samplekm & pp 1 menu- click a link aim algebra in motion fall 2008 on the menu below please find...

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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Thanks for Looking!

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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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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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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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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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Variables

For example, jean sizes are often given by waist

and leg inseam measurements.waist

measurement

leg inseammeasurement

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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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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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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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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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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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Algebraic Expressions: Examples

z9z is the variable.

- is the operation9 is a numeral and a

constant.

z9 algebraic expressionis the

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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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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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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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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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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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Evaluate an Expression:Example 1a

5xEvaluate

4x

54 )(9

when

5x

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Evaluate an Expression:Example 1b

5xEvaluate

0x

50 )(5

when

5x

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Evaluate an Expression:Example 1c

5xEvaluate

3x

53 )(8

when

5x

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Evaluate an Expression:Example 1d

5xEvaluate

5x

55 )(0

when

5x

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Evaluate an Expression: Example 2

p3Evaluate

5pa) when p3 )(53 15

2pb) when

p3 )(23 6

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Evaluate an Expression: Example 3

z95zwhen

z9

459

Evaluate

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Evaluate an Expression:

Example 4a

y5

5ywhen

y5

55

1

Evaluate

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Evaluate an Expression:

Example 4b

y5

0ywhen

y5

05

undefined

Evaluate

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Evaluate an Expression:Example 5

yx 52 4x When and 1y

yx 52 )()( 1542

)(58

13

Evaluate

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Evaluate an Expression: Example 6a

25

yxEvaluat

e

4xWhen and 1y

25

yx

2154

)()(

39

3

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Evaluate an Expression: Example 6b

25

yxEvaluat

e

8y

25

yx

2857

)()(

612

2

When and 7x

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Evaluate an Expression: Example 6c

37

yxEvaluat

e8y

37

yx

3877

)()(

50

0

When and 7x

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Evaluate an Expression: Example 6d

25

yxEvaluat

e

4xWhen and 2y

25

yx

2254

)()(

09

undefined

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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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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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English & Algebra ADDITION

The following words translate as ADDITION:

•Plus•Sum•Add•Added to•Total•More than•Increased by

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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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English & Algebra SUBTRACTION

The following words translate as SUBTRACTION:

•Minus•Difference•Subtract•Subtracted From•Take away•Less Than•Decreased by

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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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English & Algebra MULTIPLICATION

The following words translate as MULTIPLICATION:

•Multiplied by•Multiply•Product•Times•Of

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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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English & Algebra DIVISION

The following words translate as DIVISION:

•Divided by•Divide•Quotient

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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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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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“OF”

x21

“Half of a number” would be

2x

x.50or

or

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“OF”

x10030

“Thirty percent of a number” is:

x.30or

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“Twice” or “Double”

x2

•“Twice a number” is:

x2

•“Double a number” is:

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Translate a Phrase

x2

“Seven more than twice a number”

7

Seven more than

twice a number

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Translate a Phrase

x2

“Seven less than twice a number”

7

Seven less than

twice a number

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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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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.

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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That’s All for Now!

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Slope

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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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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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m

Slope

xy

2112

12 xx,xxyy

m

xinchangeyinchange

RunRise

xdeltaydelta

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Compute the Slope

(-2, -3)

(4, -1)

31

62

2431

)(m

GOING UPWARDS

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Another Slope

(-4, -3)

(0, 5)

212

48

4035

m

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Compute this Slope

(-5, 2)

(4, -1)

31

93

5421

)(m

GOING DOWNWARD

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Compute the Slope

(-5, -1)

(4, -1)

0

90

5411

)(m

HORIZONTAL

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Compute the Slope

(-4, -5)

(-4, 0)

Undefined)(m

05

4450

VERT

ICAL

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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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That’s All for Now!

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Systems of Equations

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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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Where will they meet?

32

yx

),( 32

),( 32

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How about this pair?

3242

yxyx

),( 12

),( 12

SAMPLE KM & PP 67

What if?

84242

yxyx

)x,x( 2

21

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How about these lines?

84222

yxyx

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That’s All for Now!