Download Algebra Theme Unpacked Content 03162012

 Algebra ● Unpacked Content
For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13.
What is the purpose of this document?
To increase student achievement by ensuring educators understand what the standards mean a student must know and be able to do completely and
comprehensively.
What is in the document?
Descriptions of what each standard means a student will know and be able to do. The “unpacking” of the standards done in this document is an effort to
answer a simple question “What does this standard mean that a student must know and be able to do?” and to ensure that description is helpful, specific
and comprehensive.
How do I send feedback?
We intend the explanations and examples in this document to be helpful, specific and comprehensive. That said, we believe that as this document is used,
teachers and educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us at
[email protected] and we will use your input to refine our unpacking of the standards. Thank You!
Just want the standards alone?
You can find the standards alone at www.corestandards.org.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010
1 Seeing Structure in Expressions
A-SSE
Common Core Cluster
Interpret the structure of expressions
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-SSE.1 Interpret expressions that
represent a quantity in terms of its
context.«
a. Interpret parts of an expression,
such as terms, factors, and
coefficients.
b. Interpret complicated expressions
by viewing one or more of their
parts as a single entity. For
example, interpret 𝑃(1 + 𝑟)! as
the product of P and a factor not
depending on P.
A-SSE.1a. Students manipulate the terms, factors, and coefficients in difficult expressions to explain the meaning
of the individual parts of the expression. Use them to make sense of the multiple factors and terms of the
5
expression. For example, the expression $10, 000 (1.055) represents the amount of money I have in an account.
My account has a starting value of $10,000 with a 5.5% interest rate every 5 years, where 10,000 and (1+.055) are
factors, and the $10,000 does not depend on the amount the account is increased by. More scaffolding needed for
quadratic.
(Level I)
Ex. The expression 150 + 0.10𝑆 models the income earned based on total monthly sales. Interpret the terms and
coefficients of the expression in the context of this situation.
(Level II)
Ex. The expression −4.9𝑡 ! + 17𝑡 + 0.6 describes the height in meters of a basketball t seconds after it has been
thrown vertically in the air. Interpret the terms and coefficients of the expression in the context of this situation.
(Level III)
Ex. A person is walking across a hanging bridge that is suspended over a river. A hanging bridge droops in the
middle creating a parabolic shape. The distance in feet from the person crossing the bridge to the river at any point
can be described by the expression −0.02𝑥 100 − 𝑥 + 110, where x is the horizontal distance the person has
walked from one side of the bridge. Interpret the terms, factors, and coefficients of the expression in context.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 2 A-SSE.1b Students group together parts of an expression to reveal underlying structure.
(Level I)
Ex. The expression 10 + .50(𝑚 − 5) models the cost of a taxi ride that is m miles long. What does (𝑚 − 5)
represent in the context of this situation?
(Level II)
Ex. What information related to symmetry is revealed by rewriting the quadratic formula as 𝑥 =
!!
!!
±
! ! !!!"
!!
?
(Level III)
Ex. A person is walking across a hanging bridge that is suspended over a river. A hanging bridge droops in the
middle creating a parabolic shape. The distance in feet from the person crossing the bridge to the river at any point
can be described by the expression −0.02𝑥 100 − 𝑥 + 110, where x is the horizontal distance the person has
walked from one side of the bridge. Interpret the factors −.02𝑥 and (100 − 𝑥) in the context of this situation.
A-SSE.2 Use the structure of an
expression to identify ways to rewrite
it. For example, see x4 – y4 as (x2)2 –
(y2)2, thus recognizing it as a
difference of squares that can be
factored as (x2 – y2)(x2 + y2).
A-SSE.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of
the same expression.
(Level I/II)
Ex. The expression 4000𝑝 − 250𝑝 ! represents the income at a concert, where p is the price per ticket. Rewrite
this expression in another form to reveal the expression that represents the number of people in attendance based
on the price charged.
