3. “Rational” in CCSS – High School
Number and Quantity
The Real Number System (N-RN.1, 2 & 3)
Algebra
Arithmetic with Polynomials and Rational
Expressions (A-APR)
Creating Expressions (A-CED)
Reasoning with Equations and Inequalities
(A-REI)
Functions
Interpreting Functions (F-IF.7d)
4. The Real Number
System N-RN
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational
exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for
radicals in terms of rational exponents.
2. Rewrite expressions involving radicals and rational
exponents using the properties of exponents.
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is
rational; that the sum of a rational number and an irrational
number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
5. The Real Number System N-RN
CCSS WA PE
N.RN.1 Extend the properties of
exponents to rational exponents.
WA.9-12.A1.2.C Interpret and use integer
exponents and square and cube roots,
and apply the laws and properties of
exponents to simplify and evaluate
exponential expressions.
N.RN.2 Rewrite expressions involving
radicals and rational exponents using the
properties of exponents.
WA.9-12.A1.2.C Interpret and use integer
exponents and square and cube roots,
and apply the laws and properties of
exponents to simplify and evaluate
exponential expressions.
N.RN.3 Explain why the sum or product of
rational numbers is rational; that the sum
of a rational number and an irrational
number is irrational; and that the product
of a nonzero rational number and an
irrational number is irrational.
WA.9-12.A2.2.A Explain how whole,
integer, rational, real, and complex
numbers are related, and identify the
number system(s) within which a given
algebraic equation can be solved.
6. Common Misconceptions:
N-RN.1 & 2
Students sometimes misunderstand the meaning of
exponential operations, the way powers and roots
relate to one another, and the order in which they
should be performed. Attention to the base is very
important.
Students should be able to make use of estimation
when incorrectly using multiplication instead of
exponentiation.
Source: katm.org
7. Common Misconceptions:
N-RN.3
Some students may believe that both terminating
and repeating decimals are rational numbers,
without considering nonrepeating and
nonterminating decimals as irrational numbers.
Students may also confuse irrational numbers and
complex numbers, and therefore mix their
properties. In this case, students should encounter
examples that support or contradict properties and
relationships between number sets.
Source: katm.org
8. N-RN.1, 2 & 3 Examples
Khan Academy: Level 3 Exponents Video
http://www.youtube.com/watch?feature=player_embedde
d&v=aYE26a5E1iU
Hotmath Practice Problems
“Rewrite using rational exponent notation”
Google PDF practice problems
http://educ.jmu.edu/~taalmala/235_2000post/235cont
radiction.pdf
“Given that: r is a rational number, and x is an irrational
number. Show that: r + x is irrational.”
31( )
5
9. Arithmetic with Polynomials and
Rational Expressions A-APR
Rewrite rational expressions
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.
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.
10. Arithmetic with Polynomials and
Rational Expressions A-APR
CCSS WA PE
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.
No WA PE match
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.
No WA PE match
11. Common Misconceptions:
A-APR.6
Students with only procedural understanding of
fractions are likely to “cancel” terms (rather than
factors of) in the numerator and denominator of a
fraction. Emphasize the structure of the rational
expression: that the whole numerator is divided by
the whole denominator. In fact, the word “cancel”
likely promotes this misconception.
It would be more accurate to talk about dividing the
numerator and denominator by a common factor.
Source: katm.org
13. Creating Equations
A-CED
Create equations that describe numbers or relationships
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.
14. Creating Equations A-CED
CCSS WA PE
A-CED.1 Create equations that describe
numbers or relationship. 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.
Composite Match
WA.7.1.F Write an equation that corresponds to a
given problem situation, and describe a problem
situation that corresponds to a given equation.
WA.8.1.B Solve one- and two-step linear inequalities
and graph the solutions on the number line.
WA.9-12.A1.4.A Write and solve linear equations and
inequalities in one variable.
WA.9-12.A1.1.D Solve problems that can be
represented by quadratic functions and equations.
WA.9-12.A2.2.C Add, subtract, multiply, divide, and
simplify rational and more general algebraic
expressions.
WA.9-12.A1.5 Core Content: Quadratic functions and
equations: Students study quadratic functions and
their graphs, and solve quadratic equations with real
roots in Algebra 1. They use quadratic functions to
represent and model problems and answer questions
in situations that are modeled by these functions.
Students solve quadratic equations by factoring and
computing with polynomials. The important
mathematical technique of completing the square is
developed enough so that the quadratic formula can
be derived.
15. Common Misconceptions:
A-CED.1
Students may believe that equations of linear,
quadratic and other functions are abstract and exist
only “in a math book,” without seeing the usefulness
of these functions as modeling real-world
phenomena.
Additionally, they believe that the labels and scales
on a graph are not important and can be assumed by
a reader, and that it is always necessary to use the
entire graph of a function when solving a problem
that uses that function as its model.
Source: katm.org
16. Common Misconceptions:
A-CED.1 cont’d
Students may interchange slope and y-intercept when
creating equations.
Given a graph of a line, students use the x-intercept for b
instead of the y-intercept.
