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Math 6
Expressions and Equations
EE Learning
targets
EE.D1 Model and solve problems involving whole number exponents.
EE.D2 Use mathematical properties to generate and identify equivalent
expressions.
EE.D3 Write expressions using variables from contextual problems.
EE.D4 Solve real-world problems by writing and solving equations.
Write inequalities from contextual problems and graph or describe the
solution.
Use variables to represent two quantities in a
contextual problem.
Analyze the relationship between variables using graphs and tables.
Vocabulary of this unit
Term, Expression, Equation, Variable,
Variable Structure, Coefficient,
Exponent, Factors, Simplify, Solve
EE.D1
Algebra Vocabulary
EE.D1
Term
Parts of an expression or series separated
by + or Expression:
Terms:
More Vocabulary
EE.D1
Coefficient
A numerical or constant quantity placed before and
multiplying the variable in an algebraic expression
Exponent
A number indicates the operation of repeated
multiplication of the base.
Coefficient
exponent
What’s a factor?
EE.D1
Factor
the numbers or variables used to create a product
What are the factors of:
80
125
w3
Expressions vs Equations
An Expression A group of terms added or subtracted
An Equation -
5x + 3x = 10
What’s the difference between an
expression and an equation?
EE.D1
Solving vs Simplifying
Solve – Make a true statement, find a value
that makes a true statement.
Example: Solve for n when 3 + n = 5
Simplify - Rewrite an expression so it is more
simply put.
Example: 3 + 5 => 8 or 2n + 3n => 5n
EE.D1
You try….
EE.D1
EE.D1
Exponents
How many ways can you write these
numbers using exponents?
125
27
48
81
EE.D1
Simplify: Write out factors then recombine
EE.D1
Simplify: Write out factors then recombine
EE.D1
Simplify: Write out factors then recombine
U3D1
Independent Practice
ICP GM U3D1
Must Do:
*Please set yesterday’s assignment on your desk so I
can check off that you did it!
In your notebook on the vocabulary page, write a definition and give an example of the
vocabulary we discussed so far.
Term, Expression, Equation, Variable,
Coefficient, Exponent, Factors,
Simplify, Solve
EE.D2
Use mathematical properties to generate and identify equivalent
expressions.
Let’s discuss
EE.D2
What does an equal sign mean?
EE.D2
What is an equivalent Expression?
Write an expression equivalent to:
8
6+4
n+2
EE.D2
Let’s review and discuss
Order of Operations
1. Calculations must be done from left to right.
2. Calculations in brackets (parenthesis) are done first.
When you have more than one set of brackets, do the
inner brackets first.
3. Exponents (or radicals) must be done next.
4. Multiply and divide in the order the operations occur.
5. Add and subtract in the order the operations occur.
Practice
Let’s Create some Equivalent Expressions
Expression
(y + y + y)
9(3y + n)
3(3f + g)
Equivalent Expression
EE.D2
Simplest Expression
EE.D2
What’s the difference between
n+n+n
and
nxnxn
How would you simplify each? Are they equivalent?
EE.D2
What terms can you simplify?
How do you know if you can combine terms?
n+4
4n + 4
4n + n
9) 5n + 9n
8) 1 - 3v + 10
Simplify expressions by combining like terms
10) 4
EE.D2
11) 35n - 1 + 46
10) 4b + 6 - 4
12) -
13) 30n + 8n
12) -33v - 49v
14) 7
15) 10x + 36 - 38x - 47
14) 7x + 31x
16) -
9.
10.
11.
12.
13.
14.
15.
16.
Challenging
EE.D2
EE.D2
Independent Practice
EE.D2
EE.D3
Review and Continue:
Use mathematical properties to generate and identify equivalent
expressions.
EE.D3 Write expressions using variables from contextual problems.
Write an equivalent equation
3+x=5
2n + 5 = 15
EE.D3
What does it mean to solve an equation?
3n + 4 = 16
Let’s solve it…. Using a bar model, and algebra
EE.D3
Solving simple equations using a bar model and algebraic manipulation
4n + 6 = 14
Let’s do bar model first.
How can we write equivalent equations to help us solve?
EE.D3
EE.D3
Solve this one
Discuss: Is this the same as 10/4 + 6/4? Talk about misconceptions in reducing…..
EE.D3
Writing Expressions and equations from contextual problems
Write an expression
EE.D3
Tom is 12 years old. Find his age after
a) 4 years b) 7 years c) x years
Write an expression
EE.D3
A worker’s salary is $200 less than onethird of a manager’s salary. Find the
worker’s salary when the manager’s
salary is
a) $7,200
b) $m
Write an expression
EE.D3
The price of a papaya is $2 and the price of
a mango is $1. Find the total price of
a) 4 papayas and 5 mangos
b) x papayas and y mangos
Write an expression
EE.D3
The price of a cup is $5 and the price of a
plate is $12. Find the total price of
a) 5 cups and 6 plates
b) N cups and M plates
EE.D3
Report on conclusions and the reasoning behind them
Do you agree? Prove and explain why you agree or disagree
EE.D3
Independent Practice
EE.D3
EE
EE.D4
Learning Targets
I can solve equations arising from a context by first drawing a model
and then writing an equation to represent the context.
Mathematical Practices
Make sense of problems and persevere in solving them
Model with mathematics
Resource: Singapore 3.3-7a
Review
EE.D4
The price of a cup is $5.50 and the price of
a plate is $6.75. Find the total price of
a) 5 cups and 6 plates
b) N cups and M plates
Concept Review
Order of operations
Solve when x = 3 and n = 2
EE.D4
EE.D4
Problem solving.. Use any method to solve
The average score a student earned on
11 tests was 90. He averaged a score of
88 on the first 5 tests he took, and the
average of his final 5 test scores was 94.
What score did the student earn on his
6th test?
Draw a model, write an equation
Solve
EE.D4
At birth, baby George weighs 2 pounds
lighter than his twin brother Harry.
a) Suppose Harry’s weight is x pounds,
what is George’s weight in terms of
x?
b) If the sum of their weight is 15
pounds, how much does each
weigh?
Draw a model, write an equation
Solve
EE.D4
A store is selling a notebook computer at $90
more than 4 times the price of a mobile
phone.
a) Suppose the price of the mobile phone is $m.
Express the price of the notebook in terms of
m.
b) Given that the notebook computer costs $n,
write a formula connecting m and m
Draw a model, write an equation
Solve
Stella has $11 more than Roland. Together
they have $159. How much money does
Stella have? How much money does
Roland have?
EE.D4
Draw a model, write an equation
Solve
Jared has twice as many DVDs as Eric.
Together they have 51 DVDs. How many
DVDs does Jared have? How many DVDs
does Eric have?
EE.D4
Draw a model, write an equation
Solve
The sum of three consecutive
integers is 405. What are the three
numbers?
EE.D4
Write an equation
Solve
YOU TRY!
Write an equation and solve
A corn plant was 4 inches tall. It grew 1.5 inches a
day and is now 22 inches tall. How many days did
it take?
EE.D4
EE.D5
Learning Targets
I can differentiate between like and unlike terms
I can use the distributive law and add and subtract linear equations
Mathematical Practices
Make sense of problems and persevere in solving them
Model with mathematics
Resources: Singapore 4.1 & 4.2 – 7a
Bell Work
Let x = 4, y = -2, z = 0
EE.D5
Warm Up
EE.D5
Draw a model, write an equation
EE.D5
Kylie is addressing invitations to her party. She has already
addressed 17 of them. She finishes the rest of them in 4
hours. If she addressed a total of 81 invitations, how many
invitations did she address per hour?
Like and unlike terms
EE.D5
Review vocab: Terms, Coefficient
Like terms have the same variable structure: same
variables with same exponents, but the coefficient need
not be the same.
Constants are always “Like Terms”: 5, 24, -73
EE.D5
Simplifying by adding or subtracting like terms
If two terms are “like” (have the same variable structure),
you can add or subtract them, but add the coefficient in
front of the variables. Any numbers without a variable (a
constant), can just be added or subtracted.
Examples
You Try
EE.D5
The Distributive law of multiplication over addition
Demonstrate using area models
Solve 3(4 + 5) using an area model
Solve 2(4x + 5y) using an area model
Solve -x(4y + 5y) using an area model
EE.D5
EE.D5
The Distributive Law in general
a(x + y) = ax + ay
a(x - y) = ax - ay
a(x + y + z) = ax + ay + az
You Try; use the distributive property to
remove parenthesis
EE.D5
EE.D5
Harder… use distributive property and collect like terms
(2x + 3y) – (3x + 2y)
4x – x(3 + 2y) + xy
EE.D5
Independent Practice
ICP EE.D5
EE.D6
Learning Targets
I can solve equations in one variable
Mathematical Practices
Make sense of problems and persevere in solving them
Model with mathematics
Resources: Singapore 5.1 – 7a
EE.D6
15 minute Skill Check EE.SC1
Model and write an equation
Solve
EE.D6
Sam is 4 less than twice Bill’s age. Jack
is 3 years older than Bill. If the
sum of their ages is 47, how old are
Sam, Bill & Jack?
Solving Equations Using Algebra
EE.D6
When we solve an equation using algebra, we want to undo the
operations on the unknown in order to find out what it is.
Remember! Keep the equation balanced by doing the same thing on both sides!!!
Example:
Solve 3x  48
Explain the opposite of multiplication is division, etc…
Solving Equations Using Algebra
Solve each equation
1.  2 y  68
2. 39  w  20
p
3. 32 
2
Do first problem with class, have students do 2 & 3.
EE.D6
Solving Two-Step Equations
What’s the first step in solving this equation? Why?
How about this one? Why?
p 8
 10
2
EE.D6
Solving Two-Step Equations
Solve each equation
1.
p
68
3
2.
6 y  9  30  3
3.
Do first problem with class, have students do 2 & 3.
EE.D6
A word about notation
Are these equivalent?
What about these?
EE.D6
EE.D6
Independent Practice
EE.D6
EE.D7
Learning Targets
I can solve equations involving parenthesis in one variable
I can rearrange equations to solve.
Mathematical Practices
Make sense of problems and persevere in solving them
Model with mathematics
Resources: Singapore 5.2 – 7a
Bell work
EE.D7
a) The sum of 3 consecutive numbers is 234. Find the
numbers
a) The sum of 3 consecutive even numbers is 192. Find
the numbers
a) The sum of 3 consecutive odd numbers is 111. Find the
numbers
Review
Let x = 5, y = -3, z = 0
EE.D7
Problem Solving
Kylie is addressing invitations to her graduation
party. She has already addressed 17 of them.
She finishes the rest of them in 4 hours. If she
addressed a total of 81 invitations, how many
invitations did she address per hour?
EE.D7
Problem Solving
EE.D7
104 chickens and goats in a farm have 246 legs
altogether. How many of each type of animal are
there?
Can us solve with algebra, a model, or some other method?
Solving Multi-Step Equations
Let’s review the Distributive Property for a moment…
The Distributive Property says
a(b + c) =
Simplify
 2( x  3) 
EE.D7
Solving Equations with the Distributive Property
What should we do first to solve this equation? Why?
5(2 x  1)  15
Explain we need both sides of equation as simplified as possible to make solving easier…
EE.D7
Solving Equations with the Distributive Property
Solve each equation
1.
2.
3.
2  3( x  4)  15  21
7( x  4)  21
4 x  5(2 x  4)  10
Do first problem with class, have students do 2 & 3.
EE.D7
A little more challenging examples….
How does the distributive property apply here?
EE.D7
U4D8
Independent Practice
EE.D7
EE.D7
Math 6
EE.D8
Standards Assessed:
I can solve equations using algebraic manipulation
I can create and solve equations arising from a context
Mathematical Practices:
Make sense of problems and persevere in solving them
Model with mathematics
EE.D8
U4D8
Practice
Solving proportions
EE.D8
Keep practicing!
EE.D8
Write an equation and solve. Draw a model if
necessary.
The length of a rectangle is 50 meters. This is 6
meters more than twice the width. Find the
width of the rectangle.
EE.D8
You Try!
Write an equation and solve. Draw a model if
necessary.
Twelve decreased by 8 times a number is 36.
Find the number.
EE.D8
You Try!!
Write an equation and solve. Draw a model if
necessary.
JoGrandpa is 75 years old. This is nine years less
than seven times the age of Jo. How old is Jo?
EE.D8
You Try!!
Write an equation and solve. Draw a model if
necessary.
JoGrandpa is 75 years old. This is nine years less
than seven times the age of Jo. How old is Jo?
EE.D8
Write an equation and solve
A-1 Plumbing company pays Sam $45 per
day, plus 10.50 for each hour he works. If
he earned 566.25 last week, how many
hours a day did he work.
(Sam works a 5 day week, and each day
he works the same number of hours)
EE.D8
Write an equation and solve
Lisa is bored, riding in the backseat of
her parent’s SUV on their family
vacation. She notices the mile markers
on the interstate. She adds up three
consecutive mile markers and discovers
the sum is 312. Which three mile
marker signs did her family just pass?
Independent Practice
EE.D8
EE.D9
Continue developing algebraic manipulation skills by solving inequalities
Expressing a solution to an inequality
EE.D9
Solutions to equations vs solutions to inequalities
How do we express a solution to the equations:
2n = 10
2n + 1 = 2n + 10
2n = 2n
EE.D9
Inequalities!
• Vocab: An Inequality is a
comparison of two values that
may or may not be equal.
EE.D9
The Symbols and Their Meanings
Symbol
Meaning
a>b
a is greater than b
a≥b
a is greater than or equal to b
a<b
a is less than b
a≤b
a is less than or equal to b
EE.D9
What does it all mean!?
Examples!
Graph each inequality on a number line using
1.
x2
2.
x  5
3.
x0
4. x 
1
or
2
properties ws
EE.D9
Solutions to an Inequality
• Remember that the solution to an equation
gave us a single, unique result… but for inequalities…
• The solution to an inequality is a range of
possibilities.
– Example: The solution of x  2 allows every real
number that is bigger than 2. So we get to use any
possible number we can think of as long as it is
greater than 2.
Let’s try some….
Examples!
Solve each inequality and graph the solution on a
number line.
1. x  4  6
x
3
2.
2
3. x  2  1
4. 12  3x
EE.D9
EE.D9
You Try!
• Solve each inequality and graph the solution
on a number line.
x
1.
4
3
2. x  6  8
3. 11  2x  1
EE.D10
Standards Addressed Today:
 Attend to Precision
 Reason Abstractly and Quantitatively
 Solve Inequalities Using Algebraic Manipulation
 Explain the Multiplication Property by Negative
Integers for Inequalities
Concept Review
EE.D10
• Solve each inequality and graph the solution on a
number line.
1.
2.
x  3  11
x  8  12
3.
x
 1
6
4.
7 x  35
Now for the tricky part…
• Consider the inequality 4 < 12
1. Divide both sides by 2. Is the inequality still true?
2. Now divide both sides by –2. Is the inequality still
true?
3. Divide by 4.
4. Divide by –4.
5. Complete questions 1-4 on your half sheet.
EE.D10
Solve each inequality
1.  2 x  16
2.
8  4 y
3.
x
1
3
2w
4. 12 
3
EE.D10
EE.D10
Challenge!
1. 2  2(4w  7)
4. 6  2w  2(6w  7)
4w
2. 16  8 
2
5.
2w  16
12 
3
6.
2w  8

 3 12
3. 3x 15  21  9x
EE.D10
Describe a situation
… that could be modeled by the inequality
3x  10
Work with a partner
EE.D10
• You are building a patio. You want to cover
the patio with Spanish Tile that costs $5 per
square foot. You budget for the tile is $1700.
How wide can you make the patio without
going over budget?
Independent Practice
EE.D10
EE.D10
EE.D11
EE.D11
Standards:
 Attend to Precision
 Reason Abstractly and Quantitatively
 Solve Inequalities Using Algebraic
Manipulation
 Solve Inequalities Arising From a
Context
EE.D11
Concept Review
• Solve each equation
or inequality.
1.
2.
3.
2
x  9  20
7
8(3  2 x)  2(8  3x)
17  8x  7
Graph the solution to the
inequality.
4.
 3(2 x  1)  9
5.
7 x 3x  4

9
3
6.
 3x  1
4
6
EE.D11
Concept Review
Write an inequality and solve.
• Four times a number is at least –48. What is
the number?
EE.D11
Problem Solving
Write an inequality and solve.
• Ilona is saving up to buy a digital camera that
costs $495.75. So far, she has $175 saved. She
would like to buy the camera 5 weeks from
now. At least how much must she save every
week to have enough money to purchase the
camera?
Use the definition of mean to write an
inequality and solve
• You need a mean score of at least 90 to
advance to the next round of the trivia game.
What score do you need on the fifth game to
advance?
Problem Solving
EE.D11
• For what values of x will the area of the blue
region be greater than 12 square units?
Try this on your own
a) For what values of y will the area of the
trapezoid be less than or equal to 10
square units?
EE.D11
EE.D11
In Class Practice
EE.D11 – ICP
Unit Review
Solving equations and inequalities