Chapter 1 — Integers and the Number Line

Standard: 6.NS.2 — The student will reason and use multiple strategies to represent, compare, and order integers.

By the end of this chapter you will be able to:

• Represent integers on a number line and with models, including models that come from real situations (6.NS.2a)
• Compare and order integers using a number line (6.NS.2b)
• Compare integers using the symbols <, >, and = (6.NS.2c)
• Describe the absolute value of an integer as its distance from zero (6.NS.2d)

Lessons: 1.1 Representing Integers · 1.2 Comparing and Ordering on a Number Line · 1.3 Comparing with Symbols · 1.4 Absolute Value

Lesson 1.1 — Representing Integers

Why we need new numbers

In earlier grades every number you counted with was zero or greater. But plenty of real situations need numbers on both sides of zero. A scuba diver 30 feet below the surface. A checking account overdrawn by 12 dollars. A temperature 8 degrees below zero. In each case, "how far" is not enough — you also need "which direction."

The integers are the whole numbers together with their opposites:

$$\ldots,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ \ldots$$

A positive integer is greater than zero. A negative integer is less than zero and is written with a negative sign, like $-7$. Zero is an integer, but it is neither positive nor negative — it is the dividing point.

Integers do not include fractions or decimals. So $-6$, $0$, and $19$ are integers; $2.5$ and $-\tfrac{3}{4}$ are not.

Integers on a number line

A number line is the most useful model for integers. Zero sits in the middle, positive integers extend to the right, and negative integers extend to the left.

Two features of this picture matter for everything that follows:

1. Values increase to the right. Any number is greater than every number to its left.
2. The spacing is equal. The distance from $0$ to $1$ is the same as the distance from $-6$ to $-5$. Equal spacing is what lets you compare distances later in this chapter.

Reading the negative sign. In $-7$, the sign is part of the number and is read "negative seven," not "minus seven." Save the word minus for subtraction.

Opposites

Two integers are opposites if they are the same distance from zero but on opposite sides of it.

The opposite of $4$ is $-4$, and the opposite of $-4$ is $4$. The opposite of $0$ is $0$, since zero is the only integer that is its own opposite.

Integers in context

Many real quantities have a natural zero with meaningful values on both sides. Choosing which direction is positive is a decision you make and then state.

Notice that a number line can be drawn vertically when the situation is vertical. For elevation, up is positive and down is negative, which matches how we already think about above and below sea level.

Worked examples

Example 1 — Writing an integer for a situation

A cave floor is 45 feet below sea level. Write an integer for this elevation.

Below sea level is the negative direction, and the distance is 45 feet.

Answer: $-45$

Example 2 — Interpreting an integer in context

A submarine's depth is recorded as $-120$ feet. What does this mean?

The sign says below sea level; the 120 says how far.

Answer: The submarine is 120 feet below sea level.

Example 3 — Identifying a point on a number line

On the number line in the figure above, a point sits three tick marks to the left of zero. What integer does it represent?

Each tick mark is one unit, and left of zero is negative.

Answer: $-3$

Example 4 — Opposites

Write the opposite of each integer: $9$, $-15$, $0$.

Same distance from zero, other side.

Answer: $-9$; $15$; $0$

Example 5 — Choosing a direction

A team loses 6 yards on a play, then gains 6 yards on the next. Write an integer for each play and explain what the pair shows.

A loss is negative and a gain is positive.

Answer: $-6$ and $6$. They are opposites, so together the team is back where it started at $0$.

Guided practice

1. Write an integer for each situation. a) 14 degrees below zero b) a deposit of $50 c) 7 seconds before liftoff
2. What does $-3$ mean if it describes a golf score compared to par?
3. Write the opposite of $-11$.
4. Is $-4.5$ an integer? Explain how you know.
5. A point is 5 units to the right of zero on a number line. What integer is it?

