Appendix A — Answer Key, Chapter 1: Subsets of the Real Number System

SOL 8.NS.2 · Covers textbook Chapter 1 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 102 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Convention used throughout: "name every subset" means list all of natural, whole, integer, rational, irrational, and real that apply. Every number in this chapter is real, so "real" appears in every complete answer, and no answer ever contains both "rational" and "irrational."

Lesson 1.1 — Counting Numbers, Whole Numbers, and Integers

Guided practice

1. Only $3$. Counting starts at $1$, so $0$ is not natural, and $-5$ is negative.
2. $3$ and $0$. The whole numbers are the naturals together with $0$.
3. All three: $3$, $0$, and $-5$. The integers add the opposites of the whole numbers, and $-5$ is the opposite of $5$.
4. $0$. It is the only whole number that is not a natural number.
5. Any negative integer, such as $-6$.
6. The integers were built from the whole numbers, which were built from the naturals, and each step only added members, so every natural number is still there.

Independent practice

7. a) $7$: natural, whole, integer b) $0$: whole, integer c) $-13$: integer d) $100$: natural, whole, integer
8. a) False. $0$ is a whole number but not a natural number. b) False. $-4$ is an integer but not a whole number. c) True. The whole numbers are the naturals together with $0$, so nothing was removed. d) True. $0$ is a whole number, and every whole number is an integer.
9. a) Any negative integer, such as $-2$. b) Any number with a fractional part, such as $\tfrac{1}{2}$ or $-0.75$.
10. whole numbers
11. The integers contain $-6$. It is not natural and not whole, because both of those subsets contain only $0$ and the positive counting numbers, and $-6$ is negative.
12. The integers. The whole numbers are not enough because $-2$ is negative, and the whole numbers contain no negative numbers. Every label has no fractional part and is either a whole number or the opposite of one, so the integers are the smallest of the three subsets that holds them all.
13. Counting objects starts with one object, so the counting numbers begin at $1$ and never need $0$. But you often need to report that a collection is empty — zero apples, zero points — and that report needs a number of its own, so $0$ was added to make the whole numbers.
14. Having no fraction part is not the test for a whole number; the whole numbers are $0, 1, 2, 3, \dots$ and contain no negatives. Since $-8$ is the opposite of the whole number $8$, it is an integer.

Exit ticket 1.1

15. Not natural; yes whole; yes integer.
16. Any negative integer, such as $-1$ or $-20$.
17. Natural, whole, and integer.
18. The integers include every whole number and also the opposite of each one, so the integers contain negative numbers and the whole numbers do not. Both exclude fractions and decimals with a fractional part.

Lesson 1.2 — Rational Numbers

Guided practice

19. $\dfrac{6}{1}$ (any equivalent ratio of integers, such as $\tfrac{12}{2}$, is acceptable).
20. $\dfrac{-4}{1}$
21. $0.25 = \dfrac{25}{100} = \dfrac{1}{4}$
22. $\dfrac{3}{8} = 0.375$; it terminates, because the long division reaches a remainder of $0$.
23. $\dfrac{2}{3} = 0.\overline{6} = 0.666\dots$; it repeats.
24. Yes. $49$ is a perfect square, so $\sqrt{49} = 7 = \tfrac{7}{1}$.
25. Yes. $2\tfrac{1}{2} = \dfrac{5}{2}$.
26. Yes. $-\tfrac{5}{6} = \dfrac{-5}{6}$, and $-5$ and $6$ are both integers with the denominator not zero. The definition allows the numerator to be negative.

Independent practice

27. a) $9$: natural, whole, integer b) $0$: whole, integer c) $-6$: integer d) $\tfrac{3}{4}$: none of the Lesson 1.1 subsets e) $-2.5$: none of the Lesson 1.1 subsets
28. a) $\dfrac{12}{1}$ b) $\dfrac{-7}{1}$ c) $0.4 = \dfrac{4}{10} = \dfrac{2}{5}$ d) $1.75 = \dfrac{175}{100} = \dfrac{7}{4}$ e) $\dfrac{0}{1}$
29. a) $\tfrac{5}{8} = 0.625$, terminates b) $\tfrac{1}{9} = 0.\overline{1}$, repeats c) $\tfrac{7}{20} = 0.35$, terminates d) $\tfrac{5}{6} = 0.8\overline{3}$, repeats
30. Rational: $\sqrt{36}$ (it equals $6$), $0.75$ (a terminating decimal, $\tfrac{3}{4}$), and $-11$ (equal to $\tfrac{-11}{1}$). Not rational: $\sqrt{10}$, since $10$ is not a perfect square, and $\pi$, whose decimal never terminates or repeats — both are irrational.
31. A rational number is any number that can be written as a ratio of two integers with a nonzero denominator. Given any integer $n$, write $n = \dfrac{n}{1}$: the numerator $n$ is an integer, the denominator $1$ is an integer, and $1 \neq 0$. So every integer satisfies the definition.
32. Yes, rational. $3.5 = \dfrac{35}{10} = \dfrac{7}{2}$. It is not an integer, because it has a fractional part and lies between $3$ and $4$.
33. $0.325 = \dfrac{325}{1000} = \dfrac{13}{40}$. It is rational and real, and nothing else — it is not an integer, whole number, or natural number.
34. Never ending is not the test. The test is whether the decimal terminates or repeats; either one makes the number rational. $0.\overline{6}$ repeats, and in fact $0.\overline{6} = \tfrac{2}{3}$, a ratio of two integers. It is rational and real.

Exit ticket 1.2

35. $\dfrac{-9}{1}$
36. Yes. $0.45 = \dfrac{45}{100} = \dfrac{9}{20}$.
37. Yes. $25$ is a perfect square, so $\sqrt{25} = 5$, which is $\tfrac{5}{1}$.
38. A repeating decimal can always be rewritten as a fraction of two integers — for instance $0.\overline{3} = \tfrac{1}{3}$ and $0.\overline{36} = \tfrac{4}{11}$ — and being writable as such a fraction is exactly the definition of rational. What would make a decimal irrational is never terminating and never repeating.

Lesson 1.3 — Irrational Numbers

Guided practice

39. Rational. $9$ is a perfect square and $\sqrt{9} = 3 = \tfrac{3}{1}$.
40. Irrational. $11$ is not a perfect square — it lies between $9$ and $16$ — so $\sqrt{11} = 3.31662\dots$, a decimal that never terminates and never repeats.
41. Irrational. $\pi = 3.14159265\dots$ never terminates and never repeats.
42. Irrational. It never terminates, and it has no block of digits that repeats.
43. Between $3$ and $4$, since $\sqrt{9} = 3$ and $\sqrt{16} = 4$.
44. Example: $\sqrt{2}$ (or $\pi$, or $\sqrt{5}$). Non-example: $\sqrt{25}$, because it equals the rational number $5$. (Any terminating or repeating decimal is also an acceptable non-example.)

Independent practice

45. a) rational, $\sqrt{16} = 4$ b) irrational, $17$ is not a perfect square c) rational, $-\sqrt{81} = -9$ d) irrational, $50$ is not a perfect square e) rational, $3.14$ terminates and equals $\tfrac{157}{50}$
46. The closest perfect squares are $49$ below and $64$ above, so $7 = \sqrt{49} < \sqrt{50} < \sqrt{64} = 8$ and $\sqrt{50}$ lies between $7$ and $8$. It is much closer to $7$, because $50$ is much closer to $49$ than to $64$.
47. False. The square root of a perfect square is rational: $\sqrt{36} = 6$, and $6 = \tfrac{6}{1}$.
48. a) $\tfrac{1}{8} = 0.125$, terminates b) $\sqrt{3} = 1.7320508\dots$, neither c) $\tfrac{4}{11} = 0.\overline{36}$, repeats d) $2.7$ terminates
49. $\sqrt{2} = 1.41421356\dots$ goes on forever with no repeating block. Every ratio of two integers produces a decimal that either stops or falls into a repeating block, because long division has only finitely many possible remainders and one must eventually come back around. Since the decimal for $\sqrt{2}$ does neither, no ratio of integers can produce it.
50. Irrational, because $20$ is not a perfect square. Since $16 < 20 < 25$, the side length is between $4$ and $5$ feet.
51. $\pi$ is irrational, so its decimal never terminates and never repeats; multiplying by $5$ shifts the digits but cannot create an ending or a repeating block, so $5\pi = 15.70796\dots$ is irrational too. The value $15.7$ is what you get from replacing $\pi$ with the rational approximation $3.14$. It is a terminating decimal, $\tfrac{157}{10}$, and so it is a rational number close to the exact circumference but not equal to it.
52. First error: $\pi$ does not equal $\tfrac{22}{7}$. The two disagree at the third decimal place, since $\tfrac{22}{7} = 3.142857\dots$ while $\pi = 3.141592\dots$; $\tfrac{22}{7}$ is only an approximation. Second error: even if a number is close to a fraction, that does not make it rational. Rational means the number equals a ratio of two integers, and $\pi$ equals no such ratio, because its decimal never terminates and never repeats.

