Download 7th Grade Mathematics

 7th Grade Mathematics Unit #3: Applying Algebraic
and Evaluate
Equations,
Reasoning to Write
Expressions, and Inequalities
Pacing: 42 Days
Unit Overview
This unit consolidates and expands upon students’ understanding of equivalent expressions as they apply the properties of operations to write
expressions in both standard form and in factored form. They use linear equations to solve unknown angle problems and other problems presented
within context to understand that solving algebraic equations is all about the numbers. Students use the number line to understand the properties of
inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities. They
interpret solutions within the context of problems. Students extend their sixth-grade study of geometric figures and the relationships between them
as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as
volume and surface area of right prisms. .1
1
Source: Engage NY
Prerequisite Skills
1) Order of Operations
2) Commutative, Associative, and Distributive
Properties
3) Find equivalent fractions by multiplying the
numerator and denominator by the same
number
4) Compute efficiently using all four operations
Vocabulary
Associative Property
Commutative Property
Distributive Property
Identity Property
Expression
Coefficient
Like Terms
Expand
Simplify
Expand
Linear
Equivalent
Mathematical Practices
Variable
Equation
Inequality
Solution Set
Angle
Complimentary
Supplemental
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
Common Core State Standards
Additional Standards (10%) 7.G.5: Solve Multi-­‐Step Angle Problems Major Standards (70%) 7.EE.1: Add, subtract, factor, and expand linear expressions
7.EE.2: Understand contextual equivalent expressions
7.EE.3: Solve multi-step problems with rational numbers
7.EE.B.4: Use variables to represent quantities in a real-world
or mathematical problem, and construct simple equations and
inequalities to solve problems by reasoning about the quantities.
7.EE.4a: Solve Problems with 2-Step Linear Equations
7.EE.4b: Solve Problems with 2-Step Linear Equations
According to the PARCC Model Content Framework,
Standard 7.EE.4 is an end of year fluency expectation in 7th grade:
“In solving word problems leading to one-variable equations of the form px +
q = r and p(x + q) = r, students solve the equations fluently. This will require
fluency with rational number arithmetic (7.NS.1–3), as well as fluency to some
extent with applying properties operations to rewrite linear expressions with
rational coefficients (7.EE.1).”
According to the PARCC Model Content Framework,
“Solving an equation such as 4 = 8(x – 1/2) requires students to see and make
use of structure (MP.7), temporarily viewing x – 1/2 as a single entity.”
2 | P a g e Progression of Skills
6th Grade
7th Grade
8th Grade
6.EE.6: Use variables
to
7.EE.1:
Apply
propertiesModel
of
8.EE.7: Solve
linear
According
to the
PARCC
Content
Framework,
represent numbers and
operations as strategies to add,
equations in one variable.
Standard
3.NF.2
should
serve
as an opportunity for inwrite expressions
when
subtract,
factor,
and expand
linear
solving a real-world
or
expressions
with
rational
depth focus:
mathematical problem;
understand that a variable
can represent an unknown
number, or, depending on
the purpose at hand, any
number in a specified set.
6.EE.5: Understand
solving an equation or
inequality as a process of
answering a question:
which values from a
specified set, if any, make
the equation or inequality
true? Use substitution to
determine whether a
given number in a set
makes an equation or
inequality true.
6.EE.9: Use variables to
represent two quantities
in a real-world problem
that change in
relationship to one
another; write an equation
to express one quantity,
thought of as the
dependent variable, in
terms of the other
quantity, thought of as
the independent
variable.
6.EE.9: Use variables to
represent two quantities
in a real-world problem
that change in
relationship to one
another; write an equation
to express one quantity,
thought of as the
dependent variable, in
terms of the other
quantity, thought of as
the independent variable.
coefficients.
7.EE.2: Understand that rewriting
an expression in different forms in
a problem context can shed light on
the problem and how the quantities
in it are related.
N/A
7.EE.3: Solve multi-step real-life
and mathematical problems posed
with positive and negative
rational numbers in any form
(whole numbers, fractions, and
decimals), using tools strategically.
Apply properties of operations to
calculate with numbers in any
form; convert between forms as
appropriate; and assess the
reasonableness of answers using
mental computation and estimation
strategies.
7.EE.4: Use variables to represent
quantities in a real-world or
mathematical problem, and
construct simple equations and
inequalities to solve problems by
reasoning about the quantities.
8.EE.8C: Solve real-world
and mathematical problems
leading to two linear
equations in two variables.
8.EE.C.8: Solve real-world
and mathematical problems
leading to two linear
equations in two variables.
