Download Chapter 1 Lecture Notes

Remember to read the textbook before attempting to do your homework.
Symbols of Arithmetic
Addition:
Subtraction:
Multiplication:
Division:
Computer
Notation
useful when
e-mailing
a friend
Symbol
+
–
×
•
()
÷
Symbol Name
plus sign
minus sign
times sign
raised dot
parenthesis
division sign
fraction bar
e.g. 3 × 4 = 12
e.g. 5 • 6 = 20
e.g. 7(8) = 56
(9)10 = 90
(11)(12) = 132
e.g. 12 ÷ 3 = 4
=4
e.g. 12
3
“into” symbol
e.g. 3 12 = 4
Multiplication:
*
asterisk
e.g. 6 * 7 means 6 × 7
Division:
/
slash
e.g. 15/3 means 15 ÷ 3
Exponent:
^
caret
e.g. 4^2 means 42 , and 42 = 16
Use the caret key whenever you want to write an exponent.
Warm-up: Answer the problems below. You may not use a calculator, and please do not use the
“I’m just using my calculator to check my answers” excuse. Feel free to work with your neighbors. ☺
1) 485 + 2739
2) 732 – 489
4) 4000 × 300
8) 500 × 800
5) 80 × 6000
9) 60 × 500
6) 900 × 700
10) 8000 × 900
3) 256 • 493
Check this out! If the numbers end with zeros,
then go ahead and take the multiplication shortcut!
Example: 6,000 × 800
“=” 6000 × 800 ← Look! The numbers end with zeros.
Now, first multiply the nonzero digits
“=” 48
= 6000 × 800
7) 200 × 330
11) 500 × 4000
= 4800000
= 4,800,000
Then “tag along” the appropriate
number of zeroes.
(don’t forget to put those commas)
Please start working on the homework in a timely manner – that way you’ll be on track to getting the
grade that you are willing to earn. You can check out the textbook for FREE through the Library’s
Reserve System. Instructor also has extra copies of the textbook for you to use during her office times.
Looking for more fun? Then work on the attached Pizzazz Sheet, A-23.
Math 40 || Prealgebra || Cerritos College
– Pg 1 –
Up next: Chapter 1
Chapter 1 Lecture Notes by Maria Torres
Remember to read the textbook before attempting to do your homework.
Section 1.1: An Introduction to Whole Numbers
Definitions:
▪ A set is a collection of objects. We use braces { } to enclose its elements.
▪ The set of digits is one of the group of numbers 0 to 9
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
▪ The set of NATURAL NUMBERS (aka counting numbers) is given by
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, … , 99, 100, 101, 102, 103, … , 2099, 2100, 2101, …}
▪ The set of WHOLE NUMBERS include the set of natural numbers and 0.
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, … , 99, 100, 101, 102, 103, … , 2099, 2100, 2101, …}
Number Line
Numbers can be represented by points on the number line.
To graph a number means to locate its position on the number line and highlight it with a heavy dot.
Note: The point representing the number 0 is called the origin.
0
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
4
5
6
7
8
9
10
5
6
7
8
9
10
Example 1:
a. Graph the whole numbers less than 6.
0
1
2
3
b. Graph the natural numbers less than 6.
0
1
2
3
c. Graph the natural numbers between 5 and 9.
0
1
2
3
4
Equality and Inequality Symbols
The symbol, = , is read as: “is equal to”
e.g. 10 = 10
The inequality symbol, > , is read as: “is greater than”
e.g. 5 > 3
The inequality symbol, < , is read as: “is less than”
e.g. 4 < 6
The inequality symbol, ≥ , is read as: “is greater than or equal to”
e.g. 5 ≥ 3, 5 ≥ 5
The inequality symbol, ≤ , is read as: “is less than or equal to”
e.g. 4 ≤ 6, 4 ≤ 4
Psst psst: The is less than inequality symbol, < , kind of looks like the letter L – a tired L.
L is for is less than?
Too much drama??? If worst comes to worst, then just remember that:
The inequality symbol always points to the smaller number.
Math 40 || Prealgebra || Cerritos College
– Pg 2 –
Chapter 1 Lecture Notes by Maria Torres
Example 2: Place an =, < or > symbol in the box to make a true statement
a.
53
d.
50¢
67
b.
91
$0.50
e. 50¢
87
c. 1,918
1,920
$0.96
f.
$0.99
$1
Example 3: Determine if the statement is true or false.
a.
5>8
b.
13 ≥ 13
c.
9 < 15
Example 4: Write an expression described by the word phrases.
a. Twenty-three is greater than five
b. Five equals five
d.
