Dividing Real Numbers 2-9
Quotient a divided by b or a ÷b should be written as a fraction
a/b
Definition of division --> Which we abbreviated as Def (÷)
a÷ b = a ⋅(1/b)
to divide by a non zero number, multiply by its reciprocal
a+b
c
= a/c + b/c
where c ≠ 0
and
a-b
c
= a/c - b/c
Justifying leads us to the following:
(a+b)/c
= (a+b)(1/c) --> def (÷)
=(a ⋅ 1/c) + (b ⋅ 1/c) --> DP
=a/c + b/c --> def (÷)
Friday, August 30, 2013
Math 6A( Periods 1 & 2)
The Distributive Property of Multiplication with Respect to
Addition
DP+
For any whole numbers a, b, and c
a(b + c) = ab + ac
(b+c)a = ba + ca
13(15) … well you could just multiply it out but using the distributive property you might be able to do this in your head…
13(15) = 13( 10+5) = 13(10) + 13(5) = 130 + 65 = 195
9(11) now we know
this but let’s see what happens if we try using the distributive property
9(11) = 9(10 + 1) = 90 + 9 = 99 YAY!!!
We could have completed this another way…
9(11) = (10-1)(11) = 10(11) – 1(11) = 110 – 11= 99
This last example is really
an example of the
Distributive Property of Multiplication with Respect to Subtraction
or
DP-
For any whole numbers a, b, and c
a(b - c) = ab- ac
(b - c)a = ba - ca
What should we do with the following:
(117∙ 4)
– (17∙ 4)
Last year you would multiply
(117∙ 4) and
then multiply (17∙ 4)
and then take the difference… but that is a long process
What happens if we use the distributive property?
(117∙ 4)
– (17∙ 4) = (117-17)∙4= (100)4= 400
WOW—that was so much easier
More Examples of the Distributive Property:
(7∙ 9) + (13 ∙9)
They both share a 9 so pull the 9 out
(7 + 13)9
(20)9 = 180
WOW.. look at that!
But… what happens with
57 ∙
9
There are a number of different ways we might set this up
(50 + 7)(9) = 50(9) + 7(9)
Or
57(10-1) = 57(10) – 57(1)
Or
(60-3)(9) = 60(9) – 3(9)
We could even set it up with the following but this would not help us to solve it easier…
(37 + 20)(9)… remember our discussion in class—it was to make
the multiplication easier
The results for all of the above is 513
What about 25 ∙
38
We could do 25(40 -2) = 25(40) – 25(2)
Now that’s easy…. 1000 – 50 = 950
25(30 + 8) also works but probably isn’t quite as easy
25(30) + 25(8) = 750 + 200 = 950
101(34) = (100 + 1)(34) = 100(34) + 1(34) = 3434
(89∙
5) – (19∙ 5)
(89-19)(5)
(70)(5) =350
So much easier than multiplying each separately!
BUT… what happens if you have
4(17 + 42)
Well we would first add the two numbers in the ( )
That gives us 4(59)… but that really did not make it any
easier..
So now look at
4(59) and ask yourself what can you do with that number
4(60-1) = 4(60) – 4(1) = 240 -4 = 236!!
YAY!!
Math 7 ( Period 4)
The Commutative & Associative Properties 1.7
What are
properties... They are characteristic of math operations that can be identified
Why are they your
friends? (BFF) you can count on
properties they always work they will NOT let you down... there are NO COUNTER
EXAMPLES..
Properties are the
exceptions to Aunt Sally
These properties
give you a choice when
it is ALL MULTIPLICATION or
it is ALL ADDITION
it is ALL MULTIPLICATION or
it is ALL ADDITION
(they DO not work
for subtraction or division) lots of
counterexamples for those...
10 – 2 does not equal 2- 10
Commutative
property (works for all Multiplication or All addition)
You can switch the
order and still get the same sum or product
This is the
property that you can hear because you switched the order
a+ b = b + a
or ab= ba
Commutative Big C looks like an ear...
