NAME______________________________________________
Jacoby
Fall Semester 2013
-1-
Chapter 7: Integers.
Galileo lived in Italy in the 1500s. He studied how to measure temperature. Galileo found
that a liquid would rise or fall in a glass tube, depending on the temperature. He liquid rose or
dropped, and the temperature as marked on the glass. Sounds familiar? Glass and alcohol or
mercury thermometers are really number lines, just like you use in math. They’re marked by
degrees of temperature. Alcohol or mercury expands up into the positive numbers when heated
and shrinks into the negative numbers when cooled. Those positive and negative numbers are
called integers. In math, integers can be a real hot topic.
I invented the
thermometer! I also
said the sun was the
center of the universe!
Goals for learning:
To identify the absolute value of integers
To compare the values of negative and positive
To add, subtract, multiply, and divide integers.
Watch Movie:
http://www.youtube.com/watch?v=2d5rWT6bHLU&feature=channel&safety_mode=true&persis
t_safety_mode=1
Lesson 1: The Real Number Line and Integers
NAME______________________________________________
Jacoby
Fall Semester 2013
-2-
In arithmetic, you learned to add, subtract, multiply, and divide whole numbers greater than zero.
In algebra, you will also use whole numbers less than zero.
Warm-Up:
What are some numbers less than zero? Where do you see these numbers?
You can use a number line to show the relation between positive and negative whole numbers,
also called integers.
Vocab for Understanding:
Integer:
Real Number:
Negative Integer:
Positive Integer:
Every point on the number line corresponds to a specific real number. The arrows at the ends of
the number line show that the line continues.
Numbers to the left of zero are negative integers. They are read as ‘negative one’, ‘negative
two’… and so on.
Zero is neither negative nor positive.
Numbers to the right of the zero are positive integers. They are read as ‘positive one,’ positive
two,’ and so on….
A negative number is always indicated by the______________________.
NAME______________________________________________
Jacoby
Fall Semester 2013
-3-
A positive sign is always indicated by _____________________________ or
____________________________.
The absolute value of an integer is the distance ________________________________
______________________________________________________________________.
The integer can be either to the left or to the right of zero. The symbol for absolute value is ___.
Vocab:
Opposites:
Example #1: |4| = 4
|-4| = 4
4 is 4 units from 0.
The absolute value of |4| is _______.
-4 is ________ units from 0.
The absolute value of |-4| is _____.
Every number other than 0 has an opposite number. Opposites are the same distance from zero.
Example #2
9 is the opposite of __________________.
NAME______________________________________________
Jacoby
Fall Semester 2013
-4-
-9 is the opposite of _________________.
9 is 9 units from 0.
Both -9 and 9 are _________ units from 0.
5 is the __________________________ of -5.
-5 is the opposite of _________.
5 is ____________ from 0.
Both 5 and -5 ____________________________.
Let’s Try Some Out!
Use the number line below to find the values.
1. You earn $6 from mowing the grass.
2. The wind chill is 12 below zero.
3. The Bears took a ten yard gain on the play.
4. We realized we lost five dollars.
5. A mailbox post rises 4 feet above the ground; it is set 2 feet into the ground.
In Class:
Find the absolute value:
NAME______________________________________________
Jacoby
Fall Semester 2013
-5-
1. |3|
2. |-12|
3. |-3|
Name the opposite of each integer:
4. 4
5. 8
6. -7
Solve these problems using the number line.
E
D
0
7. Which letters represent positive real numbers?
8. Which letters represent negative real numbers?
9. Which letter is the greatest distance from zero?
10. Which letter represents the greatest absolute value?
Give an integer that describes each situation.
11. A gain of 3 yards in football.
12. A loss of 15 yards in football.
13. A withdrawal of $40 from a bank account.
14. A deposit of $100 in a bank account.
15. A temperature of 13˚ F below zero.
Watch the first couple minutes of
chttp://www.youtube.com/watch?v=AQFWV0AdLE&safety_mode=true&persist_safety_mode=1
Lesson 2: Comparing Integers- Greater Than
A
B
C
NAME______________________________________________
Jacoby
Fall Semester 2013
-6-
To compare 2 numbers and determine which is greater.
To use the symbol > to identify the larger of two integers.
Warm-Up: Here is a list of basketball scores.
