Transcript Introduction to Integers

Introduction to Integers
I. The Opposite of an Integer
The 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 Approach
A 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
-3
- a =_____
2. a = -4
4
- a =_____
3. a = =
- a =_____
0
4. a = -5
5
- a =_____
Question:
Compare the given integer and it’s opposite, what
is the relationship between the two numbers and
zero?
Same distance from zero
Chip Method
Another 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
5
- a =_____
Sketch:
2. a = 3
-3
- a =_____
Sketch:
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 Value
Most 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 │ = 3, 3 units from zero
2. │-3│ = 3, 3 units from zero
3. │0│ = 0, 0 units from 0
4. │-2│ = 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.
-5
5
1. Number = ____
Absolute Value: │-5│= _____
4
4
2. Number = ____
Absolute Value: │4│= _____
0
0
3. Number = ____
Absolute Value: │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.
<
5 is to the left of -4 on the number line
>
1 is to the right of -3 on the number line
<
1. – 5 ____________________ – 4
-
2.
3.
1 ____________________ – 3
2 ____________________ 4
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
-13, -4, 4, 5, 9, 11
2. -5, 21, -3, 45, 0, -29
-29, -5, -3, 0, 21, 45
Order the given lists of integers from
greatest to least.
1. 14, -21, 0, 3, -4, 9
14, 9, 3, 0, -4, -21
2. -29, 5, -2, 7, -1, 19
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 = Positive
Grey/Red = Negative
Read 2-4: Addition with the Chip
Model
What 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 Integers
Complete 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 Integers
Using 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 Integers
Illustrate subtraction of integers using the TakeAway 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?
- 17 + 10 = -7
2. The temperature was -10 degrees Celsius and
then it rose by 8 degrees Celsius. What is the new
temperature?
-10 + 8 = -2
3. The plane was at 5000 ft and dropped 100 ft.
What is the new altitude of the plane?
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 = -5
5. 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 Integers
Complete 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 Integers
Illustrate 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
4. -3 x -4
take out 3 groups of 4 = -12
take out 3 groups of -4 = 12
IV. Division of Integers
Illustrate 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?
- 4 x 3 = -12
2. If a school lost 10 students a year, how many more
students did the school have 2 years ago?
- 10 x -2 = 20
3. Jim’s football team lost 5 yards on 2 consecutive
plays. What is the change in yards?
- 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.
20 x -4 = -80
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.
- 15 x -5= 75
6.If n is a negative integer, which of these is the
largest number?
a. 3 + n
c. 3 – n
b. 3 x n
d. 3 ÷ n