Introduction to Integers

I. The Opposite of an IntegerThe opposite of a number is better known as the additive

inverse; that is, the opposite of the number a is the number which must be added to a to produce the additive identity 0: . This quantity is often referred to as the opposite of a and is written –a. We look at two ways of investigating the opposite of an integer.

 Number Line ApproachA number line is one method of visualizing the integers.

To investigate the opposite of an integer graph both the given integer and its’ opposite on the number line provided.

 1. a = 3 - a =_____ -3

2. a = -4 - a =_____

3. a = = - a =_____    4. a = -5 - a =_____  

4

0

5

Question: Compare the given integer and it’s

opposite, what is the relationship between the two numbers and zero?

Same distance from zero

Chip MethodAnother method for visualizing integers is to

use colored chips. Typically the chips are red on one side and another color (yellow or white) on the other side which permits representing positive numbers with white or yellow chips and negative numbers with red chips. The idea of opposite seems rather natural when using these manipulatives since there are only two colors. To find the opposite of a number all that is necessary is to turn each chip to the opposite side.

 

For the following diagrams identify the number represented, sketch the opposite and state the value of the opposite.1. a = -5 - a =_____ 

Sketch: 2. a = 3 - a =_____ 

Sketch:

5

-3

For the following diagrams identify the number represented, sketch the opposite and state the value of the opposite.3. a = 0 - a =__0__ 

Sketch: 4. a = -3 - a =_3___ 

Sketch:

II. Absolute ValueMost students when asked what the absolute value

of a number is reply with what they perceive as the definition of absolute value: The absolute value of a positive number is the number itself and the absolute value of a negative number is the numbers’ opposite. In reality, the absolute value of a number is its magnitude. It is the case of real numbers the method mentioned does in fact produce the magnitude of the number but masks what is meant by finding the absolute value of a number. We look at two ways of investigating absolute value.

Number Line Approach

A number line is one method of visualizing the integers. In looking at magnitude in this context, absolute value is the distance of the given number to zero.

For the following numbers find the specified absolute value and describe how this can be illustrated using the number line. 1. │3 │   

2. │-3│   

= 3, 3 units from zero

= 3, 3 units from zero

3. │0│

4. │-2│

= 0, 0 units from 0

= 2, 2 units from 0

Chip Method

In looking at magnitude in this context, absolute value is the quantity of chips present. For our purposes the shaded/colored chips are negative and the white chips are positive.

For the following diagrams identify the number represented and find the absolute value of that number.

1. Number = ____ Absolute Value: │ │= _____

2. Number = ____ Absolute Value: │ │= _____

3. Number = ____ Absolute Value: │ │= _____

-5 -5 5

4 4 4

0 0 0

4. Number = _-3__ Absolute Value: │-3│= _3__

5. Number = _2__ Absolute Value: │2 │= _2__

III. Ordering Integers

 By convention, the number line is structured so that numbers increase from left to right. This means that numbers to the left of -3 are less than ( < ) -3, e.g. -4, -5, while numbers to the right of -3 are greater than ( > ) -3; e.g. -1, 1, 3.

  

Answer the following questions with “(is less than) < ” or “(is greater than) >” and justify your answer.

 1. – 5 ____________________ – 4

   

2. 1 ____________________ – 3   

3. 2 ____________________ 4  

<><

-5 is to the left of -4 on the number line

1 is to the right of -3 on the number line

2 is to the left of 4on the number line

Order the given lists of integers from least to greatest.

 1. 9, -13, 4, -4, 5, 11 

2. -5, 21, -3, 45, 0, -29

-13, -4, 4, 5, 9, 11

-29, -5, -3, 0, 21, 45

Order the given lists of integers from greatest to least.

 1. 14, -21, 0, 3, -4, 9 

2. -29, 5, -2, 7, -1, 19

14, 9, 3, 0, -4, -21

19, 7, 5, -1, -2, -29

Read Page 1 of the Article: When the Chips are Down…Understanding Arises

•What are the flaws when using the number line to model addition and subtraction of integers?

Read Page 2: The Manipulatives of the Chip Model

•What do you notice about the use of the chips? White = Negative

Grey = Positive

For the examples we will be going over, we will use the chips as they were at the beginning of the lesson with

White = PositiveGrey/Red = Negative

Read 2-4: Addition with the Chip ModelWhat happens when adding numbers of the

same sign with the chips?

What happens when adding numbers with different signs with chips?

Insert Video: Modeling the use of the chips AND Modeling -3 + 2 and -3 + -4 and 4 + -3 with chips

Read 4-6: Subtraction with the Chip Model•What happens if you do not have enough

to “take-away” such as in: 3 – 4?

•What happens when you do not have any to take-away such as in: -3 – 2?

•Insert video: modeling 3 – 4 and (– 3) – 2 and 2 – (-4)

Arithmetic with IntegersComplete the first two pages of the

handout involving: addition and subtraction of integers

Use colored chips to explore arithmetic with integers. All numbers should be easily identifiable and any zeros that may occur must be clearly identified as a zero.

Answers are on the following slides

I. Addition of IntegersUsing the set model of addition illustrate the following addition problems.

1. - 4 + 7 = 3

2. 3 + 2 = 5

3. - 3 + - 5 = -8

4. 5 + (-8) = -3

II. Subtraction of IntegersIllustrate subtraction of integers using the

Take-Away model of Subtraction on the following problems.

1. 7 – 4 = 3

2. - 3 – 2 = -5

3. -5 – (-2) = -3

4. 5 – 8 = -3

Problem Solving with Integers•Work through the next two slides and

think about the integer problem related to the question being asked

1. A certain stock dropped 17 points and the following day gained 10 points. What was the net change in the stock’s worth?

 2. The temperature was -10 degrees Celsius

and then it rose by 8 degrees Celsius. What is the new temperature?

 3. The plane was at 5000 ft and dropped 100

ft. What is the new altitude of the plane? 

- 17 + 10 = -7

-10 + 8 = -2

5000 + -100 = 4900

4. The temperature is 55 degrees F and is supposed to drop 60 degrees F by midnight. What is the expected midnight temperature?

 55 – 60 = -5 55 + - 60 = -55. Moses has overdraft privileges at his bank. If

he has $200 in his checking account and he wrote a $220 check, what is his balance?

200 – 220 = -20 200 + -220 = -20

Read Page 7: Multiplication with the Chip Model

What model of multiplication are we modeling when the first number is positive – such as: 2 x -3 or 2 x 3?

What do you need to do if the first number is negative – such as: -2 x 3 or -2 x -3?

•Insert video modeling 2 x -3, 2 x 3, -2 x 3, and -2 x -3

Read Page 8: Division with the Chip Model

What does the article say about dividing integers with chips?

Let’s try it anyways….Think about how you could model -6 ÷ 2 using the sharing/partition model of division

Could we use the repeated subtraction model of division to represent dividing integers?

•Insert video modeling -6 / 2, -6 ÷ -2, and 6/2

Arithmetic with IntegersComplete pages 3 & 4 of the handout

involving: multiplication and division of integers

Use colored chips to explore arithmetic with integers. All numbers should be easily identifiable and any zeros that may occur must be clearly identified as a zero.

Answers are on the following slides

III. Multiplication of IntegersIllustrate multiplication of Integers using a

Repeated Addition model for Multiplication.

1. 3 x 4 3 groups of 4 = 12

2. 3 x (-4) 3 groups of -4 = -12

3. -3 x 4 take out 3 groups of 4 = -12

4. -3 x -4 take out 3 groups of -4 = 12

IV. Division of IntegersIllustrate division of integers using the

indicated model of division. Clearly explain the answer to the division problem.

1. 12 ÷ 4 (Sharing model)

= 3, there are 3 positives in each of the 4 groups

2. -12 ÷ 4 (Sharing model)

= -3, there 3 negatives in each of the 4 groups.

3. -12 ÷ -4 (repeated-subtraction model)

= 3, there are 3 groups of -4

4. 12 ÷ -4 (either model of division)

NOT POSSIBLE

Problem Solving with Integers•Work through the next two slides and

think about the integer problem related to the question being asked

1. If I lost 4 pounds a week for 3 weeks, what is my change of weight?

2. If a school lost 10 students a year, how many more students did the school have 2 years ago?

3. Jim’s football team lost 5 yards on 2 consecutive plays. What is the change in yards?

  

- 4 x 3 = -12

- 10 x -2 = 20

- 5 x 2 = -10

4. A video is made of a train traveling 20 feet per second. If the video is played in reverse, describe the location of the train after 4 seconds.

5. A video is made of a train going in reverse at 15 feet per second. If the video is played in reverse describe the location of the train after 5 seconds.

 

20 x -4 = -80

- 15 x -5= 75

6.If n is a negative integer, which of these is the largest number?

a. 3 + n b. 3 x nc. 3 – n d. 3 ÷ n