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6th Grade Big Idea 3

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					6th   Grade Big Idea 3
       Teacher Quality Grant
Big Idea 3: Write, interpret, and use
mathematical expressions and equations.

      MA.6.A.3.1: Write and evaluate mathematical
      expressions that correspond to given situations.
      MA.6.A.3.2: Write, solve, and graph one- and two- step
      linear equations and inequalities.
      MA.6.A.3.5 Apply the Commutative, Associative, and
      Distributive Properties to show that two expressions
      are equivalent.
      MA.6.A.3.6 Construct and analyze tables, graphs, and
      equations to describe linear functions and other simple
      relations using both common language and algebraic
      notation.
Big idea 3: assessed
with Benchmarks


     Assessed with means the benchmark is present on the FCAT,
     but it will not be assessed in isolation and will follow the
     content limits of the benchmark it is assessed with.



     MA.6.A.3.3 Work backward with two-step function rules to
     undo expressions. (Assessed with MA.6.A.3.1.)



     MA.6.A.3.4 Solve problems given a formula. (Assessed with
     MA.6.A.3.2, MA.6.G.4.1, MA.6.G.4.2, and MA.6.G.4.3.)
Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big idea 3: Benchmark
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Big Idea 3: Prerequisite knowledge

      Order of Operations

      Fractions and ratios

      Decimals

      Percent
Big idea 3: Variable
video
  Writing Algebraic Expressions

     Be able to write an algebraic
expression for a word phrase or write a
    word phrase for an expression.
Although they are closely
related, a Great Dane weighs
about 40 times as much as a
Chihuahua.


When solving real-world problems, you will need
to translate words, or verbal expressions, into
algebraic expressions.

Since we do not know the can weight of the
Chihuahua we can represent it with the variable c

      =c   So then we can write the Great Dane as
           40c or 40(c).
Notes
•In order to translate a
 word phrase into an
 algebraic expression, we
 must first know some key
 word phrases for the basic
 operations.
On the back of your notes:
      Addition       Subtraction




    Multiplication    Division
  Addition Phrases:
• More than
• Increase by
• Greater than
• Add
• Total
• Plus
• Sum
  Subtraction Phrases:
• Decreased by
• Difference between
• Take Away
• Less
• Subtract
• Less than*
• Subtract from*
 Multiplication Phrases:
•Product
•Times
•Multiply
•Of
•Twice or double
•Triple
 Division Phrases:
•Quotient
•Divide
•Divided by
•Split equally
Notes
•Multiplication expressions
 should be written in side-by-side
 form, with the number always in
 front of the variable.
•3a    2t    1.5c   0.4f
Notes
Division expressions should
be written using the fraction
bar instead of the traditional
division sign.
          c x 15
           , ,
          4 24 y
       Modeling a verbal
         expression
• First identify the unknown value (the variable)

• Represent it with an algebra tile

• Identify the operation or operations

• Identify the known values and represent with
  more tiles
         Modeling a verbal
           expression
Example: Lula read 10 books. Kelly read 4 more
books then Lula.
• There is no unknown value

• More means addition

• The known values are: Lula 10 books and Kelly
  4 more

   10 books   4 books          10 +4
            Singapore math
              introduction
• Level 1

• Level 2

• Level 3

• Enrichment

• Links for other strategies
Modeling a verbal
  expression
Modeling a verbal
  expression
 Examples
• Addition phrases:
 • 3 more than x

 • the sum of 10 and a number c

 • a number n increased by 4.5
 Examples
• Subtraction phrases:
 • a number t decreased by 4

 • the difference between 10 and a
   number y

 • 6 less than a number z
   Examples
• Multiplication phrases:
 • the product of 3 and a number t

 • twice the number x

 • 4.2 times a number e
   Examples
• Division phrases:
 • the quotient of 25 and a number b

 • the number y divided by 2

 • 2.5 divide g
         Example games
• Snow man game



• Millionaire game
    Examples
• converting f feet into inches    12f


• a car travels at 75 mph for h hours 75h


• the area of a rectangle with a length
  of 10 and a width of w             10w
   Examples
                                          i
• converting i inches into feet
                                      12

