THE NEWARK PUBLIC SCHOOLS
THE OFFICE OF MATHEMATICS
Grade 6
Equations and Expressions
6.EE.1 - 4
2012 COMMON CORE STATE STANDARDS ALIGNED MODULES
Expressions and Equations 6.EE.1-4
MATH TASKS
Apply and extend previous understandings of arithmetic to algebraic expressions.
Goal: Students at this level begin to develop the ability to generalize numerical relationships and
express mathematical ideas concisely using expressions and equations (e.g., three more than a number
as x + 3, doubling as 2n, commutativity as a + b = b + a). Concrete models and pictorial representations
of algebraic expressions are used to develop understanding that the commutative, associative, and
distributive properties and order of operations apply in the same way that they did for numeric
expressions. Students use these properties and the order of operations to generate equivalent
expressions and evaluate expressions that involve positive rational numbers.
Essential Questions:
Prerequisites
How do we generalize numerical relationships and
express mathematical ideas using expressions and
equations?
Understand that algebraic expressions behave
in the same way as numerical expressions
Apply the order of operations
Apply the communicative, associative, and
distributive properties
Embedded Mathematical Practices
Office of Mathematics
THE NEWARK PUBLIC SCHOOLS
MP.1. Make sense of problems and persevere in solving them
MP.2. Reason abstractly and quantitatively
MP.3. Construct viable arguments and critique the reasoning of
others
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically
MP.6. Attend to precision
MP.7. Look for and make use of structure
MP.8. Look for and express regularity in repeated reasoning
Lesson 5: Golden Problem
6.EE.1-4
Lesson 4: Mr. Nolan’s Class
6. EE. 4 Identify and solve two expressions that
are equivalent.
Lesson 3: Equivalent Expressions
6. EE.3 Apply the orders of operation to generate equivalent
expressions.
Lesson 2: Shape Up
6. EE.2 Evaluate and write numerical expressions with
variables.
Lesson 1: Djinni in the Bottle
6. EE.1 Write and evaluate numerical expressions using
whole-number exponents.
Lesson Structure:
Assessment Task
Prerequisite Skills
Focus Questions
Guided Practice
Homework
Journal Question
Page 2 of 22
MULTIPLE REPRESENTATIONS FRAMEWORK
Concrete Representations
Algebra Tiles
Pictorial Representations
Graphic Organizers i.e. input/output charts,
tables, etc.
Pan Balance



Abstract Representations
Applying the Order of Operations
Applying Properties of Addition and Multiplication
Applying the standard algorithms for addition, subtraction, multiplication, and division
Page 3 of 22
Number Systems 6.EE.1 – Lesson 1 – Introductory Task – “Djinni in the Bottle”
Write numerical expressions using exponential notation and evaluate.
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
6. EE.1 Write and evaluate numerical expressions using whole-number exponents.
After opening an ancient bottle you find on the beach, a Djinni appears. In payment for his freedom, he gives
you a choice of either 50,000 gold coins or one magical gold coin. The magic coin will turn into two gold coins
on the first day. The two coins will turn into four coins total at the end of two days. By the end or the third day
there will be eight gold coins total. The Djinni explains that the magic coins will continue this pattern of
doubling each day for one moon cycle, 28 days. Which prize do you choose?
When you have made your choice, answer these questions:

The number of coins on the third day will be 2×2×2. Can you write another expression using exponents
for the number of coins there will be on the third day?

Write an expression for the number of coins there will be on the 28th day. Is this more or less than a
million coins?
Focus Question:
What is the difference between an algebraic expression and an equation?
Page 4 of 22
6.EE.1: Lesson 1 – “Djinni in the Bottle”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Write a variable expression that represents each word phrase.
Phrase
Expression
Other
five increased by/more than a number x
fifteen decreased by/less than a number y
seven less than number p
the product of 8 and a number q
twenty divided by a number n
Ten more than twice a number
Five times a number decreased by 4
Ten divided by twice a number
True or False:
1. (2+3)3 = 23 + 33
2. (2*5)3 = 23*53
3. 23*24 = 27
4. 25/23 = 22
Page 5 of 22
6.EE.1: Lesson 1 – “Djinni in the Bottle”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Complete the chart.
Standard
Form/Product
Exponent
25
Extended
Form
Representation
(5)(5)
Verbal
A rectangular (square) yard
represented by a length of 5ft. and a
width of 5ft is 25
A cube with a volume of 343
.
3x3x3x3x5
x
The number of people in the United States
is approximately 315,000,000 as of
January 2012.
Journal Question:
Yu-Lee has a pair of mice. The mice each have 2 babies.
The babies grow up and have two babies of their own. Each of their babies
grow up and have two babies of their own.
Create an exponential expression that reflects the last set of babies.
Page 6 of 22
6.EE.1: Lesson 1 – “Djinni in the Bottle”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Write each as an algebraic expression.
The difference of 10 and 5
u decreased by 17
x increased by 6
The sum of q and 8
5 squared
Write each scenario below as a sum and as an exponential expression; Simplify if necessary.
4
7
4
-3
7
-3
-3
3r2
3r2
3r2
-2
-2
-2
-2
-2
Page 7 of 22
6.EE.2
Lesson 2 – Shape Up
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
X
Given X


