first semester syllabus in pre calculus first quarter · grade 11: first semester syllabus in pre -...
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Grade 11: First Semester Syllabus in PRE - CALCULUS | 1
SENIOR HIGH SCHOOL
BASIC EDUCATION DEPARTMENT
Holy Angel University
Angeles City
FIRST SEMESTER SYLLABUS IN PRE – CALCULUS
FIRST QUARTER
CORE SUBJECT DESCRIPTION
The learner must be able to apply concepts and solve problems involving conic sections, systems
of nonlinear equations, series and mathematical induction, circular and trigonometric functions,
trigonometric identities, and polar coordinate system.
CONTENT STANDARD The learner demonstrates understanding of key concepts of problems involving conic sections,
systems of nonlinear equations, series and mathematical induction.
PERFORMANCE STANDARD
The learner is able to model situations appropriately and solve problems accurately using conic
sections and systems of nonlinear equations and; keenly observe and investigate patterns, and
formulate appropriate mathematical statements and prove them using mathematical induction
and/or Binomial Theorem.
TIME
FRAME TOPICS LEARNING COMPETENCIES ASSESSMENT
WEEK 1
Rectangular Coordinate System
o Distance Formula
o Midpoint Formula
o Slope
Values:
Socially Responsive Community
Leader. The concepts of the
Midpoint formula and Slope
The learner…
1. ENRICHMENT
uses the distance formula
2. ENRICHMENT
applies the midpoint formula
3. ENRICHMENT
recognize slope
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 1 (Applications of
Distance Formula, Midpoint
Grade 11: First Semester Syllabus in PRE - CALCULUS | 2
illustrate the concept of equality
versus justice in the current society.
4. ENRICHMENT
applies skills in distance formula,
midpoint formulas and slope in
solving situational problems
Formula and Slope)
Quiz 1 – 1 (Rectangular
Coordinate System)
WEEK 2
Systems of Linear and Non-
Linear Equations
Techniques in Solving Systems
of Linear and Non-Linear
Equations
o Substitution
o Elimination
o Graphing
Solving Situational Problems
Values:
Critical thinker. In any problem that
man encounters, there will always
be different ways on how to
approach it and solve it. This is the
concept shown by the various
techniques in solving systems of
linear and nonlinear systems.
The learner…
1. STEM_PC11AG-If-1
illustrates systems of linear and
nonlinear equations.
2. STEM_PC11AG-If-g-1
determines the solutions of systems
of linear and nonlinear equations
using techniques such as
substitution, elimination, and
graphing.
3. STEM_PC11AG-Ig-2
solves situational problems involving
systems of linear and nonlinear
equations.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 2 (Finding the Solutions
of Systems of Linear and
Nonlinear Equations using
Substitution, Elimination and
Graphing)
LAS 3 (Solving Situational
Problems Involving Systems of
Linear and Nonlinear
Equations)
Quiz 1 – 2 (Systems of Linear
and Nonlinear Equations)
WEEK 3
Introduction to Conics
Circle
o Standard Equation
o General Equation
The learner…
1. STEM_PC11AG-Ia-1
illustrates the different types of conic
sections: parabola, ellipse, circle,
hyperbola, and degenerate cases.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Grade 11: First Semester Syllabus in PRE - CALCULUS | 3
o Graphing
Parabola
o Equation
o Graphing
Values:
Emotionally stable individual and
Balance. A journey towards
success should be a balance from
the point of achieving and the point
of giving up, in order have the right
amount of enthusiasm. Such
balance is shown in the concept of
locus as a path traveled by a point
with equal distance from
references.
Life - long learner. The center of a
circle is likened to the core values
of a person, and the radius as one’s
knowledge and abilities. Given
these variables, one can determine
the equation of a circle, which, in
real life, is finding one’s purpose in
life.
Effective communicator. Thoughts
can be expressed in various ways,
depending on the situation, but will
still possess the same meaning.
This is implicit in the lesson on
transforming the standard form of
an equation of a parabola to its
2. STEM_PC11AG-Ia-2
defines a circle.
3. STEM_PC11AG-Ia-3
finds the equation of a circle, given
its center and radius and vice versa.
4. STEM_PC11AG-Ia-4
graphs a circle in a rectangular
coordinate system.
5. STEM_PC11AG-Ia-5
defines a parabola.
6. STEM_PC11AG-Ib-1
determines the standard form of
equation of a parabola.
7. ENRICHMENT
finds the general form of the
equation of a parabola.
8. STEM_PC11AG-Ib-2
graphs a parabola in a rectangular
coordinate system.
9. STEM_PC11AG-Ie-2
solves situational problems
involving circles and parabolas.
