CHAPTER 3 - INTEGERSLesson 1: Integers and Absolute Value

(NO CALCULATORS NEEDED)Identify and Graph IntegersIntegers can be ________________ on a number line. To graph an integer on the number line, draw a dot on the line at its location.

Example 1:Write an integer for each situation.a. An average temperature of 5 degrees below normalBecause it represents below normal, the integer is -5

b. An average rainfall of 5 inches above normal.Because it represents above normal, the integer is +5 or 5.

Got it? 1Write an integer for each situation. a. 6 degrees above normal b. 2 inches below normal

Example 2:Graph the set of integers {4, -6, 0} on a number line.

Got it? 2Graph each set of integers on a number line. a. {-2, 8, -7} b. {-4, 10, -3, 7}

Absolute Value:

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Positive Negative

On the number line in the box, notice that -5 and 5 are each 5 units from 0, even though they are on opposites sides of 0. Numbers that are the same distance from zero on the number line have the same _________________________ _______________.

Order of Operations with Absolute

Value:

1. Simplify the _________________

2. Multiply or Divide3. ____________ or ____________

Example 3:Evaluate each expression. a. -4The graph of -4 is 4 units from 0.So, -4 = 4

b. -5 - 2-5 - 25 – 23

Got it? 3Evaluate these expressions.a. 8 b. 2 + -3 c. -6 - 5

Example 4:Nick climbs 30 feet up a rock wall and then climbs 22 feet down to a landing area. The number of feet Nick climbs can be represented using the expression 30 + -22. How many feet does Nick climb?

30 + -22 = 30 + -22= 30 + 22

= 52

Guided Practice:Write an integer for each situation.1. a deposit of $16 ______________ 2. 6F below zero -_____________

Evaluate each expression.3. 18 - -10= ______________ 4. -11 - -16= _____________

5. Graph the set of integers {11, -5, -8} on a number line.

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Journal: Why is the absolute value of a number positive? Use complete sentences.

LESSON 2: ADD INTEGERSAdd Integers with the Same Sign:

(Notes)

Example 1:a. Find -3 + (-2).

Start at 0. Move 3 units down to show -3.From there, move 2 units down to show -2.

b. Find -26 + (-17).-26 + (-17) = -43

Got it? 1a. -5 + (-7) b. -10 + (-4) c. -14 + (-16)

Add Integers with Different Signs: (Notes)

If the positive number is greater, then the answer is positive. If the negative number is greater, then the answer is negative.

Example 2:a. Find 5 + (-3)So, 5 + (-3) = 2

b. Find -3 + 2So, -3 + 2 = -1

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Got it? 2a. Find 6 + (-7) b. Find -15 + 19

Example 3:a. Find 7 + (-7)7 + (-7) = 0

b. Find -8 + 3-8 + 3 = -5

c. Find 2 + (-15) + (-2)2 + (-2) + (-15)0 + (-15)= -15

Commutative Properties Associative Properties Identity Properties

Got it? 3a. 10 + (-12) b. -13 + 18 c. (-14) + (-6) + 6

Example 4:A roller coaster starts at point A. It goes up 20 feet, down 32 feet, and then up 16 feet to point B. Write an addition sentence to find the height at point B in relation to point A. Then find the sum and explain its meaning.

20 +(-32) + 16 = 20 + 16 + (-32)= 36 + (-32)

= 4Point B is 4 feet higher than point A.

Got it? 4:The temperature is -3. An hour later, it drops 6 and 2 hours later it rises 4. Write an addition expression to describe this situation. Then find the sum and explain its meaning.

Guided Practice:1. -6 + (-8) = _______ 2. -3 + 10 = ________ 3. -8 + (-4) + 12 = ________

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4. Sofia owes her brother $25. She gives her brother the $18 she earned dog-sitting. Write an addition expression to describe this situation. Then find the sum and explain its meaning.

Journal: How can you tell whether a sum is zero with actually adding?ESSON 3-3: SUBTRACT INTEGERS

Words: To subtract an integer, __________ its additive inverse.Additive Inverse = _______________________

MATCH EACH NUMBER TO ITS ADDITIVE INVERSE.

--------------------------------------------------------------------------------------------------- (Notes)

Example 1:a. Find 8 – 13.

8 – 13 = 8 + (-13)= -5

b. Find -10 – 7.-10 – 7 = -10 + (-7)

= -27

Got it? 1

5

-6

6

9.3

-9.3

2/3-2/3

a. 6 – 12 b. -20 – 15 c. -22 – 26

Example 2:a. Find 1 – (-2).

1 – (-2) = 1 + 2= 3

b. Find -10 – (-7).-10 – (-7) = -10 + 7

= -3Got it? 2a. 4 – (-12) b. -15 – (-5) c. 18 – (-6)

Example 3:a. Evaluate x – y if x = -6 and y = -5

x – y = -6 – (-5)= -6 + 5

= -1

b. Evaluate m – n if m = -15 and n= 8

m – n = -15 – 8= -15 + (-8)

= -23

Got it? 3Evaluate each expression if a = 5, b = -8, and c = -9.a. b – 10 b. a – b c. c – a

Example 4:The temperatures on the Moon vary from -173C to 127C. Find the difference between the maximum and minimum temperatures.

