Download Unit D: Linear Functions and Polynomials

Liberal Arts 1 | Unit D – Linear Functions and Polynomials
Unit Overview
Math Florida Standards
Content Standards
Major Focus: Students will understand that the graph of an equation in two variables represents the set of all its solutions.
Students will also understand the connection between two-variable notation and function notation, and will be able to
identify the domain, range, and intercepts of any given function, and the slope (rate of change) of a linear function.
Tasks:
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Students will create equations to represent the relationship between two quantities.
Students will interpret the rate of change of a function and estimate the rate of change from a graph.
Students will also apply the concepts of the domain and range of a relation and of a function, and use function
notation.
Students will perform basic operations involving polynomial expressions including adding, subtracting, and
multiplying.
Students will also use functions to represent real-world situations and solve real-world problems
Textbook Resources
Math Matters 3: An Integrated Program. (2008).
McGraw-Hill.
Sections: 2-2, 2-3, 2-6, 11-1, 11-2, 11-4
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers can
implement with their students, and rubrics that help the
teacher interpret students' responses. Teachers using
MFAS ask students to perform mathematical tasks,
explain their reasoning, and justify their solutions. Rubrics
for interpreting and evaluating student responses are
included so that teachers can differentiate instruction
based on students' strategies instead of relying solely on
correct or incorrect answers. The objective is to
understand student thinking so that teaching can be
adapted to improve student achievement of
mathematical goals related to the standards. Like all
formative assessment, MFAS is a process rather than a
test. Research suggests that well-designed and
implemented formative assessment is an effective
strategy for enhancing student learning.
http://www.cpalms.org/resource/mfas.aspx
This a working document that will continue to be revised and improved taking your feedback into consideration.
MAFS.912.A-APR.1.1
MAFS.912.F-IF.1.1
MAFS.912.F-IF.1.2
MAFS.912.F-IF.2.4
MAFS.912.F-IF.2.5
MAFS.912.F-IF.2.6
Standards for
Mathematical Practice
MAFS.K12.MP.1.1
MAFS.K12.MP.2.1
MAFS.K12.MP.3.1
MAFS.K12.MP.4.1
MAFS.K12.MP.5.1
MAFS.K12.MP.6.1
MAFS.K12.MP.7.1
*Language Arts standards
should be embedded into
instructional practices.
Other Resources
Mathematics Assessment Resource Service
Algebra Nation
Online Graphing Calculator
National Library of Virtual Manipulatives
Geogebra
Virtual Nerd
YouTube
Khan Academy—Math
Engage NY
TI Nspired Resource Center for Educators
Pearson SuccessNet
Pasco County Schools, 2014-2015
Liberal Arts 1 | Unit D – Linear Functions and Polynomials
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:
 On a coordinate plane graph three nonlinear points labeled A, B, & C. Draw segment AB. Create line L parallel to segment AB and line M perpendicular to
segment AB both intersecting at point C. Write the equation of lines L & M and state why they are parallel and perpendicular respectively to segment AB.
 Research and collect data on a topic of interest that a student thinks will be linear. Then, plot the data and find a line of best fit. Use the correlation
coefficient to discuss the trend of the data. Use the line of best fit to predict several outcomes and discuss any limitations on the domain of the line.
3.0
2.0
1.0
The Student will:
 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. (MAFS.912.A-APR.1.1)
 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph
of the equation y = f(x). (MAFS.912.F-IF.1.1)
 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
(MAFS.912.F-IF.1.2)
 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 . (MAFS.912.F-IF.2.4)
 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (MAFS.912.F-IF.2.5)
 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph. (MAFS.912.F-IF.2.6)
The student will recognize or recall specific vocabulary, such as:
 Slope, rate of change, linear equation in two variables, function, domain (input), range (output), coordinate plane, parallel, perpendicular, scatter plot
The student will perform basic processes, such as:
 Create a table of values to represent a function.
 Create an equation when given the slope and y-intercept.
 Define and understand the components of a graph.
 Understand the similarities between an equation and a function.
 Find the slope of a line that passes through two points.
 Identify the slope and intercept of a linear model.