Independent practice

6. Write an integer for each: a) a 9-yard gain b) 200 feet below sea level c) breaking even
7. Explain why zero is neither positive nor negative.
8. Name two integers that are opposites and 12 units apart in total.
9. List all integers between $-4$ and $2$.
10. A thermometer reads $-6$°F at 6 a.m. and $5$°F at noon. Write a sentence describing each temperature without using the negative sign.
11. Application. An elevator starts on the ground floor (0), goes down 2 levels to a parking garage, then up 7 levels. Write an integer for each of the two moves, and name the floor it ends on.
12. Reasoning. Marcus says $-20$ is bigger than $-2$ because 20 is bigger than 2. Explain his mistake using a number line.

Exit ticket 1.1

1. Write an integer for "35 feet below sea level."
2. What is the opposite of $8$?
3. Which of these are integers: $-3$, $0.5$, $0$, $-\tfrac{1}{2}$, $12$?
4. Explain what the sign in $-14$ tells you and what the 14 tells you.

Lesson 1.2 — Comparing and Ordering Integers on a Number Line

The left–right rule

The number line gives one rule that settles every comparison:

On a number line, the number farther to the right is always greater.

That rule holds for negatives too, which is where it does the most work. Look at $-8$ and $-3$. Since $-3$ sits to the right of $-8$, we know $-3$ is greater than $-8$ — even though 8 looks like the "bigger" digit. This is the single most common error with integers, and the number line is the fix. A debt of $8 leaves you worse off than a debt of $3.

Ordering a set of integers

To order a set of integers, plot every value, then read them off left to right for least-to-greatest.

Reading left to right: $-7$, $-3$, $0$, $2$, $5$. That is least to greatest. Reading right to left gives greatest to least.

Two patterns are worth naming, because they let you sort quickly before you plot:

• Every negative integer is less than every positive integer, and less than zero.
• Among negatives, the one with the larger digits is the smaller number. So $-9 < -4$.

Worked examples

Example 1 — Comparing two negatives

Which is greater, $-12$ or $-7$?

On a number line, $-7$ is to the right of $-12$.

Answer: $-7$ is greater.

Example 2 — Ordering least to greatest

Order $4$, $-6$, $0$, $-1$, $3$ from least to greatest.

Negatives first (most negative leftmost), then zero, then positives.

Answer: $-6$, $-1$, $0$, $3$, $4$

Example 3 — Ordering greatest to least

Order $-15$, $-2$, $-20$, $-9$ from greatest to least.

All are negative, so the one closest to zero is greatest.

Answer: $-2$, $-9$, $-15$, $-20$

Example 4 — Context comparison

On Monday the low was $-4$°F. On Tuesday it was $-11$°F. Which day was colder?

$-11$ is farther left, so it is the lesser temperature.

Answer: Tuesday was colder.

Example 5 — Between two integers

Name all integers greater than $-3$ and less than $2$.

Move right from $-3$, stopping before $2$.

Answer: $-2$, $-1$, $0$, $1$

Guided practice

1. Which is greater, $-5$ or $2$?
2. Which is less, $-14$ or $-4$?
3. Order from least to greatest: $-3$, $6$, $-8$, $1$.
4. Name the integer directly to the left of $-6$ on a number line.
5. True or false: every negative integer is less than zero. Explain.

Independent practice

6. Order from least to greatest: $-2$, $-11$, $5$, $0$, $-7$.
7. Order from greatest to least: $-1$, $-13$, $8$, $-6$.
8. List all integers between $-5$ and $-1$.
9. Which integer is greater, $-100$ or $-99$? Explain using the number line.
10. Application. Four divers are at $-18$ ft, $-42$ ft, $-7$ ft, and $-25$ ft. List them from deepest to shallowest, then state which integer is greatest and explain why "greatest" and "deepest" are not the same thing here.
11. Application. Overnight lows for five cities were $-3$°F, $8$°F, $-12$°F, $0$°F, and $-9$°F. Order them from coldest to warmest.
12. Reasoning. Explain why a number line makes comparing negative integers easier than comparing them by looking at the digits.

Exit ticket 1.2

1. Which is greater, $-9$ or $-2$?
2. Order least to greatest: $3$, $-5$, $0$, $-8$.
3. Name an integer between $-4$ and $-2$.
4. A friend says $-6 > -1$. Correct the statement and explain the error.

