Transcript Integers

Integers
Integers: the set of whole numbers and
their opposites.
The sign of an integer is positive if the
number is greater than zero and
negative if the number is less than zero.
Zero is neither positive nor negative.
Opposites
Numbers that are the same distance
from zero on a number line but in
opposite directions are called
opposites.
-4 and 4 are opposites; 11 and -11 are
opposites.
Absolute Value
A number’s distance from zero on the
number line is called its absolute value.
Absolute value is written as |x| where x
equals any number.
The absolute value of 5 is written as |5|.
The absolute value of -5 is written as |-5|.
Both 5 and -5 are five places from zero, so
the value of |5| and |-5| is 5.
Adding integers: With the
same sign
The sum of two positive integers is
positive.
The sum of two negative integers is
negative.
5+5=? -5+-5=?
5+5=10 and -5+-5= -10
Adding integers: With different
signs
To add two integers with different signs,
find the difference in their absolute
values. The sum has the sign of the
integer with the greatest absolute value.
9 + -2 = (Think “the difference in |9| and
|-2| is 7” and the sign of the greatest
number is positive, so the answer is 7.)
More addition
If we add -9+2= think “the difference in
|9| and |2| is 7 and the sign of the
largest is negative, so the answer is -7.
Can you find the answers?
Your teacher says, “ You can do it!”
Practice adding on your own paper.
1) -14 + 6 =
2) 2 + 5 =
3) 24 + -8 =
4) |-11| + 3 =
5) -16 + -3 =
Check your answers.
1) -14 + 6 = -8
2) 2 + 5 = 7
3) 24 + -8 = 16
4) |-11| + 3 = 14
5) -16 + -3 = -19
Subtracting integers
To subtract an integer, add its opposite.
(Change the subtraction sign to an addition
sign and change the sign of the following
number to its opposite.)
Example: 10 - 5 =? Should be worked as 10
+ (-5) =? And -7 -2 =? Should be worked as 7 + (-2) =?
Then, use your rules for addition.
10 + (-5) = 5 & -7 + (-2) = -9
Practice subtraction on your own paper.
1)
2)
3)
4)
5)
6)
-20 - 5 =
18 - 4 =
12 - -3 =
-7 - -3 =
|-6| -2 =
Be careful! |-9| - |-5| =
Answers to subtraction
1)
2)
3)
4)
5)
6)
-20 - 5 = -25
18 - 4 = 14
12 - -3 = 15
-7 - -3 = -4
|-6| -2 = 4
|-9| - |-5| = 4
Multiplying integers
The product of two integers with the
same sign is positive.
8 x 3 = 24 and -8 x -3 = 24
The product of two integers with
opposite signs is negative.
6 x (-5) = -30 and -6 x 5 = -30
Dividing integers
The quotient of two integers with the
same sign is positive.
56 ÷ 8 = 7 and -56 ÷ -8 = 7
The quotient of two integers with
opposite signs is negative.
36 ÷ -9 = -4 and -36 ÷ 9 = -4
Practice on your own paper.
1)
2)
3)
4)
5)
6)
7)
-3 x -5 =
-6 x 2 =
8 x -5 =
25 ÷ -5 =
-14 ÷ -2 =
-81 ÷ 9 =
|-6| x -5 =
Answers to practice
ü
ü
ü
ü
ü
ü
ü
-3 x -5 = 15
-6 x 2 = -12
8 x -5 = -40
25 ÷ -5 = -5
-14 ÷ -2 = 7
-81 ÷ 9 = -9
|-6| x -5 = -30
Integers are FUN!!
Learn the rules for operations with
integers!
Problem Solving with Integers
So by now you are probably thinking how are
integers ever used outside of math class!
Answer: In many different ways!
Negative numbers or integers can be used to
describe many “real world” occurrences.
50 degrees below zero is -50 degrees.
