Algebra 1, Quarter 4, Unit 4.1
Polynomial Operations and Factoring
Overview
Number of instructional days:
15
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Identify terms, coefficients, and degree of
polynomials.
Attend to precision.
•
Add and subtract polynomials.
•
Multiply polynomials (monomials, binomials,
trinomials including special cases) using the
Distributive Property and FOIL method.
•
Factor polynomials including trinomials of the
form f (x) = ax 2 + bx + c, for a = 1, a ≠ 1 ,
greatest common factors, perfect square
trinomials, and difference of squares.
•
•
Classify polynomials based on the number of
terms.
•
Make explicit use of degree of polynomials to
add, subtract, multiply, and factor polynomials.
Look for and make sense of structure.
Factor four-term polynomials by grouping
(e.g., 3x 3 − 12x 2 + 2x − 8 ).
•
Factor trinomials by looking for and using the
structure of the trinomial, considering
parameters a, b, and c.
•
Look for structure to factor four terms by
grouping.
Look for and express regularity in repeated
reasoning.
•
Use repeated reasoning to multiply binomials
using the Distributive Property and the FOIL
method.
•
Use shortcuts for determining the square of a
binomial and for multiplying to get a
difference of squares.
•
Use repeated reasoning to factor polynomials.
•
Use shortcuts for factoring perfect square
trinomials and differences of squares.
Essential questions
•
How are the operations and properties of real
numbers related to polynomials?
•
What characteristics of a polynomial determine
how to factor it completely?
•
How can two algebraic expressions that appear
to be different be equivalent?
•
What are the special cases and patterns used to
factor polynomials?
•
How is the factoring of polynomials related to
the multiplication of polynomials?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 37 Algebra 1, Quarter 4, Unit 4.1
Polynomial Operations and Factoring (15 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Arithmetic with Polynomials and Rational Expressions
A-APR
Perform arithmetic operations on polynomials [Linear and quadratic]
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.
Seeing Structure in Expressions
A-SSE
Interpret the structure of expressions [Linear, exponential, quadratic]
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)n as the product of P and a factor not depending on P.
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).
Common Core Standards for Mathematical Practice
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 38 Algebra 1, Quarter 4, Unit 4.1
8
Polynomial Operations and Factoring (15 days)
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them
to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically
proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In grade 6, students applied the properties of operations to generate equivalent expressions. For example,
students applied the Distributive Property to the expression 3(2 + x) to produce 6 + 3x. They also applied
the Distributive Property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y).
(6.EE.3) In grade 7, students applied properties of operations to add, subtract, factor, and expand linear
expressions with rational coefficients. (7.EE.1) In grade 8, students applied the properties of integers
(8.EE.1), and in Unit 2.3 of this course, they applied properties of exponents to rational exponents.
Current Learning
Students identify the degree, terms, and coefficients of a polynomial. They classify the polynomials based
on the number of terms and degree. Students add and subtract polynomials. Students multiply
polynomials using the distributive method, and they also square binomials. Students factor polynomials
including trinomials of the form f (x) = ax 2 + bx + c, for a = 1, a ≠ 1 , greatest common factor, perfectsquare trinomials, and difference of squares. They also factor four-term polynomials by grouping.
Future Learning
Students will use factoring in the next unit when they solve quadratic equations. They will use operations
of polynomials and factoring in this course and in later courses to factor higher degree polynomials and to
express functions in various forms such as vertex form of a quadratic and standard form of a circle,
parabola, and other conic sections. Operations of polynomials and factoring are necessary skills that will
be needed for Algebra II, Geometry, Precalculus, and Calculus.
Additional Findings
According to Algebra of Polynomials (Lausch & Nöbauer, 1974), “Polynomials are a classical subject of
mathematics. The first steps towards the abstract concept of polynomials were the investigation of
algebraic equations and the theory of real and complex functions f of the form f(x) = anxn +…+a1x + a0 .”
(p. ix)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 39 Algebra 1, Quarter 4, Unit 4.1
Polynomial Operations and Factoring (15 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 40 Algebra 1, Quarter 4, Unit 4.2
Quadratic Functions and Equations
Overview
Number of instructional days:
12
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Solve quadratic equations by factoring.
Model with mathematics.
•
Graph quadratic functions using x- and
y-intercepts and the axis of symmetry.
•
•
Transform the graph of f(x) = x2, including
translating, stretching, shrinking, and
reflecting.
Identify key features of graphs and their
relationship to the real-world situation they
model.
