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SAMPLE KM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On 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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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. Algebra

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Page 1: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 2: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 2

Thanks for Looking!

Page 3: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 4: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 9: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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….

Page 10: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 14: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 17: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 25: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 25

Evaluate an Expression:

Example 4a

y5

5ywhen

y5

55

1

Evaluate

Page 26: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 26

Evaluate an Expression:

Example 4b

y5

0ywhen

y5

05

undefined

Evaluate

Page 27: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 27

Evaluate an Expression:Example 5

yx 52 4x When and 1y

yx 52 )()( 1542

)(58

13

Evaluate

Page 28: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 28

Evaluate an Expression: Example 6a

25

yxEvaluat

e

4xWhen and 1y

25

yx

2154

)()(

39

3

Page 29: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 29

Evaluate an Expression: Example 6b

25

yxEvaluat

e

8y

25

yx

2857

)()(

612

2

When and 7x

Page 30: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 30

Evaluate an Expression: Example 6c

37

yxEvaluat

e8y

37

yx

3877

)()(

50

0

When and 7x

Page 31: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

SAMPLE KM & PP 31

Evaluate an Expression: Example 6d

25

yxEvaluat

e

4xWhen and 2y

25

yx

2254

)()(

09

undefined

Page 32: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 40: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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.

Page 50: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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.

Page 54: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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

Page 55: SAMPLEKM & PP 1 MENU- click a link AIM Algebra In Motion Fall 2008 On the MENU below please find links to three of the Power Point presentations we have

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!