Lesson 1.3 — Comparing Integers with Symbols

The three comparison symbols

The open end of $<$ or $>$ always faces the greater number, and the point always faces the lesser one. So the same fact can be written two ways: $-7 < 3$ and $3 > -7$ say exactly the same thing. Which one you write depends on which number you start with.

A reliable procedure

1. Locate both integers on a number line (mentally once you are comfortable).
2. Decide which is farther right.
3. Write the symbol so its open end faces that number.
4. Read the finished statement aloud to check it is true.

Step 4 catches most mistakes. "Negative nine is greater than negative two" sounds wrong when you say it, and it is.

Worked examples

Example 1 — Compare $-6$ and $1$ using a symbol.

$1$ is farther right, so it is greater. The open end faces the $1$.

Answer: $-6 < 1$ (equivalently $1 > -6$)

Example 2 — Compare $-3$ and $-10$.

$-3$ is farther right.

Answer: $-3 > -10$

Example 3 — Fill in the blank: $-8 \ \underline{\ \ } \ -8$.

Same number on both sides.

Answer: $-8 = -8$

Example 4 — Write two true statements comparing $-15$ and $-2$.

Answer: $-15 < -2$ and $-2 > -15$

Example 5 — Context

Kara's account balance is $-\$25$ and Dev's is $-\$40$. Write a comparison with a symbol and say who is better off.

$-25$ is farther right than $-40$, so $-25 > -40$.

Answer: $-25 > -40$; Kara is better off, since a $25 overdraft is smaller than a $40 overdraft.

Guided practice

1. Fill in $<$, $>$, or $=$: $-4 \ \underline{\ \ } \ 6$
2. Fill in: $-12 \ \underline{\ \ } \ -3$
3. Fill in: $0 \ \underline{\ \ } \ -7$
4. Rewrite $5 > -2$ using the $<$ symbol.
5. Is $-6 < -9$ true or false? Explain.

Independent practice

6. Fill in $<$, $>$, or $=$ for each: a) $-9 \ \underline{\ \ } \ -11$ b) $0 \ \underline{\ \ } \ 4$ c) $-5 \ \underline{\ \ } \ -5$ d) $7 \ \underline{\ \ } \ -7$
7. Rewrite each statement the other way: a) $-3 < 8$ b) $-1 > -20$
8. Write a true comparison statement using $-14$ and $-15$.
9. Which statements are false? Correct each one: a) $-2 > 0$ b) $-8 < -3$ c) $-30 > -29$
10. Application. Two hikers are at elevations $-15$ ft and $-6$ ft. Write a comparison using a symbol and explain which hiker is higher.
11. Application. A game scores $-3$ for a miss and $-1$ for a foul. Write a comparison and explain which penalty costs less.
12. Reasoning. Explain why $-100 < -1$ even though 100 is greater than 1.

Exit ticket 1.3

1. Fill in $<$, $>$, or $=$: $-7 \ \underline{\ \ } \ -2$
2. Fill in: $-9 \ \underline{\ \ } \ 0$
3. Rewrite $-4 < 5$ using $>$.
4. Explain how you decide which way the symbol points.

Lesson 1.4 — Absolute Value

Distance from zero

Sometimes the direction does not matter — only how far. A diver 30 feet below the surface and a kite 30 feet above the launcher are both 30 feet from the reference point.

The absolute value of an integer is its distance from zero on the number line. Because distance is never negative, absolute value is never negative.

We write absolute value with a pair of vertical bars: $|-5|$ is read "the absolute value of negative five."

Both $-5$ and $5$ sit 5 units from zero, so:

$$|-5| = 5 \qquad \text{and} \qquad |5| = 5$$

Three facts to hold onto

1. Opposites have equal absolute value. $|-12| = |12| = 12$.
2. $|0| = 0$. Zero is zero units from itself.
3. Absolute value is a distance, not a comparison. $-9 < -2$ is true, and at the same time $|-9| > |-2|$ is also true, because $9 > 2$. The lesser number can be farther from zero. Keeping these two questions separate — which is greater? versus which is farther from zero? — is the heart of this lesson.

