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MATH 1
UNIT 5
Algebra In Context
Georgia Performance Standards
High School Mathematics
Mathematics 1
Georgia Performance Standards: Curriculum Map
1st Semester
Unit 1
Unit 2
Function
Families
Algebra
Investigations
4 ½ Weeks
5 Weeks
MM1A1a, b, c,
d, e, f, g
MM1G2
MM1A2
MM1A3a
2nd Semester
Unit 3
Unit 4
Unit 5
Unit 6
The
Geometry
Chance of
Gallery
Winning
Algebra
in
Context
Coordinate
Geometry
7 Weeks
5 Weeks
6 Weeks
3 Weeks
MM1G3
MM1D1
Mm1D2
MM1D3
MM1D4
MM1A1c, d,
h, i
MM1A3
MM1G1
These units were written to build upon concepts from prior units, so later units contain tasks that depend
upon the concepts addressed in earlier units. Standards listed are key standards for the units. All units
will include the Process Standards.
Math 1 Unit 5 Algebraic Investigations Intro & Standards Page 2
MATH 1 UNIT 5 ALGEBRA IN CONTEXT
CONTENT MAP
Unit 5 – Algebra in Context (6 Weeks)
Essential Questions: How do you analyze and graph quadratic,
radical, and rational functions? How do you solve quadratic,
radical, and rational equations?
Lesson 1 –Quadratic Functions and Equations (9 Hours)
Essential Question: How do you analyze and graph quadratic functions? How do you solve
quadratic equations?
Lesson 2 – Radical Functions and Equations (9 Hours)
Essential Question: How do you analyze and graph radical functions? How do you solve
radical equations?
Lesson 3 – Rational Functions and Equations (9 Hours)
Essential Question: How do you analyze and graph rational functions? How do you solve
rational equations?
Summarizer & Evaluation of Unit 1 (3 Hours)
Math 1 Unit 5 Algebraic Investigations Intro & Standards Page 3
Mathematics I – Unit 5: Algebra in Context
INTRODUCTION:
In Units 1 of Mathematics 1 students learn properties of basic quadratic, cubic, absolute value,
square root, and rational functions. In Unit 2 students develop skill in adding, subtracting,
multiplying, and dividing elementary polynomial, rational, and radical expressions. In this unit
students extend the skills and understandings of Units 1 and 2 through further investigation of
quadratic, rational, and radical functions. The even and odd symmetry of graphs will be
explored, as well as intersections of graphs as solutions to equations. The focus is the
development of students’ abilities to solve simple quadratic, rational, and radical equations
using a variety of methods.
ENDURING UNDERSTANDINGS:
1. There is an important distinction between solving and equation and solving an applied
problem modeled by an equation. The situation that gave rise to the equation may
include restrictions on the solution to the applied problem that eliminate certain
solutions to the equation.
2. The definitions of even and odd symmetry for functions are stated as algebraic
conditions on values of functions but each symmetry has a geometric interpretation
related to reflection of the graph of through one or more of the coordinate axes.
3. For any graph, rotational symmetry of 180 degrees about the origin is the same as
point symmetry of reflection through the origin.
4. Techniques for solving rational equations include steps that may introduce extraneous
solutions that do not solve the original rational equation and, hence, require an extra
step of eliminating extraneous solutions.
5. Understand that any equation in can be interpreted as a statement that the values of
two functions are equal, and interpret the solutions of the equation domain values for
the points of intersection of the graphs of the two functions. In particular, solutions of
equations of the form f(x) = 0, where f(x) is an algebraic expression in the variable x,
correspond to the x-intercepts of the graph of the equation y = f(x).
KEY STANDARDS ADDRESSED:
MM1A1. Students will explore and interpret the characteristics of functions, using
graphs, tables, and simple algebraic techniques.
c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks,
as well as reflections across the x- and y-axes.
d. Investigate and explain characteristics of a function: domain, range, zeros, intercepts,
intervals of increase and decrease, maximum and minimum values, and end behavior.
h. Determine graphically and algebraically whether a function has symmetry and whether it
is even, odd or neither.
i. Understand that any equation in x can be interpreted as the equation f(x) = g(x), and
interpret the solutions of the equation as the x-value(s) of the intersection point(s) of the
graphs of y = f(x) and y = g(x).
Math 1 Unit 5 Algebraic Investigations Intro & Standards Page 4
MM1A2. Students will simplify and operate with radical expressions, polynomials, and
rational expressions.
a. Simplify algebraic and numeric expressions involving square root.
b. Perform operations with square roots.
c. Add, subtract, multiply, and divide polynomials.
d. Add, subtract, multiply, and divide rational expressions.
e. Factor expressions by greatest common factor, grouping, trial and error, and special
products limited to the formulas below.