7≤2
c. Six is less than eighty
Example 5: Panchito needs your help. He has been assigned a set of homework problems. Write down
every single problem that he will have to do for that assignment. Note: eoo means “every other odd”
a. 7 – 11 odd, 16 – 20 ALL, 29 – 53 eoo
b. 29 – 53 eoo
(end of Section 1.1)
Remember to read the textbook before attempting to do your homework.
Section 1.2: Adding and Subtracting Whole Numbers
Section 1.3: Multiplying Whole Numbers
Definitions:
The answer to an addition problem is called the sum.
What is the sum of 3 and 5?
The answer to a subtraction problem is called the difference.
What is the difference of 5 and 3?
The answer to a multiplication problem is called the product.
What is the product of 3 and 5?
The answer to a division problem is called the quotient.
What is the quotient of 15 and 5?
A variable is a letter used to represent an unknown number, e.g. x, y, z, a, b, c
Math 40 || Prealgebra || Cerritos College
– Pg 3 –
Chapter 1 Lecture Notes by Maria Torres
Mutlipication Symbols:
4•9
4 × 9
9
4(9)
×4
(4)9
(4)(9)
parenthesis
raised dot
Times
Note: When multiplying a number and a variable, or when multiplying two or more variables feel
free to omit the raised dot.
e.g.
13k means 13 • k, “13 times k”
xy means x • y,
“x times y”
Properties of Addition and Multiplication
If a, b, and c are real numbers, then we have:
Commutative Property of Addition:
Commutative Property of Multiplication:
a+b=b+a
a•b=b•a
e.g.
3+2=2+3
e.g.
3•2=2•3
(The order in which we add does not matter)
(The order in which we multiply does not matter)
Associative Property of Addition:
a + (b + c) = (a + b) + c
e.g.
2 + (3 + 4) = (2 + 3) + 4
Associative Property of Multiplication:
a • (b • c) = (a • b) • c
e.g.
2 • (3 • 4) = (2 • 3) • 4
The way in which the numbers are
grouped does not change their sum
The way in which the numbers are
grouped does not change their product
Distributive Property
If a, b, and c are real numbers, then we have:
Distributive Property
Distributive Property
(over addition): a(b + c) = ab + ac
(over subtraction): a(b – c) = ab – ac
e.g.
2(x + 3) = 2 • x + 2 • 3
e.g.
2(x – 3) = 2 • x – 2 • 3
= 2x + 6
= 2x – 6
Cool things about the distributive property . . .
The distributive property tells us how to multiply a with the sum of b and c.
The distributive property also tells us how to multiply a with the difference of b and c.
Example 1: Which property does each represent?
a.
2+1=1+2
b.
(4 • 5) • 9 = 4 • (5 • 9)
c.
5(6 + 2) = 5 • 6 + 5 • 2
d.
10 + (12 + 4) = (10 + 12) + 4
e.
5(6 • 2) = (5 • 6) • 2
f.
5 • (6 – 2) = 5 • 6 – 5 • 2
(end of Section 1.2 & 1.3 unit)
Don’t forget to start working on today’s homework in a timely manner – that way you’ll be on track
to getting the grade that you are willing to earn.
You can check out the textbook for FREE through the Library’s Reserve System.
Instructor also has an extra copy of the textbook for you to use during her office times.
Math 40 || Prealgebra || Cerritos College
– Pg 4 –
Chapter 1 Lecture Notes by Maria Torres
Remember to read the textbook before attempting to do your homework.
Section 1.4: Dividing Whole Numbers
Division Symbols
20 ÷ 5
20
5
5 20
20/5
divided by
fraction bar
into
slash
(computer syntax)
Properties of Division
If a is a nonzero number (a ≠ 0), then we have:
(Note: The symbol ≠ is read as “is not equal to”)
a
The quotient of a number
The quotient of a number
=1
→
→
and itself is 1
and 1 is that number
a
4
e.g.
e.g.
=1
4
The quotient of 0 and a
number is 0.
The quotient of a number
→
and 0 is undefined.
0
=0
a
0
=0
4
→
e.g.
a
=a
1
4
=4
1
a
is undefined
0
4
e.g.
= undefined
0
You can use this memory aid
when you’re dividing by 0, or when dividing into 0.
Memory aid:
aid
You can dunk your
doughnut in your coffee
but . . .
You can’t dunk your
coffee in your doughnut
Example 1: Perform the indicated operations.
a.
0 ÷ 43
b.
213
0
c.
78 ÷ 0
d.
