13.5 Math 85 Worksheet - PowerPoint by zgc58946

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									       Unit 1
SOLVING EQUATIONS AND
      EXPRESSIONS
                       Unit 1 Vocabulary
1.    absolute value                     18.    algebraic expression
2.    addition property of equality      19.    formula
3.    additive inverse
                                         20.    grouping symbols
4.    associative property of addition
5.    associative property of            21.   *identity property of addition
      multiplication                     22.   *identity property of multiplication
6.    coefficient                        23.   inequality
7.    commutative property of addition
                                         24.   inverse operation
8.    commutative property of
      multiplication                     25.   like terms
9.    constant (term)                    26.   multiplication property of equality
10.   cross-product property             27.   multiplicative inverse/reciprocal
11.   distributive property
                                         28.   operation
12.   *division by zero
13.   division property of equality      29.   order of operations
14.   equation                           30.   *solve (an equation)
15.   *equivalent equations              31.   solution (of an equation in one
16.   *equivalent expressions                  variable)
17.   evaluate                           32.   *substitution
                             Missing Terms

   division by zero: Division by zero is an undefined operation (you cannot find an
    answer) so it is disallowed.
    example: 12 ÷ 6 = 2 because 6 ∙2 =12
             12 ÷ 0 = x would mean that 0 ∙ x = 12
              But no value would work for x because 0 times any number is 0. So
       division by zero doesn't work.

   equivalent equations: equations that have the same solution.
    example: x - 6 = 5 and x - 11 = 0, both have the solution x = 11

   equivalent expressions: expressions that simplify to an equal value when
    numbers are substituted for the variables of the expression.
    example: x + 4 + 2 =x + 6 (since 4 + 2 = 6)

   identity property of addition (a.k.a. additive identity): the sum of zero and x is x.
    example:        ALGEBRA             ARITHMETIC
                    a+0=a                         6+0=6
                       More Missing Terms

   identity property of multiplication: the product of 1 and x is x.
    example:        ALGEBRA             ARITHMETIC
                    a∙1=a               6∙1=6

 solve an equation: to find all value(s) of the variable(s) that satisfies to an
  equation, inequality, or a system of equations and/or inequalities.
  example:        Solve for x.
                  x+5=9
                  So, x = 4
 substitution: a method used to evaluate algebraic expressions in which a
  variable, such as x or y, is replaced with its value.
  Example:        Evaluate the expression for a=2 and b=5
                  a+b
                  2+5=7
                 Lesson 1

        TRANSLATING BETWEEN
         WORDS AND ALGEBRA

1.   GET A NOTES OUTLINE FROM THE
     C O R N E R O F T H E F R O N T TA B L E
2.   COPY YOUR HW
3.   S TA RT O N WA R M U P ( TO P O F O U T L I N E )
  Operation terms:           Draw a table with the following 7
column headings. Make your table have 5 rows below the heading.



    +          -          x         ÷        =      (         ) < >
 increased decreased      of      quotient    is    quantity is greater
     by        by                                               than

  more      minus       product      to      are        sum      is less
  than                                                            than

   add     subtract     times     divided    was    difference is no more
                                  among                            than

   total   less than                         were              is at least/
                                                                  most

   sum     difference
Writing Math
These expressions all mean “2 times y”:
2y           2(y)
2•y          (2)(y)
2×y          (2)y
   Example 1: Translating from Algebra to Words

Give two ways to write each algebra expression in words.



A. 9 + r                         B. q – 3
    the sum of 9 and r                the difference of q and 3
    9 increased by r                  3 less than q

C. 7m                            D. j ÷ 6
   the product of m and 7             the quotient of j and 6
    m times 7                          j divided by 6
                    You Try! Example 1



Give two ways to write each algebra expression in
words.
1a. 4 - n                   1b.
    4 decreased by n              the quotient of t and 5
    n less than 4                 t divided by 5

1c. 9 + q                   1d. 3(h)
    the sum of 9 and q          the product of 3 and h
    q added to 9                3 times h
 Example 2A: Translating from Words to Algebra




John types 62 words per minute. Write an expression
for the number of words he types in m minutes.

m represents the number of minutes that John types.
62 · m or 62m      Think: m groups of 62 words
 Example 2B: Translating from Words to Algebra




Roberto is 4 years older than Emily, who is y years old.
Write an expression for Roberto’s age
y represents Emily’s age.
y+4       Think: “older than” means “greater than.”
Example 2C: Translating from Words to Algebra




Joey earns $5 for each car he washes. Write an
expression for the number of cars Joey must wash to
earn d dollars.

d represents the total amount that Joey will earn.

