samplekm & pp 1 menu- click a link aim algebra in motion fall 2008 on the menu below please find...
DESCRIPTION
SAMPLEKM & PP 3 Getting Started with Algebra What isAlgebra? is a branch of mathematics in which symbols, usually letters, are used to represent quantities that can be replaced by a number or an expression. AlgebraTRANSCRIPT
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SAMPLE KM & PP 1
MENU- click a link
AIMAlgebra In Motion
Fall 2008On the MENU below please
find links to three of the Power Point presentations we have created as part of our
AIM Project. Thanks for looking!
Kathy Monaghan & Pat Peterson
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SAMPLE KM & PP 2
Thanks for Looking!
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SAMPLE KM & PP 3
Getting Started with Algebra
What isAlgebra ?
is a branch of mathematics in which symbols, usually letters,are used to represent quantities that can be replaced by a number oran expression.
Algebra
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SAMPLE KM & PP 4
Getting Started with Algebra
Who invented Algebra ?
is a reasoning skill and language that developed and evolved along with civilization.
No one person invented
Algebra
Algebra
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SAMPLE KM & PP 5
Getting Started with Algebra
Where did the wordAlgebraoriginate ?
The word is from Kitab al-Jabr wa-l-Muqabalawhich was a book written in approximately 820 A.D. by a Persian mathematician.
Algebra
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SAMPLE KM & PP 6
Variables
A is
variablea letter used to
represent various numbers.
“x” is frequently used as the variable, but many
other letters can be used.
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SAMPLE KM & PP 7
Variables
For example, jean sizes are often given by waist
and leg inseam measurements.waist
measurement
leg inseammeasurement
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SAMPLE KM & PP 8
Define each Variable
We must always define what quantity or measurement
the letter represents.Here are three examples:
= waist measurement= leg inseammeasurement
= unknown number
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SAMPLE KM & PP 9
Constants
A is
constanta letter used to
represent a number that doesn’t change its value
in the problem. For example:
= the speed of light in Einstein’s equation E = mc2
= “pi” = 3.1416….
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SAMPLE KM & PP 10
AlgebraicExpressions
An algebraic expressionis a mathematical phrase
using variables, constants, numerals, & operation signs. An algebraic
expressionwill NOT have any of the following symbols:
= > < > <
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SAMPLE KM & PP 11
Algebraic Expressions: Examples
5xx is the variable.
+ is the operation
5 is a numeral and a constant.
5x algebraic expressionis the
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SAMPLE KM & PP 12
Algebraic Expressions: Examples
p3p is the variable.
. is the indicated operation
3 is a numeral and a constant.
p3 algebraic expressionis the
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SAMPLE KM & PP 13
Algebraic Expressions: Examples
z9z is the variable.
- is the operation9 is a numeral and a
constant.
z9 algebraic expressionis the
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SAMPLE KM & PP 14
Algebraic Expressions: Examples
y5
y is the variable.÷ is the operation
5 is a numeral and a constant.
y5 algebraic expressionis the
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SAMPLE KM & PP 15
Algebraic Expressions: Examples
yx 52
x and y are variables.
and + are operations2 and 5 are numerals and constants.yx 52 algebraic expressionis the
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SAMPLE KM & PP 16
Algebraic Expressions: Examples
25
yx
x and y are variables.
+ ÷ are operations5 and 2 are numerals and constants.
2x5x
algebraic expressionis the
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SAMPLE KM & PP 17
Substitution
When a variable is replaced with a
numerical value, that is called substitution.
Sometimes, in higher mathematics, a variable is
replaced with an expression. That is also
called substitution.
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SAMPLE KM & PP 18
Evaluate an Algebraic Expression
When a numerical value is substituted into an algebraic expression
and then simplified, that is called
evaluatingthe expression.
Evaluating means you will compute a numerical
value.
