hs ccss math rational expressions
TRANSCRIPT
Welcome to High School
Math!Building on Rational Numbers
and Fractions Maria GrossTacoma Community
u
October 2014Professional Development
Progression
Comparisons with WA Performance Expectations
Content examples MANY potential resourcesccssmath.org is a great clearinghouse
Common Misconceptions
FractionsRatios and
Proportional Relationships
Rational Numbers Rational Expressions
“Rational” in CCSS – High SchoolNumber and Quantity
The Real Number System (N-RN.1, 2 & 3)
AlgebraArithmetic with Polynomials and Rational Expressions (A-APR)Creating Expressions (A-CED)Reasoning with Equations and Inequalities (A-REI)
Functions Interpreting Functions (F-IF.7d)
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.
The Real Number System N-RN
CCSS WA PEN.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.
Common Misconceptions:
N-RN.1 & 2Students 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
Common Misconceptions:
N-RN.3Some 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
N-RN.1, 2 & 3 Examples
Khan Academy: Level 3 Exponents Videohttp://www.youtube.com/watch?feature=player_embedded&v=aYE26a5E1iU
Hotmath Practice Problems
“Rewrite using rational exponent notation”
Google PDF practice problems http://educ.jmu.edu/~taalmala/235_2000post/235contradiction.pdf
“Given that: r is a rational number, and x is an irrational number. Show that: r + x is irrational.”
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.
Arithmetic with Polynomials and Rational Expressions A-
APR CCSS WA PEA-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
Common Misconceptions: A-
APR.6Students 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
A-APR.6 & 7 Examples
Shmoop.com Worksheet:
http://www.shmoop.com/common-core-standards/handouts/a-arp_worksheet_6.pdf
with answers: http://www.shmoop.com/common-core-standards/handouts/a-arp_worksheet_6_ans.pdf
Learn Zillion online lesson:
https://learnzillion.com/lessons/1332-multiply-rational-expressions
Creating Equations
A-CEDCreate 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.
Creating Equations A-CEDCCSS 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 MatchWA.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.
Common Misconceptions: A-
CED.1Students 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
Common Misconceptions: A-
CED.1 cont’dStudents 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
A-CED.1 Example
Illustrative Mathematics Project - Basketball: https://www.illustrativemathematics.org/illustrations/702
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.
Reasoning with Equations and
Inequalities A-REI CCSS WA PEA-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.
Common Misconceptions: A-
REI.2Students 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
Common Misconceptions: A-
REI.11Students 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
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?feature=player_embedded&v=c2OVQk-FiNs
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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.
Interpreting Functions F-IF
CCSS WA PEF-IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
No WA PE Match
Common Misconceptions: F-
IF.7dStudents 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
F-IF.7d Example
Shodor.org
ResourcesAlignment Analysis: Common Core and Washington State Mathematics Standards (Hanover): https://www.k12.wa.us/CoreStandards/pubdocs/HanoverAlignmentMathematicsStandards.pdf
Alignment Analysis: Common Core and Washington State Mathematics Standards (WA educators): https://www.k12.wa.us/CoreStandards/pubdocs/WAAlignmentDocumentmathematics.pdf
CCSS Math: http://ccssmath.org/?page_id=2018References 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
More ResourcesIllustrative 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
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/