Download Integers Intro

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Integers
• Integers are whole numbers
that describe opposite ideas in
mathematics.
• Integers can either be
negative(-), positive(+) or zero.
• The integer zero is neutral. It is
neither positive nor negative, but
is an integer.
• Integers can be represented on
a number line, which can help us
understand the valve of the
integer.
Positive Integers
• Are to the right of zero
• Are valued greater than
zero.
• Express ideas of up, a gain
or a profit.
• The sign for a positive
integer is (+), however the
sign is not always needed.
• Meaning +3 is the same
value as 3.
Negative Integers
• Are to the left of zero
• Are valued less than
zero.
• Express ideas of down
or a lose.
• The sign for a negative
integer is (-). This sign
is always needed.
Zero is neither positive
or negative
Negative integers
are valued less than zero,
and are always to the left of
zero.
Positive integers
are valued more than zero,
and are always to the right
of zero.
-1
-4
+3
-3
+2
+2
+2
+2
Representing Integers
•
•
•
•
- 4 using 6 counters
+ 2 using 6 counters
0 using 6 counters
- 3 using 6 counters
The “net worth” of
opposite integers is
zero.
0
0
0
Opposite Integers
• Opposite integers always have a
“net worth” of 0. This is called
the ZERO PRINCIPAL.
• Opposite integer have the same “absolute
value”, meaning the distance from the
points on a number line to zero is the same.
• This can be referred to as the integers
magnitude.
Movement on a Number Line
Magnitude and Direction
• Every integer represents a
magnitude and a direction.
• The integer +3 describes a
movement of 3 units in a
positive direction.(right)
• The sign (+) tells you the
direction.
• The number (3) indicates
how far to move or the
MAGNIUDE( a movement of 3 units)
Directio
n
+ 3
Magnitud
e
Which integer
has a higher
value?
-4 or -8
Comparing Integers
Use your number line to help you compare each set of number.
(i.e. for the numbers 3 ,and - 2 …. 3 > -2
-2 < 3)
a) - 6, 7
b) 12, 3
c)- 5,- 8
d) 11, - 15
e) - 7, - 4
f) - 3, - 7
g) 7, - 8
h) - 13, -14
Putting Things Together
• What is the greatest valued negative
integer?
(3,5)
(4,-2)
(-1,-3)
(-2,1)
(4,5)
(-8,+3)
(-5.-1)
(-6,3)
(0,-7)
Comparing Integers
Use your number line to help you compare each set of
numbers. Copy the question and write two sentences for
each pair of numbers.
(i.e. for the numbers 3 ,and - 2 …. 3 > -2
-2 < 3)
a) - 6, 7
b) 12, 3
c)- 5,- 8
d) 11, - 15
e) - 7, - 4
f) - 3, - 7
g) 7, - 8
h) - 13, -14
i) 8, 7
j) - 8, - 7
k) 5, -1
l) 0, -2
m) 0, 3
n) - 5, 0
o) – 14, -10
p) - 9, 0
q) -7, -6
r) -1, 0
s) 4, -4
t) 0, -15
Comparing Integers Again
• For each of the
previous questions (a)
to (t), write a new
mathematical sentence
showing how much
bigger or smaller the
first number is than
the second.
• (i.e. 3, - 2 ….. 3 is 5
more than –2)
-4
+1
0
-2
Directio
n
+ 3
Magnitud
e
Comparing Integers
• -5 ___ -8
• 0 ___ -3
• 3 ___ +2
Quadrant l
(4,-5)
(-8,+3)
(-5,-1)
Outcomes
• A12 represent integers (including zero)
concretely, pictorially, and symbolically,
using a variety of models
• B11 add and subtract integers concretely,
pictorially, and symbolically to solve
problem
• B14 solve and pose problems which utilize
addition of integers
• B2 use mental math strategies for
calculations involving integers
•
Lab Performance Evaluation
• A – Student is performing beyond
expected level.
• B – Student is performing at upper range
of expected level.