Why care? it makes the math easier ... sometimes
Which would you
rather multiply
(2)(543) (5) or (2)(5)(543)
Associative
Property another friend
This friend allows
you to group all multiplication or all addition ANYWAY you choose
a + (b+ c) = ( a + b)
+ c
a(bc) = (ab)c
again.. to make
the math easier
This is the
property that you can SEE instead of hear because you use (parentheses) but you don’t change the ORDER
GIVEN
[(543)(5)](2) what would Aunt Sally say you MUST do?
but our friend the
associative property allows use to move the brackets...
(543)[(5)(2)]
which is so much easier to multiply—even in your head...
Wednesday, August 28, 2013
Algebra Honors (Periods 6 & 7)
Problem Solving Consecutive Integers 2-7
Consecutive- list in natural order from least to greatest
-2, -1, 0, 1 ..
an integer n
the next three numbers are n + 1, n+ 2, and n+ 3
the number that receded n is n-1
Let's create the following equation: The sum of four consecutive integers is 66
Let n be the first integer
n + (n + 1) + (n + 2) + (n + 3) = 66
4n + 6 = 66
n = 15
Now, if the question asked what was the first integer the solution would be {15}
but if the question asks you to state all the integers the solution would be
{15, 16, 17, 18}
If x = 15, how would you write 14 and 16 in terms of x?
14 --> x -1
16 --> x + 1
if m is odd, m + 1 is even
How do you write 4 consecutive even integers starting with x ?
x, x + 2, x + 4, x + 6
How do you write 4 consecutive odd integers, starting with x ?
x , x + 2, x + 4, x + 6
In the above cases, you defined what x was-- in the first case x was even... so you need to add multiples of 2 to continue getting an even number... AND in the 2nd case you defined that this time x was odd.. and again you need to add multiples of 2 to continue getting an odd number.
The sum of 2 consecutive integers is 43. Let x = 1st integer
x + ( x + 1) = 43
The sum of three consecutive odd integers is 40 more than the smallest.
What are the integers?
(Hint: Reread the word problem carefully!!)
Let x represent the 1st odd integer
x + (x + 2) + (x + 4) = x + 40
3x + 6 = x + 40
2x = 34
x = 17
so
{17, 19, 21} is the solution.. and if you substitute in to check, you will see you have solved the question!!
However, in our book, it asks this question a little differently, It says with the domain for the smallest given as {13, 17, 25}.
In this case you could work backwards and substitute each of the three integers to see if it works.
The Reciprocal of Real Numbers 2-8
Two numbers whose product is 1 are called reciprocals or multiplicative inverses
5 and 1/5 are reciprocals because 5⋅1/5 = 1
4/5 and 5/4 are reciprocals 4/5⋅5/4 = 1
and so are (-1.25) and (-0.8) because (-1.25) ⋅ (-0.8) = 1
look closely at (-1.25) and (-0.8) and you realize that they are really -5/4 ⋅ -4/5 = 1
If a is any real number, its reciprocal is 1/a
Property of Reciprocals
for any non zero real number a,
a⋅1/a = 1 and 1/a⋅ a = 1
-a and -1/a are also reciprocals
Notice, we discussed in class that
1/-a = -1/a = -(1/a) those are equivalent
(ab)(1/a⋅1/b) = a (1/a)⋅b(1/b) = a/a⋅b/b= 1⋅1 = 1
so try the following
1/3(42m -3v)
1/3(42m) - (1/3)(3v)
14m - v
Make sure to distribute the negative to each term
How about
-1/20( 5z -4w) - 6(-1/30w - 1/24z)
Again make sure to distribute the negative to each term!!
(-1/4)z + (1/5)w + (1/5)w + (1/4)z
or
-z/4 + w/5 + w/5 + z/4
= 2w/5 also written as (2/5)w
Consecutive- list in natural order from least to greatest
-2, -1, 0, 1 ..
an integer n
the next three numbers are n + 1, n+ 2, and n+ 3
the number that receded n is n-1
Let's create the following equation: The sum of four consecutive integers is 66
Let n be the first integer
n + (n + 1) + (n + 2) + (n + 3) = 66
4n + 6 = 66
n = 15
Now, if the question asked what was the first integer the solution would be {15}
but if the question asks you to state all the integers the solution would be
{15, 16, 17, 18}
If x = 15, how would you write 14 and 16 in terms of x?