•
Choose which form of a quadratic equation to
use when solving and interpreting different
problems.
•
Solve quadratic equations by completing the
square.
•
Graph quadratic functions using vertex form.
•
Understand that the quadratic formula is
derived from completing the square.
•
Apply the quadratic formula and give solutions
in simplified, radical form and as approximate
values.
•
Model with quadratic functions, interpret key
features (intercepts, relative maximums and
minimums, symmetries, end behavior) and
sketch graphs given a verbal description of the
relationship.
•
Interpret the domain of a quadratic function in
context of applications.
•
Use graphing technology to explore and model
quadratic relationships in real-world problem
solving.
•
Solve a simple system consisting of a linear
equation and a quadratic equation algebraically
and graphically.
Use appropriate tools strategically.
•
Choose appropriate strategies according to
task.
•
Use graphing technology to explore and model
quadratic relationships in real-world problem
solving.
Look for and make use of structure.
•
Identify and examine algebraic expressions as
single entities to evaluate characteristics of
quadratic functions.
•
Understand why the quadratic formula works
based on completing the square.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 41 Algebra 1, Quarter 4, Unit 4.2
Quadratic Functions and Equations (12 days)
Essential questions
•
What are the advantages of writing a quadratic
equation in vertex form?
•
How do you solve quadratic equations using
different methods?
•
What are the effects of a, h, and k on the graph
of y = a(x – h)2 + k?
•
What types of real-world situations can be
modeled using quadratic equations?
•
What are the different methods to solve
quadratic equations? When might one method
be more beneficial to use than another?
•
What are the characteristics of a quadratic
function?
•
What are the key features of the graph of a
quadratic function?
Written Curriculum
Common Core State Standards for Mathematical Content
Seeing Structure in Expressions
A-SSE
Write expressions in equivalent forms to solve problems [Quadratic and exponential]
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.
Reasoning with Equations and Inequalities
A-REI
Solve equations and inequalities in one variable [Linear inequalities; literal that are linear in the
variables being solved for; quadratics with real solutions]
A-REI.4
Solve quadratic equations in one variable.
a.
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.
b.
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.
Solve systems of equations [Linear-linear and linear-quadratic]
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.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 42 Algebra 1, Quarter 4, Unit 4.2
Quadratic Functions and Equations (12 days)
Interpreting Functions
F-IF
Interpret functions that arise in applications in terms of the context [Linear, exponential, and
quadratic]
F-IF.4
For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums;
★
symmetries; end behavior; and periodicity.
F-IF.5
Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives the number of person-hours it
takes to assemble n engines in a factory, then the positive integers would be an appropriate
★
domain for the function.
Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step,
piecewise-defined]
F-IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple
★
cases and using technology for more complicated cases.
a.
F-IF.8
Graph linear and quadratic functions and show intercepts, maxima, and minima.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
a.
Use the process of factoring and completing the square in a quadratic function to show
zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
Building Functions
F-BF
Build new functions from existing functions [Linear, exponential, quadratic, and absolute value; for
F.BF.4a, linear only]
F-BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
Common Core Standards for Mathematical Practice
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 43 Algebra 1, Quarter 4, Unit 4.2
Quadratic Functions and Equations (12 days)
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
Students worked with radicals and integer exponents in grade 8. They applied the properties of integer
exponents to generate equivalent numerical expressions. (8.EE.1) Students evaluated square roots of
small perfect squares and cube roots of small perfect cubes. (8.EE.2)
In Unit 4.1, students used the Distributive Property and the FOIL method to multiply polynomials
(monomials, binomials, and trinomials including special cases). They factored polynomials including
trinomials of the form f(x) = ax2 + bx + c, for a = 1, a≠1, greatest common factors, perfect square
trinomials, and difference of squares.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 44 Algebra 1, Quarter 4, Unit 4.2
Quadratic Functions and Equations (12 days)
Current Learning
Students solve quadratic equations by factoring, completing the square, and the quadratic formula. They
graph quadratic functions by determining the x- and y-intercepts and the axis of symmetry. By completing
the square, students graph quadratic functions using vertex form. They also explore and model quadratic
relations in real-world situations using graphing technology. Students transform the graph of f(x) = x2 by
translating, stretching, shrinking, and reflecting. They also solve simple systems consisting of a linear
equation and a quadratic equation.
