Ch 1 Notes

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```					                                       Lesson 1-1 #1
NOTES 1-1
VARIABLES AND EXPRESSIONS
Variable: a letter or a symbol used to represent a
value that can change

Constant: a number that does not change

Numerical Expression: contains only constants
and/or operations

Algebraic Expression: contains variables, constants,
and/or operations
+   –
Lesson 1-1 #2

x
Lesson 1-1 #3
“2 times y”            “2 divided by y”

Give two ways to write each algebraic expression
in words.
A. 9 + r                B. q – 3

C. 7m                   D. j  6
Lesson 1-1 #4
John types 62 words per minute. Write an
expression for the number of words he types in
m minutes.
m = the number of minutes that John types

Roberto is 4 years older than Emily, who is y
years old. Write an expression for Roberto’s age.
y = Emily’s age
Lesson 1-1 #5
Evaluate: substitute numbers for the variables in the
expression and simplify

Replacement Set { }: a set of numbers that can be
substituted for a variable

Evaluate     n for the replacement set {2, 3, 9}.

n                  n                n
Lesson 1-1 #6
Eighty-five bottles must be recycled to produce a
sleeping bag.
Find the number of bottles needed to make
20, 50, and 325 sleeping bags.
Lesson 1-2 #1
NOTES 1-2

The set of all numbers that can be represented on a
number line are called real numbers.

Add or subtract using a number line.
–4 + (–7) = –11

+ (–7)
–4

11 10 9 8 7 6 5 4 3              2 1 0
Lesson 1-2 #2
The absolute value of a number is the
distance from zero on a number line.
|–5| = 5      |5| = 5

5   units     5 units

-6 -5 - 4 -3 -2 -1 0 1 2 3 4 5 6
Lesson 1-2 #3
• Add their absolute values, keep the sign

3+6=                –2 + (–9) =

• Subtract their absolute values, keep the sign of the
more powerful #

–8 + 12 =            3 + (–15) =
A.                   B. –6 + (–2)

C. –13.5 + (–22.3)   D. 52 + (–68)
Lesson 1-2 #5
Two numbers are opposites if their sum is 0.

5 + (–5) = 0

same distance from zero
(same absolute value)
Lesson 1-2 #6
To SUBTRACT a number, add its opposite.
(Keep… Change… Change…)
A. 3 – 8          B. 5 – (–4)        C. –6.7 – 4.1

D.                  E.
Lesson 1-2 #7
An iceberg extends 75 feet above the sea. The
bottom of the iceberg is at an elevation of
–247 feet. What is the height of the iceberg?
elevation at top   minus   elevation at bottom

The height of the
iceberg is
____    feet.
Lesson 1-3 #1
NOTES 1-3
MULTIPLYING AND DIVIDING REAL NUMBERS
The product or quotient of two numbers with the…
SAME sign is POSITIVE

DIFFERENT sign is NEGATIVE

Find the value of each expression.

A.                    B.                     C.
Lesson 1-3 #2

Two numbers are reciprocals if their product is 1.
4   ●   1=1
4
Multiplicative
Inverses

To divide fractions… multiply by its multiplicative inverse!
Lesson 1-3 #3
Lesson 1-3 #4
0 has special properties for multiplication and division

A. 0      The quotient of zero and any nonzero # is _______.
15

B. –22  0         Division by zero is _____________.

C. –8.45(0)       The product of any number and 0 is ______.
Lesson 1-3 #5
The speed of a hot-air balloon is 3 3 mi/h. It
4
travels in a straight line for 1 1 hours before
3
landing. How many miles away from the liftoff
site will the balloon land?

Distance = Rate x Time

The balloon lands
_____ miles from the site.
Lesson 1-4 #1
NOTES 1-4
POWERS AND EXPONENTS

Exponent
Base
what you multiply   3
2
how many times you
multiply the base

Power
base and exponent
Lesson 1-4 #2

The Area of a Square                      s
s
s  s = s2 or “s squared”

s
The Volume of a Cube                  s
s
sss=    s3   or “s cubed”
Lesson 1-4 #3

Write the power represented by the geometric
models.
6
The factor 6 is used 2 times.
6      = _____

The factor 5 is used 3 times.
5
= _____
5
5
Lesson 1-4 #4

Powers are Repeated Multiplication

31 =
32 =
33 =
34 =
35 =
Lesson 1-4 #5
Write each number as a power of the given base.
64; base 8           81; base –3

–27; base –3      Remember, if the base is negative:
Lesson 1-4 #6
The exponent belongs to what is directly in front of it.
(–6)3         Exp belongs to everything inside the ( )

–102           Means “the opposite of 10²”

A. (–5)3          B. –62            C.
Lesson 1-4 #7
The PTA president calls 3 families. Each family calls 3
other families, and so on. How many families will
have been called in the 4th round of calls?

