sect 1.1 algebraic expressions variable constant variable expression evaluating the expression area...

Sect 1.1Algebraic Expressions

Variable

Constant

Variable Expression

Evaluating the Expression

Area formula Perimeter

Consist of variables and/or numbers, often with operation signs and grouping symbols.

Any symbol that represents a number…letters or

A value that never changes.

An expression that contains a variable.

To evaluate an expression, we substitute a value in for each variable in the expression and calculate the result.

A = (base)(height) = bh

A = (length)(width) = lw

P = all exterior sides added together.

Rectangle: P = 2l + 2w

Sect 1.1

added to

sum of

plus

more than

increased by

subtracted from

difference of

minus

less than

decreased by

5 pounds was added to the number

The sum of a number and 12

7 plus some number

20 more than the number

The number increased by 3

2 was subtracted from the number

difference of two numbers

8 minus some number

9 less than the number

The number decreased by 10

n + 5

x + 12

7 + m

r + 20

y + 3

w - 2

a - b

8 - c

d - 9

f - 10

Sect 1.1

multiplied by

product of

times

twice

of

divided by

quotient of

divided into

ratio of

per

the number multiplied by 4

the product of two numbers

13 times some number

twice the number

half of the number

3 divided by the number

the quotient of two numbers

8 divided into some number

the ratio of 9 to some number

There were 28 miles per g gallons

4n

xy

13z

2t

g

r

h

nm

q

28

9

8

3

x2

1

Sect 1.1

Four less than Joe’s height in inches.

Eighteen increased by a number.

A day’s pay divided by eight hours.

Half of the pallet.

Seven more than twice a number.

Six less than the product of two numbers.

Nine times the difference of a number and 3.

Eighty five percent of the enrollment.

Twice the sum of a number and 3.

The sum of twice a number and 3.

h – 4 ab – 6

18 + n 9(m – 3)

p/8 85%(e) = 0.85(e)

1/2 p 2(x + 3)

2x + 7 2x + 3

Sect 1.1

The symbol = (“equals”) indicates that the expressions on either side of the equal sign represents the same number. An equation is when two algebraic expression are equal to each other. Equations can be true or false.

In the last example, replacing the “x” with a value that makes the equation true is called a solution. Some equations have more than one solution, and some have no solutions. When all solutions have been found, we have solved the equation.

Determine whether 8 is a solution of x + 5 = 13.

3284 135 x649 True 32 = 32 False 5 = 6 We don’t know the value of

x.

8 + 5 = 13

13 = 13

True, 8 is a solution

Sect 1.1When translating phrases into expressions to equations, we need to look for the phrases “is the same as”, “equal”, “is”, and “are” for the = sign.

Translate.

What number plus 478 is 1019?

Twice the difference of a number and 4 is 24.

Three times a number plus seven is the same as the number less than one.

The Taipei Financial Center, or Taipei 101, in Taiwan is the world’s tallest building. At 1666 ft, it is 183 ft taller than the Petronas Twin Towers in Kuala Lumpur. How tall are the Petronas Twin Towers?

x + 478 = 1019

2 ( ________ – _________ )

x 4 = 24

3x + 7 = x – 1

“than” makes the terms switch around the minus sign

1666 = P + 183 – 183 – 183

1483 = P

Sect 1.2

Equivalent expressions. 4 + 4 + 4, , and 3(4)

Laws that keeps expressions equivalent.Commutative Law for Addition for Multiplication

Associative Law for Addition for Multiplication

43

abba baab

cbacba cabbca

95315 49315

switch around the plus sign

switch around the times sign

abba

Move ( )’s around new plus sign Move ( )’s around new times sign

10 + 10 =3 + 23 20 27 = 540

Sect 1.2

Use the Commutative Law and Associative Law for Addition.

Use the Commutative Law and Associative Law for Multiplication.

Use the Commutative Law for Addition and Multiplication.

37 x

yx4

37 x

37 x 37 x

x 73 x73

yx4 yx 4

xy 4 xy4

x73 73 x

Sect 1.2

Distributive Property

Factor using the Distributive Property

acabcba

2375 34 x

624 yx 19352 dcba

yx 77 4812 yx

Separate by place values & add.

1000 +

7302005

150 + 35 = 11854x – 12

8x – 4y + 24 10ab + 6ac – 18ad – 2a

yx 77

7 yx GCF leftovers

142434 yx

4 123 yx

Sect 1.2

Terms vs Factors

Term is any number, variable, or quantity being multiplied together. Be careful of the definition that terms are separated by plus or minus signs.Only if the ( )’s are simplified away!One term two terms three terms

Factors are the number, variable or quantity being multiplied together.

52 ba aab 102 432 yx

wx

72

10

Multiplying

52 ba The factors are 2, a, and (b – 5)

Sect 1.3

Review: Natural Numbers = { 1, 2, 3, 4, 5, 6, …..}

List factors of 18.