14 --> x -1
16 --> x + 1
if m is odd, m + 1 is even
How do you write 4 consecutive even integers starting with x ?
x, x + 2, x + 4, x + 6
How do you write 4 consecutive odd integers, starting with x ?
x , x + 2, x + 4, x + 6
In the above cases, you defined what x was-- in the first case x was even... so you need to add multiples of 2 to continue getting an even number... AND in the 2nd case you defined that this time x was odd.. and again you need to add multiples of 2 to continue getting an odd number.
The sum of 2 consecutive integers is 43. Let x = 1st integer
x + ( x + 1) = 43
The sum of three consecutive odd integers is 40 more than the smallest.
What are the integers?
(Hint: Reread the word problem carefully!!)
Let x represent the 1st odd integer
x + (x + 2) + (x + 4) = x + 40
3x + 6 = x + 40
2x = 34
x = 17
so
{17, 19, 21} is the solution.. and if you substitute in to check, you will see you have solved the question!!
However, in our book, it asks this question a little differently, It says with the domain for the smallest given as {13, 17, 25}.
In this case you could work backwards and substitute each of the three integers to see if it works.
The Reciprocal of Real Numbers 2-8
Two numbers whose product is 1 are called reciprocals or multiplicative inverses
5 and 1/5 are reciprocals because 5⋅1/5 = 1
4/5 and 5/4 are reciprocals 4/5⋅5/4 = 1
and so are (-1.25) and (-0.8) because (-1.25) ⋅ (-0.8) = 1
look closely at (-1.25) and (-0.8) and you realize that they are really -5/4 ⋅ -4/5 = 1
If a is any real number, its reciprocal is 1/a
Property of Reciprocals
for any non zero real number a,
a⋅1/a = 1 and 1/a⋅ a = 1
-a and -1/a are also reciprocals
Notice, we discussed in class that
1/-a = -1/a = -(1/a) those are equivalent
(ab)(1/a⋅1/b) = a (1/a)⋅b(1/b) = a/a⋅b/b= 1⋅1 = 1
so try the following
1/3(42m -3v)
1/3(42m) - (1/3)(3v)
14m - v
Make sure to distribute the negative to each term
How about
-1/20( 5z -4w) - 6(-1/30w - 1/24z)
Again make sure to distribute the negative to each term!!
(-1/4)z + (1/5)w + (1/5)w + (1/4)z
or
-z/4 + w/5 + w/5 + z/4
= 2w/5 also written as (2/5)w
Math 6A (Periods 1 & 2)
Inverse Operations 1-3
Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication &division are inverse operations
We can use these relationships between inverse operations to simplify some numerical and variable expressions
17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17
9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...
108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108
3n ÷ 3 = 3
Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0
In Algebra
a- a = 0 is called the Property of Opposites ( The Prop of Opp's)
Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1
Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0
Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'
Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication &division are inverse operations
We can use these relationships between inverse operations to simplify some numerical and variable expressions
17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17
9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...
108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108
3n ÷ 3 = 3
Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0
In Algebra
a- a = 0 is called the Property of Opposites ( The Prop of Opp's)
Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1
Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0
Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'
Math 7 (Period 4)
Powers & Exponents 1.3
A Power has two parts:
A Power has two parts:
BASE= the large
number on the bottom ( base)
EXPONENT = the
tiny number raise up
EXPONENTIAL
NOTATION = the ay you write the expression with the exponent
EXPANDED NOTATION
= showing the multiplication the appropriate number of times reflected by the
exponent
In your book this
is called “ repeated multiplication”
STANDARD NOTATION
= the actual value ( what you would expect) you find this by evaluating the
power—doing the repeated multiplication
34 is
exponential notation where 3 is the base and 4 is the exponent
(3)(3)(3)(3) is expanded notation
81 is standard
notation
The entire
expression is called 3 to the power
of 4
(4y)3 =
(4y)(4y)(4y) = 64y3
Evaluating a Power
Evaluating is
plugging and chugging when a variable is raised to a power
x4 when
x = 3 34 = 81
Some special powers
to know
There are two
special powers that have ‘nicknames”
32 can
be read “ 3 to the second power” but its
nickname is 3 squared
This is based on
finding the area of a square
If one side of the
square is 3 then the area would be ( 3)(3) which is 32
Likewise 33
can be read 3 to the third power.... but its nickname is “ 3 cubed”
This name is based
on finding the volume of a cube.