• C – Student is performing at expected
grade level
• D – Student is performing at lower range
of expected level.
• E – Student is performing below expected
level.
Areas of Evaluation
•
•
•
•
•
Organization into activity
Following directions
Presenting work neatly
Completion of work
Representing Integer sentences in
words
• Your ability to discover and represent
Integer Rules
• Making use of the Integer mat
• Working quietly and cooperative
Net Result
Positive 9
(+5) + (+4) = +9
Or
(+4) + (+5) = +9
Finding The Sum of Positive
Integers
• When finding the sum of positive
integers you add the magnitudes and
keep the positive sign.
Net Result
Negative 10
(-3) + (-7) = -10
Or
(-7) + (-3) = -10
Finding The Sum of Negative
Integers
• When finding the sum of negative
integers you add the magnitudes and
keep the negative sign.
Net Result
Positive 2
(+7) + (-5) = +2
Or
(-5) + (+7) = +2
Finding The Sum of a Positive
and a Negative Integer
• When finding the sum of a positive
and a negative integer you subtract
the magnitudes and keep the sign of
the integer with the largest
magnitude.
Net Result
Zero
(+5) + (-5) = 0
Or
(-5) + (+5) = 0
Integer Recap
• Positive symbol means
• Negative symbol means
You Have
or
You’ve
Earned
You Owe
• (+3) + (-7)
• (-5) + (-2)
• (-3) + (-6) + (+4)
• (+3) + (-2) + (+2)
• (+50) + (-100)
• (-25) + (+10)
• -60 + -20
• -20 + 15
• 30 + -5
Rules For Adding Integers
Positive Integers
To add two positive integers you add the
magnitude and keep the positive sign.
Negative Integers
To add two negative integers you add the
magnitude and keep the negative sign.
A Negative and a Positive Integer
To add a positive and a negative integer you
subtract the magnitudes and keep the sign of
the integer with the largest magnitude.
(+5) – (+3) =
(+5) – (+3) = +2
(-6) – (-2) =
(-6) – (-2) = -4
(+3) – (+5) =
(+3) – (+5) = -2
(-2) – (-6) =
(-2) – (-6) = +4
(+3) – (-2) =
(+3) – (-2) = +5
(+1) – (+4) =
(+1) – (+4) = -3
(-5) – (+3) =
(-5) – (+3) = -8
(-2) – (-5) =
(-2) – (-5) = +3
Try These
•
•
•
•
•
•
•
•
•
(-8) – (-3) =
(+4) – (-5) =
(-4) – (-5) =
(+1) – (-6) =
(-5) – (+6) =
(-2) – (-3) =
(-20) – (-10) =
(+30) – (-3) =
(-20) – (-30) =
Try These
•
•
•
•
•
•
•
•
•
(-3) – (-2) =
(+6) – (-2) =
(-1) – (-4) =
(+3) – (-2) =
(-5) – (+2) =
(-2) – (-4) =
(-30) – (-20) =
(+50) – (-10) =
(-20) – (-30) =
Try These
1.
2.
3.
4.
5.
6.
7.
8.
9.
(-5) + (+2) =
(+6) + (-2) =
(-2) – (-6) =
(+7) + (-2) =
(-5) + (+2) =
(+8) + (-4) =
(-3) – (+6) =
(+50) – (-10) =
(-20) + (-30) =
Try These
1.
2.
3.
4.
5.
6.
7.
8.
9.
(-5) + (+2) = -3
(+6) + (-2) = +4
(-2) – (-6) = +4
(+7) + (-2) = +5
(-5) + (+2) = -3
(+8) + (-4) = +4
(-3) – (+6) = -9
(+50) – (-10) = +60
(-20) + (-30) = -50
Site: www.aplusmath.com
• Go to Flashcards
• Go to Non-Java Flashcards
• Go to Adding, Subtracting, Multiplying and
Dividing With Negative Numbers
• Click on Multiplying (One by One) Use the site
to help you complete the chart
• Then, Go To Division (One by One)
(+2) x (+4) =
(+2) x (+4) =
+8
This means you have two sets of
four positive tiles or you have
earned two groups of four dollars.