Exit ticket 1.3

53. Rational. $36$ is a perfect square, so $\sqrt{36} = 6 = \tfrac{6}{1}$.
54. Irrational, since $37$ is not a perfect square. Because $36 < 37 < 49$, it lies between $6$ and $7$.
55. Example: $\pi$ (or $\sqrt{2}$, $\sqrt{11}$, $-\sqrt{7}$). Non-example: $\sqrt{9}$, which equals $3$ and is rational.
56. Write the number in decimal form. If the decimal terminates or repeats, the number is rational, because both kinds can be written as a ratio of two integers. If the decimal never terminates and never repeats, the number is irrational. Both conditions matter: $0.\overline{3}$ never ends but repeats, so it is rational.

Lesson 1.4 — The Whole System: Nesting and Classifying

Guided practice

57. $8$: natural, whole, integer, rational, real
58. $0$: whole, integer, rational, real (not natural)
59. $-15$: integer, rational, real
60. $\tfrac{3}{5}$: rational, real
61. $\sqrt{49} = 7$: natural, whole, integer, rational, real
62. $\sqrt{13}$: irrational, real
63. $-2.25$: rational, real (it equals $\tfrac{-9}{4}$)
64. $\pi$: irrational, real

Independent practice

65. Completed table.

Note the last two rows: $\sqrt{4} = 2$, a natural number, while $40$ is not a perfect square, so $\sqrt{40} = 6.324\dots$ is irrational.

66. a) $-3$ (any negative integer) b) $\tfrac{1}{2}$, or $0.7$, or $-2.5$ c) $\sqrt{2}$, $\pi$, or any irrational number d) $0$, the only such number e) $1$, $5$, $250$, or any counting number
67. a) True. The rationals are one of the two regions inside the real numbers. b) False. $\pi$ is real but not rational. c) True. The definitions are opposites: a number either can be written as a ratio of two integers or cannot, so the two regions do not overlap. d) False. Every integer $n$ equals $\tfrac{n}{1}$, so every integer is rational and no integer is irrational. e) True. Every whole number is an integer, and every integer $n$ equals $\tfrac{n}{1}$.
68. $-6$ goes in the integer region but outside the whole numbers. $\tfrac{5}{8}$ goes in the rational region but outside the integers. $\sqrt{64} = 8$ goes in the natural number region, the innermost one. $\sqrt{65}$ goes in the irrational region. $0$ goes in the whole number region but outside the naturals.
69. Rational means the number can be written as a ratio of two integers with a nonzero denominator. Irrational means it cannot. A single number cannot both be able and unable to be written that way, so the two subsets share no members. In the diagram this is why the two regions sit side by side with no overlap.
70. $12$: natural, whole, integer, rational, real. $-3$: integer, rational, real. $2.5$: rational, real. $\sqrt{2}$: irrational, real. All four are real numbers, since every one of them has a location on the number line.
71. $\sqrt{36} = 6$, which is natural, whole, an integer, rational, and real. $\sqrt{30}$ is irrational and real, because $30$ is not a perfect square; it lies between $5$ and $6$, since $25 < 30 < 36$. Only the first tile has a side that can be measured exactly in whole centimeters: $6$ cm.
72. Drawing the irrationals inside the rationals claims that every irrational number can be written as a ratio of two integers, which is the exact opposite of the definition of irrational. It is false: $\sqrt{2}$ and $\pi$ are irrational, and neither equals any ratio of two integers. The irrational region belongs beside the rational region, both inside the real numbers.

Exit ticket 1.4

73. $-11$: integer, rational, real
74. $\sqrt{81} = 9$: natural, whole, integer, rational, real
75. $\sqrt{83}$: irrational, real. ($83$ is not a perfect square; it lies between $81$ and $100$, so the value is between $9$ and $10$.)
76. The natural numbers sit inside the whole numbers, inside the integers, inside the rationals, inside the reals, so every natural number is carried all the way out to the real numbers. The reverse fails because the outer regions contain numbers the inner one does not: $-4$, $\tfrac{1}{2}$, and $\pi$ are all real but none of them is a counting number.

Chapter 1 Review

Part A — Relationships among the subsets (8.NS.2a)

77. The sketch should match fig2-subset-diagram.png: one large region labeled Real Numbers; inside it two non-overlapping regions labeled Rational Numbers and Irrational Numbers; and inside the rational region three strictly nested regions labeled Integers, then Whole Numbers, then Natural Numbers. Credit requires the nesting order to be right and the irrational region to be separate from the rational region.
78. a) whole (also integer, rational, and real are acceptable) b) integer c) rational d) real
79. Because no number is both rational and irrational. A rational number can be written as a ratio of two integers and an irrational number cannot, so the two collections share no members and their regions cannot overlap. Both are drawn inside the real numbers, since every rational and every irrational number has a location on the number line.
80. $6$: natural region. $0$: whole region, outside the naturals. $-6$: integer region, outside the whole numbers. $6.5$: rational region, outside the integers. $\sqrt{6}$: irrational region.
81. True. The integers were formed from the whole numbers by adding the opposite of each one, so every whole number is still an integer. The reverse is not true, since $-3$ is an integer and not a whole number.
82. The line should show solid points at $-4, -3, -2, -1, 0, 1, 2, 3, 4$. Any non-integer rational is acceptable for the second mark, such as $1.5$ or $-\tfrac{1}{2}$. For the irrational between $1$ and $2$, $\sqrt{2} \approx 1.41$ is the natural choice; $\sqrt{3} \approx 1.73$ also works. The irrational mark must fall strictly between $1$ and $2$ and must not land on a tick mark.
83. Each of the three number lines marks a collection of numbers, and every point marked on the natural-number line is also marked on the whole-number line, which in turn is fully marked on the integer line. That is what "subset" means: every member of the smaller collection is also a member of the larger one. It does not run backwards — the integer line marks points, such as $-3$, that the other two do not.

Part B — Classifying numbers and explaining why (8.NS.2b)

84. a) $14$: natural, whole, integer, rational, real b) $-14$: integer, rational, real c) $0$: whole, integer, rational, real d) $\tfrac{7}{8}$: rational, real e) $-3.75$: rational, real (it equals $\tfrac{-15}{4}$)
85. a) $\sqrt{100} = 10$: natural, whole, integer, rational, real b) $\sqrt{101}$: irrational, real c) $-\sqrt{16} = -4$: integer, rational, real d) $0.\overline{5} = \tfrac{5}{9}$: rational, real e) $\pi$: irrational, real
86. $121$ is a perfect square, since $11 \cdot 11 = 121$, so $\sqrt{121} = 11$. Eleven is a counting number, so it is a natural number — and therefore also whole, an integer, rational, and real.
87. $120$ is not a perfect square: it falls between $100$ and $121$, so $\sqrt{120}$ falls between $10$ and $11$ and is not a whole number. Its decimal, $10.954\dots$, never terminates and never repeats, so it cannot be written as a ratio of two integers.
88. No. The whole numbers are $0, 1, 2, 3, \dots$ and contain no negative numbers, and $-7$ is negative. It is the opposite of the whole number $7$, so the smallest subset in the diagram containing it is the integers.
89. $0.375$ terminates, so the number is rational and real, and nothing narrower — it is not an integer. As a ratio: $0.375 = \dfrac{375}{1000} = \dfrac{3}{8}$.
90. Irrational and real. The decimal never terminates and never repeats, so the number cannot be written as a ratio of two integers. (This number is $\sqrt{7}$.)
91. The natural numbers are the counting numbers $1, 2, 3, \dots$, and counting does not start at $0$, so $0$ is not natural. The complete classification of $0$ is whole, integer, rational, and real.

Part C — Describing each subset with examples and non-examples (8.NS.2c)

92. The natural numbers are the counting numbers: $1, 2, 3, 4, \dots$, going on forever, with no zero, no negatives, and no fractional parts. Examples: $1$ and $17$. Non-examples: $0$, because counting starts at $1$; and $-5$, because it is negative. ($\tfrac{1}{2}$, which is not a whole amount, is also acceptable.)
93. The whole numbers are the natural numbers together with $0$: $0, 1, 2, 3, \dots$. Examples: $0$ and $23$. Non-examples: $-1$ (negative) and $4.5$ (has a fractional part).
94. The integers are the whole numbers together with their opposites: $\dots, -2, -1, 0, 1, 2, \dots$ — every number with no fractional part, positive, negative, or zero. Examples: $-12$ and $7$. Non-examples: $\tfrac{2}{3}$ and $-0.5$, both of which have fractional parts.
95. The rational numbers are the numbers that can be written as a ratio $\tfrac{a}{b}$ of two integers with $b \neq 0$; equivalently, the numbers whose decimal form terminates or repeats. Examples: the fraction $\tfrac{3}{4}$, the decimal $0.375$ (or the repeating decimal $0.\overline{3}$), and the radical $\sqrt{81}$, which equals $9$. Non-examples: $\pi$ and $\sqrt{2}$, whose decimals never terminate and never repeat.
96. The irrational numbers are the real numbers that cannot be written as a ratio of two integers; their decimals never terminate and never repeat. Examples: $\pi$ and $\sqrt{5}$. Non-examples: $\sqrt{49}$, which equals $7$; and $3.14$, which terminates. ($\tfrac{22}{7}$, which repeats, is also acceptable.)
97. One acceptable completion.