102
54
42
56
12
59
84
76
Which ones are the winning scores?
How do you see scores posted?
Lesson:
Integers can be arranged in an increasing or decreasing order. Consecutive integers are integers
arranged from least or greatest or greatest to least without any missing integers. The integers on a
number line are consecutive integers.
You can use a number line to compare two integers. On a number line, the greater of two
numbers is the number to the right.
less
greater
Example 1: Compare 2 and -3
_________ is to the __________ of ______________ on the number line.
______ is greater than -3.
Example 2: Compare -1 and -5.
Remember: The symbol >, read “is greater than,” is used to show that one integer is greater than
another.
2 > -3
-1 > -5
Common Error:
Some students
may have a hard
time perceiving
-1 as greater than
-5. Visualize on
the number line
NAME______________________________________________
Jacoby
Fall Semester 2013
-7-
Example #3: Compare |-5| and |-3|
-5 is _________________________, so |-5| = 5.
-3 is 3 units from zero, so _______________________.
________ is farther to the right than ____, so |
|
> | -3 |.
Let’s try it out!
Compare each pair. Use > or =.
1. 5
2
2. |-2|
2
3. |8|
-3
4. |-8|
|-3|
All Class: Each student will get a note card with an integer. Try to arrange yourselves from
least on the left and the greatest on the right.
Lesson 3: Less Than
Objective:
To compare two numbers and decide which is smaller.
To use the symbol < to identify the smaller of two numbers.
Warm-Up:
What does this mean?
>
Try writing a true number sentence using that sign.
What do you think this means?
<
You can use a number line to compare to integers. On a number line, the lesser of two numbers
is the number farthest to the left.
NAME______________________________________________
Jacoby
Fall Semester 2013
-8-
less
Example #1: Compare -3 and 1.
Example #2: Compare -5 and -3
Remember the symbol ‘<’ means less than. It is used to show that one integer is less than
another.
Example #3: Compare -5 and |-3|
In Class: Compare each pair. Use < or =.
1. 2
3. 12
5
2.
|-12|
-2
4. |-1|
1
10
5. A student lists these five numbers and says that they are consecutive integers:
-2, -1, 0, 1, 2, 3.
Is she right?
6. The temperature was 0˚ F on Saturday and -11 ˚F today. Which temperature is higher?
Explain.
All Class: Get into pairs. Try to come up with a word problem comparing two different
temperatures from the data given below. You have 3 minutes.
NAME______________________________________________
Jacoby
Fall Semester 2013
-9-
January Temperatures
Monday
5˚ F
Tuesday
2˚ F
Wednesday
-3˚ F
Thursday
0˚ F
Homework:
Lesson 4: Even and Odd Integers
Objectives:
To identify integers as odd or even
To add, subtract, and multiply integers
To notes the pattern of answers obtained by operations on odd and even
numbers.
Warm-Up:
1. Count by 2s up to 20.
Even
Odd
Lesson:
Zero is an _______________________________ because it is situated between -1
and 1 on the number line.
Also, any number ending with a zero is a ______________ of 2.
Integers that have a factor of 2 without leaving a remainder are even integers. -6, -4, -2, 0, +2,
+4, and +6 are examples of even integers.
Integers that do not have a factor of 2 are odd integers. -5, -3, -1, +1, +3, and +5 are examples of
odd integers.
You can tell whether a number is even or odd by ______________________________________.
If the digit in the ones position is odd ______________________________________________.
If the digit in the ones position is even ______________________________________________.
Example #1: Even or odd?
NAME______________________________________________
Jacoby
Fall Semester 2013
- 10 -
2,478,563
3,556,430
Consecutive even integers are:
Consecutive odd integers are:
When you add or subtract integers, you can predict whether the sum or difference will be odd or
even.
Adding Integers
Even + even = even
Even + odd = odd
Odd + odd = even
Subtracting Integers
Even – even = even
Even – odd = odd
Odd – odd = even
You can also predict whether or not a product will be even:
Multiplying Integers
Even · even = even
Even · odd = even
Odd · odd = odd
Example #2: Tell whether the following numbers are odd or even:
a. 24,004
b. 35,793
c. 12,457,753,456,899
Tell whether the answer will be even or odd without solving the problem.
e. 669 – 571
d. 432 + 444
f. 24 x 36
g. 22 x 43
Tell which operation will give the desired even or off integer. Then solve the problem. There
may be more than one correct answer.
h. 4
5 =
(even)
i.