(Level III)
Ex. The height of a child’s bounce above a trampoline is given by the function 𝑦 = −16𝑡 ! + 24𝑡 − 3. Rewrite the
expression −16𝑡 ! + 24𝑡 − 3 to reveal the maximum height of the bounce and how long it takes to reach the
maximum height.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 3 Seeing Structure in Expressions
A-SSE
Common Core Cluster
Write expressions in equivalent forms to solve problems
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
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. Factor a quadratic expression to
reveal the zeros of the function it
defines.
b. Complete the square in a
quadratic expression to reveal the
maximum or minimum value of
the function it defines.
c. Use the properties of exponents to
transform expressions for
exponential functions. For
example the expression 1.15! can
!
!"
be rewritten as (1.15 )!"! ≈
(1.012)!"! to reveal the
approximate equivalent monthly
interest rate if the annual rate is
15%.
A-SSE.3a Students factor quadratic expressions and find the zeros of the quadratic function it represents. Zeroes
are the x values that yield a y value of 0. They should also explain the meaning of the zeros as they relate to the
problem.
Ex. If (3𝑚 ! − 15𝑚) is the income gathered at a rock concert, what values of m would produce an income of 0?
A-SSE.3b Students rewrite a quadratic expression in the form a- add to limitations in course 3 where limitation a =
2
1 ( x − h ) + k , with a = 1, to identify the vertex of the parabola
(h, k), and explain its meaning in context.
Ex. The quadratic expression 𝑥 ! − 24𝑥 + 55 models the height of a ball thrown vertically. Find the vertex and
interpret its meaning in this context. – context and equation don’t match (thrown ball should have a negative x
coefficient)
A-SSE.3c Use properties of exponents to write an equivalent form of an exponential function to reveal and explain
specific information about the rate of growth or decay.
Ex. The equation 𝑦 = 14000(0.8) ! represents the value of an automobile x years after purchase. Find the yearly
and the monthly rate of depreciation of the car.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 4 A-SSE.4 Derive the formula for the
sum of a finite geometric series (when
the common ratio is not 1), and use
the formula to solve problems. For
example, calculate mortgage
payments.
A-SSE.4
To derive the formula, expand the finite geometric series to show a few terms, including the last term. Create a
new series by multiplying both sides of the original series by the common ratio, r. Subtract the new series from the
original series, and solve for Sn.
Sn= (a + ar + ar2 + … + arn-1)
- r Sn= (ar + ar2 + ar3 + … + arn-1 + arn)
Sn - r Sn = a - arn
Sn(1-r) = a(1- rn)
Sn =
a(1− r n )
(1− r)
Mortgage payments can be found using the formula, P =
iA
where P represents the payment amount, A
1− (1+ i)−n
represents the loan amount, n represents the number of payments, and i is the monthly interest rate. The mortgage
payment formula can be derived from the formula for the sum of a finite geometric series because the mortgage
process can be viewed as a finite series of (Principal + Interest – Payment).
Ex. You just bought a $230,000 house, with 10% down on a 30-year mortgage with an interest rate of 8.5% per
year. What is the monthly payment? Spreadsheet connection?
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 5 Arithmetic With Polynomials and Rational Expressions
A-APR
Common Core Cluster
Perform arithmetic operations on polynomials
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-APR.1 Understand that
polynomials form a system analogous
to the integers, namely, they are
closed under the operations of
addition, subtraction, and
multiplication; add, subtract, and
multiply polynomials.
A-APR.1
The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or
product is also a polynomial. Polynomials are not closed under division because in some cases the result is a
rational expression.
(Level III)
Ex. If the radius of a circle is ( 5x − 2 ) kilometers, write an expression for the area of the circle.
Ex. Explain why (4𝑥 ! + 3)! does not equal (16𝑥 ! + 9).
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 6 Arithmetic With Polynomials and Rational Expressions
A-APR
Common Core Cluster
Understand the relationship between zeros and factors of polynomials
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-APR.2 Know and apply the
Remainder Theorem: For a
polynomial p(x) and a number a, the
remainder on division by x – a is p(a),
so p(a) = 0 if and only if (x – a) is a
factor of p(x).