Subtract the lower temperatures from the higher temperature.127 – (-173) = 127 + 173

= 300.The difference between the two temperatures is 300C.

Got it? 4Brenda had a balance of -$52 in her account. The bank charged her a fee of $10 for having a negative balance. What is her new balance?

Guided Practice:1. 14 – 17 = ___________ 2. 14 – (-10) = ____________

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3. Evaluate q – r if q = -14 and r = -6.

4. The sea surface temperatures range from -2C. Find the difference between the maximum and minimum temperatures.

Lesson 3-4: Multiply IntegersMultiply Integers with Different SignsWords: The product of two integers with different signs is _________________________.

Examples: 6(-4) = -24 -5(7) = -35

Remember that multiplication is the same as repeated addition. 4(-3) = (-3) + (-3) + (-3) + (-3) = -12

Example 1:a. Find 3(-5). 3(-5) = -15 Different signs, negative

b. Find -6(8).-6(8) = -48 Different signs, negative

Got it? 1a. 9(-2) = b. -7(4) =

Multiply Integers with the Same SignsWords: The product of two integers with same signs is __________________________.

Examples: 2(6) = 12 -10(-6) = 60

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Example 2:a. Find -11(-9)

-11(-9) = 99 Same sign, positive

b. Find (-4)2

(-4)(-4) = 16 Same signs, positive

c. Find -3(-4)(-2)-3(-4) = 12 Same signs, positive

12(-2) = -24 Different signs, negative

The product of two positive integers is positive. You can use a pattern to find the sign of products of two negative integers. Start with (2)(-3) = -6 and (1)(-3) = -3.

Each product is 3 more than the previous. This pattern can also be shown on a number line.

If you extend the pattern, the next two products are (-3)(-3) = 9 and (-4)(-3) = 12.

Got it? 2a. -12(-4) b. (-5)2 c. -7(-5)(-3)

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Example 3:A submersible is diving from the surface of the water at a rate of 90 feet per minute. What is the depth of the submersible after 7 minutes.

The submersible descends 90 feet per minute. After 7 minutes, the vessel will be at 7(-90) or -630 feet.

The submersible will be 630 feet below sea level.

Got it? 3Mr. Simon’s bank automatically deducts a $4 monthly maintenance fee from his savings account. Write a multiplication expression to represent the maintenance fees for one year. Then find the product and explain its meaning.

Guided Practice:Multiply.1. 6(-10) = ____________ 2. (-3)3 = ___________ 3. (-1)(-3)(-4) = ___________

4. Tamera owns 100 shares of a certain stock. Suppose the price of the stock drops by $3 per share. Write a multiplication expression to find the change in Tamera’s investment. Explain your answer.

Journal: Write an expression that has at least 2 negative and 2 positive and the product is positive.

Lesson 3-5: Divide IntegersComplete the table.Multiplication Sentence

Division Sentences

Same Sign or Different Sign?

Quotient

Positive or Negative?

2 x 6 = 12 12 6 = 2 Same sign 2 Positive12 2 = 6 Same sign 6 Positive

2 x (-4) = -8

-3 x 5 = -15

-2 x (-5) = 10

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Divide Integers with Different SignsWords: The quotient of two integers with different signs is ______________________.

Examples: 33 (-11) = 3 -64 8 = -8

Example 1:a. Find 80 (-10).80 (-10) = -8 Different signs, negative

b. Find −5511 .-55 11 = -5 Different signs, negative

Example 2:Use the table to find the constant rate of change in centimeters per hour. The height of the candle decreases by 2 centimeters each hour.

change∈heightchange∈hours

=−21

So the constant rate of change is -2 centimeters per hour.

Got it? 1 & 2a. 20 (-4) = b. −819 = c. -45 9 =

Divide Integers with Same SignWords: The quotient of two integers with the same signs is ____________________.

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Examples: 15 5 = 3 -64 (-8) = 8

Example 3:a. Find -14 (-7).-14 (-7) = 2 Same signs, positive

b. Find −27−3 .-27 -3 = 9 Same signs, positive

Example 4:Evaluate -16 x if x = -4.

-16 x-16 -4 = 4

Same signs, positive

Got it? 3 & 4a. -24 (-4) = b. -9 (-3) =

c. −28−7=¿ d. Evaluate a b if a = -33 and

b = -3.

Example 5One year, the estimated Australian koala population was 1,000,000. After 10 years, there were about 100,000 koalas. Find the average change in the koala population per year. Then explain its meaning.

New Population−Previous Population10 years

100,000−1,000,00010

=−900,00010

=−90,000

The koala population has changed by -90,000 per year.

Got it? 5The average temperature in January for North Pole, Alaska, is -24C. Use the expression 9C+160

5 to find this temperature in degrees Fahrenheit. Round to the nearest degree. Then explain its meaning.

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Guided Practice:Evaluate each expression if x = 8 and y = -5.1. 15 y 2. xy (-10) 3. (x + y) -3

Journal: How is dividing integers similar to multiplying integers?

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