Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 1: Kinds of Numbers
Index
Counting numbers Screen 2
Digits Screen 2
Fractions, decimal form Screen 7
Fractions, improper Screen 6
Greater than (>) Screen 3
Integers Screen 4
Irrational numbers Screen 8
Less than (<) Screen 3
Mixed numbers Screen 6
Opposite numbers Screen 4
Rational numbers Screen 5
Real numbers Screen 9
Variables Screen 10
Whole numbers Screen 3
Key Topics & Formulas
♦ Any number to the right of another number on the number line is said to be greater than the other
number. (Screen 3)
♦ Any number to the left of another number on the number line is said to be less than the other
number. (Screen 3)
♦ On the number line, negative numbers are to the left of zero, and positive numbers are to the right.
(Screen 4)
♦ Numbers that are the same distance from zero but in opposite directions are called opposite
numbers. (Screen 4)
(Screen 9)
♦ Variables (Screen 10)
ƒ Letters that represent numbers in algebra are called variables. The rules that govern how
numbers behave also govern how variables behave in algebra.
ƒ You cannot take for granted what kind of number a variable may represent.
Page 1 of 7
Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 2: Behavior of Numbers and Variables
Index
Additive identity Screen 7
Associative property, addition Screen 4
Associative property, multiplication Screen 4
Commutative property, addition Screen 2
Commutative property, multiplication Screen 3
Distributive property Screen 5
Division by zero Screen 7
Multiplicative identity Screen 7
Reciprocal Screen 3
Key Topics & Formulas
♦ Commutative property of addition: a + b = b + a
(Screen 2)
♦ Subtraction is not commutative: a − b ≠ b − a (Screen 2)
♦ Commutative property of multiplication: ab = ba
♦ Division is not commutative:
♦
a
b
is the reciprocal of
b
a
a
b
≠
b
a
(Screen 3)
(Screen 3)
(Screen 3)
♦ Associative property of addition:
(a + b) + c = a + (b + c ) = a + b + c
♦ Associative property of multiplication:
( ab ) c = a ( bc )
♦ Distributive property: a ( b + c ) = ab + ac (Screen 5)
♦ Additive Identity: a + 0 = a (Screen 7)
♦
a − 0 = a (Screen 7)
♦
a × 0 = 0 (Screen 7)
♦ Multiplicative Identity: a ⋅ 1 = a
♦ Dividing zero:
0
= 0 (Screen 7)
a
♦ Division by zero:
♦
a
=a
1
(Screen 7)
a
= undefined
0
(Screen 7)
(Screen 7)
Page 2 of 7
(Screen 4)
(Screen 4)
Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 3: Adding and Subtracting Signed Numbers and Variables
Index
Adding, a positive and a negative number Screen 4
Adding, opposites Screen 5
Adding, two negative numbers Screen 3
Adding, two positive numbers Screen 2
Signed numbers Screen 2
Subtracting, signed numbers Screen 6
Key Topics & Formulas
♦ Adding positive numbers is exactly like adding whole numbers.
(Screen 2)
♦ When adding negative numbers, the sum is always a negative number. (Screen 3)
♦ Adding positive and negative numbers is a little trickier. Ignoring the signs, first find the difference
between the larger number and the smaller number. Then, give your sum the sign of the number
farther from zero. (Screen 4)
♦ When you add positive and negative numbers (or variables) that are the same distance from zero,
the sum is 0 . Another way to state this rule is “The sum of a number and its opposite is zero.”
( −a ) + a = 0 and a + ( −a ) = 0 (Screen 5)
♦ To subtract a number, add the number being subtracted but use its opposite sign. In other words,
change the sign of the subtracted number and change the problem to an addition problem. You can
think of “subtracting” as “adding the opposite” to help you remember this rule. (Screen 6)
ƒ
a − b = a + ( −b )
ƒ
a − ( −b ) = a + b
ƒ
−a − b = −a + ( −b )
ƒ
−a − ( −b ) = − a + b
Page 3 of 7
Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 4: Multiplying and Dividing Signed Numbers and Variables
Index
Dividing, two numbers with opposite signs Screen 4
Dividing, two numbers with the same sign Screen 4
Multiplying and dividing with variables
Screen 5
Multiplying, an even number of negative numbers Screen 2
Multiplying, an odd number of negative numbers Screen 3
Multiplying, two numbers with opposite signs Screen 2
Multiplying, two numbers with the same sign Screen 2
Key Topics & Formulas
♦ The product (or the quotient) of two numbers with the same sign is positive:
+⋅+ = +
↔
−⋅− = +
↔ "An enemy of an enemy is a friend."
"A friend of a friend is a friend."
(Screen 2, 4)
♦ The product (or the quotient) of two numbers with different signs is negative:
+⋅− = −
↔ "A friend of an enemy is an enemy."
−⋅+ = −
↔ "An enemy of a friend is an enemy."
(Screen 2, 4)
♦ Multiplying an even number of negative numbers gives a positive product.
♦ Multiplying an odd number of negative numbers gives a negative product.
(Screen 2)
(Screen 3)
♦ To simplify expressions with both numbers and variables, we handle them separately, starting with
the numbers. The variables just “come along for the ride,” so to speak. (Screen 5)
Page 4 of 7
Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 5: Operations with Fractions and Variables
Index
Adding and subtracting fractions with like denominators Screen 7
Adding and subtracting fractions with unlike denominators Screen 8
Common denominators Screen 8
Dividing fractions Screen 5
Multiplying fractions Screen 2
Negative sign in fractions Screen 5
Reducing a fraction to lowest terms Screen 2
Simplify Screen 2
Key Topics & Formulas
♦ Multiplication of fractions:
a c
ac
⋅ =
b d
bd
(Screen 2)
a
c
a d
ad
÷
= ⋅ =
♦ Division of fractions:
b d
b c
bc
♦ Negative fractions:
a
−a
a
=
=−
−b
b
b
and
a
b = a ⋅ d = ad
c
b c
bc
d
(Screen 5)
(Screen 5)
♦ Adding and subtracting fractions with like denominators:
a c
a+c
+ =
b b
b
and
a c
a−c
− =
b b
b
(Screen 7)
♦ Adding and subtracting fractions with unlike denominators: Multiply the fractions by one, written
as
a
, in such a way that all the resulting fractions to be added or subtracted have the same
a
denominator. Then add or subtract the numerators, over their common denominator.
Page 5 of 7
(Screen 8)
Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 6: Squares, Square Roots, and Absolute Value
Index
Absolute value Screen 8
Estimating square roots Screens 6, 7
Perfect squares Screen 4
Principal root Screen 4
Radical Screen 4
Radicand Screen 4
Square root of a negative number Screen 6
Square root sign Screen 4
The square of a number Screen 2
The square root of a number Screen 4
The square root of a variable Screen 10
Key Topics & Formulas
♦ For any real number, a : a 2 = a × a . (Screen 2)
♦ Finding the square root of a number means finding a factor that, when multiplied by itself, equals
that number. (Screen 4)
♦ The square root of a negative real number cannot be any real number. (Screen 6)
♦ “The absolute value of x ”, x , is the distance from the point x on the number line to 0 . (Screen 9)
♦ If a represents any non-negative real number, then
♦ If a represents any real number, then
a 2 = a . (Screen 10)
a 2 = a . (Screen 10)
Page 6 of 7
Chapter 2: Types of Numbers and Their Behavior
Lesson Index & Summary
Section 7: Order of Operations
Index
Order of multiplication and division Screen 2
Order of addition and subtraction Screen 2
Grouping symbols Screen 4
Key Topics & Formulas
♦ Multiply and divide (from left to right) before adding and subtracting (from left to right). (Screen 2)
♦ Grouping symbols include parentheses, ( ), brackets, [ ], or braces, { }, as well as the radical sign,
, and the fraction bar. (Screen 4)
♦ Perform operations within parentheses and other grouping symbols first. (Screen 5)
♦ When an expression contains more than one grouping symbol, perform the operations within the
innermost grouping symbol first. Then work toward the outermost grouping symbol. (Screen 6)
Page 7 of 7