Reflect: Anong Hugot Mo?
Summative:
LAS 4 (Equations and Graphs
of Circles)
LAS 5 (Equations and Graphs
of Parabolas)
Quiz 1 – 3 (Circle and
Parabola)
Grade 11: First Semester Syllabus in PRE - CALCULUS | 4
general form.
WEEK 4
Ellipse
o Equation
o Graphing
Hyperbola
o Equation
o Graphing
Values:
Christ – centered individual.
Ellipses with vertex at the center is
like having God as the center of
one’s life, which makes things feel
lighter and easier, because He is
always there.
Life – long learner. The figure of a
hyperbola itself is a constant
reminder that there will always be
opposites, whether it be positive or
negative. Life becomes a mirror of
one’s dreams and aspirations
versus one’s achievements, where
the axis of symmetry is determined
by hard work and prayer.
The learner…
1. STEM_PC11AG-Ic-1
defines an ellipse.
2. STEM_PC11AG-Ic-2
determines the standard form of
equation of an ellipse.
3. STEM_PC11AG-Ic-3
graphs an ellipse with center at the
origin in a rectangular coordinate
system.
4. STEM_PC11AG-Id-1
defines a hyperbola.
5. STEM_PC11AG-Id-2
determines the standard form of
equation of a hyperbola.
6. ENRICHMENT
find the vertices, foci and
asymptotes of a hyperbola given an
equation
7. STEM_PC11AG-Id-3
graphs a hyperbola in a rectangular
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 6 (Equations and Graphs
of Ellipse)
LAS 7 (Equations and Graphs
of Hyperbola)
Quiz 1 – 4 (Ellipse and
Hyperbola)
Performance Task 1: Conic
sections on Earth!
Grade 11: First Semester Syllabus in PRE - CALCULUS | 5
coordinate system.
8. STEM_PC11AG-Ie-2
solves situational problems
involving ellipse and hyperbolas.
9. STEM_PC11AG-Ie-1
recognizes the equation and
important characteristics of the
different types of conic sections.
WEEK 5
Examination Week
1ST Mid - Quarter Exam
WEEK 6
Sequence
Series
o Sigma notation
Values:
Critical and systematic thinker.
Sequence is a continuous or
connected series that comes one
after the other. The idea conveys
that man should have an organized
and structured mind set to have a
smooth pace of life and an untangled
life path.
The learner…
1. STEM_PC11SMI-Ih-1
illustrates a series.
2. STEM_PC11SMI-Ih-2
differentiates a series from a
sequence.
3. ENRICHMENT
find the general and nth term of
the sequence
4. STEM_PC11SMI-Ih-3
uses the sigma notation to
represent a series.
5. ENRICHMENT
state the difference between
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 8 (Sequence)
LAS 9 (Sigma Notation for
Series)
Quiz 1 – 5 (Sequence and
Series)
Grade 11: First Semester Syllabus in PRE - CALCULUS | 6
arithmetic, geometric and
harmonic sequence and series.
WEEK 7
Pascal’s Triangle
Binomial Theorem
Values:
Life – long learner. Pascal’s triangle
has “1” on its peak and it is the
foundation of everything and the
triangle is made accumulatively
starting from the peak. Hence, man
should always look back before
moving forward in life. He has to use
his past experience to pass through
the challenges that comes along his
way.
The learner…
1. STEM_PC11SMI-Ii-2
illustrates Pascal’s Triangle in the
expansion of 𝑥 + 𝑦 𝑛 for small positive
integral values of 𝑛.
2. STEM_PC11SMI-Ii-3
proves the Binomial Theorem.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 10 (Pascal’s Triangle)
Quiz 1 – 6 (Pascal’s Triangle)
WEEK 8
Mathematical Induction
Values:
Gender sensitive and Person with
High integrity. Mathematical
induction shows that a sigma
notation is equal to its expanded
form. In life, man should not be
blinded of the differences of man
physically. Man should treat
everybody else fairly.
The learner…
1. STEM_PC11SMI-Ih-4
illustrates the Principle of
Mathematical Induction.
2. STEM_PC11SMI-Ih-i-1
applies mathematical induction in proving
identities.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 11 (Mathematical
Induction)
Quiz 1 – 7 (Mathematical
Induction)
Grade 11: First Semester Syllabus in PRE - CALCULUS | 7
WEEK 9
Expanding Binomial Theorem
Solving Problems using
Mathematical Induction and
Binomial Theorem
Values:
Critical thinker and Life – long
learner. Binomial Theorem is solved
using combination and its quite
complex in appearance but on the
latter it’s as easy as pie. Man
shouldn’t be afraid to face the
challenges in his life because not
everything that looks difficult is
difficult at all. Man should also learn
to limit or control himself. He
shouldn’t do everything that he can
do but only those what he is ask to.