(3)(3)(3) = 33
Tuesday, August 27, 2013
Math 7 (Period 4)
Expressions & Variables 1-2
(we are skipping
1.1 for now—will pick it up with a unit
on data and stats)
LOTS of VOCAB
term: a number and
or variable added or subtracted to another number and or variable
factors : a number
and or variable multiplied by another number and or variable
addition words:
sum, total plus, more than greater than addends
altogether both
subtraction words:
difference minus, less than, less
minuend subtrahend
multiplication
words: product, times, factors, each
division words:
quotient, dividend divisor , shared equally
expression: at
least one term ( with NO equal sign)
equation: equal
expressions ( it must have an = sign)
numerical: all
numbers ( called constants)
variable: a letter
( or symbol) used to represent a number
algebraic (or
variable): has at least one variable
simplify a
numerical expression: do order of operations you will get a simpler expression
evaluate- given
the value of the variable you “plug and chug” in an algebraic expression and
simplify using operations
You should be able
to write a numerical expression in words:
3 + 7 is The sum of three and seven
You should be able to
write a numerical equation in words:
3 + 7 = 10
The sum of three
and seven is ten
Evaluating VARIABLE ( or
algebraic) expressions
Evaluate 5x + y
when x = 1.2 and y = 0.8
5(1.2) + 0.8 = 6.8
Math 6A (Periods 1 & 2)
Properties of Addition &Multiplication 1-2
1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers
Some important properties are found using whole numbers:
Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a
Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba
You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!
However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18
Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)
Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)
You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!
Addition Property of Zero from the book
also known as the Identity Property of Addition ( ID+)
for any whole number a
a + 0 = a and 0 + a = a
Multiplication Property of One
also know as the Identity Property of Multiplication (IDx)
For any whole number a
a(1) = a and 1(a) = a
Multiplication Property of Zero MPZ
for any whole number a
0(a) = 0 and 0(a) = 0
So how do we justify?
16 + 19 + 34 --- given
16 + 34 + 19 ---C+
(16 + 34) + 19--- A+
50 + 19---simplify
69--- simplify
2 (14)(15)--- given
2 (15) (14) --- Cx
[2(15)] (14) --- Ax
30 (14)--- simplify
420--- simplify
how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 --- given
14 ⋅ 8 ⋅ 5⋅ 25 --- Cx
14 ⋅ (8 ⋅ 5)⋅ 25 --->Ax
14 ⋅ 40⋅ 25---simplify
14 ⋅ (40⋅ 25) --- Ax
14 ⋅ 1000 --- simplify
14000 --- simplify
Make sure you justify each step of tonight's homework!!
1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers
Some important properties are found using whole numbers:
Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a
Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba
You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!
However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18
Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)
Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)
You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!
Addition Property of Zero from the book
also known as the Identity Property of Addition ( ID+)
for any whole number a
a + 0 = a and 0 + a = a
Multiplication Property of One
also know as the Identity Property of Multiplication (IDx)
For any whole number a
a(1) = a and 1(a) = a
Multiplication Property of Zero MPZ
for any whole number a
0(a) = 0 and 0(a) = 0
So how do we justify?