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SAMPLE KM & PP 19
Evaluate an Expression:Example 1a
5xEvaluate
4x
54 )(9
when
5x
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SAMPLE KM & PP 20
Evaluate an Expression:Example 1b
5xEvaluate
0x
50 )(5
when
5x
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SAMPLE KM & PP 21
Evaluate an Expression:Example 1c
5xEvaluate
3x
53 )(8
when
5x
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SAMPLE KM & PP 22
Evaluate an Expression:Example 1d
5xEvaluate
5x
55 )(0
when
5x
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SAMPLE KM & PP 23
Evaluate an Expression: Example 2
p3Evaluate
5pa) when p3 )(53 15
2pb) when
p3 )(23 6
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SAMPLE KM & PP 24
Evaluate an Expression: Example 3
z95zwhen
z9
459
Evaluate
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SAMPLE KM & PP 25
Evaluate an Expression:
Example 4a
y5
5ywhen
y5
55
1
Evaluate
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SAMPLE KM & PP 26
Evaluate an Expression:
Example 4b
y5
0ywhen
y5
05
undefined
Evaluate
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SAMPLE KM & PP 27
Evaluate an Expression:Example 5
yx 52 4x When and 1y
yx 52 )()( 1542
)(58
13
Evaluate
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SAMPLE KM & PP 28
Evaluate an Expression: Example 6a
25
yxEvaluat
e
4xWhen and 1y
25
yx
2154
)()(
39
3
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SAMPLE KM & PP 29
Evaluate an Expression: Example 6b
25
yxEvaluat
e
8y
25
yx
2857
)()(
612
2
When and 7x
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SAMPLE KM & PP 30
Evaluate an Expression: Example 6c
37
yxEvaluat
e8y
37
yx
3877
)()(
50
0
When and 7x
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SAMPLE KM & PP 31
Evaluate an Expression: Example 6d
25
yxEvaluat
e
4xWhen and 2y
25
yx
2254
)()(
09
undefined
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SAMPLE KM & PP 32
Application: Area of a Rectangle
The AREA of a RectangleArea = length x width
A=l.w
cmw 6
cml 16
cmcmA 616296cmA
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SAMPLE KM & PP 33
Translating: English into Algebra
In order to solve problems, English phrases must be
translated into the language of algebra.
The following slides list keywords which can
help us translate.
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SAMPLE KM & PP 34
English & Algebra ADDITION
The following words translate as ADDITION:
•Plus•Sum•Add•Added to•Total•More than•Increased by
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SAMPLE KM & PP 35
X + 7
The following phrases would translate to :
•A number plus seven•The sum of a number and seven•Add a number and seven•Seven added to a number•The total of seven and a number•Seven more than a number• A number increased by seven
7x
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SAMPLE KM & PP 36
English & Algebra SUBTRACTION
The following words translate as SUBTRACTION:
•Minus•Difference•Subtract•Subtracted From•Take away•Less Than•Decreased by
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SAMPLE KM & PP 37
X - 7
The following phrases would translate to :
•A number minus seven•The difference of a number and seven•Subtract a number and seven•Seven subtracted from a number•Seven take away a number•Seven less than a number• A number decreased by seven
7x
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SAMPLE KM & PP 38
English & Algebra MULTIPLICATION
The following words translate as MULTIPLICATION:
•Multiplied by•Multiply•Product•Times•Of
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SAMPLE KM & PP 39
7x
The following phrases would translate to : x7
•A number multiplied by seven•Multiply seven and a number •The product of a number and seven•The product of seven and a number •Seven times a number
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SAMPLE KM & PP 40
English & Algebra DIVISION
The following words translate as DIVISION:
•Divided by•Divide•Quotient
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SAMPLE KM & PP 41
x/7
The following phrases would translate to :
7x
•A number divided by seven• Divide a number by seven•The quotient of a number and seven
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SAMPLE KM & PP 42
7/x
The following phrases would translate to :
x7
•Seven divided by a number• Divide a seven by a number•The quotient of seven and a number
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SAMPLE KM & PP 43
“OF”
x21
“Half of a number” would be
2x
x.50or
or
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SAMPLE KM & PP 44
“OF”
x10030
“Thirty percent of a number” is:
x.30or
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SAMPLE KM & PP 45
“Twice” or “Double”
x2
•“Twice a number” is:
x2
•“Double a number” is:
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SAMPLE KM & PP 46
Translate a Phrase
x2
“Seven more than twice a number”
7
Seven more than
twice a number
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SAMPLE KM & PP 47
Translate a Phrase
x2
“Seven less than twice a number”
7
Seven less than
twice a number
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SAMPLE KM & PP 48
Translate a Phrase (watch for the
comma)
2x
“the quotient of seven anda number increased by two”
7
2x7
“the quotient of seven and a number,
increased by two”
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SAMPLE KM & PP 49
Salary Increase?