A-APR.2
The Remainder Theorem states that if a polynomial, 𝑝(𝑥) is divided by a monomial, (𝑥 – 𝑐), the remainder is the
same as if you evaluate the polynomial for 𝑐, i.e. calculate 𝑝(𝑐). If the remainder when dividing by (𝑥 – 𝑐) is 0, or
𝑝(𝑐) = 0, then (𝑥 – 𝑐) is a factor of the polynomial.
If 𝑓(𝑎) = 0, then (𝑥 − 𝑎) is a factor of 𝑓(𝑥), which means that 𝑎 is a root of the function 𝑓(𝑥). This is known as
the Factor Theorem.
(Level III)
Ex. Given 𝑓 𝑥 = 2𝑥 ! + 6𝑥 − 20, determine whether −5 is a root of the function, then write the function in
factored form.
Ex. Compare the process of synthetic division to the process of long division for dividing polynomials.
Ex. Assume that (𝑥 − 𝑐) is a factor of 𝑓, which means that 𝑓 is divisible by 𝑥 − 𝑐 . Explain why it must be true
that 𝑓 𝑐 = 0.
Ex. Assume we know that 𝑓 𝑐 = 0. Explain why it must be true that (𝑥 − 𝑐) is a factor of 𝑓.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 7 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.
A-APR.3 Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph.
(Level II)
Ex. Given the function 𝑦 = 2𝑥 ! + 6𝑥 – 3, list the zeros of the function and sketch the graph.
(Level II)
Ex. Sketch the graph of the function 𝑓 𝑥 = 𝑥 + 5 ! . What is the multiplicity of the zeros of this function? How
does the multiplicity relate to the graph of the function?
(Level III)
Ex. For a certain polynomial function, 𝑥 = 3 is a zero with multiplicity two, 𝑥 = 1 is a zero with multiplicity three,
and 𝑥 = −3 is a zero with multiplicity one. Write a possible equation for this function and sketch its graph.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 8 Arithmetic With Polynomials and Rational Expressions
A-APR
Common Core Cluster
Use polynomial identities to solve problems
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-APR.4 Prove polynomial identities
and use them to describe numerical
relationships. For example, the
polynomial identity (x2 + y2)2 = (x2 –
y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
A-APR.4 Prove polynomial identities algebraically by showing steps and providing reasons or explanations. The
following examples are meant to be investigated by students considering analogous problems, and trying special
cases and simpler forms of the original problem in order to gain insight into its solution(s).
A-APR.5 (+) Know and apply the
Binomial Theorem for the expansion
of (x + y)n in powers of x and y for a
positive integer n, where x and y are
any numbers, with coefficients
determined for example by Pascal’s
Triangle.1
A-APR.5
The Binomial Theorem describes the algebraic expansion of powers of a binomial. There are patterns that develop
with the coefficients and the variables when expanding binomials. Pascal’s triangle is a triangular array that
identifies the coefficients of an expanded binomial. The numbers in Pascal’s triangle are also evaluations of
combinations, nCr. The values of the combinations correspond with the coefficients of the expanded binomial,
which indicates how many times that term will appear in the completely expanded form. This is a connection
between Probability and Algebra that should be made explicit.
For example, when squaring the binomial (𝑎 + 𝑏), note that the product 𝑎𝑏 occurs twice:
1
The Binomial Theorem can be
proved by mathematical induction or
by a combinatorial argument.
(Level III)
Ex. Is (2𝑥 − 3)! − 64 equivalent to (2𝑥 − 11)(2𝑥 + 5)? Explain why or why not.
Ex. Jessie claims that (𝑥 + 𝑦)! = 𝑥 ! + 2𝑥𝑦 + 𝑦 ! . Is he correct? Prove why or why not.
Ex. Prove 𝑥 ! − 𝑦 ! = 𝑥 − 𝑦 (𝑥 ! + 𝑥𝑦 + 𝑦 ! ). Justify each step.
!
Ex. Solve the quadratic 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 Justifying each step. What was interesting about the result?
(a + b)2 = a 2 + ab + ab + b 2 = a 2 + 2ab + b 2
Using combinatorics, the coefficient of the second term would be 2C1 = 2.