      Think: How many groups of $5 are in d?
                  You try! Example 2a



Lou drives at 65 mi/h. Write an expression for the
number of miles that Lou drives in t hours.
t represents the number of hours that Lou drives.

65t     Think: number of hours times rate per hour.
                  You Try! Example 2b



Miriam is 5 cm taller than her sister, who is m
centimeters tall. Write an expression for Miriam’s height
in centimeters.

m represents Miriam’s sister's height in centimeters.


m+5     Think: Miriam's height is 5 added to her sister's
        height.
                You Try! Example 2c



Elaine earns $32 per day. Write an expression for the
amount she earns in d days.


d represents the amount of money Elaine will earn each
day.

32d     Think: The number of days times the amount Elaine
        would earn each day.
What is an Algebraic Expression?




                          Important:
                          Expressions DO
                          NOT have equal
                          (=) signs
       Evaluating Expressions


To evaluate an expression means to
 find its value.

To evaluate an algebraic expression…
1. Substitute numbers for the variables
   in the expression
2. Apply order of operations.
3. Simplify the expression.
 Example 3: Evaluating an Algebraic Expression

Evaluate each expression for a = 4, b =7, and c = 2.


A. b – c
   b–c=7–2             Substitute 7 for b and 2 for c.
       =5              Simplify.
B. ac
    ac = 4 ·2          Substitute 4 for a and 2 for c.
       =8              Simplify.
                 You Try! Example 3


 Evaluate each expression for m = 3, n = 2, and p = 9.



a. mn
   mn = 3 · 2          Substitute 3 for m and 2 for n.
      =6               Simplify.
b. p – n
   p–n=9–2             Substitute 9 for p and 2 for n.
        =7             Simplify.
c. p ÷ m
   p÷m=9÷3             Substitute 9 for p and 3 for m.
         =3            Simplify.
     Problem Solving: Recycling Application


Approximately eighty-five 20-ounce plastic
bottles must be recycled to produce the fiberfill
for a sleeping bag.
Write an expression for the number of bottles
needed to make s sleeping bags.

The expression 85s models the number of
bottles to make s sleeping bags.
          Recycling Application Continued


Approximately eighty-five 20-ounce plastic
bottles must be recycled to produce the fiberfill
for a sleeping bag.
Find the number of bottles needed to make
20, 50, and 325 sleeping bags.
Evaluate 85s for s = 20, 50, and 325.

   s               85s              To make 20 sleeping bags 1700
                                    bottles are needed.
  20          85(20) = 1700         To make 50 sleeping bags 4250
  50          85(50) = 4250         bottles are needed.
                                    To make 325 sleeping bags
 325        85(325) = 27,625        27,625 bottles are needed.
       Lesson 2

    REVIEW OF INTEGER
OPERATIONS AND EXPONENTS
The absolute value of a number is the distance
from zero on a number line. The absolute value
of 5 is written as |5|.

          5 units         5 units


 - 6 - 5 - 4 - 3 - 2 -1 0 1 2 3 4 5 6

         |–5| = 5        |5| = 5
      Review: Operations with Integers

Adding Integers
Same Sign (pos + pos or neg + neg)= Add, Keep the Sign
Different Signs (pos + neg) = Subtract, Keep the “bigger” number’s
  sign (absolute value)

Subtracting Integers
Change subtraction to addition, change the sign of the 2nd number
  and follow addition rules

Multiplying and Dividing Integers (same rules)
Same Sign = Positive
Different Signs = Negative
         Example 1A: Adding Real Numbers


Add.

   y + (–2) for y = –6

  y + (–2) = (–6) + (–2)   First substitute –6 for y.

                           When the signs are the same,
           (–6) + (–2)     find the sum of the absolute
                           values: 6 + 2 = 8.