1   Understand the problem
• The president calls 3 families.
• Each family calls 3 more families.
President
2   Make a Plan
• Draw a diagram.                  1st round

2nd round
Lesson 1-4 #8
3   Solve
1st round: 1  3 = 3 or 31

2nd round: 3  3 = 9 or 32

3rd round: 9  3 = 27 or 33
4th round: 27  3 = 81 or 34 families

Or 3  3  3  3 = 34 = 81 families
Lesson 1-5 #1
NOTES 1-5
ROOTS AND IRRATIONAL NUMBERS

Square Root: a number multiplied by itself to form
a product

= “radical sign”, used to represent square roots

49 = 7 or -7

**Positive numbers have two square roots
Lesson 1-5 #2
Principal Square Root:    Always Positive
36 =            25 =

Negative Square Root: –     Always Negative

– 81 =          – 121 =
Lesson 1-5 #3
Squaring and taking the square root are opposite
operations… they undo each other.

Perfect Squares: a number whose positive square
root is a whole number
1     4     9     16    25    36   49
1²    2²    3²    4²    5²    6²   7²
Lesson 1-5 #4
Index: tells you how many times to multiply the root

What # do you multiply by itself 3 times
to get 8?

What # do you multiply by itself 4 times
to get 16?
Lesson 1-5 #5
Find each root.
Lesson 1-5 #6
Irrational Numbers: any number that cannot be
written as a fraction, “wacky numbers”
Non-terminating non-repeating decimals (pi)
Irrational square roots ( )

REAL NUMBERS include all rational and irrational #s.
(All numbers you have learned about so far…)
Lesson 1-5 #7

Rational Numbers: any number that can be written
as a fraction, includes all numbers that “make sense”

Natural Numbers: counting numbers
{1, 2, 3, 4 . . . }
Whole Numbers: counting numbers plus 0
{0, 1, 2, 3, 4 . . .}
Integers: Whole numbers and their opposites
{ . . . -3, -2, -1, 0, 1, 2, 3 . . . }

Also fractions and decimals (terminating and repeating)
Lesson 1-5 #8

Note the symbols for the sets of numbers…
R: real #s      Q: rational #s
Z: integers     W: whole #s      N: natural #s
Lesson 1-5 #9
Write all classifications that apply to each real #.
A. –32                       B.

C. 7                        D.
Lesson 1-6 #1
NOTES 1-6
PROPERTIES OF REAL NUMBERS
ComMUtative Property – (Mixed Up)
You can add or multiply real #s in any order.
a+b=b+a                     ab = ba

AsSOciative Property – (Same Order)
Changing the grouping does not change the sum
(a + b) + c = a + (b + c)       (ab)c = a(bc)
Lesson 1-6 #2
Name the property that is illustrated in each
equation.
A. 7(mn) = (7m)n

B. (a + 3) + b = a + (3 + b)

C. x + (y + z) = x + (z + y)
Lesson 1-6 #3
Distributive Property: also works with subtraction
because subtracting is “adding the opposite”
a(b + c) = ab + ac              a(b – c) = ab – ac

Write and simplify each product using the Distributive
Property:       5(71)                  4(38)
Lesson 1-6 #4
Counterexample - example that disproves a
statement, or shows that it is false.

Statement             Counterexample

No month has fewer than   February has fewer than 30
30 days.                  days, so the statement is
false.

Every integer that is     18 is divisible by 2 but not
divisible by 2 is also    by 4, so the statement is
divisible by 4.           false.
Lesson 1-6 #5
Closure Property: Real numbers are “closed” if the
result of the operation on any two numbers in the set
is also in the set

The set of even numbers is closed under addition.
4 + 6 = 10
(When you add any two even #s,
the result must be even)
Even #s
Lesson 1-6 #6
Find a counterexample to show that the statement is
false.

The prime numbers are closed under addition.
Find two prime #s such that their sum is not prime.

Since ___ is not a prime number, the statement is _____.
Lesson 1-6 #7
Find a counterexample to show that the statement is
false.
The set of odd numbers is closed under subtraction.
Find two odd #s such that the difference is not odd.

___ and ___ are odd numbers, but __________, which is
not an odd number, so the statement is ________.
Lesson 1-7 #1
NOTES 1-7
SIMPLIFYING EXPRESSIONS

Order of Operations tells you which operation to
perform first in an expression.

Order of Operations

G    Grouping Symbols - ( ), [ ], { }, fraction bar,   ,

E    Exponents

M/D   Mult. and Div. from left to right

A/S   Add. and Sub. from left to right

Goodbye PEMDAS and hello GEMDAS!!!
Lesson 1-7 #2
Simplify each expression.
15 – 2  3 + 1        12 + 32 + 10 ÷ 2
Lesson 1-7 #3
grouping symbol and work
outward…

100 – [(3+4)² – 6]
Lesson 1-7 #4
Terms: the parts that are added or subtracted
Terms

4x – 3x + 2           Constant:
numerical term
Like Terms    Constant with no variable

Like Terms: terms that have the same variables and
exponents
Lesson 1-7 #5
Coefficient: a number multiplied by a variable
Coefficients

1x2 + 3x
Combining Like Terms –
• Add or subtract the coefficients
• Keep the variables and exponents the same

Alphabetical order with constants last!
Lesson 1-7 #6
A. 72p – 25p        B.             C. 0.5m + 2.5n

D. 14x + 4(2 + x)        E. 2a + 4b + 6(a – 3) – 5b

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