Traveling at elevations below sea level are
negative numbers. Parts of the Grand
Canyon may be at -75 feet elevations.
Problem Solving with Integers
Submarines always travel at negative
elevations when they are underwater.
Unfortunately, you can sometimes have
“negative” amounts of money!
Spending money can be considered negative.
If you buy a hot dog for $2 that could be
considered as negative 2 dollars because you
don’t have it any more.
Problem Solving with Integers
More examples with money:
If you owe your brother or sister money,
guess what? You have negative dollar.
Owing $5 is like having -5 dollars.
Also, if you withdraw money from your bank
you are taking money out of your account and
your balance decreases. Again, this is an
example of negative money. Withdrawing $5
is considered $-5.
Problem Solving with Integers
There is good news, though! If you
deposit $5, you are adding money to
your account which is considered
positive $5!!!
Maybe another rule of integers should
be that saving money gives us more
positive value or worth than spending!
That’s a rule you will definitely want to
learn and remember as you grow older.
Problem Solving with Integers
Also, if businesses lose money and don’t
have a profit, they are “in the red”. That
means they are spending more than they are
making and have negative money and
earnings. If it continues, they won’t be
around long.
Finally when a team like the Dallas Cowboys
is losing to the Redskins by 35 points, the
integer, -35, could be used to describe the
loss.
Steps for Solving Integer Word
Problems
When solving a problem with integers, 1st
decide which numbers are needed and
whether they should be positive or negative.
Then decide which operation (=, -, x, ÷)
should be used with those integers to solve
the problem.
Then use the rules of integers to solve the
problem.
Problem 1
Dallas lost each of their first six games by 14
points. What integer represents how much
they lost by in the first six games combined?
Step 1: We need to use 6 & 14. 6 games
would be positive. Losing by 14 points would
be negative. So, we have +6 and -14.
Step 2: We must find the total amount, so
multiplication or addition should be used.
Problem 1 continued You could add -14 six times or it would be
easier to multiply the two numbers 6 x (-14)
=
You know the rule to multiply integers with
different signs, right?
Right, the answer is NEGATIVE!
6 x (-14) = (-84) Ouch! Their 6 total losses
can be written as -84. They should be
embarrassed!
Problem 2:
An English teacher went shopping and
spent $29 on a purple and pink shirt, $43 on a
green and yellow pair of pants, and $15 on a
red hat to complete the outfit. What number
represents how much was spent?
Solve it on your own. Use the steps.
1st: Find the numbers needed to solve the
problem and decide if each is positive or
negative.
2nd: Choose the correct operation to use.
Problem 2 continued Step 1: -29, -43, -15 They are all negative
because money was spent on each item.
Step 2: -29 + (-43) + (-15) Add them to find
the total spent on the outfit.
-29 + (-43) + (-15) = -87
Remember, when adding integers with the
same sign, keep the same sign in your
answer. GREAT, now try these two.
Mountain Problem:
Alease went mountain climbing and
reached a mountain top at 456 feet
above sea level. On the same day, her
sister, Jill, explored caves in an old
dried up river valley. Jill was at an
elevation of 157 feet below sea level.
How much higher was Alease than Jill?
Mountain Problem Solution
Step 1: 456 above sea level is +456
157 feet below sea level is -157.
Step 2: “How much higher. . .” means
find the difference. 456 - (-157) =
Use the rules and solve to find the
correct answer of 613 feet.
Only 1 problem left! (Would that be -1 or
+1?)
Problem Solving with Integers
Kevin went to basketball camp for
seven days. The total cost of camp was
$917. What number best represents
the cost each day?
Problem Solving with Integers
Step 1: 7 days is +7; a cost of $917 is -917.
Step 2: To find the cost of “each” day, you
should divide. -917 ÷ 7 =
Use the rule of dividing integers with different
signs. -917 ÷ 7 = (-131).
-131 describes the cost per day.
CONGRATULATIONS! You have finished this
integer presentation!