Chapter 5, MA318 Notes, Overmann Fall 2009
Sections 5.1 – 5.2
-1. Homework 1: Integer Pretest
0. Manipulatives
1. Two-color Counters: Examples of three, what number is this?
2. Why do humans need integers? Why aren’t the counting numbers enough?
3. Preliminary Integer Definitions
a. Definition: Two integers are said to be opposites if they are the same distance from zero on the number line,
but in opposite directions.
b. Definition: The integers are the set of whole numbers and their opposites, a set of numbers that describes
whole number quantities and a sign which indicates a directional property of the number. The sign and the
whole number indicate how many whole units a quantity is from zero and which direction on the number line
that distance is measured. The integers are notated I  Z  ..., 3, 2, 1,0,1, 2,3,... .
c. Definition: The absolute value of a number is the number’s distance from zero on the number line.
d. Properties: i. Zero is neither positive or negative.
ii. The opposite of zero is zero.
iii. The sum of opposites is zero.
4. Models of Integers for learners:
a. Two color counter (red chip/black chip, red bean/white bean) set or discrete (charged particle) model.
b. Mail Carrier Stories
c. Number line model
d. Number Patterns
5. Addition and Subtraction of integers Models
a. Two color counter (red chip/black chip, red bean/white bean, charged particle) set or discrete
model.
Illustrate Adding Join, Subtracting Integers Take Away, Adding the Opposite, Missing Addend
http://nlvm.usu.edu/
http://connectedmath.msu.edu/CD/Grade7/Chip/index.html
Examples:
53
5  (3)
5  3
5  (3)
Examples:
53
5  (3)
5  3
5  (3)
b. Mail Carrier Stories
Adding, Subtracting Integers – Karmos Worksheets 1 – 3
c. Number line model:
Recall the definition of opposites: The vector (arrow) points right if n is positive; –n points left.
For addition, walk the first vector; face in the direction of the second vector, walk forward.
For subtraction, walk the first vector; face in the direction of the second vector, walk backward.
OR
For addition, place the first vector; place second tail to first head; read at second head.
For subtraction, place the first vector, place second head to first head; read at second tail.
Examples:
Examples:
53
53
5  (3)
5  (3)
5  3
5  3
5  (3)
5  (3)
http://www.teacherlink.org/content/math/interactive/flash/IntegerCars/IntCars4_b.html
Dolan, Williamson, Muri, Activity 5: Clown on a Tightrope
d. Pattern Model
Dolan, Williamson, Muri Activity 3 and 6: Addition, Subtraction Patterns
6. Further Addition and Subtraction of Integers Items
a. Properties for addition of integers, page 298
Which of these properties do not hold for subtraction?
b. Ways to think about Subtraction of Integers
Take away
Adding the opposite: change subtraction to addition of the opposite.
Missing addend method: if a, b, and c are integers, then a – b = c if and only if a = b + c.
c. The ‘–’ symbol and how to read it
–7 can be read ‘negative seven’ or ‘opposite of seven’ or ‘additive inverse of seven’
–a can be read ‘opposite of a’ OR ‘additive inverse of a’, but avoid ‘negative a’. Why?
a – b is read ‘a minus b’ or ‘the difference of a and b’
d. Subtraction word problems are difficult for students
Find the difference of 2 and 7.
Subtract 2 from 7.
Find 2 less than 7.
Find 7 less 2.
Take 2 away from 7.
Note this is not an exhaustive list of examples – there are other wordings for subtraction problems.
Teach your students how to read them carefully and give them experiences solving each type.
Chapter 5, MA318 Notes, Overmann Fall 2009
Sections 5.3, Multiplying and Dividing Integers
1. Integer Multiplication
a. Repeated addition: Set, Mail Carrier, Number Line Models
Dolan, Williamson, Muri Activity 7: Multiplication/Division Patterns
Karmos Worksheets 4 – 6
3 4
3  (4)
What property gives us 4  (3) ?
How do we justify an answer to 3  (4) ?
 Pattern approach
 Distributive Property justification/Additive inverse justification 0  (4  4)(3) 
Multiplication Rules
1. a  0  0  a  0
2. a  b is positive if a and b have the same sign
3. a  b is negative if a and b have different signs
Multiplication Properties
1. closure
2. commutative
3. associative
4. identity
5. distributive property of multiplication over addition
Other theorems important to integer multiplication:
1. a(1)  a
2. (a)(b)  (ab)
3. (a)(b)  ab
4. Multiplicative cancellation property: If c does not equal zero, then ac = bc
if and only if a = b.
5. Zero divisor (factor) property – if ab = 0, then either a = 0, b = 0, or both a and b are 0.
2. Division of integers
a  b  c if and only if a  bc , for b not equal to 0.
Rules
1. a  1  a for any integer a
2. a  b is positive if a and b have the same sign
3. a  b is negative if a and b have different signs
4. 0  b = 0 for b not equal to zero and b  0 is undefined
Division Properties?
Beans and Cups Activity
3. Negative exponents – laws of exponents
Product rule for exponents: a m  a n  a mn
4 7
Examples: x x 
 2 x y  3x y  
4
3
2
Zero exponent: If a does not equal 0, then a 0  1 .
0
Examples: 25 
 25
0

250 
am
Quotient rule for exponents: n  a m n
a
15
x
x9

Examples: 9 
x
x15
Examples: f 7 
Negative Exponents: a  n 
x8

x8
a 5b 7 c5

d 8e2
Power rules for exponents:
1. (a )  a
m n
b g a b
mn
1
1
, also  n  a n
n
a
a
9 2 5
25 x y z

5 x 3 y 7 z 8
2. ab
m
m m
aI a
F
G
Hb J
K b
n
3.
n
n
Scientific Notation a 10n
a is called the mantissa and n is called the characteristic.
4. Ordering integers
Number line approach
Addition approach: A < B if and only if there exists a positive integer P such that
A + P = B.
Properties of Ordering Integers.
1. Transitive property
2. Addition (Subtraction) property of inequalities
3. Multiplication (division) property of inequalities when the factor is positive
4. Multiplication (division) property of inequalities when the factor is negative