DCA Erie 1 BOCES
Mathematics: 2012 - 2013
Updated on: 3/1/2013
CCLS Algebra
Module 4: Expressions and Equations
Essential Questions:
Common Core Standards
Content
A-SSE.1. Interpret expressions that
represent a quantity in terms of its
context.
A-SSE.1.a. Interpret parts of an
expression, such as terms, factors, and
coefficients.
(70%)
Skills

Parts of an expression

Vocabulary
Resources
I can identify the parts of an
expression by defining
appropriate vocabulary such
as: terms, factors,
coefficients, constant,
monomial, binomial,
trinomial, polynomial,
exponent, degree, variables,
standard form, index, etc.
I can explain how different
parts of an expression effect
the expression.
Rigorous Sample Tasks
1)Give an explanation, in terms of the structure of the expression below, why it halves in value when
n is quadrupled:
𝒔
Scaffolded sample tasks
1)When you simplify
, why do you
take the cube root of 27 separately from the
cube root of
?
√𝒏
Most savings accounts advertise an annual interest rate, but they actually compound that interest at
regular intervals during the year. That means that, if you own an account, you’ll be paid a portion of the
interest before the year is up, and, if you keep that payment in the account, you’ll start earning interest
on the interest you’ve already earned.
For example, suppose you put $500 in a savings account that advertises 5% annual interest. If that
interest is paid once per year, then your balance B after t years could be computed using the equation
𝑩 = 𝟓𝟎𝟎(𝟏. 𝟎𝟓)𝒕 , since you’ll end each year with 100% + 5% of the amount you began the year with.
2) a)Write an algebraic expression to model the
following phrase:
the price of a meal plus a 15% tip for the meal.
b) What could the expression
represent, including units?
c) What if the tip is 20% instead of 15% and 3
On the other hand, if that same interest rate is compounded monthly, then you would compute your
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
balance after t years using the equation
𝑩 = 𝟓𝟎𝟎(𝟏 +
Updated on: 3/1/2013
CCLS Algebra
people are sharing the cost evenly. How can you
represent the amount each person pays with a
simplified algebraic expression? Identify the units
for the expression.
.𝟎𝟓 𝟏𝟐𝒕
)
𝟏𝟐
1.
Why does it make sense that the equation includes the term
.05 by 12?
2.
3.
4.
.𝟎𝟓
𝟏𝟐
? That is, why are we dividing
How does this equation reflect the fact that you opened the account with $500?
.𝟎𝟓
.𝟎𝟓
What do the numbers 1 and represent in the expression (𝟏 + )?
𝟏𝟐
𝟏𝟐
What does the “12t ” in the equation represent?
A-SSE.1.b. Interpret complicated
expressions by viewing one or more of
their parts as a single entity. For
example, interpret P(1+r)^n as the
product of P and a factor not
depending on P.
(70%)

Interpreting expressions

I can identify the factors of an
expression.
I can describe the effects of
the parts of an expression
(examples: the interest
formula where p is a
coefficient and when
factoring a quadratic
equation, the GCF does not
affect the roots).
Rigorous Sample Tasks
Suppose P and Q give the sizes of two different animal populations, where Q>P . In (a)–(d), say which
of the given pair of expressions is larger. Briefly explain your reasoning in terms of the two populations.
1.
2.
3.
P+Q and 2P
𝑷
𝑷+𝑸
and
𝑷+𝑸
(𝑸−𝑷)
𝟐
and
𝟐
𝑸−𝑷
𝟐
Scaffolded sample tasks
3) Given the expression:
a)What is the degree of this expression?
b)What is the largest coefficient?
DCA Erie 1 BOCES
4.
Mathematics: 2012 - 2013
Updated on: 3/1/2013
CCLS Algebra
𝑷 + 𝟓𝟎𝒕 and 𝑸 + 𝟓𝟎𝒕
c) What is the leading coefficient of the
polynomial in standard form?
d) How many terms are in the polynomial?
A-SSE.2. Use the structure of an
expression to identify ways to rewrite
it. For example, see x^4 - y^4 as (x^2)^2
- (y^2)^2, thus recognizing it as a
difference of squares that can be
factored as (x^2 - y^2)(x^2 + y^2).
(70%)

