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TEACHING GUIDE FOR SENIOR HIGH SCHOOL
Precalculus
CORE SUBJECT
This Teaching Guide was collaboratively developed and reviewed by educators from public and private schools, colleges, and universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Commission on Higher Education, K to 12 Transition Program Management Unit - Senior High School Support Team at [email protected]. We value your feedback and recommendations.
Commission on Higher Education
in collaboration with the Philippine Normal University
Published by the Commission on Higher Education, 2016 Chairperson: Patricia B. Licuanan, Ph.D.
Commission on Higher Education K to 12 Transition Program Management Unit Office Address: 4th Floor, Commission on Higher Education, C.P. Garcia Ave., Diliman, Quezon City Telefax: (02) 441-1143 / E-mail Address: [email protected]
DEVELOPMENT TEAM Team Leader: Dr. Ian June L. Garces
Writers: Dr. Jerico B. Bacani, Dr. Richard B. Eden, Mr. Glenn Rey A. Estrada, Dr. Flordeliza F. Francisco, Mr. Mark Anthony J. Vidallo
Technical Editors: Dr. Maria Alva Q. Aberin, Dr. Flordeliza F. Francisco, Dr. Reginaldo M. Marcelo
Copy Reader: Naomi L. Tupas
Cover Artists: Paolo Kurtis N. Tan, Renan U. Ortiz
CONSULTANTS THIS PROJECT WAS DEVELOPED WITH THE PHILIPPINE NORMAL UNIVERSITY. University President: Ester B. Ogena, Ph.D. VP for Academics: Ma. Antoinette C. Montealegre, Ph.D. VP for University Relations & Advancement: Rosemarievic V. Diaz, Ph.D.
Ma. Cynthia Rose B. Bautista, Ph.D., CHED Bienvenido F. Nebres, S.J., Ph.D., Ateneo de Manila University Carmela C. Oracion, Ph.D., Ateneo de Manila University Minella C. Alarcon, Ph.D., CHED Gareth Price , Sheffield Hallam University Stuart Bevins, Ph.D. , Sheffield Hallam University
SENIOR HIGH SCHOOL SUPPORT TEAM CHED K TO 12 TRANSITION PROGRAM MANAGEMENT UNIT Program Director: Karol Mark R. Yee
Lead for Senior High School Support: Gerson M. Abesamis
Lead for Policy Advocacy and Communications: Averill M. Pizarro
Course Development Officers: Danie Son D. Gonzalvo, John Carlo P. Fernando
Teacher Training Officers: Ma. Theresa C. Carlos, Mylene E. Dones
Monitoring and Evaluation Officer: Robert Adrian N. Daulat
Administrative Officers: Ma. Leana Paula B. Bato, Kevin Ross D. Nera, Allison A. Danao, Ayhen Loisse B. Dalena
This Teaching Guide by the Commission on Higher Education is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. This means you are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material. The licensor, CHED, cannot revoke these freedoms as long as you follow the license terms. However, under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. NonCommercial — You may not use the material for commercial purposes. ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. Printed in the Philippines by EC-TEC Commercial, No. 32 St. Louis Compound 7, Baesa, Quezon City, [email protected]
1
Introduction
As the Commission supports DepEd’s implementation of Senior High School (SHS), it upholds the vision and mission of the K to 12 program, stated in Section 2 of Republic Act 10533, or the Enhanced Basic Education Act of 2013, that “every graduate of basic education be an empowered individual, through a program rooted on...the competence to engage in work and be productive, the ability to coexist in fruitful harmony with local and global communities, the capability to engage in creative and critical thinking, and the capacity and willingness to transform others and oneself.”
To accomplish this, the Commission partnered with the Philippine Normal University (PNU), the National Center for Teacher Education, to develop Teaching Guides for Courses of SHS. Together with PNU, this Teaching Guide was studied and reviewed by education and pedagogy experts, and was enhanced with appropriate methodologies and strategies.
Furthermore, the Commission believes that teachers are the most important partners in attaining this goal. Incorporated in this Teaching Guide is a framework that will guide them in creating lessons and assessment tools, support them in facilitating activities and questions, and assist them towards deeper content areas and competencies. Thus, the introduction of the SHS for SHS Framework.
The SHS for SHS Framework
The SHS for SHS Framework, which stands for “Saysay-Husay-Sarili for Senior High School,” is at the core of this book. The lessons, which combine high-quality content with flexible elements to accommodate diversity of teachers and environments, promote these three fundamental concepts:
SAYSAY: MEANING Why is this important?
Through this Teaching Guide, teachers will be able to facilitate an understanding of the value of the lessons, for each learner to fully engage in the content on both the cognitive and affective levels.