10 0
e.
0 98
f.
23, 467 ÷ 0
Math 40 || Prealgebra || Cerritos College
– Pg 5 –
Chapter 1 Lecture Notes by Maria Torres
Definition:
Divisibility: One number is divisible by another if, when we divide them, the remainder is 0.
In other words: Divisibility means that one number goes into another number without a remainder.
Example 2: Is 10 divisible by 5?
Solution:
Example 3: Is 11 divisible by 5?
Solution:
2
5 10
Look!
2 R1
5 11
Look!
5 goes into 10
2 times with no remainder.
5 does not goes into 11 without a
remainder.
Answer: No, 11 is not divisible by 5.
Answer: Yes, 10 is divisible by 5. ☺
Tests (Rules) For Divisibility
(Divisibility Rules: Can I divide this number by that number without leaving a remainder?)
A number is divisible . . .
By 2: if the last digit is even (0, 2, 4, 6, or 8).
By 5: if the last digit is 0 or 5.
By 10: if the last digit is 0.
58: 8 is even, so 58 is divisible by 2
235: the last digit is 5, so 235 is divisible by 5.
390: the last digit is 0, so 390 is divisible by 10.
By 3: if the sum of the digits is divisible by 3.
Note: For large numbers, digits may be
summed iteratively.
1,269 → 1 + 2 + 6 + 9 = 18, and
18 → 1 + 8 = 9, and since 9 is divisible by 3,
then 1,269 is divisible by 3.
Example 4: By which of the following are the given numbers divisible by? Circle your response.
If you’d like, you’re more than welcomed to cross out the numbers that the
given number is not divisible by.
Please wait for instructor to do these with you before you start having fun.
Strategy: Don’t work going across, just go straight down.
First check to see which numbers are divisible by 2, 5, or 10 – check for 2, then for 5, then for 10.
Why??? Because all of these involve looking at the last digit.
Then find the sum of the digits for each number and check to see which are divisible by 3.
Why??? Because this test involves looking at the sum of the digits
Number
Divisible by . . .
a.
320
2
3
5
10
b.
1,095
2
3
5
10
c.
37,798
2
3
5
10
Math 40 || Prealgebra || Cerritos College
– Pg 6 –
Chapter 1 Lecture Notes by Maria Torres
Example 5: By which of the following are the given numbers divisible by? Circle your response.
Number
Divisible by . . .
a.
87
2
3
5
10
b.
108
2
3
5
10
c.
477
2
3
5
10
d.
210
2
3
5
10
e.
400,002
2
3
5
10
f.
37
2
3
5
10
g.
42
2
3
5
10
h.
48,060
2
3
5
10
(End of Section 1.4)
Bonus Problem: By which of the following are the given numbers divisible by? Circle your response.
Number
Divisible by . . .
1.
39,758
2
3
5
10
2.
99,995
2
3
5
10
3.
29,063
2
3
5
10
4.
5,334
2
3
5
10
5.
98
2
3
5
10
6.
684,289
2
3
5
10
7.
5,331
2
3
5
10
Math 40 || Prealgebra || Cerritos College
– Pg 7 –
Chapter 1 Lecture Notes by Maria Torres
Remember to read the textbook before attempting to do your homework.
Section 1.5: Exponents and Prime Factors
Definitions:
Factors: Whole numbers that multiply to form a product are called factors.
In English: Numbers that are multiplied together are called factors.
For example, since 2 • 7 = 14, then 2 and 7 are factors of 14.
Prime number: A whole number greater than 1, that only has 1 and itself as factors.
Alternative definition: A number is prime if it is only divisible by 1 and itself.
Note: If a number is not prime, then it is composite, e.g. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
Example 1: List all the prime numbers less than 30.
Example 2: Identify each of the numbers below as either a prime or composite number. If composite,
give at least one divisor (factor) other than 1 and the number itself.
a.
29
b.
13
c.
24
d.
50
prime number
prime number
prime number
prime number
composite number
composite number
composite number
composite number
other factors:
other factors:
other factors:
other factors:
Exponents
An exponent is used to indicate repeated multiplication. It tells us how many times the base is used as
a factor.
Using exponents, we can rewrite 2 • 2 • 2 • 2 as 24 (read as “2 raised to the fourth power”),
where 2 is called the base, and 4 is called the exponent.
In general, x to the nth power, written as xn means x • x • x • … • x = xn
n-times
Special note for numbers with an exponent of 2 or 3:
▪ If a number is raised to the second power, then we can say that the number is squared.
e.g. 52 may be read as: 5 raised to the second power, or just 5 squared.
▪
If a number is raised to the third power, then we can say that the number is cubed.
e.g. 53 may be read as: 5 raised to the third power, or just 5 cubed.
Example 3: Which student has the correct answer to the problem below?
23
Student #1:
23
Student #2:
23
= 6
= 8
Math 40 || Prealgebra || Cerritos College
– Pg 8 –
Chapter 1 Lecture Notes by Maria Torres
Example 4:
a. Find the prime factorization of each number – Write each number as a product of prime factors.
If a number is prime, so state.
b. List all the factors of each number. You may not use a calculator to answer.
a. 24
b. 7
c. 48
d. 56
e. 60
f. 90
BP: 210
Example 4: Write each number in prime-factored form. Use exponent for repeated factors.
Hint: Use a tree diagram (or factor tree).
From the internet:
• Prime factorization can be found using a tree diagram or a ladder diagram.
o Remember, 1 is not prime so it is not a valid number for a tree diagram or for prime
factorization!
a. 180
b. 600
Example 5: The prime factorization of a number is given. Find the number.
a. 32 • 2
b. 2 • 2 • 2 • 7
Example 6: Evaluate the expression. OK to use a calculator.
a. 514
b. 123 • 152
(Example 4 concludes the end of Section 1.5. Up next: Section 1.8 & 1.9 combo unit)
Math 40 || Prealgebra || Cerritos College
– Pg 9 –
Chapter 1 Lecture Notes by Maria Torres
Remember to read the textbook before attempting to do your homework.
Section 1.8: Solving Equations by Adding or Subtracting
Section 1.9: Solving Equations by Multiplying or Dividing
Definitions:
An equation is a math sentence that contains the equals symbol, =.
w =4
e.g. x + 3 = 10,
p – 13 = 15,
3x = 9,
2
The solution to an equation is the number (or numbers) that when substituted for the variable make the
equation a true statement.
Goal:
To solve equations. In other words, the goal of this section is to find all values of the variable
that make the equation true (you’re finding the solutions).
How?
By isolating the variable. This means that you want to get the variable by itself.
(variable) = (constant)
x = #
Example 1: Determine whether the given number is a solution of the equation.
x
+ 5 = 15 ?
a. Is 20 a solution of
b. Is 9 a solution of n – 9 = 9 ?
2
Example 2: Find the number that makes the equation a true statement. If you can answer mentally,
then go for it! Please write answer in the space provided. You may not use a calculator to answer.
a.
+ 7 = 15
d.
=5
8
Math 40 || Prealgebra || Cerritos College
b.
–4=6
– Pg 10 –
c. 3 •
= 27
Chapter 1 Lecture Notes by Maria Torres
Properties of Equality
Let
a=b
If a, b, and c represent real numbers, where c ≠ 0, then we have the following properties:
The addition property of equality states that adding the same
Addition Property of Equality
number to both sides of an equation does not change its
If
a = b,
then a + c = b + c
solution.
The subtraction property of equality states that subtracting the
Subtraction Property of Equality
same number from both sides of an equation does not change its
If
a = b,
then a – c = b – c
solution.
Multiplication Property of Equality The multiplication property of equality states that multiplying the
same number to both sides of an equation does not change its
If
a = b,
then a • c = b • c
solution.
The division property of equality states that dividing both sides
Division Property of Equality
of an equation by a nonzero number does not change its
If
a = b,
then a ÷ c = b ÷ c
solution.
Guidelines to Help You Solve Equations
1. Isolate the variable on one side of the equation.
Fix it so you have the variable on one side, and all the non-variables on the other side.
Question: What can I do to isolate the variable?
Answer: Add, subtract, multiply, or divide both sides of the equation by the same number.
2. Check.
Substitute your answer into the original equation and verify that it gives you a true statement.
If your answer does not check out, then find your mistake and fix it.
Question: Help! How do you know when you are supposed to add, subtract, multiply, or divide
by a number?
Solution: Oh that’s easy! Just go ahead and do the inverse operation.
1.
The inverse operation of …..
2.
The inverse operation of …..
3.
The inverse operation of ….. multiplication ….. is
4.
The inverse operation of …..
addition
….. is
….. is
….. is
Example 3: Solve. Check your answer. You may not use a calculator to answer.
Ran out of space???
(Example 4 concludes the end of Section 1.8 & 1.9)
Don’t worry! Just take out a sheet of paper and continue writing.
Math 40 || Prealgebra || Cerritos College
– Pg 11 –
Chapter 1 Lecture Notes by Maria Torres