                   –8      Both numbers are negative, so the
                           sum is negative.
       Example 1B: Adding Real Numbers



                  When the signs of numbers are different,
Add.              find the difference of the absolute values:




                   Use the sign of the number with the greater
                   absolute value.

                   The sum is negative.
                      You Try! Example 1A




Add.

       –5 + (–7)


 –5 + (–7) = 5 + 7              When the signs are the same, find the
                                sum of the absolute values.

         5 + 7 = 12

                                Both numbers are negative, so the sum
              –12
                                is negative.
                      You Try! Example 1b




Add.

    –13.5 + (–22.3)


–13.5 + (–22.3)                 When the signs are the same, find the
                                sum of the absolute values.
   13.5 + 22.3


        –35.8                   Both numbers are negative so,
                                the sum is negative.
                         Check It Out! Example 1c




       Add.


x + (–68) for x = 52                   First substitute 52 for x.


x + (–68) = 52 + (–68)                 When the signs of the numbers are
                                       different, find the difference of the
                                       absolute values.

          68 – 52

                                       Use the sign of the number with the
                                       greater absolute value.
               –16

                                       The sum is negative.
              Example 2A: Subtracting Real Numbers




Subtract.

 –6.7 – 4.1


 –6.7 – 4.1 = –6.7 + (–4.1)    To subtract 4.1, add –4.1.

                               When the signs of the numbers
                               are the same, find the sum of the
                               absolute values: 6.7 + 4.1 = 10.8.

              = –10.8          Both numbers are negative, so the sum is
                               negative.
                Example 2B: Subtracting Real Numbers




Subtract.

     5 – (–4)


    5 − (–4) = 5 + 4             To subtract –4 add 4.


            9                     Find the sum of the absolute values.
                   You Try! Example 2a



Subtract.

      13 – 21


      13 – 21             To subtract 21 add –21.

      13 + (–21)          When the signs of the numbers are
                          different, find the difference of the
                          absolute values: 21 – 13 = 8.



            –8            Use the sign of the number with the greater
                          absolute value.
                      You Try! Example 2b


Subtract.

       x – (–12) for x = –14


       x – (–12) = –14 – (–12)   First substitute –14 for x.

      –14 + (12)                 To subtract –12, add 12.

                                 When the signs of the numbers are different,
                                 find the difference of the absolute values: 14 –
                                 12 = 2.



                                 Use the sign of the number with the greater
             –2                  absolute value.
      Example 3: Multiplying and Dividing Signed Numbers




Find the value of each expression.

                             The product of two numbers
 A.
                             with different signs is negative.


         –5
 B.


                             The quotient of two numbers
         12                  with the same sign is positive.
Example 3C: Multiplying and Dividing Signed Numbers




 Find the value of the expression.




                                     First substitute   for x.




                                     The quotient of two numbers
                                     with different signs is negative.
                        You Try! Example 3



Find the value of each expression.


 A. 35  (–5)
                             The quotient of two numbers
           –7                with different signs is negative.

 B. –11(–4)
                             The product of two numbers
           44
                             with the same sign is positive.
  C. –6x for x = 7

        –6x = –6(7)         First substitute 7 for x.

              = –42         The product of two numbers with different
                            signs is negative.
Properties of Zero
                         Multiplication by Zero
                 The product of any number and 0 is 0.
   WORDS



                     1
  NUMBERS                ·0=0             0(–17) = 0
                     3




  ALGEBRA            a·0=0                   0·a=0
Properties of Zero
                     Zero Divided by a Number
                 The quotient of 0 and any nonzero number is 0.
   WORDS



                     0                          2
  NUMBERS                    =0          0÷         =0
                     6                          3



                         0
  ALGEBRA                     =0          0÷a=0
                         a
Properties of Zero
                         Division by Zero

   WORDS         Division by 0 is undefined.



                     12 ÷ 0                    –5
  NUMBERS
                               Undefined       0



                                                a
  ALGEBRA             a÷0
                                                0
                               Undefined
      Lesson 3

ORDER OF OPERATIONS AND
 EVALUATING EXPRESSIONS
                   Warm Up

Simplify this expression.