HUSAY: MASTERY
How will I deeply understand this? Given that developing mastery goes beyond memorization, teachers should also aim for deep understanding of the subject matter where they lead learners to analyze and synthesize knowledge.
SARILI: OWNERSHIP
What can I do with this? When teachers empower learners to take ownership of their learning, they develop independence and self- direction, learning about both the subject matter and themselves.
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The Parts of the Teaching Guide
This Teaching Guide is mapped and aligned to the DepEd SHS Curriculum, designed to be highly usable for teachers. It contains classroom activities and pedagogical notes, and integrated with innovative pedagogies. All of these elements are presented in the following parts:
1. INTRODUCTION - Highlight key concepts and identify the essential questions - Show the big picture - Connect and/or review prerequisite knowledge - Clearly communicate learning competencies and objectives - Motivate through applications and connections to real-life 2. MOTIVATION - Give local examples and applications - Engage in a game or movement activity - Provide a hands-on/laboratory activity - Connect to a real-life problem 3. INSTRUCTION/DELIVERY - Give a demonstration/lecture/simulation/hands-on activity - Show step-by-step solutions to sample problems - Give applications of the theory - Connect to a real-life problem if applicable 4. PRACTICE - Provide easy-medium-hard questions - Give time for hands-on unguided classroom work and discovery - Use formative assessment to give feedback 5. ENRICHMENT - Provide additional examples and applications - Introduce extensions or generalisations of concepts - Engage in reflection questions - Encourage analysis through higher order thinking prompts - Allow pair/small group discussions - Summarize and synthesize the learnings 6. EVALUATION - Supply a diverse question bank for written work and exercises - Provide alternative formats for student work: written homework, journal, portfolio, group/ individual projects, student-directed research project
7
On DepEd Functional Skills and CHED’s College Readiness Standards
As Higher Education Institutions (HEIs) welcome the graduates of the Senior High School program, it is of paramount importance to align Functional Skills set by DepEd with the College Readiness Standards stated by CHED.
The DepEd articulated a set of 21st^ century skills that should be embedded in the SHS curriculum across various subjects and tracks. These skills are desired outcomes that K to 12 graduates should possess in order to proceed to either higher education, employment, entrepreneurship, or middle-level skills development.
On the other hand, the Commission declared the College Readiness Standards that consist of the combination of knowledge, skills, and reflective thinking necessary to participate and succeed - without remediation - in entry-level undergraduate courses in college.
The alignment of both standards, shown below, is also presented in this Teaching Guide - prepares Senior High School graduates to the revised college curriculum which will initially be implemented by AY 2018-2019.
College Readiness Standards Foundational Skills DepEd Functional Skills Produce all forms of texts (written, oral, visual, digital) based on:
- Solid grounding on Philippine experience and culture;
- An understanding of the self, community, and nation;
- Application of critical and creative thinking and doing processes;
- Competency in formulating ideas/arguments logically, scientifically, and creatively; and
- Clear appreciation of one’s responsibility as a citizen of a multicultural Philippines and a diverse world;
Visual and information literacies Media literacy Critical thinking and problem solving skills Creativity Initiative and self-direction
Systematically apply knowledge, understanding, theory, and skills for the development of the self, local, and global communities using prior learning, inquiry, and experimentation
Global awareness Scientific and economic literacy Curiosity Critical thinking and problem solving skills Risk taking Flexibility and adaptability Initiative and self-direction
Work comfortably with relevant technologies and develop adaptations and innovations for significant use in local and global communities;
Global awareness Media literacy Technological literacy Creativity Flexibility and adaptability Productivity and accountability
Communicate with local and global communities with proficiency, orally, in writing, and through new technologies of communication;
Global awareness Multicultural literacy Collaboration and interpersonal skills Social and cross-cultural skills Leadership and responsibility
Interact meaningfully in a social setting and contribute to the fulfilment of individual and shared goals, respecting the fundamental humanity of all persons and the diversity of groups and communities
Media literacy Multicultural literacy Global awareness Collaboration and interpersonal skills Social and cross-cultural skills Leadership and responsibility Ethical, moral, and spiritual values 8
Lesson 1.1. Introduction to Conic Sections and Circles
Time Frame: 4 one-hour sessions
Learning Outcomes of the Lesson
At the end of the lesson, the student is able to:
(1) illustrate the different types of conic sections: parabola, ellipse, circle, hyper- bola, and degenerate cases;
(2) define a circle;
(3) determine the standard form of equation of a circle;
(4) graph a circle in a rectangular coordinate system; and
(5) solve situational problems involving conic sections (circles).