Future Learning
Quadratics have many applications to problems in physics and engineering that students will encounter in
future math courses and careers. The study of quadratics prepares students for working with higher order
polynomials, the Fundamental Theorem of Algebra, and the Rational Root Theorem.
Additional Findings
In relation to quadratics, John Allen Paulos wrote in Beyond Numeracy, “Many situations in physics,
engineering, and elsewhere lead to such equations.” (p. 198)
Relative to using graphical representations to solve equations, A Research Companion to Principals and
Standards for School Mathematics states, “One cannot simply expect students to be able to read these
representations in the ways they are intended. The process of learning to read such representations is
complex and requires teaching and learning.” (p. 131)
PARCC Model Content Frameworks for Mathematics notes that “fluency in transforming expressions and
chunking (seeing parts of an expression as a single object) is essential in factoring, completing the square
and other mindful calculations.” (p. 52)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 45 Algebra 1, Quarter 4, Unit 4.2
Quadratic Functions and Equations (12 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 46 Algebra 1, Quarter 4, Unit 4.3
Operations with Radicals
Overview
Number of instructional days:
12
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Write an expression with a rational exponent in
radical form.
Attend to precision.
•
Write a radical in exponential form.
•
Simplify expressions involving rational
exponents.
•
Simplify radical expressions (square roots).
(not in the CCSS)
•
Add, subtract, and multiply radical monomials,
expressing the solutions in simplified form
(square roots). (not in the CCSS)
•
•
State the meaning of the radical symbol and
interpret it in terms of rational exponents.
Look for and make use of structure.
•
Review the properties of exponents to find
structure in examples and apply the structure
to simplifying rational expressions with
exponents.
•
Understand that the set of irrational numbers
is closed under addition, but not under
multiplication.
Investigate the products and sums of two
rational numbers, two irrational numbers, and a
rational and irrational number (Closure
Property).
Essential questions
•
What type of number(s) results from the sum of
a rational number and an irrational number?
•
How do you use rational exponents to represent
radicals?
•
What type of number(s) results from the
product of a nonzero rational and an irrational
number?
•
How do you know when a radical expression is
in simplest form?
•
How do you know when an expression is in
simplified rational exponent form?
•
What type of number(s) results from the sum of
two irrational numbers?
•
What type of number(s) results from the
product of two irrational numbers?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 47 Algebra 1, Quarter 4, Unit 4.3
Operations with Radicals (12 days)
Written Curriculum
Common Core State Standards for Mathematical Content
The Real Number System
N-RN
Extend the properties of exponents to rational exponents.
N-RN.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. For example, we define 51/3 to be the cube root of 5 because we want
(51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
N-RN.2
Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
Use properties of rational and irrational numbers.
N-RN.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.
Common Core Standards for Mathematical Practice
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 48 Algebra 1, Quarter 4, Unit 4.3
Operations with Radicals (12 days)
Clarifying the Standards
Prior Learning
Students worked with radicals and integer exponents in grade 8. They applied the properties of integer
exponents to generate equivalent numerical expressions. (8.EE.1) Students evaluated square roots of
small perfect squares and cube roots of small perfect cubes. (8.EE.2)
In Unit 4.1, students used the Distributive Property and the FOIL method to multiply polynomials
(monomials, binomials, and trinomials including special cases). In Unit 4.2, students used the quadratic
formula to simplify expressions with radicals.
Current Learning
Students extend their knowledge of exponents to include rational exponents. (N-RN.2) They rewrite
expressions involving radicals and rational exponents using the properties of exponents. Students add,
subtract, and multiply radical expressions and realize that the set of irrational numbers is closed under
addition but not multiplication. (N-RN.3)
Future Learning
In Geometry, students will simplify radicals, work with trigonometric ratios, and solve special right
triangles. (G-SRT.8) In Algebra II and advanced algebra courses, students will connect the closure
properties of irrational numbers to operations of complex number solutions for polynomials.
(N-CN.3, 7, 8)
Additional Findings
Principles and Standards for School Mathematics notes that “high school algebra should provide students
with insights into mathematical abstraction and structure. In grades 9–12, students should develop an
understanding of algebraic properties that govern the manipulation of symbols in expressions, equations,
and inequalities.” It continues by adding that students “should become fluent in performing such
manipulations by appropriate means—mentally, by hand, or by machine—to solve equations and
inequalities, to generate equivalent forms of expressions or functions, or to prove general results.”
(p. 297)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 49 Algebra 1, Quarter 4, Unit 4.3
Operations with Radicals (12 days)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 50