Pre-Calculus Unit I: Introduction to Calculus
Unit Overview
Math Florida Standards
Content Standards
Precalculus students will not have previous experience with limits, but students will have previous experience with reading
and interpreting graphs, which will be incorporated into the ability to understand and find limits. In this unit students will:
estimate limits from graphs and tables of values; find limits by substitution; find limits of sums, differences, products, and
quotients; find limits of rational functions that are undefined at a point; find one-sided limits; decide if a function is
continuous at a point; find the types of discontinuities of a function; find relative/absolute maximum(s) and minimum(s); and
apply the Intermediate Value Theorem and Extreme Value Theorem.
Students must understand the concepts of limit and continuity. Students must also Understand the Intermediate Value
Theorem and Extreme Value Theorem.
Textbook Resources
Glencoe McGraw Hill Precalculus copyright 2011
Connect Ed McGraw Hill
Sections:
12.1, 12.2, 1.3, 1.4, *12.4 Extreme Value Theorem
*Use the other resources link for sample questions on the
Extreme Value Theorem.
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.
MAFS.912.C.1.1
MAFS.912.C.1.2
MAFS.912.C.1.3
MAFS.912.C.1.4
MAFS.912.C.1.5
MAFS.912.C.1.9
MAFS.912.C.1.10
MAFS.912.C.1.11
MAFS.912.C.1.12
MAFS.912.C.1.13
Standards for
Mathematical Practice
MAFS.K12.MP.1.1
MAFS.K12.MP.6.1
MAFS.K12.MP.7.1
Other Resources
Paul’s Online Math Notes
Sample Question on Extreme Value Theorem
Khan Academy Precalculus
Mathematics Formative Assessment System Tasks (MFAS)
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Pasco County Schools, 2014-2015
Pre-Calculus Unit I: Introduction to Calculus
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:


3.0
2.0
Apply the limit concept given an abstract concept.
Create a context that has a practical application of the Intermediate Value Theorem and/or the Extreme Value Theorem.
The Student will:
 Understand the concept of limit and estimate limits from graphs and tables of values. (MAFS.912.C.1.1)

Find limits by substitution. (MAFS.912.C.1.2)


Find limits of sums, differences, products, and quotients. (MAFS.912.C.1.3)
Find limits of rational functions that are undefined at a point. (MAFS.912.C.1.4)

Find one-sided limits. (MAFS.912.C.1.5)

Understand continuity in terms of limits. (MAFS.912.C.1.9)

Decide if a function is continuous at a point. (MAFS.912.C.1.10)

Find the types of discontinuities of a function. (MAFS.912.C.1.11)

Understand and use the Intermediate Value Theorem on a function over a closed interval. (MAFS.912.C.1.12)

Understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval.
(MAFS.912.C.1.13)
The student will recognize or recall specific vocabulary, such as:
 Limit, continuous, discontinuous, Intermediate Value Theorem, Extreme Value Theorem
The student will perform basic processes, such as:
 Factor polynomials
 Find zeros of a function
 Evaluate functions
1.0
With help, partial success at 2.0 content but not at score 3.0 content
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Pasco County Schools, 2014-2015
Pre-Calculus Unit I: Introduction to Calculus
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: Calculus
Cluster: Limits and Continuity
Standard: Find limits of sums, differences, products, and quotients. (MAFS.912.C.1.3)
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Limits can be found for sums, differences, products, and quotients using various methods.
Know
Declarative knowledge: Facts, vocab., information

Limit
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Comprehension
 Find limits of sums, differences, products, and quotients
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Closed/open interval, continuous, discontinuous, evaluate functions
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Pre-Calculus Unit I: Introduction to Calculus
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.
Common Core State Standard:
Score
Learning Progression
I can…
 Apply the limit concept given an abstract context.
Sample Tasks
The graphs of functions f and g are shown below. Evaluate each limit using the
graphs provided. Show the computations that lead to your answer.
a. lim ( f ( x ) + 4)
x®1
4.0
3.5
5
g ( x)
b.
lim
c.
lim ( f ( x ) × g ( x ))
d.
lim-
x®3
x®2
x®3
f ( x ) æ Assume that f and g are ö
ç
÷
g ( x ) -1 çè linear on the interval [ 2, 3]. ÷ø
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
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Pasco County Schools, 2014-2015
Pre-Calculus Unit I: Introduction to Calculus
I can…
 Find limits of sums, differences, products, and quotients
3.0
2.5
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 Evaluate functions
 Find a limit from a graph
 Factor a polynomial
2.0
1.0
Find each limit.
a.
lim 3 2x 2 + x -10
b.
lim ( x cos x )
c.
lim
x®3
x®p
x®1
x 3 -1
x -1
Given f ( x ) shown below. Evaluate:
a.
f ( 0)
b.
f (1)
c.
lim f ( x )
d.
lim f ( x )
x®1
x®-1
I need prompting and/or support to complete 2.0 tasks.
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Pasco County Schools, 2014-2015
Pre-Calculus Unit I: Introduction to Calculus
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 Mathematical Content Standard(s)
Decide if a function is continuous at a point. (MAFS.912.C.1.10)
Design Question 1; Element 1
MAFS Mathematical Practice(s)
Reason abstractly and quantitatively. (MAFS.K12.MP.2.1)
Design Question 1; Element 1
Marzano’s Taxonomy
Analysis
Teacher Notes
The intent of this problem is for students to understand the concept of continuity at a point and take limits of abstract
functions. Students might need probing questions about what it means for a piecewise function to be continuous.
Questions to develop mathematical thinking,
possible misconceptions/misunderstandings,
how to differentiate/scaffold instruction,
anticipate student problem solving strategies
Let a and b represent real numbers. Defined below is f ( x ) .
Task
*These tasks can either be teacher created or
modified from a resource to promote higher
order thinking skills. Please cite the source for
any tasks.
a.
Find the values of a and b such that f is continuous everywhere.
b.
Evaluate lim f ( x )
c.
Let g ( x ) =
ì ax 2 + x - b if x £ 2
ï
f ( x ) = í ax + b
if 2 < x < 5
ï2ax - 7
if x ³ 5
î
x®3
f ( x)
x -1
, evaluate lim g ( x ).
x®1
Calculus of a Single Variable (10e) Larson & Edwards
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015