Careful with the bars. The expression $|-5|$ asks for a distance and equals $5$. It does not "cancel" the sign as a rule to memorize; it reports how far $-5$ is from zero.

Worked examples

Example 1 — Find $|-14|$.

$-14$ is 14 units from zero.

Answer: $14$

Example 2 — Find $|23|$ and $|0|$.

Answer: $23$ and $0$

Example 3 — Which is farther from zero, $-8$ or $6$?

$|-8| = 8$ and $|6| = 6$. Since $8 > 6$:

Answer: $-8$ is farther from zero.

Example 4 — Name all integers whose absolute value is $7$.

Two integers are 7 units from zero, one on each side.

Answer: $7$ and $-7$

Example 5 — Comparing value and distance

For $-11$ and $4$: which integer is greater, and which is farther from zero?

$4$ is farther right, so $4 > -11$. But $|-11| = 11$ and $|4| = 4$, so $-11$ is farther from zero.

Answer: $4$ is greater; $-11$ is farther from zero.

Example 6 — Context

Two temperatures differ from freezing (0°C) by these amounts: $-6$°C and $9$°C. Which is farther from freezing?

$|-6| = 6$ and $|9| = 9$.

Answer: $9$°C is farther from freezing.

Guided practice

1. Find $|-9|$.
2. Find $|15|$.
3. Find $|0|$.
4. Name both integers whose absolute value is $3$.
5. Which is farther from zero, $-4$ or $2$?

Independent practice

6. Evaluate: a) $|-21|$ b) $|8|$ c) $|-1|$ d) $|-100|$
7. Name all integers with absolute value $12$.
8. True or false, with a reason: $|-6| = |6|$.
9. Order $|-7|$, $|3|$, $|-2|$, $|5|$ from least to greatest.
10. Is there an integer whose absolute value is $-4$? Explain.
11. Application. Two divers are at $-24$ ft and $-31$ ft. Use absolute value to say which is farther from the surface, then write an inequality comparing the two elevations. Explain why the two answers point in opposite directions.
12. Reasoning. Lena says that if $a < b$ then $|a| < |b|$ must be true. Give a counterexample with integers and explain what went wrong.

Exit ticket 1.4

1. Find $|-13|$.
2. Name both integers whose absolute value is $9$.
3. Which is farther from zero, $-15$ or $12$?
4. Explain why absolute value can never be negative.

Chapter 1 Review

Vocabulary. integer · positive integer · negative integer · opposites · number line · absolute value

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

1. Write an integer for each: a) 22 feet below sea level b) a gain of 14 yards c) 5 minutes before the start
2. Write the opposite of each: a) $-17$ b) $6$ c) $0$
3. Which of these are integers: $-8$, $3.2$, $0$, $-\tfrac{2}{5}$, $41$?
4. A point on a number line is 6 units left of zero. Name the integer.

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

5. Order from least to greatest: $-4$, $7$, $-12$, $0$, $-1$.
6. Order from greatest to least: $-8$, $-19$, $2$, $-3$.
7. Fill in $<$, $>$, or $=$: a) $-6 \ \underline{\ \ } \ -13$ b) $-2 \ \underline{\ \ } \ 0$ c) $-10 \ \underline{\ \ } \ -10$
8. Rewrite $-5 < 9$ using $>$.
9. List all integers between $-6$ and $-2$.

Part C — Absolute value (6.NS.2d)

10. Evaluate: a) $|-16|$ b) $|0|$ c) $|-45|$
11. Name both integers whose absolute value is $20$.
12. Which is farther from zero, $-9$ or $7$?

Part D — Mixed application and reasoning

13. Overnight lows: $-7$°F, $4$°F, $-15$°F, $0$°F. Order coldest to warmest, then name which is farthest from zero.
14. A submarine at $-90$ ft and a drone at $60$ ft. Which is farther from sea level? Which elevation is greater? Explain why these have different answers.
15. Explain, using a number line, why the integer with the larger digits is not always the greater integer.
16. Two integers are opposites. One is 13 units from zero. Name both integers and write a comparison statement using a symbol.

Standards coverage check — Chapter 1

Answer keys for every set in this chapter are in Appendix A.