(Level III)
Ex. Explain how to generate a row of Pascal’s triangle.
Ex. What are the coefficients of the expanded terms of (𝑎 + 𝑏)5?
Ex. Using the binomial theorem, expand (𝑎 + 𝑏)5.
Ex. Why are the coefficients of a binomial expansion equal to values of nCr?
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 9 Arithmetic With Polynomials and Rational Expressions
A-APR
Common Core Cluster
Rewrite rational expressions
Common Core Standard
A-APR.6 Rewrite simple rational
expressions in different forms; write
a(x)/b(x) in the form q(x) + r(x)/b(x),
where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x)
less than the degree of b(x), using
inspection, long division, or, for the
more complicated examples, a
computer algebra system.
Unpacking
What does this standard mean that a student will know and be able to do?
A-APR.6 Rewrite rational expressions,
a( x)
r ( x)
, in the form q ( x) +
using long division, synthetic division or
b( x )
b( x )
with expressions that pose difficulty by hand, use a computer algebra system such as the TI Inspire CAS or Ipad
applications.
When dividing a polynomial by a polynomial, the new form is the quotient plus the remainder divided by the
divisor. This process should be connected to dividing with numbers. The quotient represents the number of times
something will divide, plus the parts or pieces remaining. Know that the degree of the quotient is less than the
degree of the dividend. Connect division of polynomials to the remainder theorem when 𝑏(𝑥) is in the form
(𝑥 − 𝑐).
(Level III)
!"#
Ex. We know from arithmetic, that a fraction like
indicates the division of 327 by 10. The result can be
expressed 32 R 7 or as 32 +
!
!"
!"
. Use division of polynomials to show that
equivalent expression in the form of 𝑞 𝑥 +
!(!)
!!!
!! ! !!!!!
!!!
can be written with an
.
Ex. Divide. Write the answer in the form of quotient plus remainder/divisor.
𝑥 ! + 3𝑥
𝑥! − 4
Ex. Use a computer algebra system to rewrite the following rational expression in quotient and remainder form
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 10 A-APR.7 (+) Understand that
rational expressions form a system
analogous to the rational numbers,
closed under addition, subtraction,
multiplication, and division by a
nonzero rational expression; add,
subtract, multiply, and divide rational
expressions.
9𝑥 ! + 9𝑥 ! − 𝑥 + 2
2
𝑥+
3
A-APR.7 When performing any operation on a rational expression, the result is always another rational expression,
which is the Closure Property for rational expressions. Compare this to the Closure Property for polynomials.
Perform operations with rational expressions, division by nonzero rational expressions only.
Ex. A rectangle has an area of
rational expression in terms of 𝑥.
High School Mathematics Common Core State Standards
(! ! !!!!)
!!
sq. ft. and a height of
!!
(!!!)
ft. Express the width of the rectangle as a
Unpacked Content December 10, 2010 11 Creating Equations
A-CED
Common Core Cluster
Create equations that describe numbers or relationships
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
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 and exponential functions.
A-CED.1 From contextual situations, write equations and inequalities in one variable and use them to solve
problems. Include one-variable equations that arise from functions by the selection of a particular target y-value.
For example, in the radioactive decay problem below, 25 would be substituted for y in the equation 𝑦 = 100
which results in the one-variable equation 25 = 100
using a table or graph. See A-REI.11.
! !
!
! !
!
,
. Note, the resulting equation can be solved in Level I
(Level I)
Ex. The Tindell household contains three people of different generations. The total of the ages of the three family
members is 85.
a. Find reasonable ages for the three Tindells.
b. Find another reasonable set of ages for them.
c. One student, in solving this problem, wrote C + (C+20)+ (C+56) = 85
1. What does C represent in this equation?
2. What do you think the student had in mind when using the numbers 20 and 56?
3. What set of ages do you think the student came up with?
(Level I)
Ex. A salesperson earns $700 per month plus 20% of sales. Write an equation to find the minimum amount of
sales needed to receive a salary of at least $2500 per month.