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
(x + y)(x - y) = x2- y2
(x + a)(x + b) = x2 + (a + b)x + ab
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x - y)3 = x3 - 3x2y + 3xy2 – y3
MM1A3. Students will solve simple equations.
a. Solve quadratic equations in the form ax2 + bx + c = 0, where a = 1, by using factorization
and finding square roots where applicable.
b. Solve equations involving radicals such as, using algebraic techniques. x  b  0.
c. Use a variety of techniques, including technology, tables, and graphs to solve equations
resulting from the investigation of .x2 + bx + c = 0.
d. Solve simple rational equations that result in linear equations or quadratic equations with
leading coefficient of 1.
RELATED STANDARDS ADDRESSED:
MM1P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
MM1P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
MM1P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers,
and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
Math 1 Unit 5 Algebraic Investigations Intro & Standards Page 5
MM1P4. Students will make connections among mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to
produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
MM1P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical
ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical
phenomena.
UNIT OVERVIEW:
Prior to this unit, students need to have worked extensively with operations on integers,
rational numbers, and square roots of nonnegative integers as indicated in the grade 6 – 8
standards for Number and Operations.
In the unit students will apply and extend all of standards for algebra listed as key standards
addressed in Units 1 and 2 and all of the supporting standards from Grades 6 – 8 referenced in
the overviews for Units 1 and 2. In working with symmetry of graphs, students will apply the
concepts of similarity and transformations of geometric figures inherent in the Grade 7
standards for geometry.
The unit begins with applied problems that can be modeled by quadratic equations. The need
to solve such equations provides motivation for the topic of solving quadratic equations by
factoring and, hence, for learning to factor. As indicated by the standards for this unit, the
quadratic equations to be solved are limited to those which are equivalent to equations of the
form x2 + bx + c = 0. Students see application of adding, subtracting, and multiplying
polynomials as they take a variety of quadratic equations and put them in this standard form. In
work with solving quadratic equations by factoring, it is intended that students learn to factor
expressions of the form x2 + bx + c by applying the special product
(x + a)(x + b) = x2 + (a + b)x + ab. This method gives a strong foundation for learning to factor
other trinomials by the grouping method, a method that allows students who do not yet grasp
the “big picture” of factoring to be successful in factoring trinomials that would have a very
large number of cases if approached as factoring by trial and error.
The beginning task of the unit also introduces the concept of viewing solutions of equations as
first coordinates of points of intersection of the graphs of appropriate functions. This concept is
revisited throughout the unit so that students gain understanding of the variety of ways in
which the concept can be applied. The unit introduces techniques for solving simple radical
equations and motivates the need to solve such equations in application that requires finding
intersection points of graphs. Thus, students see the task of finding the intersection points of
graphs from two perspectives: (1) as an important question that often requires algebraic
techniques for exact solution and (2) as a method for interpreting and verifying algebraic
solutions of equations. The treatment of algebraic techniques for radical equations is limited to
Math 1 Unit 5 Algebraic Investigations Intro & Standards Page 6
equations of the form. x  b  0. Solving such equations does introduce the basic techniques
for solving radical equations. Solution of more advanced equations involving radicals is
addressed Mathematics III. (See standard MM3A3.)
The unit also includes the techniques for solving rational equations. The need for solving such
equations is introduced by a topic from physical science, the concept of resistance in an
electrical circuit, but other applications are included. The presentation focuses on the
techniques for solving rational equations and reinforces the topic of solving quadratic
equations since rational equations that lead to both linear and quadratic equations are
included. For this unit, the denominators that occur in the expressions are limited to rational
numbers and first degree polynomials. Solution of rational equations involving higher degree
denominators is addressed in Mathematics IV. (See standard MM4A1.)
The unit emphasizes the connections among the solution of equations and graphs of functions.
This approach gives students many opportunities to consolidate their understandings of the
topics from Units 1 and 2. Additionally, the unit includes an in-depth discussion of even and
odd symmetry of graphs of functions and transformations of graphs by reflection in the
coordinate axes. These topics and the discussion of solving rational equations and quadratic
equations of the form x2– c = 0, c ≥ 0, reinforce topics from geometry, especially the topics of
symmetry and transformation of geometric figures, similar triangles, and the Pythagorean
Theorem.
Students need extensive practice with multiple representations of the same mathematical
concept. This unit focuses in integration of algebraic and graphical viewpoints but includes
many opportunities to use verbal, tabular, algebraic, and geometric representations to
organize, record, and communicate mathematical ideas. Throughout the unit it is important to:
 Promote student use of multiple representations of concepts and require students to
explain how to translate information from one representation to another. Such activities
especially include requiring students to explain how their equations represent the
physical situation they are intended to model and how graphs represent algebraic
equalities.
