. A and B intersect in three squares. Example:
6                 6
3   25
5  61 6
22    21
A                                                           5    6 5 3
3. The      expectation of 21
indicates that, on average,
3
the player expects to move 1 place in a negative
B                                  direction for every 3 turns taken.
9
4. a. E(game) 5 A 1 B 3 (15) 1 A 5 B 3 (21) 5 0
6                 6
b. P(A or B) 25 .36                                           b. The expectation of 0 indicates that, on
7. a. B is a subset of A. A and B intersect in four                 average, a player does not expect to advance
squares. Example:                                             or fall behind over a long period of play.
A                                            After many turns, the player should still be in
the START box, although the marker may
have moved to the left and to the right during
different turns.
5. Answers will vary. To win on this board in an
average of 24 turns, a player must advance
8 spaces to the right in that time. Therefore,
B                                       E(game) should equal 18 , or 11 . Here is one
8                                                                       24     3
b. P(A or B) 25 .32                                           possible set of rules to obtain E(game) 1:   3
8. P(A or B) is found by either:                                 (1) If a player guesses correctly, move 7
(1) counting the number of square units in the                    spaces in the positive direction (to the
5-by-5 grid that are shaded by A, by B, or                    right).
by both, and then dividing this area by 25,               (2) If a player guesses incorrectly, move 1
which is the area of the grid; or                             space in the negative direction (to the
(2) using the formula P(A or B) P(A) P(B)                         left).
P(A and B)                                                Thus:
E(game) A 1 B 3 (17) 1 A 5 B 3 (21) 5 2 5 1
9. No. P(A and B) cannot exceed P(A), and since
4                                                                   6                6               6 3
P(A) 25 .16, the statement P(A) .16 is false.

278
Enrichment Activity 15-12: A “Pick Six”                                 Enrichment Activity 16-7: The
Lottery                                                                 Median-Median Line
C ?     C    1 ? 1         1                                    1. Yes
1. a. 6 6 C48 0 5 25,827,165 5 25,827,165
54 6
b. .0000000387                                                                 100
6C5   ? 48C1       6 ? 48             288
2. a.         5                       5                                               95
54C6         25,827,165        25,827,165
b. .0000112                                                                        90
C ?     C       15 ? 1,128         16,920
3. a. 6 4 C48 2 5 25,827,165 5 25,827,165                                             85
54 6

Posttest
b. .000655                                                                         80
C ? C ?       47C2       20 ? 1 ? 1,081        21,620                         75
4. a. 6 3 1 C 1
5     25,827,165     5 25,827,165
54 6
b. .000837                                                                         70
5. a. Add the solutions to Exercises 1–4 part a:                                      65
38,829                                                                       60
25,827,165
b. Add solutions to Exercises 1–4 part b: .0015                                    55
6. Yes, the claim is essentially correct:                                             50
1
333 5 .003
By choosing two sets of 6 numbers, the probabil-                                   0
50 55 60 65 70 75 80 85 90 95 100
ity of winning a prize is doubled, so we multiply                                                Pretest
the probability found in Exercise 5 by two:                          2. See graph             3. (58, 71); see graph
2 .0015 .003, which is close to the given                            4. (69, 84); see graph   5. (85, 91); see graph
probability.                                                         6.
100
Enrichment Activity 16-1:Taking a Survey:                                              95
Designing a Statistical Study                                                                                     +
90
In 1–4, student responses will vary.
85              +
Posttest

Enrichment Activity 16-4: Locating the                                                 80
Median Value                                                                           75
a.    4th                                                                              70        +
b.    5th
65
c.    8th
d.    20th                                                                             60
e.    41st                                                                             55
f.   119th                                                                            50
N11
g.     2
h.    4th and 5th                                                                      0    50 55 60 65 70 75 80 85 90 95 100
i.   5th and 6th                                                                                        Pretest
j.   9th and 10th                                                      In 7 and 8, answers will vary but should be close to
k.    21st and 22nd                                                     those given.
l.   44th and 45th                                                      7. Using the points (55, 70) and (75, 85),
m.    115th and 116th                                                       y 0.75x 28.75
N      N
n.    2 and 2 1 1                                                        8. a. 0.75         b. 28.75
General rule: Arrange the data values in order. Let the                  9. a. 82           b. 74              c. 92 or 93
number of data values be N.                                             10. The regression equation (values rounded to the
If N is odd, the median is the value that is N 2 1 from
1                            nearest hundredth) is y 0.75x 30.24, which is
either end.                                                                 close to the median-median line.
If N is even, the median is the average of the values                   11. Student results will vary. The median-median line
that are N and N 1 1 from either end.
2     2
should be close to the regression line.

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