Part D — Mixed application and reasoning

98. $45$ is not a perfect square, so $\sqrt{45}$ is irrational and real. Since $36 < 45 < 49$, the side length lies between $6$ and $7$ feet, closer to $7$. The builder writes $6.7$ feet because the exact value $6.7082\dots$ never ends, and a saw and tape measure cannot use an unending decimal; $6.7$ is a rational approximation accurate enough to cut the patio.
99. $-45.50$ is rational and real. Its decimal terminates, and $-45.50 = \dfrac{-91}{2}$. It is not an integer because it has a fractional part — half a dollar — so it sits strictly between the integers $-46$ and $-45$ on the number line.
100. No. Every integer $n$ can be written as $\tfrac{n}{1}$, a ratio of two integers, so every integer is rational. An irrational number is by definition one that cannot be written as such a ratio. In the diagram, the integer region lies entirely inside the rational region, and the rational and irrational regions do not overlap, so nothing can be in both.
101. The radical sign says "take the square root," and that operation gives a rational result whenever the number underneath is a perfect square. $\sqrt{144} = 12$, a natural number and therefore rational. By contrast $145$ is not a perfect square — it falls between $144$ and $169$ — so $\sqrt{145} = 12.0415\dots$ never terminates or repeats and is irrational. The test is the number under the radical, not the radical sign.
102. $\pi$ is irrational, and multiplying it by $16$ cannot make its decimal stop or fall into a repeating block, so the exact area $16\pi = 50.265\dots$ square inches is irrational. The value $50.24$ comes from replacing $\pi$ with the rational approximation $3.14$, since $16(3.14) = 50.24$. That result is a terminating decimal, equal to $\tfrac{1256}{25}$, so it is rational — and it is close to the exact area without equaling it.

Workbook-only items

Page 2, fill in the blanks. The natural numbers are also called the counting numbers. The whole numbers are the naturals together with $0$. The integers are the whole numbers together with their opposites. The symbol $\subset$ is read "is a subset of." The chain is $\text{natural} \subset \textbf{whole} \subset \textbf{integers}$.

Page 2, circle table. Row 1 (natural): circle $3$ only. Row 2 (whole): circle $3$ and $0$. Row 3 (integers): circle $3$, $0$, and $-5$.

Page 6, fill in the frames. $7 = \dfrac{7}{\mathbf{1}}$, $-4 = \dfrac{\mathbf{-4}}{1}$, $0 = \dfrac{0}{\mathbf{1}}$ (any nonzero integer denominator works for $0$). A decimal that ends is a terminating decimal. A decimal with a block repeating forever is a repeating decimal. Before classifying a radical, always simplify it first.

Page 10, fill in the frames. $0.375$ terminates, so it is rational. $0.\overline{36}$ repeats, so it is rational. $1.41421356\dots$ never terminates and never repeats, so it is irrational. The trapping frame is $\mathbf{3} = \sqrt{9} < \sqrt{11} < \sqrt{16} = \mathbf{4}$. The comparison frame is $\pi = 3.1415\mathbf{9}\dots$, which differs from $\tfrac{22}{7} = 3.142857\dots$ at the third decimal place.

Page 14, labeling and chain. Labels from outside in: Real Numbers; then Rational Numbers beside Irrational Numbers; then Integers, Whole Numbers, and Natural Numbers nested inside the rationals. The chain is $\text{natural} \subset \textbf{whole} \subset \textbf{integer} \subset \textbf{rational} \subset \text{real}$. The irrational region is drawn beside the rational region, never inside it, because no number is both.

Page 21, item 92 frame. The natural numbers are the counting numbers $1, 2, 3, \dots$, with no zero, no negatives, and no fractional parts. See item 92 above for examples and non-examples.

Page 23, item 101 frames. $\sqrt{144} = \mathbf{12}$, which is a natural number, and so rational. $\sqrt{145}$ is irrational, because $145$ is not a perfect square, so its decimal never terminates and never repeats.