16 + 19 + 34 --- given
16 + 34 + 19 ---C+
(16 + 34) + 19--- A+
50 + 19---simplify
69--- simplify
2 (14)(15)--- given
2 (15) (14) --- Cx
[2(15)] (14) --- Ax
30 (14)--- simplify
420--- simplify
how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 --- given
14 ⋅ 8 ⋅ 5⋅ 25 --- Cx
14 ⋅ (8 ⋅ 5)⋅ 25 --->Ax
14 ⋅ 40⋅ 25---simplify
14 ⋅ (40⋅ 25) --- Ax
14 ⋅ 1000 --- simplify
14000 --- simplify
Make sure you justify each step of tonight's homework!!
Algebra Honors ( Periods 6 & 7)
Chapter 2 Highlights Continued
Substitution Principle-- an expression may be replaced by another expression that has the same value.
JUSTIFY THE FOLLOWING:
b + (a -b)
= b + [a +(-b)] Definition of subtraction ( we used def-)
= b + [-b + a] C+
=(b + -b) + a A+
= 0 + a Property of Opposites ( How about Prop of Op for this one!!)
= a Id+
Check out the 5 step plan on Page 27... use the index in your textbook to find many concepts and explanations...
Multiplication Property of One a(1) = a and 1(a) = a IDx 1 is the identity element for multiplication
Multiplication Property of Zero A90) = 0 and 0(a) = 0 Ox
Multiplicative Prop of -1
a(-1) = -a and (-1)a = -a
Justify the following
a(-1) + a
= a(-1) + a(1) ID x
= a(-1 +1) DP
=a(0) Prop of Op
=0 Ox
Property of Opposites in Products or the Prop of Opposites in Products or just POP
(-a)b= -ab
a(-b) = -ab
(-a)(-b) = ab
JUSTIFY
a(b-c)
=a[b + (-c)] Def-=ab + a(-c) DP
ab + (-ac) POP
ab -ac Def -
-3(7c + d) - 2(10d-c)
-21c -3d -20d + 2c
-19c -23d
Substitution Principle-- an expression may be replaced by another expression that has the same value.
JUSTIFY THE FOLLOWING:
b + (a -b)
= b + [a +(-b)] Definition of subtraction ( we used def-)
= b + [-b + a] C+
=(b + -b) + a A+
= 0 + a Property of Opposites ( How about Prop of Op for this one!!)
= a Id+
Check out the 5 step plan on Page 27... use the index in your textbook to find many concepts and explanations...
Multiplication Property of One a(1) = a and 1(a) = a IDx 1 is the identity element for multiplication
Multiplication Property of Zero A90) = 0 and 0(a) = 0 Ox
Multiplicative Prop of -1
a(-1) = -a and (-1)a = -a
Justify the following
a(-1) + a
= a(-1) + a(1) ID x
= a(-1 +1) DP
=a(0) Prop of Op
=0 Ox
Property of Opposites in Products or the Prop of Opposites in Products or just POP
(-a)b= -ab
a(-b) = -ab
(-a)(-b) = ab
JUSTIFY
a(b-c)
=a[b + (-c)] Def-=ab + a(-c) DP
ab + (-ac) POP
ab -ac Def -
-3(7c + d) - 2(10d-c)
-21c -3d -20d + 2c
-19c -23d
Monday, August 26, 2013
Math 6A (Periods 1 & 2)
Mathematical Expressions 1-1
A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.
An expression, such as 3 x n, that involves a variable is called a variable expression.
Expressions, such as 3 x 2, that name a certain number are called numerical expressions
When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab
In numerical expressions for products a multiplication symbol must be used to avoid confusion.
9 x 7 may be written as 9 ∙ 7 or even 9(7)
When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression
When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48
148 ÷ 4 =
148/4
37
if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6
If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x
Now substitute in your values
4(18) / 3(8)
72/24 = 3
A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.
An expression, such as 3 x n, that involves a variable is called a variable expression.
Expressions, such as 3 x 2, that name a certain number are called numerical expressions
When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab
In numerical expressions for products a multiplication symbol must be used to avoid confusion.