Prime Numbers are Natural numbers that have 2 different factors, 1 and itself.

{2, 3, 5, 7, 11, 13, 17, 19, 23, …}

Composite Numbers are Natural numbers that have 3 or more factors.

{4, 6, 8, 9, 10, 12, 14, 15, …}

Notice that “1” is not in either set!

The factors are 1, 2, 3, 6, 9, 18

Sect 1.3

List the prime factorization of 48.

Tree method Staircase Method

Division Rules

2: any even number

3: sum of the digits is divisible by 3

5: ends in 0 or 5

48

4 12

2 2 2 6

2 3

3222248

24482

Always start with smallest prime numbers and work up to largest prime number.

2

122

62

3

3222248

The prime number outside the upside down division boxes should be all the prime numbers.

Sect 1.3Fraction notation.

Fraction Properties

Notation for 1 Notation for 0 Undefined

b

a numeratordenominator

1a

a 00

b

undefineda

0

Why do we use the undefined term?We have to define Multiplication and Division with the same numbers.

Example1535

5315 We start with 5 and finish with 5 when we multiply by 3 and divide by 3.

005 500 Multiplying by 0 and divide by 0 doesn’t

return to the original value, not defined.

Sect 1.3

Fraction multiplication.

Multiplicative inverse (Reciprocal)

Fraction Division

Multiplicative Identity

bd

ac

d

c

b

a

1a

b

b

a

bc

ad

c

d

b

a

d

c

b

a

Tops together and bottoms together.

15

4

8

3

10

1

1012

112

120

12

158

43

Another technique is to Simply first.

15

4

8

3

10

1

52

11

53

4

24

3

11

We don’t divide by fractions, but Multiply by the reciprocal of the fraction that we are dividing by.

48

35

12

7

35

48

12

7

57

412

12

7

11

5

4

b

a

b

a1

bc

ac

c

c

b

a

Use this property to get common denominators.

Sect 1.3

Simplify the fraction by multiplication rules.

Canceling errors!

Addition and Subtraction of Fractions (same Denominators)

40

1524

36

2

32

72

9

24

14

b

ca

b

c

b

a

b

ca

b

c

b

a

85

35

8

3 212

312

2

3

89

19

8

1

Can’t cancel with addition or subtraction!1

3 2

1

12

1

12

5

8

5

8

11

12

6

2

1

1

28

6

4

3

3

4

Sect 1.3Addition and Subtraction of Fractions (with different Denominators)

bd

bcad

bd

bc

bd

ad

b

b

d

c

d

d

b

a

d

c

b

a

bd

bcad

bd

bc

bd

ad

b

b

d

c

d

d

b

a

d

c

b

a

12

11

8

7

6

5

8

9

Rule that works every time, however, can create huge numbers!

8

8

12

11

12

12

8

7

96

88

96

84

96

172

244

434

24

43

We can work with smaller numbers and prior knowledge…staircase method.

12

11

8

7

,8 124,2 3

Multiply the outsides for the LCD = 4(2)(3) = 24 12

11

8

7

3

3

242424

2

2

21 22 43

,8 62,4 3

Multiply the outsides for the LCD = 2(4)(3) = 24 6

5

8

9

3

3

242424

4

4

27 20 7Notice cross multiplying = 24

Sect 1.4 Positive and Negative Real NumbersReview: The Set of Numbers

0

REAL NUMBERSAny number on the number line.

IRRATIONAL NUMBERSNumbers that can’t be written

as a fraction

...73205.13,: Examples

RATIONAL NUMBERSNumbers that CAN be written

as a fraction24,3.0,,9: 4

3 Examples

INTEGERS NUMBERS… -3, -2, -1, 0, 1, 2, 3, …

WHOLE NUMBERS0, 1, 2, 3, …

NATURAL NUMBERS1, 2, 3, …

Less Than, Greater Than

< >

Less Than or Greater Than or

Equal to, Equal to

< >

To compare decimal numbers, both numbers need to have the same number of decimal places. Add a 0 to the end of the left number and compare place values until different.

0309.2___0310.210 > 9

To compare fractions, we need common denominators. Multiply the other denominators to the numerators and compare the products.

___1211

126

1112

117

77___72

12

7

11

6

Sect 1.4 Positive and Negative Real Numbers

Absolute Value The POSITIVE distance a number is away from zero on the number line.

5 75 77

-5 -4 -3 -2 -1 0 1 2

5 units long

Convert a repeating decimal to fraction.

3.0 Step 1. Set the repeating decimal = x 3.0x

Step 2. Get the decimal point to the left of the repeating digits. Already done.

3.0x

Step 3. Get the decimal point to the right of the repeating digits. Multiply by 10’s to both sides of the equation. This moves the decimal point one place for each 10.