9
3 =
(even)
NAME______________________________________________
Jacoby
Fall Semester 2013
- 11 -
j. 6
3 =
(odd)
Make a table for division:
See if there are rules for division.
Homework:
Lesson 5: Adding Positive Integers
Objectives:
To visualize sums of integers on the number line.
To add positive integers to positive or to negative integers.
Warm- Up:
NAME______________________________________________
Jacoby
Fall Semester 2013
- 12 -
0
-1
-2
-3
-4
-5
-6
-7
-8
The temperature of the fluid is -4 degrees. Mark this on a number line.
If you add heat to the fluid and raise it by 3 degrees. What is the new temperature of the fluid?
You can use the number line to help you find and understand how to add positive integers to
other positive and negative integers.
Example 1: Find the sum of 3 + 5
Begin at 3.
Move to the right 5 units.
3+5=
Rule: Adding a positive to a positive gives a result farther to the right on the number line.
Example 2: Find the sum of -3 + 5
Begin at -3.
Then move ______________________________.
-3 + ________ = ___________.
Rule: Adding a positive to a negative gives a result farther to the right on the number line.
Examples:
Find each sum.
1. -12 + 4
2. -7 + 7
3. -6 + 8
NAME______________________________________________
Jacoby
Fall Semester 2013
- 13 -
Calculator practice:
1. 652 + 978
2. -57 + 784
3. -441 + 239
Here is a trick!
Think of
-7 + 7 as 7 – 7
Further Understanding:
-6 + 8 as 8 – 6
You can do this do to
the commutative
property
Lesson 6: Adding Negative Numbers
Objectives:
To move left along the number line to represent the sum of a positive or negative
integer and a negative integer.
To add negative integers to positive or negative integers.
Warm-Up:
“When could you add two numbers and get a sum with a value smaller than wither addend? “
Lesson:
You can use the number line to help you find and understand how to add negative integers to
other positive or negative integers.
RULE: Adding a negative to a positive gives a result farther to the left on the number line.
Example 1: Find the sum of 3 + (-5).
Begin at 3. Then move to the left 5 units.
+
NAME______________________________________________
Jacoby
Fall Semester 2013
- 14 -
Example 2: Find the sum of -3 + (-5)
Begin at ________________________.
Then move _____________________ 5 units.
1. -2 + (-7)
which number is the biggest?
What will our sign be for our answer?
2. -4 + (-5)
3. 2 + (-3)
4. 5 + (-2)
Let’s Step It Up!
1. Find the number that is the
biggest.
The sign that is with the biggest
number will be your sign for your
answer.
2. If the numbers are both
negative, treat it as an addition
problem and carry down the
negative sign.
3. If the numbers are of opposite
sign with the bigger number
being negative, treat the problem
like a subtraction problem. Take
the smaller number away from
the larger number. But carry
down the negative sign.
4. If the numbers are of opposite
signs with the bigger number
being positive, treat the problem
like a subtraction problem. Take
NAME______________________________________________
Jacoby
Fall Semester 2013
- 15 -
Try some out!
1. -2 + (-5)
2. 12 + (-32)
3. 28 + (-32)
4. -6 + (-6)
5. 6 + (-6)
6. 6 + 6
Try these harder ones with a calculator.
1. -359 + (-243)
2. 233 + (-873)
3. -649 + (-573)
Word Problems:
1. An airplane flying at 25,000 feet climbs an additional 2,000 feet. What is the final altitude of
the airplane?
2. At midnight, the temperature was 29˚C. During the day the temperature fell 4 degrees. What is
the final temperature?
Homework:
Lesson 7: Subtracting Positive & Negative Integers
Objectives:
To model subtraction of positive and negative integers on the number line.
To convert subtraction expressions to addition expression
To subtract positive and negative integers from positive and negative integers.
Warm-Up: white board,
+++ --- 3 5 2
NAME______________________________________________
Jacoby
Fall Semester 2013
- 16 -
Lesson:
A number line can be used to show that subtracting a number gives the same result as adding the
opposite of the number.
Example #1
Find the difference of 3 – (+5)
Begin at 3.