• the cost for tickets if you purchase 5 adult
    tickets at x dollars each             5x


• the cost for tickets if you purchase 3
   children’s tickets at y dollars each        3y
 Examples
• the total cost for 5 adult tickets and
  3 children’s tickets using the dollar
  amounts from the previous two
  problems

      5x + 3y = Total Cost
                    Example
Great challenge problems are located on the website
bellow:

Challenges
PROBLEM
SOLVING


What is the
role of the
teacher?
“Through problem solving, students can
experience the power and utility of
mathematics. Problem solving is central
to inquiry and application and should be
interwoven throughout the mathematics
curriculum to provide a context for
learning and applying mathematical
ideas.”
                        NCTM 2000, p. 256
Instructional programs from
prekindergarten through grade 12
should enable all students to-

•build new mathematical knowledge
 through problem solving;
•solve problems that arise in
 mathematics and in other contexts;
•apply and adapt a variety of
 appropriate strategies to solve
 problems;
•monitor and reflect on the process of
 mathematical problem solving.
Teachers play an important role in
developing students' problem-solving
dispositions.

1. They must choose problems that engage
   students.
2. They need to create an environment that
   encourages students to explore, take
   risks, share failures and successes, and
   question one another.

In such supportive environments, students
develop the confidence they need to explore
problems and the ability to make
adjustments in their problem-solving
strategies.
• Three Question Types
  – Procedural
  – Conceptual
  – Application
• Procedural questions require
  students to:
  – Select and apply correct operations
    or procedures
  – Modify procedures when needed
  – Read and interpret graphs, charts,
    and tables
  – Round, estimate, and order
    numbers
  – Use formulas
• Sample Procedural Test Question

  A company’s shipping department is
    receiving a shipment of 3,144 printers that
    were packed in boxes of 12 printers each.
    How many boxes should the department
    receive?
• Conceptual questions require
  students to:
  – Recognize basic mathematical concepts
  – Identify and apply concepts and principles
    of mathematics
  – Compare, contrast, and integrate concepts
    and principles
  – Interpret and apply signs, symbols, and
    mathematical terms
  – Demonstrate understanding of
    relationships among numbers, concepts,
    and principles
• Sample Conceptual Test
  Question
 A salesperson earns a weekly salary of $225
 plus $3 for every pair of shoes she sells. If
 she earns a total of $336 in one week, in
 which of the following equations does n
 represent the number of shoes she sold that
 week?
  (1) 3n + 225 = 336
  (2) 3n + 225 + 3 = 336
  (3) n + 225 = 336
  (4) 3n = 336
  (5) 3n + 3 = 336
• Application/Modeling/Problem
  Solving questions require students
  to:
  – Identify the type of problem represented
  – Decide whether there is sufficient
    information
  – Select only pertinent information
  – Apply the appropriate problem-solving
    strategy
  – Adapt strategies or procedures
  – Determine whether an answer is reasonable
• Sample Application/Modeling/Problem
  Solving Test Question
 Jane, who works at Marine Engineering, can
 make electronic widgets at the rate of 27 per
 hour. She begins her day at 9:30 a.m. and
 takes a 45 minute lunch break at 12:00 noon.
 At what time will Jane have made 135
 electronic widgets?
  (1) 1:45 p.m.
  (2) 2:15 p.m.
  (3) 2:30 p.m.
  (4) 3:15 p.m.
  (5) 5:15 p.m.
According to Michael E. Martinez
 There is no formula for problem solving

   How people solve problems varies
   Mistakes are inevitable
   Problem solvers need to be aware of the total
    process
   Flexibility is essential
   Error and uncertainty should be expected
   Uncertainty should be embraced at least
    temporarily
What steps should we take when
solving a word problem?

1. Understand the problem

2. Devise a plan


3. Carry out the plan.


4. Look back
   Reads the problem carefully
   Defines the type of answer that is required
   Identifies key words
   Accesses background knowledge regarding a
    similar situation
   Eliminates extraneous information
   Uses a graphic organizer
   Sets up the problem correctly
   Uses mental math and estimation
   Checks the answer for reasonableness
      K               W                E                S
What do you     What does the   Is there an      What steps did
KNOW from the   question WANT   EQUATION or      you use the
word problem?   you to find?    model to solve   SOLVE the
                                the problem?     problem?
UNDERSTAND THE PROBLEM

Ask yourself….

 •What am I asked to find or show?
 •What type of answer do I expect?
 •What units will be used in the answer?
 •Can I give an estimate?
 •What information is given?
 •If there enough or too little information
 given?
 •Can I restate the problem in your own
 words?
      K                 W                 E                S
What do you      What does the     Is there an      What steps did
KNOW from the    question WANT     EQUATION or      you use the
word problem?    you to find?      model to solve   SOLVE the
                                   the problem?     problem?

Pattern:           What are the next 1              The amount
1, 3, 6, 10, 15, … 4 numbers?        1+2=3          being added
                                     3+3=6          increases by 1
                                     6+4=10         each time so:
                                     10+5=15        15+6=21
                                                    21+7=28
                                                    28+8=36
                                                    36+9=45
       K               W                    E                S
What do you     What does the        Is there an      What steps did
KNOW from the   question WANT        EQUATION or      you use the
word problem?   you to find?         model to solve   SOLVE the
                                     the problem?     problem?

 Number of      What fraction of     What you want     Green Chips
chips:          the total chips is
3 green         green?                   Total         Total Chips
4 blue
1 red
8 total chips                                                3
                                                         8
   Solve problems out loud
   Explain your thinking process
   Allow students to explain their thinking
    process
   Use the language of math and require
    students to do so as well
   Model strategy selection
   Make time for discussion of strategies
   Build time for communication
   Ask open-ended questions
   Create lessons that actively engage learners
         Jennifer Cromley, Learning to Think, Learning to Learn
LOOK BACK
This is simply checking all steps and
calculations. Do not assume the
problem is complete once a solution
has been found. Instead, examine the
problem to ensure that the solution
makes sense.
   Hierarchical diagramming

   Sequence charts

   Compare and contrast charts
   Algebra          Geometry

             MATH



Calculus            Trigonometry
 Compare and Contrast
                       Category

Illustration/Example    What is it?    Properties/Attributes



                       Subcategory




                       Irregular set
  What are some                         What is it like?
  examples?
 Compare and Contrast - example
                       Numbers

Illustration/Example   What is it?   Properties/Attributes

   6, 17, 25, 100                        Positive Integers
                       Whole
   -3, -8, -4000       Numbers           Negative Integers



   0                                     Zero

                       Fractions
  What are some                       What is it like?
  examples?
                   Prime Numbers

                   5             7
                       11   13

               2                 3

Even Numbers
                                     Multiples of 3
4       6
    8 10                    6        9      15    21
Right                      Equiangular

  3 sides                           3 sides

  3 angles                          3 angles

  1 angle = 90°                     3 angles = 60°
                  TRI-
                  ANGLES




Acute                      Obtuse
  3 sides                                3 sides
  3 angles                               3 angles
 3 angles < 90°                          1 angle > 90°
Word     =     Category    +   Attribute

               =                 +


Definitions: ______________________
________________________________
________________________________
 Word        =     Category       +    Attribute

                   =                     +
                                      4 equal sides &
Square           Quadrilateral
                                      4 equal angles (90°)

 Definition: A four-sided figure with four equal
  sides and four right angles.
1. Word:        2. Example:




4. Definition   3. Non-example:
1. Word: semicircle       2. Example:




4. Definition             3. Non-example:

A semicircle is half of
a circle.
   Divide into groups
   Match the problem sets with the appropriate
    graphic organizer
   Which graphic organizer would be most suitable
    for showing these relationships?
   Why did you choose this type?
   Are there alternative choices?
Parallelogram               Rhombus
Square                           Quadrilateral
Polygon                       Kite
Irregular polygon     Trapezoid
Isosceles Trapezoid      Rectangle
Counting Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .
Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1
Reals: all numbers
Irrationals: π, non-repeating decimal
Addition                 Multiplication
a+b                      a times b
a plus b                 axb
sum of a and b    a(b)
                             ab

Subtraction          Division
a–b                     a/b
a minus b               a divided by b
a less b
                 a÷b
Use the following words to organize into
categories and subcategories of
Mathematics:
NUMBERS, OPERATIONS, Postulates, RULE,
Triangles, GEOMETRIC FIGURES, SYMBOLS,
corollaries, squares, rational, prime, Integers,
addition, hexagon, irrational, {1, 2, 3…},
multiplication, composite, m || n, whole,
quadrilateral, subtraction, division.
                      POLYGON


Square, rectangle,                          Parallelogram:
rhombus                                     has 2 pairs of
                                            parallel sides


                                            Trapezoid: has 1
Trapezoid,               Quadrilateral
isosceles trapezoid
                                            set of parallel
                                            sides


                                            Kite: has 0 sets of
Kite Kite
                                            parallel sides
                       Irregular: 4 sides
                       w/irregular shape
REAL NUMBERS
Addition                   Subtraction
____a + b____                    ____a - b_____

___a plus b___                   __a minus b___

Sum of a and b                   ___a less b____
                  Operations



Multiplication
                           Division
___a times b___                ____a / b_____
____a x b_____
_____a(b)_____                 _a divided by b_
_____ab______
                               _____a  b_____
                 Mathematics

                                                    Geometric
   Numbers                                          Figures
             Operations        Rules      Symbols

Rational     Addition         Postulate   m║n        Triangle
Prime
             Subtraction                  √4
                              Corollary
Integer                                              Hexagon
             Multiplication
Irrational
             Division                                Quadrilateral
Whole

Composite

{1,2,3…}
Mike, Juliana, Diane, and
Dakota are entered in a 4-
person relay race. In how
many orders can they run
the relay, if Mike must run
list? List them.
Mrs. Stevens earns $18.00 an
hour at her job. She had
$171.00 after paying $9.00 for
subway fare. Find how many
hours Mrs. Stevens worked.
Try solving this problem by
working backwards.
Use the work backwards strategy
to solve this problem.

A number is multiplied by -3. Then
6 is subtracted from the product.
After adding -7, the result is -25.
What is the number?
Big Idea 3: Patterns and Equations

      Analyzing patterns and sequences (lesson ENLVM)
Properties of
 Addition &
Multiplication
Why do we
need rules or
properties in
math?

Lets see what
can happen if
we didn’t have
rules.
  Before We Begin…
• What is a VARIABLE?
A variable is an unknown amount
in a number sentence represented
as a letter:
   5+n    8x   6(g)     t+d=s
   Before We Begin…
• What do these symbols mean?
  ( ) = multiply: 6(a) or group: (6 + a)
  * = multiply
   · = multiply
  ÷ = divide
  / = divide
     Algebra tiles and counters
      • Represent the following expressions
      with algebra tiles or counters:
      1.3 + 4 and 4 + 3
      2.3 - 4 and 4 – 3
      3. 3 4 and 3  4



      
Algebra tiles and counters
 • Represent the following expressions
 with algebra tiles or counters:
 1.9x+ 2 and 2 + 9x


 1.9x - 2 and 2 – 9x
Commutative Property
• To COMMUTE something is to
change it
• The COMMUTATIVE property
says that the order of numbers in a
number sentence can be changed
• Addition & multiplication have
COMMUTATIVE properties
Commutative Property
• One way you can remember this
is when you commute you don’t
move out of you community.
Commutative Property
Examples: (a + b = b + a)

7+5=5+7
9x3=3x9
Note: subtraction & division DO NOT
have commutative properties!
a       b       As you can see,
                when you have
                two lengths
                a and b, you get
                the same length
    b       a   whether you put
                a first or b first.
              b                         a
  a
                                    b
The commutative property of
multiplication says that you may
multiply quantities in any order
and you will get the same result.
When computing the area of a
rectangle it doesn’t matter which
side you consider the width, you
will get the same area either
way.
Commutative Property
Practice: Show the commutative
 property of each number sentence.
1. 13 + 18 =
2. 42 x 77 =
3. 5 + y =
4. 7(b) =
Commutative Property
Practice: Show the commutative
 property of each number sentence.
1. 13 + 18 = 18 + 13
2. 42 x 77 = 77 x 42
3. 5 + y = y + 5
4. 7(b) = b(7) or (b)7
You can change +         to +
You can change          to 
       And the result will not change

Keep in mind the  and      do not have to be
numbers.