Big Ideas
•
•
•
•
•
How can I formulate
and use different
strategies to solve
one and two‐step
equations?
Students Will…
•
•
How can I use the
properties of
equality to express
an equation in a
different but
equivalent way?
•
What is the
difference between
expressions,
equations and
inequalities?
•
What is the
difference between
an algebraic solution
and an arithmetic
solution?
How do you graph
the solution set of
inequalites in the
form px + q > r or px
+ q < r?
3 | P a g e •
•
•
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•
Know/Understand
What it means to expand or simplify a linear
expression.
Linear expressions can be expanded and simplified
by using the properties of operations with addition,
subtraction, combining like terms, and factoring.
Different expressions are equivalent if they can be
simplified or rewritten to have the same value.
Expressions can be decomposed and recomposed in
different ways to generate equivalent expressions.
That rewriting an expression in different forms can
better explain or highlight different parts of the
relationship between quantities in an expression.
That estimation and mental fluency can be used to
determine the reasonability of answers.
Variables are letters or symbols are used to represent
an unknown quantity in equations and inequalities.
Real world and mathematical problems may be
modeled using equations or inequalities.
Equations are used to model real world or
mathematical problems when trying to find the values
of two expressions that are equal and may take the
form of px + q = r and/or p(x+q)=r, where p, q, and
rare rational numbers.
When solving an equation it is possible to use inverse
operations (algebraic) or substitution (arithmetic) by
following the sequence of operations.
Inequalities are used to model real world or
mathematical problems when trying to find the values
of two expressions that are not equal and may take
the form of px + q > r and/or p(x+q)<r, where p, q,
and r are rational numbers.
•
•
•
•
•
•
•
•
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•
Be Skilled At…
Writing a linear expression following a given
sequence of operations.
Adding, subtracting, factoring and expanding linear
expressions with rational coefficients using the
properties of operations, including but not limited to
the distributive, commutative, and associative
properties.
Using properties of operations in a linear equation to
identify equivalent expressions or to simplify an
expression.
Justify the reasonableness of answers by using
fluency and estimation
Writing equations (px + q = r and p(x + q) = r) and
inequalities (px + q > r and p(x + q) < r), using
variables to represent unknown quantities.
Using inverse operations fluently to solve equations
and inequalities.
Finding the value of the variable in an algebraic
equation quickly and accurately using inverse
operations and substitution
Comparing the solutions found to an equation when
using inverse operations (algebraic) versus
substitution (arithmetic).
Writing equations (px + q = r and p(x + q) = r) and
inequalities (px + q > r and p(x + q) < r) based off of
real world problems.
Solving word problems that are in the form of
equations (px + q = r and p(x + q) = r) and
inequalities (px + q > r and p(x + q) < r).
Unit Sequence
1
Student Friendly Objective
SWBAT…
Evaluate and write algebraic
expressions
•
Key Points/
Teaching Tips
Note: since is largely an
opportunity to review key
prerequisite skills, take some time
to explicitly practice writing
expressions to match a sequence
given in words (i.e. see exit ticket
and provided links)
2
Write algebraic expressions to
describe and extend sequences
34
Create equivalent forms of
expression to represent realworld scenarios
•
Pacing: 2 days
56
Apply properties of operations
to simplify expressions and
generate equivalent expressions
•
•
Pacing: 2 days
The My Math lesson listed here is
intended to be used either for
additional practice or a
remediation resource; the Engage
NY lessons should be at the heart
of your planning for these two
days
Exit Ticket
Write an expression to describe each
sequence of operations.
Instructional
Resources
My Math Chapter 5
Lesson 1
(a) Add 3 to x, subtract the result from 1,
then double what you have.
http://www.ixl.com/ma
th/grade-7/writevariable-expressions
(b) Add 3 to x, double what you have,
then subtract 1 from the result.
http://www.mathgoodie
s.com/lessons/vol7/exp
ressions.html
My Math Chapter 5
Lesson 2
For parts (a)-(e), select Yes or No to
indicate whether each of these
expressions is equivalent to 2(2x + 1).
a) 4x + 2
___ Yes ___ No
b) 2(1 = 2x)
___ Yes ___ No
c) 2(2x) + 1
___ Yes ___ No
d) 2x + 1 + 2x + 1
___ Yes ___ No
e) x + x + x + x + 1 + 1 ___ Yes ___ No
Engage NY
Module 2
Lessons 18-19
(Appendix C)
Engage NY
Module 3 Lessons 1 – 2
(Appendix C)
Remediation /Extra
Practice Resource:
My Math Chapter 5
Lesson 3
http://www.coolmath.c
om/prealgebra/06properties/index.html
4 | P a g e 78
9
10
Use area models and the
distributive property to write
products as sums and sums as
products
Apply the identity properties of
0 and 1 and the existence of
inverses (opposites and
reciprocals) to write equivalent
expressions
Simplify algebraic expressions
by combining like terms.