 Identify whether two lines are parallel, perpendicular, or neither given their slopes.
With help, partial success at 2.0 content but not at score 3.0 content
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Liberal Arts 1 | Unit D – Linear Functions and Polynomials
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Functions: Interpreting Functions
Cluster: Understand the concept of a function and use function notation. (Major)
Standard: MAFS.912.F-IF.1.1: (Understand) that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly
one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
equation y = f(x).
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Understand that a function from one set to another set assigns to each element of the domain exactly one element of the range, where x is an element of its domain, and f(x)
denotes the output of f corresponding to the input x, and can be represented by the graph y = f(x).
Know
Declarative knowledge: Facts, vocab., information
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Vocabulary: function, domain (input), range
(output), element (of the domain and range), y =
f(x) (function notation)
Understand if f is a function and x is an element of its domain, and f(x) denotes the output of f corresponding to the input
x.
Understand that the graph of f is the graph of the equation y = f(x).
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of
the domain exactly one element of the range.
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Identify domain and range, mapping, graphing an equation
Learning Goals:
Students will understand if f is a function and x is an element of its domain, and f(x) denotes the output of f corresponding to the input x.
Students will understand that the graph of f is the graph of the equation y = f(x).
Students will understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the
range.
Moving Beyond:
This work of understanding the basics of functions will be used in Algebra II when function families are explored.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Liberal Arts 1 | Unit D – Linear Functions and Polynomials
Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
MAFS.912.F-IF.1.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of
the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Learning Progression
Sample Tasks
I can…
 Evaluate a set of data in terms of a function—identify the domain and
range, verify that each element in the domain is assigned to only one
element in the range, and the graph of the data can be represented
by y = f(x).
1. Find two examples of data by researching a written or online newspaper.
One set of data should be a relation that is also a function, and one set should
be a relation that is not a function. For each data set:
a. List the elements of the domain and range and describe the domain and
range in terms of its context.
b. Graph the data on a scatterplot. Make sure to attend to precision on
axes, labels, and units.
c. Classify the data as being a function or not a function and interpret your
classification in terms of the context of the data.
Score
4.0
3.5
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
I can…
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3.0
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
2.5
2.0
1.0
Understand that a function from one set to another set assigns to
each element of the domain exactly one element of the range.
Understand that if f is a function and x is an element of its domain,
then f(x) denotes the output of f corresponding to the input x.
Understand that the graph of f is the graph of the equation y = f(x).
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 Identify domain and range in a relation.
 Identify whether a relation is a function by inspecting graphs, maps,
or sets of points.
I need prompting and/or support to complete 2.0 tasks.
1. State the domain and range of the relation {(2, 1), (4, 3), (-3, 2), (4, 5)}.
Represent the relation graphically and by mapping the elements. Determine
whether the relation is a function.
2. a. Graph y = x2 – 4x.
b. Determine if the graph represents a function.
c. If it is a function, rewrite the equation using function notation.
1. Given a graph of a curve or points, determine whether the relation is a
function.
2. Given a set of ordered pairs, state the domain and range.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Liberal Arts 1 | Unit D – Linear Functions and Polynomials
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS.912.F-IF.1.1: Understand that a function from one set (called the domain) to another set (called the range) assigns
MAFS Mathematical Content Standard(s)
to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then
Design Question 1; Element 1
f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
MAFS.K12.MP.2.1: Reason abstractly and quantitatively.
MAFS Mathematical Practice(s)
MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.
Design Question 1; Element 1
MAFS.K12.MP.6.1: Attend to precision.
Marzano’s Taxonomy
Level 3—Analysis: “Matching/Comparative Analysis”
Questions:
In “f(0)”, where is the x?
Can you defend your answer with mathematical reasoning?
Can you find a point that would prove the statement incorrect?
Misconceptions:
Although the axes are not labeled, there are specific values that can be found given the information.
Teacher Notes
The inequality symbol is not being used here as an area problem; the inequality is just used to compare both functions.
Differentiation:
Have extending students should create a new function and write 4 new statements that are true based on their new function.
Struggling students should get axes that have hash marks.
Task
Design Question 3; Element 17
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This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015