Download LP.092514

_______North Seattle College____________________________________________________________
Math and Sciences Division
Mr. Thomas
Math 084: Lesson Plan
Date: Thurs, Sep 25, 2014
Homework: 1.1 Page 43 – 45 Problems 19, 25, 37, 51, 57, 65, 71, 73, 77, 89, 93,
99, 107, 123, 125
Due Friday 9/25
Agenda:






Attendance (2min)
Warm-up (3min)
Introduction to the Real Numbers (15min)
Identifying the Real Numbers (10min)
Evaluate expressions according to the “Order of Operations” (10min)
Practice Problems (10min)
Warm-up: Can you get the number 24 from
7 4
6 6
? or try
?
3 3
6 6
The set of Real Numbers is an infinite set. There are some very important subsets
of the Real Numbers. Some of the important subsets of the Real Numbers are:
The Rationals, The Irrationals, The Integers, The Whole Numbers, and The Natural
Numbers.
_______North Seattle College____________________________________________________________
Math and Sciences Division
The Natural Numbers are sometimes also called the Counting Numbers. All
members of the Natural Numbers are real numbers. That’s why the Natural
Numbers are a subset of the Reals. The Natural Numbers = { 1, 2, 3, 4, 5, 6, 7,….}
without ever ending.
The Whole Numbers are sometimes just written as the symbol W. Whole
Numbers and Natural Numbers are very similar. W= { 0, 1, 2, 3, 4, 5, 6, ….}. The set
W is also infinite like the set of Natural Numbers. What is the difference between
the set of Whole Numbers and the set of Natural Numbers? In terms of sets, what
is the relationship between the sets of Naturals and the Wholes? Well, it turns out
that the Naturals are a subset of the Wholes. Why might this be?
The Integers are sometimes called signed numbers. Z stands for the set of
Integers. Z = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}. The set of Integers have both
Positive and Negative Whole Numbers.
The Rationals are symbolized by Q. The Integers is a subset of Q. All
fractions with an Integer for the Numerator and for the Denominator is called a
Rational Number and belongs to the set Q.
Here are (5) rational numbers.
I.
-5.352
II.
81
III.
4.32323232…….
IV.
0.777777777…….
V.
3
16
_______North Seattle College____________________________________________________________
Math and Sciences Division
An Irrational Number is any number that is not Rational. That is, any
number that cannot be written as a fraction where the Numerator and the
Denominator is some Integer.
Most irrationals are nonterminating nonrepeating decimal numbers. Here are (5)
irrational numbers to serve as examples:
I.
0.8347533611423….
II.
-35.873456123277342….
2
III.
IV.

V.
4.12112111211112111112….
The Irrationals and the Rationals are both Infinite sets and they make up all
of the Real Numbers.
More Practice:
Real
-8.1
3/4
-2
√5
Rational
Irrational
Integer
Whole
_______North Seattle College____________________________________________________________
Math and Sciences Division
The Order of Operations is greatly misunderstood. When evaluating numerical or
algebraic expressions, there is a correct orderly fashion that has been decided
upon by Mathematicians on how to evaluate. Many were taught the acronym
PEMDAS, (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction)
but this is only partially correct. PEMDAS can help you remember the correct
order but one must remember that Multiplication and Division have the same
order, and Addition and Subtraction have the same order.
The correct official “Order of Operations” when evaluating expressions is as
follows:
1. Parenthesis: [Simplify and perform all operations inside the parenthesis
accordingly]
2. Exponents or Radicals: [Evaluate all exponents, radicals]
3. Multiplication or Division: [Perform operations from left to right]
4. Addition or Subtraction: [Perform operations from left to right]
Notice that Multiplication and Division are 3rd according to the correct order,
which ever operation comes first, when looking from left to right, you do first.
Also, note the 4th order, Addition and Subtraction have the same order, you add
or subtract from left to right respectively.
Example:
Evaluate the following expressions.
1.) 3+4  2 =
_______North Seattle College____________________________________________________________
Math and Sciences Division
Answer: 3+4  2 =
3+8 = 11. [Based on the order of operations, we must multiply before
add, so 4 by 2 is 8, then 3 plus 8 is 11.]
we
2.) 6+3  3-5
[We multiplied 1st, then added and subtracted from L to R.]
=6+9-5
=15-5
=10
3.) 16  2  4
=8  4
[Looking from L to R, division comes first then multiplication.]
=32
4.) 4-3  2+(4+8  2)=
4-3  2+(4+4)= [Evaluate inside the Parenthesis, and then Division comes
before addition]
4-3  2+8=
4-6+8=
[Multiply before Subtraction or Addition]
[Operating from L to R]
-2+8=
6.
5.) 16-8  2+32
=16-8  2+9
addition]
[Evaluate exponents before division, subtraction and
_______North Seattle College____________________________________________________________
Math and Sciences Division
=16-4+9
=12+9
=21
[Operating from L to R, step by step….]
_______North Seattle College____________________________________________________________
Math and Sciences Division
Divisible
by:
If:
Examples:
2
The last digit is even (0,2,4,6,8)
128 is
129 is not
3
The sum of the digits is divisible by 3
4
The last 2 digits are divisible by 4
1312 is (12÷4=3)
7019 is not
5
The last digit is 0 or 5
175 is
809 is not
381 (3+8+1=12, and 12÷3 = 4) Yes
217 (2+1+7=10, and 10÷3 = 3 1/3)No
114 (it is even, and 1+1+4=6 and 6÷3
= 2) Yes
6
The number is divisible by both 2 and 3
308 (it is even, but 3+0+8=11 and
11÷3 = 3 2/3) No
If you double the last digit and subtract it from
the rest of the number and the answer is:
7

0, or

divisible by 7
(Note: you can apply this rule to that answer
again if you want)
672 (Double 2 is 4, 67-4=63, and
63÷7=9) Yes
905 (Double 5 is 10, 90-10=80, and
80÷7=11 3/7) No
109816 (816÷8=102) Yes
8
The last three digits are divisible by 8
The sum of the digits is divisible by 9
9
(Note: you can apply this rule to that answer
again if you want)
216302 (302÷8=37 3/4) No
1629 (1+6+2+9=18, and again,
1+8=9) Yes
2013 (2+0+1+3=6) No
_______North Seattle College____________________________________________________________
Math and Sciences Division
10
11
The number ends in 0
220 is
221 is not
If you sum every second digit and then subtract
all other digits and the answer is:
1364 ((3+4) - (1+6) = 0) Yes

0, or

divisible by 11
3729 ((7+9) - (3+2) = 11) Yes
25176 ((5+7) - (2+1+6) = 3) No
648
(By 3? 6+4+8=18 and 18÷3=6 Yes.
By 4? 48÷4=12 Yes) Yes
12
The number is divisible by both 3 and 4
524
(By 3? 5+2+4=11, 11÷3= 3 2/3 No.
Don't need to check by 4.) No