Algebra Unpacked ContentFor 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.)
Seeing Structure in Expressions A-SSE
Common Core Cluster
Interpret the structure of expressions
Common Core Standard
UnpackingWhat 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 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 expression. For example, the
expressionrepresents 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 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 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 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.
A-SSE.1b Students group together parts of an expression to
reveal underlying structure.
(Level I)
Ex. The expression models the cost of a taxi ride that is m
miles long. What does represent in the context of this
situation?
(Level II)
Ex. What information related to symmetry is revealed by
rewriting the quadratic formula as ?
(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 where x is the horizontal distance the person has
walked from one side of the bridge. Interpret the factors and 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 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 childs bounce above a trampoline is given by
the function . Rewrite the expression to reveal the maximum height
of the bounce and how long it takes to reach the maximum
height.
Seeing Structure in Expressions A-SSE
Common Core Cluster
Write expressions in equivalent forms to solve problems
Common Core Standard
UnpackingWhat 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 can be rewritten
as 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 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 = 1 , with a = 1,
to identify the vertex of the parabola
(h, k), and explain its meaning in context.
Ex. The quadratic expression models the height of a ball thrown
vertically. Find the vertex and interpret its meaning in this
context. context and equation dont 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 represents the value of an automobile x years
after purchase. Find the yearly and the monthly rate of
depreciation of the car.
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 =
Mortgage payments can be found using the formula, where P
represents the payment amount, A 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?
Arithmetic With Polynomials and Rational Expressions A-APR
Common Core Cluster
Perform arithmetic operations on polynomials
Common Core Standard
UnpackingWhat 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 kilometers, write an expression
for the area of the circle.
Ex. Explain why does not equal.
Arithmetic With Polynomials and Rational Expressions A-APR
Common Core Cluster
Understand the relationship between zeros and factors of
polynomials
Common Core Standard
UnpackingWhat 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, is divided by
a monomial,, the remainder is the same as if you evaluate the
polynomial for , i.e. calculate. If the remainder when dividing by
is 0, or , then is a factor of the polynomial.
If , then is a factor of , which means that is a root of the
function . This is known as the Factor Theorem.
(Level III)
Ex. Given , determine whether 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 is a factor of , which means that is divisible
by Explain why it must be true that
Ex. Assume we know that . Explain why it must be true that is a
factor of .
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 , list the zeros of the function and
sketch the graph.
(Level II)
Ex. Sketch the graph of the function 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, is a zero with
multiplicity two, is a zero with multiplicity three, and is a zero
with multiplicity one. Write a possible equation for this function
and sketch its graph.
Arithmetic With Polynomials and Rational Expressions A-APR
Common Core Cluster
Use polynomial identities to solve problems
Common Core Standard
UnpackingWhat 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).
(Level III)
Ex. Is equivalent to ? Explain why or why not.
Ex. Jessie claims that . Is he correct? Prove why or why
not.
Ex. Prove . Justify each step.
Ex. Solve the quadratic Justifying each step. What was
interesting about the result?
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 Pascals Triangle.1
1The Binomial Theorem can be proved by mathematical induction or
by a combinatorial argument.
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. Pascals
triangle is a triangular array that identifies the coefficients of
an expanded binomial. The numbers in Pascals 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 , note that the product
occurs twice:
Using combinatorics, the coefficient of the second term would be
2C1 = 2.
(Level III)
Ex. Explain how to generate a row of Pascals triangle.
Ex. What are the coefficients of the expanded terms of 5?
Ex. Using the binomial theorem, expand 5.
Ex. Why are the coefficients of a binomial expansion equal to
values of nCr?
Arithmetic With Polynomials and Rational Expressions A-APR
Common Core Cluster
Rewrite rational expressions
Common Core Standard
UnpackingWhat does this standard mean that a student will know
and be able to do?
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.
A-APR.6 Rewrite rational expressions,, in the form using long
division, synthetic division or 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 is in the form .
(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 .
Use division of polynomials to show that can be written with an
equivalent expression in the form of .
Ex. Divide. Write the answer in the form of quotient plus
remainder/divisor.
Ex. Use a computer algebra system to rewrite the following
rational expression in quotient and remainder form
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.
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 sq. ft. and a height of ft.
Express the width of the rectangle as a rational expression in
terms of .
Creating Equations A-CED
Common Core Cluster
Create equations that describe numbers or relationships
Common Core Standard
UnpackingWhat 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, which results in the one-variable equation . Note,
the resulting equation can be solved in Level I using a table or
graph. See A-REI.11.
(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.
(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)
(x cmx cm16 cm30 cm)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 the maximum volume?
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 kickers 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 womans 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.
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.
(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 companys profit from calculator sales.
c. How many of each type of calculator should the company
produce to maximize profit using the stock on hand?
A-CED.4 Rearrange formulas to highlight a quantity of interest,
using the same reasoning as in solving equations. For example,
rearrange Ohms 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. If , solve for T2
Reasoning with Equations and Inequalities A-REI
Common Core Cluster
Understand solving equations as a process of reasoning and
explain the reasoning
Common Core Standard
UnpackingWhat 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.
(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-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 II)
Ex. Solve for x.
Ex. Mary solved 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 for x.
Reasoning with Equations and Inequalities A-REI
Common Core Cluster
Solve equations and inequalities in one variable
Common Core Standard
UnpackingWhat 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. Grandmas 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 Grandmas.
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.
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.
A-REI.4a Transform a quadratic equation written in standard form
to an equation in vertex form, - , 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 , where a and b are real numbers.
Students should understand that the solutions are always complex
numbers of the form . Real solutions are produced when b = 0, and
pure imaginary solutions are found when a = 0. The value of the
discriminant, , 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 was
his result.
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?
Reasoning with Equations and Inequalities A-REI
Common Core Cluster
Solve systems of equations
Common Core Standard
UnpackingWhat does this standard mean that a student will know
and be able to do?
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
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
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
(Level I)
Ex. Solve the following system graphically and symbolically:
x2 + y2 = 1
y = x
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
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.
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
Common Core Standard
UnpackingWhat 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
a. (4, 0)
b. (0, 10)
c. (-1, 7.5)
d. (2.3, 5)
How many solutions does this equation have? Justify your
conclusion.
A-REI.11 Explain why the x-coordinates 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.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. Johns earnings are represented by the equation, p(x) =
2x and Jerrys 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 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.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?
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
inequalitys 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
1
High School Mathematics Common Core State Standards Unpacked
Content December 10, 2010
2
High School Mathematics Common Core State Standards Unpacked
Content December 10, 2010
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