The learner…
1. STEM_PC11SMI-Ij-1
determines any term of (𝑥+𝑦)𝑛,
where 𝑛 is a positive integer,
without expanding.
2. STEM_PC11SMI-Ij-2
solves problems using
mathematical induction and the
Binomial Theorem.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 12 (Expand me!)
LAS 13 (Mathematical
Induction and Binomial
Theorem in the Present!)
Quiz 1 – 8 (Problem solving in
Mathematical Induction and
Binomial Theorem)
Performance Task 2
WEEK 10 Examination Week
1st Quarter Examination
Grade 11: First Semester Syllabus in PRE - CALCULUS | 8
SENIOR HIGH SCHOOL
BASIC EDUCATION DEPARTMENT
Holy Angel University
Angeles City
FIRST SEMESTER SYLLABUS IN PRE – CALCULUS
SECOND QUARTER
CORE SUBJECT DESCRIPTION
The learner must be able to apply concepts and solve problems involving conic sections, systems of nonlinear equations, series and mathematical induction, circular and trigonometric functions, trigonometric identities, and polar coordinate system.
CONTENT STANDARD
The learner demonstrates understanding of key concepts of circular and trigonometric functions, trigonometric identities, inverse trigonometric functions and the polar coordinate system.
PERFORMANCE STANDARD
The learner is able to formulate and solve situational problems involving circular functions, trigonometric identities, trigonometric functions and the polar coordinate system.
TIME
FRAME TOPICS LEARNING COMPETENCIES ASSESSMENT
WEEK 11
Unit Circle
Conversion of Degree and Radian Measure
Angles o Standard Position o Co-terminal Angles
Circular Functions o Reference Angles
Values Life – long learner. The unit circle,
conversion of degree measure to radian measure, co-terminal angles,
The learner… 1. STEM_PC11T-IIa-1
illustrates the unit circle and the relationship between the linear and angular measures of a central angle in a unit circle.
2. STEM_PC11T-IIa-2 converts degree measure to radian measure and vice versa.
3. STEM_PC11T-IIa-3 illustrates angles in standard position
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 1 (Applications of unit circle, conversion of degree to radian measure and vice versa)
Grade 11: First Semester Syllabus in PRE - CALCULUS | 9
and reference angles are the foundation of trigonometry. These lessons serve as a gateway to more trigonometry and other mathematical topics that will be studied more in depth as the class proceeds. All of us started learning the basic skills we needed to survive each day. Weak foundation results to a chaotic and unorganized living. Whereas, a life established with a strong foundation yields a harmonious cycle of living.
and co-terminal angles.
4. STEM_PC11T-IIb-1 illustrates different circular functions.
5. STEM_PC11T-IIb-2 uses reference angles to find exact values of circular functions.
LAS 2 (Basic Circular Functions)
Quiz 2 – 1 (Unit Circle and Conversion)
WEEK 12
Circular Functions
o Domain and Range Six Circular Functions
o Amplitude o Period o Phase Shift
Problem Solving involving Circular Functions
Values: Life – long learner. The circular
functions, also being repetitive, are very useful in analyzing such periodic phenomena.
In the circle of life, certain behaviors must be practiced repeatedly in order for them to be reinforced.
The learner… 1. STEM_PC11T-IIc-1
determines the domain and range of different circular functions.
2. STEM_PC11T-IIc-d-1 graphs the six circular functions (a) amplitude, (b) period and, (c) phase shift.
3. STEM_PC11T-IId-2 solves problems involving circular functions.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 3 (Circular Functions and Problem Solving)
Quiz 2 – 2 (Circular Functions)
WEEK 13
Conditional Equations
Fundamental Trigonometric
The learner… 1. STEM_PC11T-IIe-1
Formative:
Recitation: How Far Can You
Grade 11: First Semester Syllabus in PRE - CALCULUS | 10
Identities o Sum and Difference of Angles o Double and Half-Angle
Formulas Values: Critical thinker and Globally
competitive. Trigonometric expressions are non-routine appearing problems. They are unfamiliar because the language of trigonometry looks foreign and complicated. In order to learn how to simplify or reduce the complexity of trigonometric expressions, we first need to examine the identities we need to utilize. Part of the history and story of a mankind are unfamiliar situations that may either be treated as struggles or challenges that can define a person’s future. In life, we always have to face these foreign circumstances for these can be tools for the betterment of one’s life.
determines whether an equation is an identity or a conditional equation.
