Download per of less than more ratio twice decreased increased

Lesson 1.1 (Part 1)
Translating to Algebraic Expressions
Key Words
Addition
Subtraction
Multiplication
Division
add
sum of
plus
increased
more
subtract
difference
minus
decreased
less than
multiply
product of
times
twice
of
divide
divided
quotient
ratio
per
Examples:
Let n = a number Write each as an algebraic expression.
1)
twelve less than twice some number
2)
eight more than the ratio of twice a number and five
3)
the difference of 45% of a number and one
Exponential Notation
The expression an, in which n is a counting number means
a • a • a • a..... • a
n factors
In a , a is called the base and n is called the exponent, or power. When no
exponent appears, it is assumed to be 1. Thus a1 = a.
n
Rules for Order of Operations
1.
Simplify within any grouping symbols.
2.
Simplify all exponential expressions.
3.
Perform all multiplication and division working from left to
right.
4.
Perform all addition and subtraction working from left to right.
1
Examples:
1)
4 2 + 6(7) − 4(1 + 2) =
2)
2(12)
+ 4(6 + 1) 2 − 4(3 − 1) =
3 +1
To EVALUATE an expression or formula, substitute the given value(s) for the
variable(s) and follow the order of operations.
3)
8 xz − y for x = 2, y = 7, z = 3
2
4)
2 ( x + 3) − 12 ÷ x 2 for x = 2.
5)
4 x 2 + 2 xy − z for x = 3, y = 2, z = 8
2
6)
3a 2 m − 4am 2 for a = 3, m = 2
7)
x 2 − [2(a + b) − ab] 2 for x = 12, a = 2, b = 1
8)
The base of a triangle is 10 feet and the height is 3.1 feet. Find the area of the
triangle.
1
A = bh
2
3.1
10
9)
Find the area of a trapezoid if b1 = 12, b2 = 18, h = 8
b1
A=
h
1
h(b1 + b2 )
2
b2
Sets of Numbers (Part 2 of lesson)
1. Natural Numbers: {1, 2, 3, ...} These are sometimes called the Counting
Numbers.
2. Whole Numbers: {0, 1, 2, 3, ...} The Whole Numbers are the Natural Numbers
plus zero.
3
3. Integers: {..., -3, -2, -1, 0, 1, 2, 3,...} The integers include all whole numbers
and their opposites.
The set notation used in the previous number sets is called roster set notation. Roster
notation simply lists the numbers included in the set. If there is a pattern in either the
positive infinity or negative infinity directions, the ‘...’ is used to show the pattern.
p

4. Rational Numbers:  | p is an integer, q is a non-zero integer  This set of
q

numbers includes any type of number that can be represented as a fraction. These
types of number include the following
Fractions (proper, improper, or mixed)
Integers (denominator of 1)
Terminating Decimals
Repeating Decimals
Types of Rational Numbers
−2, 5,17, 0 (integers)
 9 7 11
−1 , , −
(fractions)
Examples include the following:  11 8 2
−2.34, 0.0456 (terminating decimals)

1.2,3.4787878... (repeating decimals)
The set notation used in the above number set is called Set-Builder Notation. The
one above is read ‘all numbers of the form p over q such that p is an integer and q is a
non-zero integer’. In this set notation a description is used. The vertical bar is read
‘such that’ or ‘where as’. There will be a variable or an expression with variables at
the front.
5. Irrational Numbers: Irrational numbers include numbers that cannot be
expressed as a fraction. Most of the irrational numbers you will encounter will be
roots (such as 5 or 3 11 ) or the number pi (π). When written as a decimal,
irrational numbers will not terminate or repeat. For example, an approximation
for π is 3.14, but 3.14 does not exactly equal π. A number such as 49 is
rational, not irrational, since it can be simplified to 7.
6. Real Numbers: {x | x is rational or irrational}
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10)
Use this list of numbers and identify which numbers are in the following sets.
5
−3
10
6.2
5
2
3
π
−
2
4
5
0.457
0.234
−
16
0
400
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Whole Numbers:
Integers:
Rational Numbers:
Irrational Numbers.
There is a good diagram found on the course web page or in your textbook that shows the
relationship among the sets of numbers.
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Roster Notation: {2, 4, 6,8}
Types of Set Notation
Set-Builder Notation: {x | x is an even number between 1 and 9}
Ex 11) Write the following using roster notation.
{x | x is an even number between 5 and 12}
{x | x is a natural number at least 7}
Ex 12) Write the following using set-builder notation.
{9,11,13,15,17}
{24, 28,32}
Elements & Subsets
B = {1,3,5, 7} The elements of set B are 1, 3, 5, and 7
3 ∈ B read '3 is an element of B '
Students must know how to
read and use the symbols
∈, ∉, and ⊆ ,.
4 ∉ B read '4 is not an element of B '
When all numbers of a first set A are members of a second set B, we can write A ⊆ B ,
which is read ‘A is a subset of B’.
A = {1,3}
B = (1, 3,5, 7}
A⊆ B
13) Use the following sets to determine if the statements below are true or false.
N = Natural Numbers
W = Whole Numbers
Z = Integers
Q = Rational Numbers
H = Irrational Numbers
R = Real Numbers
2.3 ∈ W
6 =Q
4
∉H
5
Q⊆R
H ⊄Z
W⊆N
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