•
•
•
•
5 | P a g e “Learning Task:
Distributing and
Factoring Area”
(Appendix C)
Pacing: 2 days
Note: use the “Learning Task”
resource as the heart of your
lesson for day 1, then use Engage
NY (you may choose only 1 or
select problems from each) and
My Math lesson as additional
practice after students have
conceptually connected area
models to the distributive property
Attend to precision to describe
each component of an expression
(terms, like terms, coefficients,
constants, etc)
It is recommended to use the “real
world link” from page 387 of My
Math as the hook for the lesson, so
that students can conceptualize
like terms by modeling a real
world scenario; the remaining My
Math resources in this lesson
should be used for additional
practice and/or
reteach/remediation, as the Engage
NY practice problems are more
appropriate for 7th grade.
Engage NY
Module 3 Lessons 3-4
(Appendix C)
Additional Practice:
My Math
Chapter 5 Lesson 4
Engage NY
Module 3 Lesson 5
(Appendix C)
1) Simplify the following expressions:
1.
2.
3.
4.
p+q+p+q+p
5c + 2d – 3c – 4d
xy + yx
2xy – 4ac + 5yx + 4ac – 2
Engage NY
Module 3 Lesson 6
(Appendix C)
My Math Chapter 5
Lesson 5
http://learnzillion.com/l
essons/810
https://learnzillion.co
m/lessons/812simplify-anexpression-with-afraction-by-addingor-subtracting-termswith-fractions
11
Use substitution and the
properties of operations to
determine whether two
expressions are equivalent
•
Students must understand that two
different expressions are
equivalent if they can be
simplified or rewritten to have the
same value.
€
1) If we multiply
+ 3.
x 3
+ by 4, we get 2x
2 4
Is 2x + 3 and equivalent expression to
x 3 €
+ ? Why or why not?
2 4
http://www.cpalms.org/
Public/PreviewResourc
e/Preview/49255
https://learnzillion.com/
lessons/816
2) Look at each expression. Is it
x + 3y
equivalent to
?
2
Select Yes or No for each expression.
€
3) The students in Mr. Sanchez's class
are converting distances measured in
miles to kilometers. To estimate the
number of kilometers, Abby takes the
number of miles, doubles it, then
subtracts 20% of the result. Renato first
divides the number of miles by 5, then
multiplies the result by 8.
(a) Write an algebraic expression that
shows the process each student used.
6 | P a g e 12
13
Apply properties of operations
to add linear expressions
•
•
14
Apply properties of operations
to subtract linear expressions
•
•
7 | P a g e Flex Day (Instruction Based on Data)
Recommended Resources:
My Math Chapter 5 Mid-Chapter Check (Page 386)
My Math Problem Solving Investigation (Pages 383 – 385)
“Learning Task: Area and Algebra” (Appendix C)
“Guess My Number” (Appendix C)
“Algebra Magic” (Appendix C)
1) Which of the following is equivalent
Consider beginning the lesson
with a problem students will need to the expression below?
(-3m + 5) + (m-11)
to come back to after the lesson so
they are able to use the lesson in
A. 4m – 16
order to create their linear
B. -4m – 6
expression
Encourage students to think about C. 2m – 16
D. -2m – 6
the importance of order of
operations/sequence by asking
them to explain how changing the 2) Which expression represents the sum
of (2x – 5y) and (x + y)?
sequence of operations for an
expression affects the final result.
A. 3x – 4y
B. 3x – 6y
C. x – 4y
D. x – 6y
Consider beginning the lesson
When 5 x +1 1 is subtracted from
8
3
with a problem students will need
to come back to after the lesson so 1 1 x − 5 1 , the result is…
4
6
they are able to use the lesson in
order to create their linear
€
A. 5 x + 3 5
expression
8
6
Encourage students to think €
about B. 5
5
x+3
the importance of order of
8
6
5
5
operations/sequence by asking €
C. − x + 3
8
6
them to explain how changing the
5
1
sequence of operations for an €
D. − x + 6
8
2
expression affects the final result.