Structure of expressions

I can rewrite an expression
using laws of exponents.
I can factor an expression by
grouping (GCF).
Rigorous Sample Tasks
Factoring by
grouping
Laws of
exponents
Difference of
squares and
cubes
Structure
Scaffolded sample tasks
Rewrite the polynomial
such that it has more
terms than the original expression.
Why is the expression
not the correct factored form for
?
You purchase n cans of tennis balls at $4.50 per can from an online retailer. There is a tax of 8% on your
order. There are also shipping costs of $7.00 per order. Write the expression that can be used to find
the total cost. Rewrite the expression in an equivalent form.
Simplify the expression
ways. Justify each step.
in two different
You burn calories at a rate of 15 calories per
minute when running and 6 calories per minute
when walking. Suppose you exercise for 60
minutes by running for r minutes and walking for
the remaining time. The expression 15r+6(60-r)
represents the calories burned.
a) What units are associated with r? What units
are associated with 15? What does 15r
represent? Use unit analysis to explain.
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
Updated on: 3/1/2013
CCLS Algebra
b)What does 60-r represent? What units are
associated with 6(60-r)?
c) Simplify 15r+6(60-r) . What units are
associated with the expression? Use unit analysis
to explain.
A-SSE.3. Choose and produce an
equivalent form of an expression to
reveal and explain properties of the
quantity represented by the
expression.
A-SSE.3.a. Factor a quadratic expression
to reveal the zeros of the function it
defines.
(20%)
Factoring Quadratics
Rigorous Sample Tasks

I can factor quadratic
expressions (Greatest
Common Factor, Difference
of two perfect squares, and
reverse distribution).
Reverse
distribution
Scaffolded sample tasks
What critical features of the function
will be revealed when
written in factored form?
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
Updated on: 3/1/2013
CCLS Algebra
Given the following expressions, rewrite them in
an equivalent form.
a)
b)
c)
A student deposits $500 into an account that
earns 3.5% interest compounded quarterly.
Describe and correct the student’s error in
calculating the account balance after two years.
IV) Does the function
represent
exponential growth or decay? Justify your
reasoning.
V) Write the function
form with a different base.
A-SSE.3.b. Complete the square in a
quadratic expression to reveal the
maximum or minimum value of the
function it defines.
(20%)
Complete the square



I can factor quadratic
expressions by completing
the square.
I can explain what different
forms of a quadratic
expression reveal about the
function.
I can write a quadratic
Complete the
square
Vertex form
Standard form
in an equivalent
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
Updated on: 3/1/2013
CCLS Algebra
expression in vertex form to
reveal the maximum or
minimum value of the
function.
Rigorous Sample Tasks
Scaffolded sample tasks
What critical features of the function
will be revealed while
completing the square?
A-SSE.3.c. Use the properties of
exponents to transform expressions for
exponential functions. For example the
expression 1.15^t can be rewritten as
(1.15^(1/12))^12t approximately equals
1.012^12t to reveal the approximate
equivalent monthly interest rate if the
annual rate is 15%.
(20%)

Rewriting exponential
expressions

Rigorous Sample Tasks
A-APR.1. Understand that polynomials
(70% and Fluency)
I can rewrite exponential
expressions using laws of
exponents.
I can rewrite exponential
functions to compare and
contrast them with other
exponential functions.
Scaffolded sample tasks

I can define the system of
Closure
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
form a system analogous to the
integers, namely, they are closed under
the operations of addition, subtraction,
and multiplication; add, subtract, and
multiply polynomials.
Operations with
polynomials





Rigorous Sample Tasks
A.APR.3 Identify zeros of polynomials
when suitable factorizations are
available, and use the zeros to
construct a rough graph of the function
defined by the polynomial.
polynomials as closed under
addition, subtraction, and
multiplication.
I can explain why the
operation of division is not
defined as closed.
I can explain that polynomials
can be operated on in the
same way as integers.
I can add polynomials.
I can subtract polynomials.
I can multiply polynomials.
Analogous
Scaffolded sample tasks
(20%)

Factor quadratics


Rigorous Sample Tasks
A-CED.1. Create equations and
inequalities in one variable and use
them to solve problems. Include
equations arising from linear and
quadratic functions, and simple rational
Updated on: 3/1/2013
CCLS Algebra
I can factor a quadratic
function.
I can identify the zeros of a
quadratic function
algebraically.
I can sketch the graph of a
quadratic function using the
zeros.
Zeros
Sketch
Factorization
Scaffolded sample tasks
(70%)

Writing quadratic
equations


I can write a linear equation
from a word problem.
I can write a linear inequality
from a word problem.
I can write a quadratic
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
and exponential functions.