2. STEM_PC11T-IIe-2 derives the fundamental trigonometric identities.
3. STEM_PC11T-IIe-3
derives trigonometric identities involving sum and difference of angles.
4. STEM_PC11T-IIf-1 derives the double and half-angle formulas.
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo? Summative:
LAS 4 (Fundamental Trigonometric Identities)
Quiz 2 – 3 (Conditional; Equations and Fundamental Trigonometric Identities)
WEEK 14
Trigonometric Expressions o Simplifying o Proving
Problem Solving Values: Creative and Innovative Individual.
Simplifying trigonometric expressions is comparable to
The learner... 1. STEM_PC11T-IIf-2
simplifies trigonometric expressions.
2. STEM_PC11T-IIf-g-1 proves other trigonometric identities.
3. STEM_PC11T-IIg-2 solves situational problems involving
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo? Summative:
Grade 11: First Semester Syllabus in PRE - CALCULUS | 11
making life easier. Simplified living means creating peace and harmony in one’s life and making each day as easy as possible; not escaping from the complexities of life but facing difficulties with enthusiasm resulting to a well-evaluated life.
trigonometric identities. LAS 5 (Trigonometric Expressions and Problem Solving)
Quiz 2 – 4 (Trigonometric Expressions)
Performance Task 3
WEEK 15
Examination Week 2nd Mid - Quarter Examination
WEEK 16
Inverse Trigonometric Functions o Domain and Range o Evaluating Expressions
Values: Emotionally stable Individual. Inverse
trigonometric functions are used to find the unknown measure of an angle when two side lengths are known. This concept is similar to the story of Sadness in the movie Inside Out. She may be negative in nature but she still has an important purpose in her world. Likewise in life, unnecessary events happen for a certain purpose. They may be in the form of struggles, but still, these can make a person wiser.
The learner... 1. STEM_PC11T-IIh-1
illustrates the domain and range of the inverse trigonometric functions.
2. STEM_PC11T-IIh-2 evaluates an inverse trigonometric expression.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo? Summative:
LAS 6 (Inverse Trigonometric Functions)
Quiz 2 – 5 (Inverse Trigonometric Functions)
WEEK 17
Trigonometric Equations
Problem Solving Values:
The learner... 1. STEM_PC11T-IIh-i-1
solves trigonometric equations.
Formative:
Recitation: How Far Can You
Go?
Grade 11: First Semester Syllabus in PRE - CALCULUS | 12
Critical thinker. Solving trigonometric equations include a method of simplification for each side of the equation. This is true in life, wherein, people must know how to simplify complicated situations in order to reach a balanced life.
2. STEM_PC11T-IIi-2 solves situational problems involving inverse trigonometric functions and trigonometric equations.
Board work: What’s the
Point?
Reflect: Anong Hugot Mo?
Summative:
LAS 7 (Trigonometric Equations and Problem Solving)
Quiz 2 – 6 (Trigonometric Equations)
WEEK 18
Polar Coordinate System
Converting Coordinates from Rectangular to Polar
Values: Socially responsive Community
leader. Polar coordinate system is a rule in mapping pair of points. This is magnified in the philosophy that “No man is an island.” where the need for interdependence is implied. Every person needs another, because there will always come a time that there will be a need from another person and vice versa.
The learner... 1. STEM_PC11T-IIj-1
locates points in polar coordinate system.
2. STEM_PC11T-IIj-2
converts the coordinates of a point from rectangular to polar system and vice versa.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo? Summative:
LAS 8 (Rectangular Coordinates to Polar Coordinates)
Quiz 2 – 7 (Polar Coordinate System)
WEEK 19
Problem Solving involving Polar Coordinate System
Values: Person with high Integrity and
Effective communicator. The points of a polar coordinate system are distance and angle. These pairs of
The learner... 1. STEM_PC11T-IIj-3
solve situational problems involving polar coordinate system.
Formative:
Recitation: How Far Can You
Go?
Board work: What’s the
Point?
Reflect: Anong Hugot Mo?
Grade 11: First Semester Syllabus in PRE - CALCULUS | 13
concepts are applied to real life as reminders that too much distance from other people will not allow you to see the perspectives or angles of others.
Summative:
LAS 9 (Problem Solving Involving Polar Coordinate System)
Quiz 2 – 8 (Problem solving in Polar Coordinate System)
Performance Task 4
WEEK 20 Examination Week
2nd Quarter Examination
Prepared by: Checked by:
Mr. Jeric Arrevado Ms. Brielene T. Nacpil Coordinator, S.T.E.M.
Ms. Cindy R. Dizon
Ms. Rachel Anne T. Medina
Ms. Helen Mae S. Moreno Teachers, Pre - Calculus