Write an expression using exponential notation to express the area of each shaded figure.
Use the expressions you generated to evaluate the area of each figure when x = 2.
Focus Question
Answers
How can a variable expression help with problem
solving?
Page 8 of 22
6.EE.2
Lesson 2 – “Shape Up”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
1. Some of the students at Kahlo Middle School like to ride their bikes to and from school. They always
ride unless it rains.
 Let d be the distance in miles from a student's home to the school. Write two different
expressions that represent how far a student travels by bike in a four week period if there is one
rainy day each week.
2. The equation 15d = 1245 where d represents the number of payments to be made. The bill was 1245
dollars and each payment was 15 dollars. How many payments were made?
Write the following as numerical or variable expressions.
1. The area of a square with a perimeter of 12ft
2. The area of a square with a side length of 8m
3. Jayce makes $25 per hour.
4. Nine less than the total of a number and two
Consider the following expression: x2 + 5y + 3x + 6. Choose from the chart below to answer each question.
1




2
3
5
6
What are the variables?
What are the terms?
What are the coefficients of each term?
Which is the only constant?
x
x2
3x
y
y2
5y
Focus Question
Can you identify the parts of an expression
using the correct mathematical terms (sum,
term, product, factor, quotient, coefficient)?
Page 9 of 22
6.EE.2
Lesson 2 – Shape Up
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Write each of the following statements as a variable expression. Solve when possible.
1. The volume of a cube with a side length of 5 cm
2. The length of a football field is 30 yards more than its width. Express the length of the field in terms of
its width w.
3. Twenty gallons of crude oil were poured into two containers of different size. Express the amount of
crude oil poured into the smaller container in terms of the amount g poured into the larger container.
4. A wallet contains the same number of pennies, nickels, and dimes. Express the content of the wallet
as a variable expression.
5. The sum of two consecutive numbers is 15
6. 33% of a number
7.
2-times another number plus 4 is equivalent to 40. What number could possibly make this a true
statement?
Journal Question
To compute the perimeter of a rectangle you add the length (l) and width, (w) and double this sum.


Write an expression for the perimeter of a rectangle.
Use the expression to find the perimeter of a rectangle with a width of 30 and length of 75.
Page 10 of 22
6.EE.2
Lesson 2 –Shape Up
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Write the following as variable expressions and evaluate.
1. Jane and her three college friends are going to be sharing the cost of a 3 bedroom apartment. The cost
of rent is n dollars.
a. What expression can you write that will tell you what Jane's share is?
b. What is Jane’s share if the rent is $1260.00 per month?
2. John is on page 25 of a book with n pages.
a. How many more pages does John have to read?
b. How many more pages does John have to read when n = 744 pages.
3. Linda sold 12 more adult tickets (a) than children’s tickets (c); each adult ticket costs $10 and each
children’s ticket costs $8.
a. What expression can you write, using only one variable, that will tell you the total amount of
Linda’s sales?
b. What is the total cost of sales when c = 133?
4. What expression can you write that will tell you the number of months in n years?
5. The volume of a cube is V = s3. What numerical expression can you write that will tell you the volume
of a cube with a face area of 49
?
Page 11 of 22
6.EE.3 - Lesson 3 – “Equivalent Expressions”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
6.EE.3
Which of the following expressions are equivalent? Why? If an expression has no match, create 2 equivalent
expressions to match it. (Use equal groupings to help.)
1. 2(x+4)
2. 8+2x
X
3. 2x+4
4. 3(x+4) - (4+x)
X
5. x+4
X
X
X
X
X
X
X
X
Focus Question
How can the distributive property be used to generate equivalent
expressions?
Page 12 of 22
6.EE.3
Lesson 3 –“Equivalent Expressions”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Evaluate each expression.
1. 23 + (5 + 3)2 - 43
2. (3·3)2
3. 72 + 72
4. (8·8·8·8·8·3·3·3·3) ÷ 84·32
Which of the expressions below are equivalent? How do you know?
4m + 8
4(m+2)
3m + 8 + m
2 + 2m + m + 6 + m
8m+4
Journal Question
How do you combine like terms?
Page 13 of 22
6.EE.3
Lesson 3 – “Equivalent Expressions”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Match the expression in Column A with the equivalent expression in Column B.
COLUMN A
COLUMN B
a. 5(-2x + 5)
1. 2(1 + x/2)
b. 0.5(4 +2 x)
2. 4x + 20
c. 3x + x + 5x + 3
3. 4x + 2
d. x ( -7 + -4)
4.
e. 100x + 4
5. –x(11)
f.
6(x + 2) + 2(4 – x)
+ 10x -2(10x)
6. 3x -1
g. 5x + 3 -2(x -2)
7. 0.25(400x - -16)
h. (20x +10) / 5
8. 3 + 9x
Page 14 of 22
6.EE.3
Lesson 3 – “Equivalent Expressions”
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Simplify each variable expression.
A. 2 (3 - 2 x) =
B. 0.25(4 + 16y) =
C. z + 5z - 5 + 2z -1 =
D. y (1 + 1 + 1)=
E. 7 + 2m + 3m + 3 - 6m =
F. q + 24/8 –4q =
G. 2(k + 7) – 3(4k – 3) =
Page 15 of 22
6.EE.4
Introductory Task
Lesson 4 – “Mr. Nolan’s Class”
Guided Practice
Collaborative Work
Homework
Assessment
The students in Mr. Nolan's class are writing expressions for the perimeter of a rectangle of side length
ℓ and width w. After they share their answers, the following expressions are on the board:




Sam: 2(ℓ+w)
Joanna: ℓ+w+ℓ+w
Kiyo: 2ℓ+w
Erica: 2w+2ℓ
l
w
Which of the expressions are correct and how might the students have been thinking about finding the perimeter
of the rectangle?
Page 16 of 22
6.NS.4
Lesson 4 – Mr. Nolan’s Class
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
1. The rectangle below has a perimeter of 30 inches. Its width is half as long as its length. Write an
algebraic expression to find both the length and width of the rectangle. Use only one variable.
l
w
2. The square below has a perimeter of 30 inches. Write an algebraic expression to find the length of
a side. Use only one variable.
3. Find the value of
.
600
Page 17 of 22
6.NS.4
Lesson 4 – Mr. Nolan’s Class
Introductory Task
Guided Practice
Collaborative Work
Homework
Ms. Esposito asked her students to generate an expression that describes the number of
term as an expression of the stage (s).
Stage 1
Stage 2
Assessment
‘s in each
Stage 3
Stage 1
Stage 1
Four students each provided Ms. Esposito with a different expression to describe the number of
in a stage.
‘s
Student A: 5 + 4(s – 1)
Student B: 1 + 4(s)
Student C: s + s + s + 2s + 1
Student D: 6s - 1



Which of the students provided the correct expression?
Which expressions are equivalent?
Select the expression provided by one student and explain how it relates to the pattern.
Page 18 of 22
6.NS.4
Lesson 4 – Mr. Nolan’s Class
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
Mr. DeVries asked his 7th grade students to generate an expression that describes the number of
each term as an expression of the stage (s).
Stage 1
Stage 2
’s in
Stage 3
Stage 1
Stage 1
Four students each provided Mr. DeVries with a different expression to describe the number of f
stage.
’s in a
Student A: s2 - s
Student B: (S + 2)2 - s
Student C: 2s
Student D: 2(s + 2) + 2



Which of the students provided the correct expression?
Which expressions are equivalent?
Select the expression provided by one student and explain how it relates to the pattern.
Page 19 of 22
Golden Problem (Non-Calculator)
Introductory Task
Guided Practice
Collaborative Work
Homework
Assessment
A clothing store sells t-shirts for one price and sweatshirts for a different price. Jonas
spent the same amount on 10 t-shirts as he spent on 4 sweatshirts. He spent a total of
$230.00 on the 10 T-shirts and 4 sweatshirts.
 What was the cost of each sweatshirt?
 What was the cost of each t-shirt?
 Be sure to show all of your work.
Page 20 of 22
Golden Problem Rubric:
3-Point Response





The student indicates that one sweatshirt costs $28.75 and one t-shirt costs $11.50.
OTHER CONSIDERATIONS
The student’s written work reflects an expression for the sweatshirts (4s) and the t-shirts (10t).
AND
The student’s work reflects the relationship between the cost of the sweatshirts and t-shirts (i.e. 4s = 10t).
AND
The student’s work reflects a replacement of one variable expression for another to generate an equation for
the total cost (i.e. 4s + 4s = $230.00; 8s = $230.00; 10t + 10t = $230.00; or 20t = $230.00).
AND
The student’s work indicates a check of the final answers.
2-Point Response


The student shows correct work/procedures but does not provide the correct answers.
OR
The student commits a significant error but provides a correct response based on their incorrect work
with clear explanations.
1-Point Response

The student only begins to provide a solution.
0-Point Response


The response demonstrates insufficient understanding of the problem’s essential mathematical
concepts. The procedures, if any, contain major errors. There may be no explanation of the required
solutions, or the explanation may not be understandable. How decisions were made may not be
readily understandable.
OR
The student shows no work or justification.
Page 21 of 22
Vocabulary
A term is a part of an expression separated from others by an addition or subtraction sign. The number of terms
in an expression is often counted. For example, the expression 5 + 7x consists of two terms: 5 and 7x.
A coefficient is the numeric factor in a product that accompanies a variable. For example, in the expression
above, 7 is the coefficient in the product 7x. In the equation 5y + 10 = 30, 5 is the coefficient in 5y.
A constant is a numeric term, in contrast to a variable term. For example, in the expression x+3, 3 is a constant
while x is a variable.
A factor is a number or algebraic expression that divides another number or expression evenly (with no
remainder).
A variable is a symbol that represents a quantity in an algebraic expression.
Expression vs. Equation and Numerical Expression vs. Algebraic Expression
Numerical
Expressions
5+8
Equations
5 + 8 = 13
Algebraic
5+x
5 + x = 13
Page 22 of 22