(+2) x (-4) =
(+2) x (-4) = -8
This means you have two sets of four
negative tiles or you have two bills that
you owe,each bill is for four dollars.
(-2) x (-4) =
(-2) x (-4) = +8
This means you don’t have two sets of
four negative tiles or you don’t owe
two bills, each bill is for four dollars.
(-2) x (+4) =
(-2) x (+4) = -8
This means you don’t have two sets
of four positive tiles or you don’t
have two groups of four dollars.
Try These
•
•
•
•
•
•
(+3) x (-2) =
(-2) x (-2) =
(+5) x (-2) =
(-3) x (+2) =
(+3) x (+4) =
(+3) x (-2) =
Try These
•
•
•
•
•
•
(-91) x (-101) =
(+152) x (-21) =
(-19) x (+203) =
(-69) x (-102) =
(-62) x (-11) =
(-128) x (+12) =
Try These
•
•
•
•
•
•
(-91) x (-101) =
(+152) x (-21) =
(-19) x (+203) =
(-69) x (-102) =
(-62) x (-11) =
(-128) x (+12) =
Multiplying Integers
FACTOR
FACTOR
PRODUCT
+
+
+
_
_
+
_
+
_
+
_
_
Dividing Integers
DIVIDEND
DIVISOR
QUOTIENT
+
+
+
_
_
+
_
+
_
+
_
_
Try These
•
•
•
•
•
•
(-1) x (+1) x (-1) =
(+1) x (+1) x (-1) =
(-1) x (-1) x (+1) =
(-1) x (-1) x (-1) =
(-1) x (-1) x (+1) x (-1) x (+1) =
(-1) x (+1) x (+1) x (-1) x (+1) =
Short Cuts For Multiplying
Several Integer Factors
a. (-1) x (+1) x (-1) = +1
b. (+1) x (+1) x (-1) = -1
If there is an even
number of
negative signs,
the product is
positive
c. (-1) x (-1) x (+1) = +1
d. (-1) x (-1) x (-1) = -1
If there is an odd
number of
negative signs,
the product is
negative
Short Cuts For Multiplying
Several Integer Factors
a. (-1) x (+1) x (-1) x (+1) =
b. (+1) x (+1) x (-1) x(-1) =
c. (-1) x (+1) x (-1) x (-1) x (+1) =
d. (-1) x (-1) x (-1) x (-1) x (+1) x (-1) =
e. (1) x (+1) x (-1) x (-1) x (+1) x (-1) =
f. (-1) x (-1) x (-1) x (-1) x (-1) x (-1) =
g. (-2) x (-3) x (-2) x (+1) =
h. (-1) x (-3) x (-2) x (-2) x (-3) =
Try These
•
•
•
•
•
(-2) x (+2) x (-1)(-3)=
(+1) x (+4) x (-5) =
(-17) x (-2) x (+2) =
(-2) x (-3) x (-6) x 4 =
(-2) x (-3) x (-3)
(+2) x (+4) =
(+2) x (+4) =
+2
Positive and Negative Integers
• For each of the
following numbers,
write down an
example of where it
could be used and
what it means in that
situation.
• -3 -100m +15
•
+3050m -$45.83
Order of Operations With
Integers
3 x (–7) + 4 x (-5)
15 + (+5)2 x 2
(-18) -- 32 – 9 x 2
Practice for Problem Solving
• Fiona spends $5 per week on bus
fare. How much does she spend in 2
weeks?
• Lucy spends 2 per week on snacks.
How much does she spend in 4 weeks?
• Anton earns $8 each week for babysitting. How much does he earn in 3
weeks?
Practice for Problem Solving
• Lional pays $3 per day for bus
transportation. How much does she
pay in a school week?
• Jill has $100 in the bank. She owes
3 of her friends $10 dollars each.
What is her net worth?