(Level I)
Ex. A scientist has 100 grams of a radioactive substance. Half of it decays every hour. Write an equation to find
how long it takes until 25 grams are left.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 12 (Level II)
Ex. A pool can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the pool is full, it can be drained by
pipe C in 4 hours. If the pool is initially empty and all three pipes are open, write an equation to find how long it
takes to fill the pool.
(Level III)
Ex. A cardboard box company has been contracted to manufacture open-top rectangular storage boxes for small
hardware parts. The company has 30 cm x 16 cm cardboard sheets. They plan to cut a square from each corner of
the sheet and bend up the sides to form the box. If the company wants to make boxes with the largest possible
volume, what should be the dimensions of the square to be cut out? What are the dimensions of the box? What is
x cm the maximum volume?
x cm 16 cm 30 cm A-CED.2 Create equations in two or
more variables to represent
relationships between quantities;
graph equations on coordinate axes
with labels and scales.
A-CED.2 Given a contextual situation, write equations in two variables that represent the relationship that exists
between the quantities. Also graph the equation with appropriate labels and scales. Make sure students are
exposed to a variety of equations arising from the functions they have studied.
(Level I)
Ex. The height of a ball t seconds after it is kicked vertically depends upon the initial height and velocity of the ball
and on the downward pull of gravity. Suppose the ball leaves the kicker’s foot at an initial height of 0.7 m with
initial upward velocity of 22m/sec. Write an algebraic equation relating flight time t in seconds and height h in
meters for this punt.
(Level I)
Ex. In a woman’s professional tennis tournament, the money a player wins depends on her finishing place in the
standings. The first-place finisher wins half of $1,500,000 in total prize money. The second-place finisher wins
half of what is left; then the third-place finisher wins half of that, and so on.
a. Write a rule to calculate the actual prize money in dollars won by the player finishing in nth place, for any
positive integer n.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 13 b. Graph the relationship that exists between the first 10 finishers and the prize money in dollars.
c. What pattern do you notice in the graph? What type of relationship exists between the two variables?
(Level II)
Ex. The intensity of light radiating from a point source varies inversely as the square of the distance from the
source. Write an equation to model the relationship between these quantities given a fixed energy output.
need Level III
A-CED.3 Represent constraints by
equations or inequalities, and by
systems of equations and/or
inequalities, and interpret solutions as
viable or non- viable options in a
modeling context. For example,
represent inequalities describing
nutritional and cost constraints on
combinations of different foods.
A-CED.3 When given a problem situation involving limits or restrictions, represent the situation symbolically using
an equation or inequality. Interpret the solution(s) in the context of the problem.
When given a real world situation involving multiple restrictions, develop a system of equations and/or inequalities
that models the situation. In the case of linear programming, use the Objective Equation and the Corner Principle to
determine the solution to the problem.
A-CED.4 Rearrange formulas to
highlight a quantity of interest, using
the same reasoning as in solving
equations. For example, rearrange
Ohm’s law V = IR to highlight
resistance R.
A-CED.4 Solve multi-variable formulas or literal equations, for a specific variable. Explicitly connect this to the
process of solving equations using inverse operations.
(Level II)
Ex. Imagine that you are a production manager at a calculator company. Your company makes two types of
calculators, a scientific calculator and a graphing calculator.
a. Each model uses the same plastic case and the same circuits. However, the graphing calculator requires
20 circuits and the scientific calculator requires only 10. The company has 240 plastic cases and 3200
circuits in stock. Graph the system of inequalities that represents these constraints.
b. The profit on a scientific calculator is $8.00, while the profit on a graphing calculator is $16.00. Write an
equation that describes the company’s profit from calculator sales.
c. How many of each type of calculator should the company produce to maximize profit using the stock on
hand?
(Level II)
Ex. If H =
kA (T1 − T2 )
, solve for T2
L
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 14 Reasoning with Equations and Inequalities
A-REI
Common Core Cluster
Understand solving equations as a process of reasoning and explain the reasoning
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-REI.1 Explain each step in solving
a simple equation as following from
the equality of numbers asserted at the
previous step, starting from the
assumption that the original equation
has a solution. Construct a viable
argument to justify a solution method.
