# 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:

Multiplication    Division
• More than
• Increase by
• Greater than
• 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
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
books then Lula.
• There is no unknown value

• 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

Modeling a verbal
expression
Modeling a verbal
expression
Examples
• 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
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
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
• 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
– 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
   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

•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
   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
   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 &
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
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
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,
POLYGON

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

Trapezoid: has 1
isosceles trapezoid
set of parallel
sides

Kite: has 0 sets of
Kite Kite
parallel sides
Irregular: 4 sides
w/irregular shape
REAL NUMBERS
____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

Prime
Subtraction                  √4
Corollary
Integer                                              Hexagon
Multiplication
Irrational
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
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
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
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
\$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
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

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
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
This is the property that allows you to add the same number
to both sides of an equation.

STATEMENT                    REASON
x=y                       given
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
undoes subtraction)
Subtraction is opposite of addition (subtraction
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