They can be expressions that evaluate to a number.
Lets see why
subtraction
and division
are NOT
commutative.
The commutative property: a + b = b + a    and   a*b=b*a
                    7 + 3 = 3 + 7 and     7*3 = 3*7
                       10 = 10              21= 21




Try this subtraction:   8–4 = 4–8           8÷4 = 4÷8
and division              4 ≠ -4              2 ≠ 0.5
  Associative Property
Practice: Show the associative
 property of each number sentence.
1. (7 + 2) + 5 = 7 + (2 + 5)
2. 4 x (8 x 3) = (4 x 8) x 3
3. 5 + (y + 2) = (5 + y) + 2
4. 7(b x 4) = (7b) x 4 or (7 x b)4
     Identity property
Multiplication:
1. 4 x 1 = 4
         2 6
2. why is 
         3 9
Division:
1. 10 1  10
   
 Distributive Property
• To DISTRIBUTE something is give it out
or share it.
• The DISTRIBUTIVE property says that
we can distribute a multiplier out to each
number in a group to make it easier to
solve
• The DISTRIBUTIVE property uses
MULTIPLICATION and ADDITION!
    Distributive Property
Examples: a(b + c) = a(b) + a(c)

2 x (3 + 4) = (2 x 3) + (2 x 4)
5(3 + 7) = 5(3) + 5(7)
Note: Do you see that the 2 and the 5 were shared
(distributed) with the other numbers in the group?
 Distributive Property
Practice: Show the distributive
 property of each number sentence.
1. 8 x (5 + 6) = (8 x 5) + (8 x 6)
2. 4(8 + 3) = 4(8) + 4(3)
3. 5 x (y + 2) = (5y) + (5 x 2)
4. 7(4 + b) = 7(4) + 7b
Ella sold 37 necklaces for
$20.00 each at the craft
fair. She is going to
donate half the money
she earned to charity.
Use the Commutative
Property to mentally find
how much money she will
donate. Explain the steps
you used.
                Use the Associative
  1         1   Property to write two
4         5
  2         2   equivalent
                expressions for the
                perimeter of the
              triangle
      6
Six Friends are going to
the state fair. The cost
of one admission is
$9.50, and the cost for
one ride on the Ferris
wheel is $1.50. Write
two equivalent
expressions and then
find the total cost.
Identity and Inverse Properties
   Identity Property of
 Addition
The Identity Property of Addition states that
for any number x, x + 0 = x




   5+0=5                 27 + 0 = 27

   4.68 + 0 =            ¾+0=¾
Identity Property of Multiplication


The Identity Property of Multiplication states
that for any number x, x (1) = x




Remember the number 1 can be in ANY
form.
The number 1 can be in ANY form. In
this case 3/3 is the same as 1.




   2 3 6 2
                   
   33 9 3
              same
Inverse Property of Addition

The inverse property of addition states that
for every number x, x + (-x) = 0


4 and -4 are considered opposites.
4 + -4 = 0
             -4
                  +4
What number can be added to 15 so
that the result will be zero?
                           -15
What number can be added to -22
so that the result will be zero?

                            22
Inverse Property of Multiplication


The Inverse Property of Multiplication states
for every non-zero number n, n (1/n) = 1


The non-zero part is important or else we
would be dividing by zero and we CANNOT
do that.
Properties of Equality

 In all of the following properties

 Let a, b, and c be real numbers
 Properties of Equality
 Addition property:
  If a = b, then a + c = b + c
 Subtraction property:
  If a = b, then a - c = b – c
 Multiplication property:
  If a = b, then ca = cb
 Division property:
                a b for c ≠ 0
  If a = b, then 
                c c
           Addition Property
This is the property that allows you to add the same number
to both sides of an equation.