A-REI.1 Relate the concept of equality to the concrete representation of the balance of two equal quantities.
Properties of equality are ways of transforming equations while still maintaining equality/balance. Assuming an
equation has a solution, construct a convincing argument that justifies each step in the solution process with
mathematical properties.
A-REI.2 Solve simple rational and
radical equations in one variable, and
give examples showing how
extraneous solutions may arise.
A-REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous
solutions arise. Add context.
(Level I)
Ex. Solve the following equations for x. Use mathematical properties to justify each step in the process.
a. 5(x+3)-3x=55
b. a ± 0i
(Level II)
Ex. Solve 5 − − ( x + 4) = 2 for x.
Ex. Mary solved x = 2 − x for x and got x=-2, and x=1. Are all the values she found solutions to the equation?
Why or why not?
Ex. Solve
3
x
3
=
− for x.
x −3 x −3 2
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 15 Reasoning with Equations and Inequalities
A-REI
Common Core Cluster
Solve equations and inequalities in one variable
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-REI.3 Solve linear equations and
inequalities in one variable, including
equations with coefficients
represented by letters.
A-REI.3 Solve linear equations in one variable, including coefficients represented by letters.
(Level I)
Ex. Solve, Ax +B =C for x. What are the specific restrictions on A?
Ex. What is the difference between solving an equation and simplifying an expression?
Ex. Grandma’s house is 20 miles away and Johnny wants to know how long it will take to get there using various
modes of transportation.
a. Model this situation with an equation where time is a function of rate in miles per hour.
b. For each mode of transportation listed below, determine the time it would take to get to Grandma’s.
Mode of Transportation
Rate of Travel in mph Time of Travel hrs.
bike
12mph
car
55mph
walking
4mph
A-REI.3 Solve linear inequalities in one variable, including coefficients represented by letters.
(Level I/II)
Ex. A parking garage charges $1 for the first half-hour and $0.60 for each additional half-hour or portion thereof.
Write an inequality and solve it to find how long you can park if you have only $6.00 in cash.
Ex. Compare solving an inequality to solving an equation. Compare solving a linear inequality to solving a linear
equation.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 16 A-REI.4a 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.
2
A-REI.4a Transform a quadratic equation written in standard form to an equation in vertex form, (x - p ) = q , by
completing the square.
(Level II)
Ex. Write the quadratic equation, y= -2x2 – 16x -20 in vertex form. What is the vertex of the graph of the
equation?
A-REI.4a Derive the quadratic formula by completing the square on the standard form of a quadratic equation.
Add context or analysis.
(Level II/III)
Ex. Solve y = ax2 +bx+c for x. What do you notice about your solution?
A-REI.4b Solve quadratic equations
by inspection (e.g., for x2 = 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 ± bi for
real numbers a and b.
A-REI.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and
completing the square. Add context or analysis
(Level II)
Ex. Find the solution to the following quadratic equations:
a. x2 – 7x -18 = 0
b. x2 = 81
c. x2- 10x + 5 = 0
A-REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula always produces
solutions. Write the solutions in the form a ± bi , where a and b are real numbers.
Students should understand that the solutions are always complex numbers of the form a ± bi . Real solutions are
produced when b = 0, and pure imaginary solutions are found when a = 0. The value of the discriminant,
b 2 − 4ac , determines how many and what type of solutions the quadratic equation has.
(Level II) Ex. Ryan used the quadratic formula to solve an equation and x =
8 + (−8)2 − 4(1)(−2)
was his result.
2(1)
a. Write the quadratic equation Ryan started with.
b. Simplify the expression to find the solutions.
c. What are the x-intercepts of the graph of this quadratic function?
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 17 Reasoning with Equations and Inequalities
A-REI
Common Core Cluster
Solve systems of equations
Common Core Standard
A-REI.5 Prove that, given a system
of two equations in two variables,
replacing one equation by the sum of
that equation and a multiple of the
other produces a system with the
same solutions. Common
Unpacking
What does this standard mean that a student will know and be able to do?
A-REI.5 Solve systems of equations using the elimination method (sometimes called linear combinations).
Since equations that are multiples of each other have the same solution, one equation in a system can
be multiplied by a constant to produce another equation with the same solution.
For example, in the system. Core plus: Jenna Conrad and Andrea
3x + 2y = 6
x - 4y = 2
the first equation can be multiplied by 2 to generate 6x + 4y = 12. Then the two equations are added to
eliminate y.
6x + 4y = 12
x - 4y = 2
7x
= 14
7
7
x = 2
Then the x-coordinate of 2 can be substituted in either equation to retrieve the y-coordinate, resulting in
(2, 0) as the solution for this system.
(Level II)
Ex. Solve the system:
-3x + 5y = 6
2x + y = 6
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 18 A-REI.6 Solve systems of linear
equations exactly and approximately
(e.g., with graphs), focusing on pairs
of linear equations in two variables.
A-REI.6 Solve systems of equations using graphs.
(Level I/II)
Ex. The equations y = 18 + .4m and y = 11.2 + .54m Give the lengths or two different springs in centimeters, as
mass is added in grams, m, to each separately.
a. Graph each equation.
b. When are the springs the same length?
c. When is one spring at least 10cm longer than the other?
d. Write a sentence comparing the two springs.
A-REI.7 Solve a simple system
consisting of a linear equation and a
quadratic equation in two variables
algebraically and graphically. For
example, find the points of
intersection between the line y = –3x
and the circle x2 + y2 = 3.
A-REI.7 Solve a system containing a linear equation and a quadratic equation in two variables (conic sections
possible) graphically and symbolically. Add context or analysis
A-REI.8 (+) Represent a system of
linear equations as a single matrix
equation in a vector variable.
A-REI.8 Write a system of linear equations as a single matrix equation. Add context or analysis
(Level I)
Ex. Solve the following system graphically and symbolically:
x2 + y2 = 1
y=x
Ex. Write the system of equations as a matrix equation.
x+2y-z=1
2x-y+3z=2
2x+y+z=-1
A-REI.9 (+) Find the inverse of a
matrix if it exists and use it to solve
systems of linear equations (using
technology for matrices of dimension
3 × 3 or greater).
A-REI.9 Find the inverse of the coefficient matrix in the equation, if it exits. Use the inverse of the coefficient
matrix to solve the system. Use technology for matrices with dimensions 3 by 3 or greater.
The inverse of a matrix exists if and only if the matrix is square and the determinate is not 0. The process of
finding the inverse of a 2x2 matrix is easily done by hand, however technology should be used to find the
inverse of 3x3s or greater.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 19 Solving a matrix equation is similar to solving linear equations in that we multiply both sides by the inverse
matrix to find the solution. However the inverse matrix must be multiplied on the left on each side, because
matrix multiplication is not commutative. After multiplying, the left side results in the identity matrix
multiplied by the variable matrix and the right side yields the solutions to the variables in the system of
equations. The product of the identity matrix and any other matrix with the same dimensions is that
matrix.
Ex. AX = B
A-1A
Reasoning with Equations and Inequalities
A-REI
Common Core Cluster
Represent and solve equations and inequalities graphically
Unpacking
Common Core Standard
What does this standard mean that a student will know and be able to do?
A-REI.10 Understand that the graph
of an equation in two variables is the
set of all its solutions plotted in the
coordinate plane, often forming a
curve (which could be a line).
A-REI.10 The solutions to equations in two variables can be shown in a coordinate plane where every ordered pair
that appears on the graph of the equation is a solution. Understand that all points on the graph of a two-variable
equation are solutions because when substituted into the equation, they make the equation true. Add context or
analysis
Ex. Which of the following points lie on the graph of the equation −5𝑥 + 2𝑦 = 20?
a. (4, 0)
b. (0, 10)
c. (-1, 7.5)
d. (2.3, 5)
How many solutions does this equation have? Justify your conclusion.