Given a graph, students incorrectly compute slope as run
over rise rather than rise over run.
Students do not correctly identify whether a situation
should be represented by a linear, quadratic, or
exponential function.
Students often do not understand what the variables
represent.
Source: katm.org
18. Reasoning with Equations
and Inequalities A-REI
Understand solving equations as a process of reasoning and explain
the reasoning
2. Solve simple rational and radical equations in one variable, and
give examples showing how extraneous solutions may arise.
Represent and solve equations and inequalities graphically
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.
19. Reasoning with Equations and
Inequalities A-REI
CCSS WA PE
A-REI.2 Understand solving equations as a process of
reasoning and explain the reasoning. Solve simple
rational and radical equations in one variable, and give
examples showing how extraneous solutions may
arise.
WA.9-12.M3.6 Core Content: Algebraic properties:
Students continue to use variables and expressions to
solve both purely mathematical and applied
problems, and they broaden their understanding of
the real number system to include complex numbers.
Students extend their use of algebraic techniques to
include manipulations of expressions with rational
exponents, operations on polynomials and rational
expressions, and solving equations involving rational
and radical expressions.
A-REI.11 Represent and solve equations and
inequalities graphically. 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.
Partial Match: The WA does not account for
inequalities.
WA.9-12.A1.5.B Sketch the graph of a quadratic
function, describe the effects that changes in the
parameters have on the graph, and interpret the x-
intercepts as solutions to a quadratic equation.
20. Common Misconceptions:
A-REI.2
Students may believe that solving an equation such
as 3x + 1 = 7 involves “only removing the 1,” failing to
realize that the equation 1 = 1 is being subtracted to
produce the next step.
Additionally, students may believe that all solutions
to radical and rational equations are viable, without
recognizing that there are times when extraneous
solutions are generated and have to be eliminated.
Source: katm.org
21. Common Misconceptions:
A-REI.11
Students may believe that the graph of a function is
simply a line or curve “connecting the dots,” without
recognizing that the graph represents all solutions to
the equation.
Students may also believe that graphing linear and
other functions is an isolated skill, not realizing that
multiple graphs can be drawn to solve equations
involving those functions.
Additionally, students may believe that two-variable
inequalities have no application in the real world.
Source: katm.org
22. A-REI.2 & 11 Examples
IXL:
http://www.ixl.com/math/algebra-
1/solve-radical-equations
Common Core support video:
http://www.youtube.com/watch?fe
ature=player_embedded&v=c2OVQ
k-FiNs
10/4/2014
Algebra 1 > FF.3 Solve radical equations I
MATH LANGUAGE ARTS REPORTS AWAR
Solve for s.
1
2
= s
Write your answer as a fraction.
s =
Submit
Search topics and skills
23. Interpreting Functions
F-IF
Analyze functions using different representations
7. Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using
technology for more complicated cases.
d. (+) Graph rational functions, identifying zeros and
asymptotes when suitable factorizations are available,
and showing end behavior.
24. Interpreting Functions F-IF
CCSS WA PE
F-IF.7d (+) Graph rational functions,
identifying zeros and asymptotes when
suitable factorizations are available, and
showing end behavior.
No WA PE Match
25. Common Misconceptions:
F-IF.7d
Students may believe that each family of functions
(e.g., quadratic, square root, etc.) is independent of
the others, so they may not recognize commonalities
among all functions and their graphs.
Additionally, student may believe that the process of
rewriting equations into various forms is simply an
algebra symbol manipulation exercise, rather than
serving a purpose of allowing different features of
the function to be exhibited.
Source: katm.org
27. Resources
Alignment Analysis: Common Core and Washington State
Mathematics Standards (Hanover):
https://www.k12.wa.us/CoreStandards/pubdocs/HanoverAlign
mentMathematicsStandards.pdf
Alignment Analysis: Common Core and Washington State
Mathematics Standards (WA educators):
https://www.k12.wa.us/CoreStandards/pubdocs/WAAlignmen
tDocumentmathematics.pdf
CCSS Math: http://ccssmath.org/?page_id=2018
References multiple websites for lesson plans, etc.
Kent State University and Kansas Association of Teachers of
Mathematics (katm.org): flip books with lesson plans and
misconceptions
28. More Resources
Illustrative Mathematics (HS Standards):
https://www.illustrativemathematics.org/standards/hs
Khan Academy: https://www.khanacademy.org/
Hotmath.com: practice problems/answers online
Shmoop.com: worksheets/answers for classroom use
LearnZillion.com: online lessons
ixl.com: online practice (submit answers for step-by-step
explanation)
Shodor.org: online grapher and activities
29. Even More Resources
Desmos calculator:
https://www.desmos.com/calculator
NCTM > Illuminations >Interactives:
http://illuminations.nctm.org/
National Library of Virtual Manipulatives:
http://nlvm.usu.edu/
Editor's Notes
I learned a couple of years ago about Desmos and their incredibly useful online graphing calculator. It is worth a look, I think.
( https://www.desmos.com/calculator )