My Math
Chapter 5 Lesson 7
http://www.coolmath.c
om/prealgebra/06properties/index.html
My Math
Chapter 5 Lesson 7
http://www.coolmath.c
om/prealgebra/06properties/index.html
€
€
15
Factor and expand linear
expressions
Review the root words of the
properties to help students
understand them and how to
identify them
Briefly review order of operations
and different answers that can be
produced when operations are
moved and grouped differently
1) Why is the following true? 2(x +
y) = 2x + 2y
2) Rearrange, using the Associative
Property: 2(3x)
3) Why is it true that 3(4x) =
(4x)(3)?
4) Simplify 3a – 5b + 7a. Justify
your steps.
• Application Problem:
You are the manager of a store and
business is a little slow. The best way
to attract more customers is to have a
sale. Here are the sales you are
considering:
1) Leo bought a used car for x dollars.
One year later the value of the car was
0.88x. Which expression is another way
to describe the change in the value of the
car?
•
•
16
Write expressions to represent
and solve real world problems
involving taxes, discounts and
mark-ups
a. Buy one, get one of equal
value half off
b. Buy three of equal value for
the price of two
c. Apply a 40% discount on the
item
As a business owner, which sale
should you run? Write an expression
to represent each sale, then use
substitution to plug in different values
to help you make your decision:
8 | P a g e A.
B.
C.
D.
0.12% decrease
0.88% increase
12% decrease
88% increase
My Math
Chapter 5 Inquiry Lab
Chapter 5 Lesson 8
https://learnzillion.co
m/lessons/1128-factorlinear-expressions
https://learnzillion.co
m/lessons/815rewrite-anexpression-byexpanding-it
https://learnzillion.com/
lessons/3441-write-anexpression-to-find-thecost-of-an-item-withtax
https://learnzillion.com/
lessons/3442-write-apercent-markupexpression
https://learnzillion.com/
lessons/813-write-apercent-increaseproblem-as-a-productof-the-original-amount
https://learnzillion.com/
lessons/814-write-apercent-decreaseproblem-as-a-productof-the-original-amount
17
18
Flex Day (Instruction Based on Data)
Recommended Resources:
My Math Chapter 5 21st Century Career (Pages 423 -424)
My Math Chapter 5 Review (Pages 425 – 428)
Deduce that if a number
sentence is true, then adding,
subtracting, multiplying or
dividing each value by the same
number will result in the same
outcome
Engage NY
Module 2 Lesson 21
(Appendix C)
• Students understand that if a
number sentence is true and we
make any of the following changes
to the number sentence, the
resulting number sentence will be
true:
If a = b, then a + c = b + c
If a = b, then a - c = b – c
If a = b, then a( c )= b( c )
19
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Apply if-then reasoning to solve • We will use this language/thought
one-step addition and
process in lesson 22 (see it’s
subtraction equations
accompanying Engage NY
resource). To lay the foundation
have students think through
problems using the same if/then
reasoning we discussed yesterday.
Example:
1) The difference between a number
b and 7.4 is -6.8. Write as an
algebraic equation to represent
this scenario, then solve for b
2) Solve for x:
-6.5 + x = -4.12
If x + 6 = 4, then x + 6 – 6 = 4 – 6
9 | P a g e http://www.mrmaisonet
.com/index.php?/PreAlgebraNotes/Identify-PartsOf-An-Equation.html
3) Solve for p:
1
3
4 + p = −5
2
4
X+6=4
Therefore, x = -2
My Math
Chapter 6 Lesson 1
€
20
Apply if-then reasoning to solve •
one-step multiplication and
division equations
Example:
1) Solve for h:
2h = -57
My Math
Chapter 6 Lesson 2
7x = 63
“If 7x = 63, then 7x divided by 7 =
63 divided by 7. Therefore, x = 9”
21
22
23
Apply if-then reasoning to solve
one-step equations with rational
coefficients
Use algebra and if/then
reasoning to solve two-step
equations (of the form
𝑝𝑥+_𝑞=_𝑟 _and 𝑝(_𝑥+_𝑞)_=_𝑟)
1) Solve for the unknown:
My Math
Chapter 6 Lesson 3
b
-7 =4
•
Pacing: 2 days
€
True or False: The following equations
are true:
1)
2)
3)
4)
3 + 4 = 12 – 7
12 + 3 * 2 / 6 = 9 + 4
4b = b * 4
2(-3 + d) = -6 – 2d
My Math
Chapter 6 Lessons 4-5
Engage NY
Module 2 Lesson 22
(Appendix C)
http://www.phschool.c
om/iText/math/sample_
chapter/Ch02/0202/PH_Alg1_ch0202_Obj1.html
10 | P a g e