Then move to the ___________ 3 units.
RULE: _______________________________________________________________________
Example #2
Find the difference of -3 – (-5)
Begin at -3.
Then move to the right _____ units.
RULE: _____________________________________________________________________
___________________________________________________________________________.
Example #3
Find the difference of 3 – (-5)
Begin at 3.
Then move to the ___________ 5 units.
NAME______________________________________________
Jacoby
Fall Semester 2013
- 17 -
RULE: ___________________________________________________________________
Example #4
Find the difference of -3 – (+5)
Begin at __________
Then move to the left __________________.
RULE: ___________________________________________________________________
Look again at addition of positive and negative integers. Compare the results with subtraction of
integers
Subtraction
Addition
3 – (+5) = -2
is the same as
is the same as
-3 + 5 = 2
3 – (-5) = 8
is the same as
is the same as
-3 + (-5) = -8
Subtraction
is the same as
RULE:____________________________________________________
Try some out!
1. 4 – (+5)
2. -6-4
3. 8 – 4
4. -2 – (+4)
5. 10 – (+7)
8. -9 – (-11)
NAME______________________________________________
Mental Math In-Class Work 10-9, 10-8, 10- (-1), 10- (-2)…
Jacoby
Fall Semester 2013
- 18 -
Word Problems:
9. The highest point on Earth is Mt. Everest, 29,028 feet above sea level. The shore of the Dead
Sea is the lowest place on Earth’s surface at 1,312 feel below sea level. What I the vertical
distance from the shore of the Dead Sea to the top of Mount Everest?
10. What is the difference between Alaska’s record low temperature of -80˚F and Utah’s record
low of -69˚F?
Homework:
Lesson 8 & 9: Multiplying by Positive and Negative Numbers
Objectives: To identify the sign of a product as positive or negative.
To multiply positive and negative integers.
Example #1 Find the product of (+4) and (+3).
Rule:_______________________________________________________________________
NAME______________________________________________
Jacoby
Fall Semester 2013
- 19 -
Example #2
Find the product of (-4) (3).
Rule: ______________________________________________________________________
Try some out!
1. (4)(5)
2. (4)(-5)
3. (-6)(3)
4. (6)(3)
Example #3
Find the product of (-4)(-3)
Rule:_________________________________________________________________________
To be successful in Algebra, you will need to memorize these rules.
Try some more out; look at your rules for help.
5. (-2)(6)
6. (-10)(-11)
7. (23)(30)
8. (-2)(-6)
Word Problems:
Every year for the past five years; the forest preserve has lost 16 trees due to disease or weather.
How many trees have been lost in the past five years?
NAME______________________________________________
Jacoby
Fall Semester 2013
- 20 -
At a local movie theater, senior citizens receive a $2.00 discount. How much less will a senior
pay to attend 15 movies in a year?
Homework:
Lesson 10: Dividing Positive and Negative Integers
Objectives: To model division of negative and positive integers on the number
line.
To divide negative and positive integers and identify the sign of a quotient.
Warm-Up:
(-8)(4)
(4)(3)
(-2)(-3)
Because multiplication and division are opposite operations, the rules for multiplying positive
and negative integers are similar to those of division.
Complete the table below:
Multiplication
(3)(4) =
Rule:
Division
and 12 ÷ 4 =
Rule:
NAME______________________________________________
Jacoby
Fall Semester 2013
- 21 -
(3)(-4) =
Rule:
(-3)(4)=
Rule:
(-3)(-4)
Rule:
and 12 ÷ (-4) =
Rule:
and (-12) ÷ (4) =
Rule:
and (-12) ÷ (-4) =
Rule:
A general statement can be made that covers all the multiplication and division rules:
Try some out! Use the rules to help you out:
1. 4 ÷ 2
2. 32 ÷ (-4)
3. (-24) ÷ (4)
4. (30) ÷ (-5)
5. (-20) ÷ (-4)
6. (-28) ÷ (-4)
7. The theater complex has 1,440 seats in six theaters. All theaters have the same number of
seats. How many seats does each theater have?
NAME______________________________________________
Jacoby
Fall Semester 2013
- 22 -
8. A submarine moving underwater at -390/meters. It dives to that level in 13 minutes. How
many meters does it dive per minute?
Homework: