Appendix A — Answer Key, Chapter 1: Integers and the Number Line

SOL 6.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. Reasoning answers show an acceptable response, not the only wording.

Lesson 1.1 — Representing Integers

Guided practice

1. a) $-14$ b) $50$ c) $-7$
2. Three strokes under par.
3. $11$
4. No. Integers are whole numbers and their opposites; $-4.5$ falls between $-5$ and $-4$, so it is not a whole number or the opposite of one.
5. $5$

Independent practice

6. a) $9$ b) $-200$ c) $0$
7. Zero is the dividing point between positive and negative. It is not greater than zero, so it is not positive; it is not less than zero, so it is not negative.
8. $6$ and $-6$. Each is 6 units from zero, so they are 12 units apart in total. (Any pair $n$ and $-n$ with $n = 6$ works; this is the only such pair.)
9. $-3$, $-2$, $-1$, $0$, $1$
10. At 6 a.m. it was 6 degrees below zero. At noon it was 5 degrees above zero.
11. Down 2 is $-2$; up 7 is $7$. It ends on floor $5$.
12. He compared the digits instead of the positions. On a number line $-20$ sits farther left than $-2$, and anything farther left is less. So $-20 < -2$. In context, owing $20 is worse than owing $2.

Exit ticket 1.1

1. $-35$
2. $-8$
3. Integers: $-3$, $0$, $12$
4. The negative sign tells you the direction — below zero, or the left side of the number line. The $14$ tells you how far from zero it is.

Lesson 1.2 — Comparing and Ordering Integers

Guided practice

1. $2$
2. $-14$
3. $-8$, $-3$, $1$, $6$
4. $-7$
5. True. Every negative integer lies to the left of zero, and anything to the left is less.

Independent practice

6. $-11$, $-7$, $-2$, $0$, $5$
7. $8$, $-1$, $-6$, $-13$
8. $-4$, $-3$, $-2$
9. $-99$ is greater, because it lies to the right of $-100$ on the number line.
10. Deepest to shallowest: $-42$, $-25$, $-18$, $-7$. The greatest integer is $-7$. "Greatest" and "deepest" are opposite here because greater elevation means closer to the surface — the deepest diver has the least elevation.
11. Coldest to warmest: $-12$°F, $-9$°F, $-3$°F, $0$°F, $8$°F
12. Digits alone are misleading for negatives, because larger digits mean farther from zero, which on the left side means smaller. A number line removes the guesswork: you only have to read positions left to right.

Exit ticket 1.2

1. $-2$
2. $-8$, $-5$, $0$, $3$
3. $-3$
4. The correct statement is $-6 < -1$. The error is comparing 6 to 1 instead of comparing positions; $-6$ is farther left, so it is less.

Lesson 1.3 — Comparing Integers with Symbols

Guided practice

1. $-4 < 6$
2. $-12 < -3$
3. $0 > -7$
4. $-2 < 5$
5. False. $-6$ lies to the right of $-9$, so $-6 > -9$.

Independent practice

6. a) $-9 > -11$ b) $0 < 4$ c) $-5 = -5$ d) $7 > -7$
7. a) $8 > -3$ b) $-20 < -1$
8. $-14 > -15$ (or equivalently $-15 < -14$)
9. a) False → $-2 < 0$ b) True c) False → $-30 < -29$
10. $-15 < -6$. The hiker at $-6$ ft is higher, since $-6$ is the greater elevation.
11. $-1 > -3$. The foul costs less, because losing 1 point leaves a greater score than losing 3.
12. Because $-100$ sits 100 units to the left of zero and $-1$ sits only 1 unit to the left. Farther left means less, so the larger digits produce the smaller number.

Exit ticket 1.3

1. $-7 < -2$
2. $-9 < 0$
3. $5 > -4$
4. Find which number is farther right on the number line; that one is greater, and the open end of the symbol faces it.

Lesson 1.4 — Absolute Value

Guided practice

1. $9$
2. $15$
3. $0$
4. $3$ and $-3$
5. $-4$, since $|-4| = 4$ and $|2| = 2$.

Independent practice

6. a) $21$ b) $8$ c) $1$ d) $100$
7. $12$ and $-12$
8. True. Opposites are the same distance from zero, and absolute value reports that distance.
9. $|-2| = 2$, $|3| = 3$, $|5| = 5$, $|-7| = 7$ → $2$, $3$, $5$, $7$
10. No. Absolute value is a distance from zero, and a distance cannot be negative.
11. $|-31| = 31$ and $|-24| = 24$, so the diver at $-31$ ft is farther from the surface. Comparing elevations, $-31 < -24$. The answers point opposite ways because being farther from zero downward makes the elevation less, not greater.
12. Counterexample: $a = -9$, $b = 4$. Then $a < b$ is true, but $|a| = 9$ and $|b| = 4$, so $|a| > |b|$. Her claim fails because $<$ compares position while absolute value compares distance from zero, and a negative number can be far from zero while still being small.

Exit ticket 1.4

1. $13$
2. $9$ and $-9$
3. $-15$, since $|-15| = 15$ and $|12| = 12$.
4. It measures distance from zero, and distance is never negative. The smallest it can be is $0$, for the number $0$ itself.

Chapter 1 Review

Part A — Representing integers (6.NS.2a)

1. a) $-22$ b) $14$ c) $-5$
2. a) $17$ b) $-6$ c) $0$
3. Integers: $-8$, $0$, $41$
4. $-6$

Part B — Comparing and ordering (6.NS.2b, 6.NS.2c)

5. $-12$, $-4$, $-1$, $0$, $7$
6. $2$, $-3$, $-8$, $-19$
7. a) $-6 > -13$ b) $-2 < 0$ c) $-10 = -10$
8. $9 > -5$
9. $-5$, $-4$, $-3$

Part C — Absolute value (6.NS.2d)

10. a) $16$ b) $0$ c) $45$
11. $20$ and $-20$
12. $-9$, since $|-9| = 9$ and $|7| = 7$.

Part D — Mixed application and reasoning

13. Coldest to warmest: $-15$°F, $-7$°F, $0$°F, $4$°F. Farthest from zero: $-15$°F, because $|-15| = 15$ is the greatest distance.
14. The submarine is farther from sea level, since $|-90| = 90$ and $|60| = 60$. The greater elevation is the drone's $60$ ft, because $60 > -90$. The answers differ because distance from zero ignores direction while comparing elevations does not.
15. On a number line, values increase left to right. Negative integers with larger digits sit farther to the left, so they are less. For example $-8$ has larger digits than $-2$ but lies farther left, so $-8 < -2$.
16. $13$ and $-13$. A comparison: $-13 < 13$.

Workbook-only items

Page 2, fill in the blanks. Positive integers are greater than zero and sit to the right of zero. Negative integers are less than zero and sit to the left of zero. Zero is neither positive nor negative.

Page 2, table. $-12$; $75$; $-40$; $8$; $-6$; $0$

Page 3, opposites table. $7$; $-14$; $25$; $0$; $-9$; $100$

Page 3, circle the integers. Integers: $-6$, $0$, $18$, $-41$, $7.0$ (equals 7). Not integers: $2.5$, $-\tfrac{3}{4}$, $\tfrac{1}{2}$.

Page 3, explain. Zero is its own opposite because it is $0$ units from itself, so there is no other number the same distance from zero on the other side.

Page 4, context table. Temperature: zero means zero degrees; $-20$ means 20 degrees below zero. Elevation: zero means sea level; $-20$ means 20 units below sea level. Bank account: zero means a balance of nothing owed or held; $-20$ means overdrawn by $20. Yards: zero means no gain or loss; $-20$ means a 20-yard loss.

Page 4, sentences. $-9$°F: nine degrees below zero. $-55$ ft: fifty-five feet below sea level. $-30$ dollars: overdrawn by thirty dollars.

Page 4, challenge. Move 1: $-3$. Move 2: $8$. Ends on floor $5$.

Page 6, circle the greater. a) $2$ b) $-7$ c) $0$ d) $-1$ e) $6$ f) $-29$

Page 6, watch out. $-2$ is greater. Larger digits on the negative side mean farther left of zero, and farther left is less.

Page 7, plot and label. Left to right: $-6$, $-2$, $0$, $3$, $5$ — already least to greatest.

Page 9, fill in. a) $<$ b) $<$ c) $>$ d) $>$ e) $=$ f) $>$ g) $>$ h) $<$ i) $<$

Page 9, rewrite. $-3 < 8$ becomes $8 > -3$. $5 > -2$ becomes $-2 < 5$.

Page 10, error hunt. $-2 > 0$ false → $-2 < 0$. $-8 < -3$ true. $-30 > -29$ false → $-30 < -29$. $-5 < 1$ true. $-7 = 7$ false → $-7 < 7$.

Page 10, apply. $-15 < -6$; the hiker at $-6$ ft is higher.

Page 12, evaluate. a) $9$ b) $15$ c) $0$ d) $21$ e) $8$ f) $1$ g) $100$ h) $45$

Page 12, both integers. $3$ and $-3$; $12$ and $-12$; $20$ and $-20$.

Page 13, two questions table.

Page 13, order. $2$, $3$, $5$, $7$

Page 13, challenge. $a = -9$, $b = 4$ (any negative with larger absolute value than a positive $b$ works). Then $a < b$ but $|a| > |b|$.