Rigorous Sample Tasks
A-CED.2. Create equations in two or
more variables to represent
relationships between quantities; graph
equations on coordinate axes with
labels and scales.
Updated on: 3/1/2013
CCLS Algebra
equation from a word
problem.
I can write an exponential
equation from a word
problem.
I can solve a linear equation
written from a word problem
and interpret its solution
within the context.
I can solve a linear inequality
written from a word problem
and interpret its solution
within the context.
I can solve a quadratic
equation written from a word
problem and interpret its
solution within the context.
I can solve an exponential
equation written from a word
problem and interpret its
solution within the context.
All with integer exponents.
Scaffolded sample tasks
(70%)

Writing equations


I can write linear equations
using two or more variables.
I can rewrite an equation in
terms of one variable (y =
form).
I can identify parts of the
coordinate plane (axes and
DCA Erie 1 BOCES
Mathematics: 2012 - 2013







Rigorous Sample Tasks
A-CED.4. Rearrange formulas to
highlight a quantity of interest, using
the same reasoning as in solving
(70%)
Literal equations
Updated on: 3/1/2013
CCLS Algebra
quadrants).
I can graph systems of
equations with correct labels
and scales from a word
problem.
I can write quadratic
equations using two or more
variables.
I can rewrite a quadratic
equation in terms of one
variable (y = form).
I can graph a quadratic
equation from a table of
values with an appropriate
scale and labels from a word
problem.
I can write an exponential
equation using two or more
variables.
I can rewrite an exponential
equation in terms of one
variable (y = form).
I can graph an exponential
equation from a table of
values with an appropriate
scale from a word problem.
Scaffolded sample tasks

I can rewrite literal equations
in terms of a different
variable using real world
DCA Erie 1 BOCES
Mathematics: 2012 - 2013
equations. For example, rearrange
Ohm's law V = IR to highlight resistance
R.

Rigorous Sample Tasks
A-REI.4. Solve quadratic equations in
one variable.
A-REI.4.a. Use the method of
completing the square to transform any
quadratic equation in x into an
equation of the form (x - p)^2 = q that
has the same solutions. Derive the
quadratic formula from this form.
(70%)
formulas.
I can rewrite equations with
squared variables in terms of
a different variable.
Scaffolded sample tasks

Quadratic formula

Rigorous Sample Tasks
A-REI.4.b. Solve quadratic equations by
inspection (e.g., for x^2 = 49), taking
square roots, completing the square,
the quadratic formula and factoring, as
appropriate to the initial form of the
equation. Recognize when the
quadratic formula gives complex
solutions and write them as a plusminus bi for real numbers a and b.
Updated on: 3/1/2013
CCLS Algebra
I can rewrite a quadratic
equation to the form (x - p)^2
= q by using "completing the
square" method.
I can derive the quadratic
formula from "completing
the square" method.
Quadratic
formula
Derive
Scaffolded sample tasks
(70%)

Solving quadratic
equations




I can solve quadratic
equations by "completing the
square" method.
I can solve quadratic
equations by factoring.
I can solve quadratic
equations by applying the
quadratic formula.
I can solve quadratic
equations by inspection (e.g.
square rooting both sides).
I can explain why quadratic
Non-real
solution
Complex
solution
DCA Erie 1 BOCES
Mathematics: 2012 - 2013

Rigorous Sample Tasks
Updated on: 3/1/2013
CCLS Algebra
equations have two solutions.
I can identify when a
quadratic equation gives nonreal/complex solution(s).
**Solving and writing
complex solutions is not
required in Algebra I.
Scaffolded sample tasks