Suppose you will get a salary increase of 3%.
Let s represent your old salary.
The increase is 3% of your current salary, so 0.03s is the
increase.Your new salary will be the sum
of the old salary and the increase.
So, s + 0.03s is your new salary.
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SAMPLE KM & PP 50
Discount?
Suppose the bookstore has all merchandise on sale for 15%
off. Let p represent the regular price.
The discount is 15% of the regular price, so 0.15p is the
discount.The sale price will be the
difference of the regular price and the discount
So, p - 0.15p is the sale price.
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SAMPLE KM & PP 51
That’s All for Now!
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SAMPLE KM & PP 52
Slope
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SAMPLE KM & PP 53
Definition: Slope
The slope of the linecontaining points
P1(x1, y1) and P2(x2, y2) is given by
2112
12 xx,xxyym
The denominator
can’t be zero.
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SAMPLE KM & PP 54
A Slope Triangle
2112
12 xx,xxyym
P1(x1, y1)
RUNx2 - x1
RISEy2 - y1
P2(x2, y2)
RISEy2 - y1
x2 - x1RUN
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SAMPLE KM & PP 55
m
Slope
xy
2112
12 xx,xxyy
m
xinchangeyinchange
RunRise
xdeltaydelta
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SAMPLE KM & PP 56
Compute the Slope
(-2, -3)
(4, -1)
31
62
2431
)(m
GOING UPWARDS
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SAMPLE KM & PP 57
Another Slope
(-4, -3)
(0, 5)
212
48
4035
m
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SAMPLE KM & PP 58
Compute this Slope
(-5, 2)
(4, -1)
31
93
5421
)(m
GOING DOWNWARD
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SAMPLE KM & PP 59
Compute the Slope
(-5, -1)
(4, -1)
0
90
5411
)(m
HORIZONTAL
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SAMPLE KM & PP 60
Compute the Slope
(-4, -5)
(-4, 0)
Undefined)(m
05
4450
VERT
ICAL
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SAMPLE KM & PP 61
SLOPEBASICS
POSITIVE SLOPE The line rises from left to right.
ZERO SLOPE The line is HORIZONTAL
NEGATIVE SLOPEThe line falls from left to right.
UNDEFINED SLOPEThe line is VERTICAL.
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SAMPLE KM & PP 62
That’s All for Now!
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SAMPLE KM & PP 63
Systems of Equations
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SAMPLE KM & PP 64
Systems of Equations
In order to solve a system of equations by graphing:
•Graph each of the lines using the best method.
• Plot Points• Plot x and y-intercepts• Use Point Slope
•The point(s) where the lines intersect are in the solution set.
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SAMPLE KM & PP 65
Where will they meet?
32
yx
),( 32
),( 32
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SAMPLE KM & PP 66
How about this pair?
3242
yxyx
),( 12
),( 12
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SAMPLE KM & PP 67
What if?
84242
yxyx
)x,x( 2
21
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SAMPLE KM & PP 68
How about these lines?
84222
yxyx
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SAMPLE KM & PP 69
That’s All for Now!