33.01010 x

3.310 x

Step 4. Subtract Step 3 – Step 2 and solve for x.

3.0x39 x

31

93 x

31



Step 2. Get the decimal point to the left of the repeating digits. Already done.

63.0x


6363.0100100 x

63.63100 x


63.0x6399 x

117

9963 x

117



Step 2. Get the decimal point to the left of the repeating digits. Multiply by 10.

66.110 x


66.1101010 x

66.16100 x


66.110 x1590 x

61

9015 x

61

Sect 1.5 and 1.6 Add & Subtract sign numbers

Add & Subtract with number line.

3 Step rule. Any two signed numbers.

1. Remove all double signs.

2. Keep the sign of the largest number ( absolute value ).

3. a. Same Signs Sum

b. Different Signs Difference (subtract)

a + ( - b )

a – b

a – ( - b )

a + b

+Large – small = Positive answer Small – Large = Negative answer

- a – b = - (a + b)

+ a + b = + (a + b) +Large – small = + (Large – small)

– Large + small = – ( Large – small)


-12 + (-7) -15 + 9

-16 – 18 23 + (-11)

-32 – (-4) 19 – (-7)

-9 + (-7) – (-4) + 3 – 8 – (-12)

Law of Opposites: a + (-a) = 0

3 Same signs

SUM

2 Sign of Largest number1. Double signs

-12 – 7 = – 19

Add all positive numbers 1st and negative numbers 2nd.

1. Double signs

NONE

3 Different signs

Difference LG - sm

2 Sign of Largest number

1. Double signs

-9 – 7 + 4 + 3 – 8 + 12

= – 6

1. Double signs

NONE

3 Same signs

SUM

2 Sign of Largest number

= – 34


23 – 11 = + 12 3 Different signs

Difference LG - sm

1. Double signs

-32 + 4 = – 28 2 Sign of Largest number

3 Different signs

Difference LG - sm 3 Same signs

SUM


19 + 7 = + 26

19 – 24 2 Sign of Largest number

= – 5

3 Different signs

Difference LG - smGood to use this property when adding a long list of sign numbers…canceling is good!


Combine Like Terms

Defn. 1. Must have the same variables in the individual terms.

2. The exponents on each variable must be the same.

Identify the like terms. 7x + 3y – 5 + 2x – 9y – 8x + 10

Now Combine them.

Combine Like Terms

2a + (- 3b) + (-5a) + 9b 2xy + 3x – 7y + 5 – 8x – 2 + y

x – terms

7x + 2x – 8xy – terms

+ 3y – 9yconstants

– 5 + 10

x – 6y + 5

2a – 3b – 5a + 9b

– 3a + 6b

2xy + 3x – 7y + 5 – 8x – 2 + y

2xy – 5x – 6y + 3

Sect 1.7 Mult and Division of sign numbers

2 steps

1. Determine the sign.

Even number of Negatives being multiplied or divided = Positive answer.

Odd number of Negatives being multiplied or divided = Negative answers.

2. Multiply or divide the values.

Multiply by 0 rule.

Sign on the fraction rule.

341523 31454

1723

102 3 4 1

36

360 20

1

27301115103 0

b

a

b

a

b

a

Sect 1.8 Exponential Notation & Order of Operations

Exponential notation is a short cut to writing out repetitive multiplication.

Simplify.

6aaaaaaa

43 4343

32x32x

3333

81

99

3333

81

99

43133331

81

991

xxx 222 38x

xxx 232x

aa 1Negative quantities are defined as a -1 multiplied to the positive quantity.

Sect 1.8 Exponential Notation & Order of Operations

Order of Operation P.E.MD.AS

1. P = ( )’s which means all grouping symbols. ( ), { }, [ ], | |, numerators, denominators, square roots, etc.

2. E = Exponents. All exponential expressions must be simplified.

3. MD = Multiply or Divide in order from Left to Right

4. AS = Add or Subtract in order from Left to Right

35215 3532948 34 23

547912

31015

35

8

32292

8292

1692

72

5

Top Bottom

547912 34 23

20212 881

2024 89

44

89

44

Sect 1.8 Exponential Notation & Order of OperationsSimplify

54295 xx xxxx 5237 22

When variables are present, remove ( )’s by the Distributive Property and Combine Like Terms.

10895 xx

x13 1

Include the sign

xxxx 5637 22

210x x

Original Expression

Our Answer

2, STO> button, X, enter

57 yx 243 xx tttt 9425 22

237 x 81527 33 xx

571 yx

57 yx

243 xx2 x

tttt 9425 22

tt 119 2

637 x

x31

85527 33 xx

3527 33 xx

3527 33 xx

12 3 x

sect 1.1 algebraic expressions variable constant variable expression evaluating the expression area...

Documents

number difference

number half

p slide

solution of x

solution slide

w slide

value of

equivalent expressions