9 x 7 may be written as 9 ∙ 7 or even 9(7)
When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression
When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48
148 ÷ 4 =
148/4
37
if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6
If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x
Now substitute in your values
4(18) / 3(8)
72/24 = 3
Algebra Honors (Periods 6 & 7)
Chapter 1 Highlights
Variable→ symbol used to represent one or more numbers
Variable expression contains a variable
Numerical expression names a particular number
Simplifying the expression→Replacing a numerical expression by the simplest name for its value
Grouping symbols→ parentheses, brackets, fraction bar, etc used to enclose an expression that should be simplified first.
Equation→ two numerical or variable expressions that are equal. Represented by an equals sign placed between the two sides of the equation.
Open Sentences→ contain variables, such as 5x – 1 =9
or y + 2 = 2 + y
The given set of numbers that a variable represents is called the domain of the variable.
Use brackets { } to show a set of numbers. A short way to write “the set whose members are 1, 2, and 3” is {1, 2, 3}
Any value of a variable that turns an open sentence into a true sentence is a solution or root of the sentence.
The set of all solutions of an open sentence is called the solution set of the sentence.
Some equations have only one solution, and some equation have no solutions. The sentence y + 2 = 2 + y has an infinite number. The solution set is the set of all numbers. If, however, you are asked to solve of the domain {0,1,2,3}, you state that the solution set is the domain itself {0, 1, 2, 3}.
Another way to express this is
Read the above as
“y belongs to the set whose members are 0, 1, 2, 3”
Real Numbers
Natural Numbers or Counting Numbers→ 1, 2, 3, 4, 5….
Whole Numbers (Natural Numbers + 0) → 0, 1,2, 3, 4, 5 …
Integers (Natural Numbers , their opposites, + 0) → …-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Rational Numbers (quotient of two integers)
So any number that is either positive, negative, or zero is called a real number.
Each number is a pair, such as 4 and -4 is called the opposite of the other number. The opposite of a is written –a.
The numerals -4(lowered minus sign) and -4 (raised minus sign) name the same number.
Caution –a read “the opposite of a” is NOT necessarily a negative number.
For example, if a= -2 then –a would be –(-2) = 2
Absolute value is a distance concept. It my be thought of as the distance between the graph of a number and the origin on a number line. The graphs of -4 and 4 are both 4 units from the origin.
Variable→ symbol used to represent one or more numbers
Variable expression contains a variable
Numerical expression names a particular number
Simplifying the expression→Replacing a numerical expression by the simplest name for its value
Grouping symbols→ parentheses, brackets, fraction bar, etc used to enclose an expression that should be simplified first.
Equation→ two numerical or variable expressions that are equal. Represented by an equals sign placed between the two sides of the equation.
Open Sentences→ contain variables, such as 5x – 1 =9
or y + 2 = 2 + y
The given set of numbers that a variable represents is called the domain of the variable.
Use brackets { } to show a set of numbers. A short way to write “the set whose members are 1, 2, and 3” is {1, 2, 3}
Any value of a variable that turns an open sentence into a true sentence is a solution or root of the sentence.
The set of all solutions of an open sentence is called the solution set of the sentence.
Some equations have only one solution, and some equation have no solutions. The sentence y + 2 = 2 + y has an infinite number. The solution set is the set of all numbers. If, however, you are asked to solve of the domain {0,1,2,3}, you state that the solution set is the domain itself {0, 1, 2, 3}.
Another way to express this is
“y belongs to the set whose members are 0, 1, 2, 3”
Real Numbers
Natural Numbers or Counting Numbers→ 1, 2, 3, 4, 5….
Whole Numbers (Natural Numbers + 0) → 0, 1,2, 3, 4, 5 …
Integers (Natural Numbers , their opposites, + 0) → …-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Rational Numbers (quotient of two integers)
So any number that is either positive, negative, or zero is called a real number.
Each number is a pair, such as 4 and -4 is called the opposite of the other number. The opposite of a is written –a.
The numerals -4(lowered minus sign) and -4 (raised minus sign) name the same number.
Caution –a read “the opposite of a” is NOT necessarily a negative number.
For example, if a= -2 then –a would be –(-2) = 2
Absolute value is a distance concept. It my be thought of as the distance between the graph of a number and the origin on a number line. The graphs of -4 and 4 are both 4 units from the origin.
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