 Regularly use graphing technology to explore graphs of functions and verify
calculations. While graphing functions by hand is necessary for developing
understanding of many situations, students can deepen their understanding of topics
through the use of graphing technology, sometimes by viewing a calculator, or
computer, drawn version of a graph that they have already drawn by hand. The unit
includes many situations where graphing by hand would so time consuming that it
would be a major distraction and hindrance to the focus of the activity. For example,
exploration of graphs with technology lead to student conjectures that must be verified
algebraically.
 Continue to be extremely careful in the use of language of functions as indicated in the
Unit 1 overview.
 In discussing solutions of equations, emphasize the concept of finding equivalent
equations, exceptions to this concept, and the concept of solutions set of an equation so
that students put particular techniques for solving quadratic, radical, or rational
equations in the context of the general theory of solving equations.
Math 1 Unit 5 Algebraic Investigations Intro & Standards Page 7
TASKS:
The remaining content of this framework consists of student tasks or activities presented in a
real-world context. Tasks 1 – 4 are designed to allow students to learn by active exploration of
the topic in a context and are denoted Learning Tasks. The first task is intended to launch the
unit. It introduces the two themes of the unit, techniques for solving non-linear equations and
connections among algebraic statements and graphs of functions. The second task explores
graph symmetry and odd and even functions. It also includes solving simple radical equations.
The other learning tasks focus on techniques for solving rational equations and quadratic
equations by finding square roots. The last task is designed to demonstrate the type of
assessment activities students should be comfortable with by the end of the unit. Thorough
Teacher’s Guides which provide solutions, discuss teaching strategy, and give additional
mathematical background are available to accompany each task.
RESOURCES NEEDED BY THE TEACHER FOR THE LESSONS IN THIS
UNIT:
Geometers Sketchpad, Elmo or Overhead Projector for each teacher, Classroom set of
Graphing Calculators, Classroom set of Algebra Tiles, Classroom set of small cubes such as
Algeblocks, Classroom set of individual marker boards (blank on one side and a grid on the
other) and markers for students to use, Linguini, Soda Straws, Classroom set of compasses,
Classroom set of protractors, Classroom set of rulers, Miras, Patty Paper, Coordinate Grids,
Colored Pencils, Rulers, Masking Tape, Markers, Roll of Graph Paper with Inch Squares, Pad
of Quad Paper, Glue Sticks, Scissors, Post-it Notes, Construction Paper, Poster Board, Copies
of all Handouts for Students, Copies of the Standards for Students, Large Copy of the
Standards to Post on the Wall
RESOURCES NEEDED BY THE STUDENTS FOR THE LESSONS IN THIS
UNIT:
Notebook with at least 10 dividers for the introduction, individual lessons, and culminating
activities, pencils, notebook paper, graph paper
Note: A copy of the standards for this unit should be given to the students with discussion to
be held throughout the unit concerning their meaning and relation to the learning tasks of the
day. Students will need individual copies of all handouts in the lessons of the unit. These
should be kept in a math notebook for ease in use.
Math 1 Unit 3 Geometry Intro and Standards Page 8
Student Learning Map for Math 1 Unit 5
Topic: Algebra in Context
Unit Enduring Understandings:
1. There is an important distinction between solving and equation and solving an applied problem
modeled by an equation. The situation that gave rise to the equation may include restrictions on
the solution to the applied problem that eliminate certain solutions to the equation.
2. The definitions of even and odd symmetry for functions are stated as algebraic conditions on
values of functions but each symmetry has a geometric interpretation related to reflection of the
graph of through one or more of the coordinate axes.
3. For any graph, rotational symmetry of 180 degrees about the origin is the same as point
symmetry of reflection through the origin.
4. Techniques for solving rational equations include steps that may introduce extraneous solutions
that do not solve the original rational equation and, hence, require an extra step of eliminating
extraneous solutions.
5. Understand that any equation in can be interpreted as a statement that the values of two
functions are equal, and interpret the solutions of the equation domain values for the points of
intersection of the graphs of the two functions. In particular, solutions of equations of the form
f(x) = 0, where f(x) is an algebraic expression in the variable x, correspond to the x-intercepts of
the graph of the equation y = f(x).
 Algebraic
equations
can be identities
that express
operations
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real numbers.
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1. Even Function
1. Extraneous Solutions
2. Odd Function
Notes:
Notes:
Unit Essential Questions:
How do you analyze and graph quadratic,
radical, and rational functions? How do
you solve quadratic, radical, and rational
equations?
Instructional Tools Needed for Unit 3:
1. Elmo or Overhead Projector
2. Graphing Calculators
3. Graph Paper
4. Mira
5. Patty Paper
6. Rulers
Concept 3: Rational Functions and Equations
Lesson Essential Questions
1. How do you analyze and graph rational
functions?
2. How do you solve rational equations?
Vocabulary
1. Restricted Domain
Notes: