Transcript 6th Grade Big Idea 3 - Math GR. 6-8

th
6
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
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
Big idea 3: Benchmark
Item Specifications
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
• converting i inches into feet
i
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)
(2)
(3)
(4)
(5)
1:45
2:15
2:30
3:15
5:15
p.m.
p.m.
p.m.
p.m.
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
What do you
KNOW from the
word problem?
W
What does the
question WANT
you to find?
E
Is there an
EQUATION or
model to solve
the problem?
S
What steps did
you use the
SOLVE the
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
What do you
KNOW from the
word problem?
W
What does the
question WANT
you to find?
E
Is there an
EQUATION or
model to solve
the problem?
Pattern:
What are the next 1
1, 3, 6, 10, 15, … 4 numbers?
1+2=3
3+3=6
6+4=10
10+5=15
S
What steps did
you use the
SOLVE the
problem?
The amount
being added
increases by 1
each time so:
15+6=21
21+7=28
28+8=36
36+9=45
K
W
What do you
KNOW from the
word problem?
What does the
question WANT
you to find?
Number of
chips:
3 green
4 blue
1 red
8 total chips
What fraction of
the total chips is
green?

E
Is there an
EQUATION or
model to solve
the problem?
What you want
Total

S
What steps did
you use the
SOLVE the
problem?
GreenChips
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
Geometry
Algebra
MATH
Calculus
Trigonometry
Compare and Contrast
Category
Illustration/Example
What is it?
Properties/Attributes
Subcategory
Irregular set
What are some
examples?
What is it like?
Compare and Contrast - example
Numbers
Illustration/Example
What is it?
6, 17, 25, 100
-3, -8, -4000
Properties/Attributes
Positive Integers
Whole
Numbers
0
Negative Integers
Zero
Fractions
What are some
examples?
What is it like?
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°
TRIANGLES
Acute
Obtuse
3 sides
3 sides
3 angles
3 angles
3 angles < 90°
1 angle > 90°
Word
=
Category
=
+
Attribute
+
Definitions: ______________________
________________________________
________________________________
Word
Square
=
Category
=
Quadrilateral
+
Attribute
+ sides &
4 equal
4 equal angles (90°)
Definition: A four-sided figure with four equal
sides and four right angles.
1. Word:
4. Definition
2. Example:
3. Non-example:
1. Word: semicircle
4. Definition
A semicircle is half of
a circle.
2. Example:
3. Non-example:


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
Square
Polygon
Irregular polygon
Isosceles Trapezoid
Rhombus
Quadrilateral
Kite
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
a+b
a plus b
sum of a and b
Multiplication
a times b
axb
a(b)
ab
Subtraction
a–b
a minus b
a less b
Division
a/b
a divided by 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
Parallelogram:
has 2 pairs of
parallel sides
Square, rectangle,
rhombus
Trapezoid,
isosceles trapezoid
Quadrilateral
Trapezoid: has 1
set of parallel
sides
Kite: has 0 sets of
parallel sides
Kite Kite
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
Multiplication
___a times b___
____a x b_____
_____a(b)_____
_____ab______
Operations
___a less b____
Division
____a / b_____
_a divided by b_
_____a  b_____
Mathematics
Numbers
Rational
Prime
Operations
Addition
Subtraction
Integer
Rules
Symbols
Postulate
m║n
Corollary
√4
Geometric
Figures
Triangle
Hexagon
Multiplication
Irrational
Division
Whole
Composite
{1,2,3…}
Quadrilateral
Mike, Juliana, Diane, and
Dakota are entered in a 4person 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
b
a
As you can see,
when you have
two lengths
a and b, you get
the same length
whether you put
a first or b first.
b
a
a
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.
b
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
numbers.
 do not have to be
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
7 + 3 = 3 + 7 and
10 = 10
Try this subtraction:
and division
8–4 = 4–8
4 ≠ -4
and
a*b=b*a
7*3 = 3*7
21= 21
8÷4 = 4÷8
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.
1
4
2
1
5
2

6
Use the Associative
Property to write two
equivalent
expressions for the
perimeter of the
triangle
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
If a = b, then  for c ≠ 0
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
x=5
3+x=y
3+5=y
REASON
given
given
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 onevariable on a number line you would put a dot (point) on the
answer.
x – 3 = -7
x – 3 + 3 = -7 + 3
x = -4