          5 + (8 – 6)2 ⋅ 3

Which operation do you perform first?
         Review: Evaluating Exponents

 An exponent is a number that tells how many times the base number is
   used as a factor. For example, 34 indicates that the base number 3 is
   used as a factor 4 times. To determine the value of 34, multiply 3*3*3*3
   which would give the result 81.
** Be careful – many students get confused and instead of multiplying 3
   times itself 4 times, they multiply 3 times 4 and get 12. This is
   incorrect!

 Exponents are written as a superscript number (ex. 34). When a number
  is written with an exponent, this is called exponential notation.

Some important facts about exponents:
 Zero raised to any power is zero (ex. 05 = 0)
 One raised to any power is one (ex. 15 = 1)
 Any number raised to the zero power is one (ex. 70 = 1)
 Any number raised to the first power is that number (ex. 71 = 7)
When a numerical or algebraic expression contains
more than one operation symbol, the order of
operations tells which operation to perform first. Using order
of operations correctly, everyone should get the same answer.


                 Order of Operations
  First:     Perform operations inside grouping symbols.

 Second:     Evaluate powers.


 Third:      Perform multiplication and division from left to right.


 Fourth:     Perform addition and subtraction from left to right.
Grouping symbols include parentheses ( ),
brackets [ ], and braces { }. If an expression
contains more than one set of grouping symbols,
evaluate the expression from the innermost set
first.
    Helpful Hint
The first letter of these words can help you
remember the order of operations.
       Geez                    Grouping
       Excuse                   Symbols
       My                       Exponents
       Dear                     Multiply
       Aunt                     Divide
       Sally                    Add
                                Subtract
                      Example 1


Simplify each expression.
A. 15 – 2 · 3 + 1
   15 – 2 · 3 + 1         There are no grouping symbols.
     15 – 6 + 1           Multiply.
          10              Subtract and add from left to right.
B. 12 – 32 + 10 ÷ 2
                          There are no grouping symbols.
   12 – 32 + 10 ÷ 2
                          Evaluate powers. The
    12 – 9 + 10 ÷ 2       exponent applies only to the 3.
                          Divide.
      12 – 9 + 5          Subtract & add from left to right.
            8
  Example 2: Evaluating Algebraic Expressions



Evaluate the expression for the given value of x.


10 – x · 6 for x = 3
     10 – x · 6                  First substitute 3 for x.
     10 – 3 · 6                  Multiply.
       10 – 18                   Subtract.
          –8
  Example 2B: Evaluating Algebraic Expressions



Evaluate the expression for the given value of x.

   42(x + 3) for x = –2

      42(x + 3)                  First substitute –2 for x.
      42(–2 + 3)
                                 Perform the operation
         42(1)                   inside the parentheses.
         16(1)                   Evaluate powers.
          16                     Multiply.
                         You Try!



Evaluate the expression for the given value of x.


     14 + x2 ÷ 4 for x = 2

     14 + x2 ÷ 4

     14 + 22 ÷ 4               First substitute 2 for x.

     14 + 4 ÷ 4                Square 2.

        14 + 1                 Divide.
           15                  Add.
                          You Try!


Evaluate the expression for the given value of x.

  (x · 22) ÷ (2 + 6) for x = 6

  (x · 22) ÷ (2 + 6)

  (6 · 22) ÷ (2 + 6)             First substitute 6 for x.

   (6 · 4) ÷ (2 + 6)             Square two.

      (24) ÷ (8)                 Perform the operations inside
                                 the parentheses.
            3                    Divide.
Fraction bars, radical symbols, and absolute-value symbols
can also be used as grouping symbols. Remember that a
fraction bar indicates division.
Example 3: Simplifying Expressions with Other
             Grouping Symbols


 Simplify.
   2(–4) + 22
                The fraction bar acts as a grouping
   42 – 9       symbol. Simplify the numerator and the
                denominator (separately) before dividing.

                Multiply to simplify the numerator.

                Evaluate the power in the denominator.
                Add to simplify the numerator. Subtract to
                simplify the denominator.

                Divide.
      Lesson 5

EQUIVALENT EXPRESSIONS
               What is equivalence?

 Equivalent means the same thing as equal, so
  when items are equivalent, their value is the
  same.
 When expressions are equivalent they simplify
  to an equal value when numbers are substituted
  for the variables.
 example: x + 4 + 2 =x + 6 (since 4 + 2 = 6)
 When equations are equivalent, they have the
 same solution.
 example: x - 6 = 5 and x - 11 = 0, both have the solution x = 11
             QUIZ WEDNESDAY!

Review the following for the quiz:

 Lesson 1 – Translating between words and
             Algebra
 Lesson 2 – Operations with Integers (Review
             from last year)
 Lesson 3 – Order of Operations and Evaluating
             Expressions (Review from last year)
 Lesson 4 – Equivalent Expressions
                    Lesson Quiz: Part I


Give two ways to write each algebraic expression in words.
1. j – 3
2. 4p

3. Mark is 5 years older than Juan, who is y years old. Write an
expression for Mark’s age.
                 Lesson Quiz: Part II


Evaluate each expression for c = 6, d = 5, and e = 10.

4. e                             5. c + d
   d


Shemika practices basketball for 2 hours each day.
6. Write an expression for the number of hours she
practices in d days.

7. Find the number of hours she practices in 5, 12, and 20
days.
                 Lesson Quiz: Part III


Use the following terms to complete the blanks
 below: variable, expression, constant, coefficient


8. 4x + 7 is an algebraic _________________.

9. The “4” in problem number 8 is the ________________.

10. The “x” in problem number 8 is the _________________.

11. The “7” in problem number 8 is the _________________.
                   Lesson Quiz: Part I


Give two ways to write each algebraic
expression in words.
1. j – 3 The difference of j and 3; 3 less than j.
2. 4p   4 times p; The product of 4 and p.

3. Mark is 5 years older than Juan, who is y years old. Write
an expression for Mark’s age. y + 5
                 Lesson Quiz: Part II


Evaluate each expression for c = 6, d = 5, and e = 10.

4. e   2                          5. c + d   11
   d

Shemika practices basketball for 2 hours each day.

6. Write an expression for the number of hours she
practices in d days. 2d

7. Find the number of hours she practices in 5, 12,
and 20 days. 10 hours; 24 hours; 40 hours
                 Lesson Quiz: Part III


Use the following terms to complete the blanks
 below: variable, expression, constant, coefficient

                            expression
8. 4x + 7 is an algebraic _________________.

                                        coefficient
9. The “4” in problem number 8 is the ________________.

                                         variable
10. The “x” in problem number 8 is the _________________.

                                         constant
11. The “7” in problem number 8 is the _________________.
      Lesson 4

SIMPLIFYING EXPRESSIONS
    What is a SIMPLIFIED EXPRESSION?

Simplified means to put something in a more
  compact, sometimes easier to read form (
  basically…it makes it “simpler”.
A simplified algebraic expression will…
 not have operations that have not been
  performed
 not have grouping symbols
 not have any like terms that are not combined
The Commutative and Associative
Properties of Addition and Multiplication
allow you to rearrange an expression to
simplify it.
Example 1A: Using the Commutative and Associative Properties




  Simplify.




                           Use the Commutative Property.


                           Use the Associative Property to make groups
                           of compatible numbers.


          11(5)

              55
Example 1B: Using the Commutative and Associative Properties




  Simplify.


45 + 16 + 55 + 4

  45 + 55 + 16 + 4         Use the Commutative Property.


(45 + 55) + (16 + 4)       Use the Associative Property to make groups
                           of compatible numbers.


      (100) + (20)


              120
Helpful Hint

Compatible numbers help you do math
mentally. Try to make multiples of 5
or 10. They are simpler to use when
multiplying.
                 You Try! Example 1a




Simplify.




                       Use the Commutative Property.

                       Use the Associative Property to make groups
                       of compatible numbers.




            21
                        You Try! Example 1b




 Simplify.

 410 + 58 + 90 + 2


  410 + 90 + 58 + 2           Use the Commutative Property.

                              Use the Associative Property to make groups
(410 + 90) + (58 + 2)         of compatible numbers.


        (500) + (60)


                560
                     You Try! Example 1c




Simplify.

1
         • 7•8
2

 1
         • 8•7         Use the Commutative Property.
 2

     1
(2        • 8   )7     Use the Associative Property to make groups
                       of compatible numbers.

         4 • 7

          28
The Distributive Property is used with Addition to Simplify Expressions.




The Distributive Property also works with subtraction because subtraction is the
same as adding the opposite.
Example 2A: Using the Distributive Property with Mental Math




 Write the product using the Distributive Property. Then simplify.



           5(59)

        5(50 + 9)                  Rewrite 59 as 50 + 9.

      5(50) + 5(9)                 Use the Distributive Property.

         250 + 45                  Multiply.

             295                   Add.
Example 2B: Using the Distributive Property with Mental Math




 Write the product using the Distributive Property. Then simplify.



           8(33)

         8(30 + 3)                 Rewrite 33 as 30 + 3.

       8(30) + 8(3)                Use the Distributive Property.

         240 + 24                  Multiply.

              264                  Add.
                      You Try! Example 2a




Write the product using the Distributive Property. Then simplify.



          9(52)

       9(50 + 2)                  Rewrite 52 as 50 + 2.

     9(50) + 9(2)                 Use the Distributive Property.

       450 + 18                   Multiply.

            468                   Add.
                      You Try! Example 2b




Write the product using the Distributive Property. Then simplify.



       12(98)

     12(100 – 2)                   Rewrite 98 as 100 – 2.

  12(100) – 12(2)                  Use the Distributive Property.

       1200 – 24                   Multiply.

          1176                     Subtract.
Distributive Property…algebraically speaking

To us the distributive property to simplify an
  algebraic expression…
  Multiply the number outside the parenthesis by
  each number inside the parenthesis
  SEPARATELY.
Examples:
5(4 + x)         (x - 2)4         ½(4x + 10)
Punchline Worksheet 2.10

Get rid of parenthesis by
applying distributive
property…do not simplify
further.

Show all work on the back
of the sheet.
The terms of an expression are the parts to be added
or subtracted. Like terms are terms that contain the
same variables raised to the same powers. Constants
are also like terms.
         Like terms   Constant



          4x – 3x + 2
A coefficient is a number multiplied by a variable.
Like terms can have different coefficients. A variable
written without a coefficient has a coefficient of 1.

               Coefficients




              1x2 + 3x
              Like Terms Activity

 You need the sticky note at your table.
 Notice the options hanging around the room.
 Stick your term with a like term.
 Be able to defend why your term is like the one
 you chose.
                 Example 3A: Combining Like Terms



Simplify the expression by combining like terms.




        72p – 25p


         72p – 25p                  72p and 25p are like terms.


             47p                    Subtract the coefficients.
                 Example 3B: Combining Like Terms

Simplify the expression by combining like terms.




                                 A variable without a coefficient has a
                                 coefficient of 1.


                                         and       are like terms.


                                 Write 1 as


                                  Add the coefficients.
                 Example 3C: Combining Like Terms



Simplify the expression by combining like terms.




      0.5m + 2.5n



      0.5m + 2.5n                  0.5m and 2.5n are not like terms.



      0.5m + 2.5n                  Do not combine the terms.
                      You Try! Example 3
Simplify by combining like terms.

3a. 16p + 84p

     16p + 84p                 16p + 84p are like terms.

         100p                  Add the coefficients.

 3b. –20t – 8.5t2

     –20t – 8.5t2              20t and 8.5t2 are not like terms.
     –20t – 8.5t2               Do not combine the terms.

 3c. 3m2 + m3

     3m2 + m3                  3m2 and m3 are not like terms.

     3m2 + m3                  Do not combine the terms.
           Example 4: Simplifying Algebraic Expressions



Simplify 14x + 4(2 + x). Justify each step.

               Procedure                         Justification

  1.        14x + 4(2 + x)

  2.    14x + 4(2) + 4(x)                Distributive Property

  3.         14x + 8 + 4x                 Multiply.

                                          Commutative Property
  4.         14x + 4x + 8

  5.        (14x + 4x) + 8                Associative Property

  6.            18x + 8                   Combine like terms.
                           You Try! Example 4a

Simplify 6(x – 4) + 9. Justify each step.


             Procedure                         Justification

1.       6(x – 4) + 9


2.       6(x) – 6(4) + 9                Distributive Property

3.         6x – 24 + 9                  Multiply.
                                        Combine like terms.
4.            6x – 15
Finish Punchline
Worksheet 2.10

Since you already the used
the distributive property to
get rid of ( ) yesterday,
today combine like terms
to simplify the expression
completely.
Solve the riddle if you’d
like to.

								
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