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 20 A-REI.11 Explain why the xcoordinates of the points where the
graphs of the equations y = f(x) and y
= g(x) intersect are the solutions of
the equation f(x) = g(x); find the
solutions approximately, e.g., using
technology to graph the functions,
make tables of values, or find
successive approximations. Include
cases where f(x) and/or g(x) are linear,
polynomial, rational, absolute value,
exponential, and logarithmic
functions.★
A-REI.12 Graph the solutions to a
linear inequality in two variables as a
half- plane (excluding the boundary in
the case of a strict inequality), and
graph the solution set to a system of
linear inequalities in two variables as
the intersection of the corresponding
half-planes.
A-REI.11 Construct an argument to demonstrate understanding that the solution to every equation can be found by
treating each side of the equation as separate functions that are set equal to each other, f(x) = g(x). Allow y1=f (x)
and y2= g(x) and find their intersection(s). The x-coordinate of the point of intersection is the value at which these
two functions are equivalent, therefore the solution(s) to the original equation. Students should understand that this
can be treated as a system of equations and should also include the use of technology to justify their argument using
graphs, tables of values, or successive approximations.
(Level 1/II)
Ex. John and Jerry both have jobs working at the town carnival. They have different employers, so their daily
wages are calculated differently. John’s earnings are represented by the equation, p(x) = 2x and Jerry’s by
g(x) = 10 + 0.25x.
a. What does the variable x represent?
b. If they begin work next Monday, Michelle told them that Friday would be the only day they made the same
amount of money. Is she correct in her assumption? Explain your reasoning.
c. When will Jerry earn more money than John? When will John earn more money than Jerry? During what
day will their earnings be the same? Justify your conclusions.
A-REI.12 By graphing a two variable inequality, students understand that the solutions to this inequality are all the
ordered pairs located on a portion or side of the coordinate plane that, when substituted into the inequality, make
the equation true. Students should be able to graph the inequality, specifying whether the points on the boundary
line are also solutions by using a dotted or solid line. Using a variety of methods, which include selecting and
substituting test points into the inequality, students should be able to determine which portion or side of the graph
contains the ordered pairs that are the solutions to the original inequality.
(Level III)
Ex. Michelle has $80 total for shopping. In her favorite store, each shirt costs $12 and each pair of earrings cost $4.
a. Write an inequality representing the total number of shirts and earrings she can purchase without spending
more than $80.
b. Graph this inequality on a coordinate plane.
c. Are the points on the boundary line included in the solution set? Explain why or why not.
d. Which portion of the plane contains the solutions to the inequality? Demonstrate this appropriately on
your graph.
e. William claims that all of the solutions to this inequality are reasonable solutions. Do you agree with him?
Development an argument to support your position?
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 21 A-REI.12 When given a system of linear inequalities, students will understand the need to graph the linear
inequalities separately, finding the solutions for each. When shading each inequality’s half-plane or solution set,
students understand that the over-lapping region of the half-planes represents all the ordered pairs shared by both
linear inequalities. Therefore, when these ordered pairs are substituted into each inequality, their values will be
congruent, and therefore, these ordered pairs are solutions to both linear inequalities.
(Level III)
Ex. The Flatbread Pizza Palace makes gourmet flatbread pizzas for sale to hotel chains. They only sell vegetarian
and pepperoni pizzas to the hotels. Their business planning has the following constraints and objective:
• Each vegetarian pizza takes 15 minutes of labor and each pepperoni pizza takes 8 minutes of labor. At most,
the plant has 4,800 minutes of labor available each day.
• The restaurant freezer can hold a total of at most 580 pizzas per day.
• The pepperoni flatbread pizza is more popular than the vegetarian pizza, so the plant makes at most 290
vegetarian pizzas each day.
• Each pepperoni pizza sold earns Flatbread Pizza Palace $4 profit and each vegetarian pizza sold earns them
$3.25 profit.
a. What are the variables in this situation?
b. Write a system of linear equations representing each constraint.
c. Write the objective function that will maximize the profit for Flatbread Pizza Palace.
d. Graph the constraints
High School Mathematics Common Core State Standards
Unpacked Content December 10, 2010 22