Lesson Outline
(1) Introduction of the four conic sections, along with the degenerate conics
(2) Definition of a circle
(3) Derivation of the standard equation of a circle
(4) Graphing circles
(5) Solving situational problems involving circles
Introduction
We introduce the conic sections, a particular class of curves which sometimes appear in nature and which have applications in other fields. In this lesson, we discuss the first of their kind, circles. The other conic sections will be covered in the next lessons.
1.1.1. An Overview of Conic Sections
We introduce the conic sections (or conics), a particular class of curves which oftentimes appear in nature and which have applications in other fields. One of the first shapes we learned, a circle, is a conic. When you throw a ball, the trajectory it takes is a parabola. The orbit taken by each planet around the sun is an ellipse. Properties of hyperbolas have been used in the design of certain telescopes and navigation systems. We will discuss circles in this lesson, leaving parabolas, ellipses, and hyperbolas for subsequent lessons.
- Circle (Figure 1.1) - when the plane is horizontal
- Ellipse (Figure 1.1) - when the (tilted) plane intersects only one cone to form a bounded curve
- Parabola (Figure 1.2) - when the plane intersects only one cone to form an unbounded curve
- Hyperbola (Figure 1.3) - when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the hyper- bola)
Figure 1.1 Figure 1.2 Figure 1. We can draw these conic sections (also called conics) on a rectangular co- ordinate plane and find their equations. To be able to do this, we will present equivalent definitions of these conic sections in subsequent sections, and use these to find the equations.
There are other ways for a plane and the cones to intersect, to form what are referred to as degenerate conics: a point, one line, and two lines. See Figures 1.4, 1.5 and 1.6.
Figure 1.4 Figure 1.5 Figure 1.
1.1.2. Definition and Equation of a Circle
A circle may also be considered a special kind of ellipse (for the special case when the tilted plane is horizontal). For our purposes, we will distinguish between these two conics.
(3) circle in Figure 1.
(4) circle A in Figure 1.
(5) circle B in Figure 1.
(6) center (5, −6), tangent to the y-axis
(7) center (5, −6), tangent to the x-axis
(8) has a diameter with endpoints A(− 1 , 4) and B(4, 2)
Figure 1.
Solution. (1) x^2 + y^2 = 16
(2) (x + 4)^2 + (y − 3)^2 = 7
(3) The center is (3, 1) and the radius is 5, so the equation is (x − 3)^2 + (y − 1)^2 =
(4) By inspection, the center is (− 2 , −1) and the radius is 4. The equation is (x + 2)^2 + (y + 1)^2 = 16.
(5) Similarly by inspection, we have (x − 3)^2 + (y − 2)^2 = 9.
(6) The center is 5 units away from the y-axis, so the radius is r = 5 (you can make a sketch to see why). The equation is (x − 5)^2 + (y + 6)^2 = 25.
(7) Similarly, since the center is 6 units away from the x-axis, the equation is (x − 5)^2 + (y + 6)^2 = 36.
(8) The center C is the midpoint of A and B: C =
2 ,^
4+ 2
2 ,^3
. The radius is then r = AC =
+ (4 − 3)^2 =
29 4.^ The circle has equation
x − (^32)
Seatwork/Homework 1.1.
Find the standard equation of the circle being described in each item.
(1) With center at the origin, radius
11 Answer: x^2 + y^2 = 11 (2) With center (− 6 , 7), tangent to the y-axis Answer: (x + 6)^2 + (y − 7)^2 = 36 (3) Has a diameter with endpoints A(− 3 , 2) and B(7, 4) Answer: (x − 2)^2 + (y − 3)^2 = 26
1.1.3. More Properties of Circles
After expanding, the standard equation
( x −
can be rewritten as x^2 + y^2 − 3 x − 6 y − 5 = 0,
an equation of the circle in general form.
If the equation of a circle is given in the general form
Ax^2 + Ay^2 + Cx + Dy + E = 0, A #= 0,
or x^2 + y^2 + Cx + Dy + E = 0,
we can determine the standard form by completing the square in both variables.
Completing the square (^) Teaching Notes Recall the technique of completing the square. This was introduced in Grade 9.
in an expression like x^2 + 14x means determining the term to be added that will produce a perfect polynomial square. Since the coefficient of x^2 is already 1, we take half the coefficient of x and square it, and we get 49. Indeed, x^2 + 14x + 49 = (x + 7)^2 is a perfect square. To complete the square in, say, 3x^2 + 18x, we factor the coefficient of x^2 from the expression: 3(x^2 + 6x), then add 9 inside. When completing a square in an equation, any extra term introduced on one side should also be added to the other side.
Example 1.1.2. Identify the center and radius of the circle with the given equa- tion in each item. Sketch its graph, and indicate the center.
(1) x^2 + y^2 − 6 x = 7