    STATEMENT                    REASON
        x=y                       given
   x+3=y+3                Addition property of
                                equality
      Subtraction Property
This is the property that allows you to subtract the same
number to both sides of an equation.

    STATEMENT                      REASON
        a=b                          given

    a-2=b-2               Subtraction property of
                                 equality
     Multiplication Property
This is the property that allows you to multiply the same
number to both sides of an equation.


    STATEMENT                       REASON
        x=y                          given


       3x = 3y            Multiplication property of
                                   equality
          Division Property
This is the property that allows you to divide the same
number to both sides of an equation.


   STATEMENT                      REASON
       x=y                          given

    x/3 = y/3         Division property of equality
More Properties of Equality
  Reflexive Property:
      a=a

  Symmetric Property:
      If a = b, then b = a

  Transitive Property:
      If a = b, and b = c, then a = c
     Substitution Property of
             Equality
If a = b, then a may be substituted for b in any equation
or expression.

You have used this many times in algebra.

       STATEMENT            REASON
          x=5                 given
        3+x=y                 given
        3+5=y              substitution
                       property of equality
Solving One-Step
   Equations
            Definitions

Term: a number, variable or the
product or quotient of a number
and a variable.
examples:
   12       z    2w       c
                          6
Terms are separated by addition (+)
or subtraction (-) signs.


    3a – ¾b + 7x – 4z + 52
 How many Terms do you see?

               5
          Definitions

Constant: a term that is a number.


Coefficient: the number value in
 front of a variable in a term.
         3x – 6y + 18 = 0

What are the coefficients? 3 , -6


What is the constant?    18
 Solving One-Step Equations
A one-step equation means you only have to
perform 1 mathematical operation to solve it.
You can add, subtract, multiply or divide to
solve a one-step equation.
The object is to have the variable by itself on
one side of the equation.
Example 1: Solving an addition equation
                       t + 7 = 21
To eliminate the 7 add its opposite to both sides of the
  equation.

                       t + 7 = 21
                   t + 7 -7 = 21 - 7
                       t + 0 = 21 - 7
                             t = 14
Example 2:
Solving a subtraction equation
                x – 6 = 40
To eliminate the 6 add its opposite to both
sides of the equation.
                 x – 6 = 40
             x – 6 + 6 = 40 + 6
                     x = 46
Example 3:
Solving a multiplication equation
                  8n = 32
To eliminate the 8 divide both sides of the
equation by 8. Here we “undo” multiplication
by doing the opposite – division.

                8n = 32
                 8    8
                  n=4
Example 4:
Solving a division equation
                    x
                       11
                   9
To eliminate the 9 multiply both sides of the
equation by 9. Here we “undo” division by doing
the opposite – multiplication.
                    
                  x
                    11
                  9
                   x
                9  (11)(9)
                   9

                 x  99
Identify operations
 Undo operations
Balance equation
  Repeat steps
Solve for variable
 Check solution
Identify Operations
                       Minus sign means subtraction

                x
                  38
                2
Fraction bar means division
         Use Opposite Operations
          or “undo” Operations
Addition is opposite of subtraction (addition
undoes subtraction)
Subtraction is opposite of addition (subtraction
undoes addition)
Multiplication is opposite of division
(multiplication undoes division)
Division is opposite of multiplication (division
undoes multiplication)
Keep Equation Balanced




What ever you do to one side of the equation
you do to the other side of the equation.
Repeat these steps until the equation is solved.
 1-step equations

2-step equations
Example:

    7x + 15 = 85
7x +15 – 15 = 85 - 15
           7x = 70
           7     7
            x = 10
Example:
             2
               x  6  28
             3
          2
            x  6  6  28  6
          3      2
                   x  28
               3
            3 2       3
                  x  28
            2 3       2

                 x  42
   Graphing a Linear Equation
When graphing the solution to a linear equation with one-
variable on a number line you would put a dot (point) on the
answer.
                   x – 3 = -7
               x – 3 + 3 = -7 + 3
                       x = -4

				
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posted:8/13/2011
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