Pre-Calculus-SHS-Training.pdf guide deped

Guide k-12 precalculus

1. Session Guide for SHS
Training
PRE-CALCULUS
Bureau of Curriculum Development
March 2017

2. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
OBJECTIVES
Explain the components of the SHS
Curriculum Guide.
Distinguish the content & performance
standards and competencies.

3. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
CURRICULUM GUIDE
Summarizes what learning should be achieved in
what grades or over certain grade spans. It plots the
road map that each learner must follow where
content standards, performance standards, and
competencies are given emphasis
Covers competencies where strategies and activities
can be anchored on

4. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

5. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Refers to the topical coverage of a particular subject
in a grade level in a given period.
CONTENT

6. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

7. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
CONTENT STANDARDS
Identifies and set the essential understandings that
should be learned in a specific period
Covers specified scope of sequential topics within
each learning strand, domain, theme, or component.
Content standards answer the question: What
should the learners know, do and understand?

8. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

9. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
PERFORMANCE STANDARDS
Answer the following questions:
What can learners do with what they know?
How well must learner do their work?
How well do learners use their learning or understanding
in different situations?
How do learners apply their learning or understanding in
real-life contexts?
What tools and measures should learners used to
demonstrate what they know?

10. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

11. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Refer to the knowledge, understanding, skills
and attitudes that students need to demonstrate
in every lesson or learning activity.
LEARNING COMPETENCIES

12. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

13. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
CODING EXAMPLE
PRE-CALCULUS
GRADE 11
Week 2
ANALYTIC
GEOMETRY Quarter I
COMPETENCY 1

14. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
PRE-CALCULUS
FIRST QUARTER

15. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

16. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

17. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
a.How are the Content Standards congruent with the
Performance Standards?
b. How are these standards aligned with the Learning
Competencies?
c. What are the possible assessment tools and strategies
that you think will be applicable to the competencies?
After browsing the CG…

18. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Ensure that the
Standardsare
ACHIEVED

19. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
AVP ON CONIC FORMATION

20. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Remember…

21. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
IDENTIFYING CONICS!
P
C
E
E
C
H
P
H

22. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
CONVERTING CONICS!
9????????????
2
+16????????????
2
−126????????????+64????????????−71=0
????????????
2
+????????????
2
−8????????????−10????????????+16=0
????????????
2
+6????????????+8????????????−7=0
4????????????
2
−5????????????
2
+32????????????+30????????????−1=0

23. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
A circleis a set of all points in a plane equidistant from a fixed point.
The fixed point is called the centerand the constant equal distance is
called the radius of the circle.
The standard form of the circle is
(????????????−????????????)
????????????
+(????????????−????????????)
????????????
=????????????
????????????

24. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
A parabola is the set of points in the plane equidistant from a
fixed line and a fixed point not on the line.
Axis of Symmetry
Latus Rectum
Vertex
Directrix
Focus

25. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
An ellipse is the set of all points in the plane, the sum of
whose distance from two fixed points is constant.
Major and Minor Axis
Latera Recta
Vertices
Directrix
Foci

26. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
A hyperbola is the set of all points in the plane, the difference
of whose distance from two fixed points is a positive constant.
Transverse Axis
Conjugate Axis
Center
Foci
Vertices

27. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
1. Determine the standard form of the circle with center at (– 2, 3)
and tangent to the line x = - 10
CONICS CONDITION

28. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
2. Find the standard equation of the circle concentric with the circle
x
2
+ y
2
–8x –10y = – 16 and 4 times its area.
CONICS CONDITION

29. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
3. Find the standard equation of the parabola whose vertex is
(–5, –7), with horizontal axis of symmetry and passes through the
point (7, 11).
CONICS CONDITION

30. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
4. A parabola has focus F(− 2 ,− 5) and directrixx = 6. Find the
standard equation of the parabola.
CONICS CONDITION

31. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
5. Find the standard equation of the ellipse whose foci are F
1(− 3, 0)
and F
2(3, 0), such that for any point on it, the sum of its distances
from the foci is 10.
CONICS CONDITION

32. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
CONICS CONDITION

33. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
7. A hyperbola has vertices (1, 9) and (13, 9), and one of its foci is
(−2, 9). Find its standard equation.
CONICS CONDITION

34. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
8. The foci of a hyperbola are (− 5,− 3 ) and (9,− 3 ). For any point on
the hyperbola, the absolute value of the difference of its of its
distances from the foci is 10. Find the standard equation of the
hyperbola.
CONICS CONDITION

35. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
1. A seismological station is located at (0, –3), 3 km away from a straight
shoreline where the x-axis runs through. The epicenter of an earthquake was
determined to be 6 km away from the station.
a. Find the equation of the curve that contains the possible location of the
epicenter.
b. If furthermore, the epicenter was determined to be 2 km away from the
shore, find its possible coordinates (rounded off to two decimal places).
REAL-LIFE CONICS
(±3.32, 2)x
2
+ (y + 3)
2
= 36Answer

36. 12 m
DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
2. A ferriswheel is elevated 1 m above ground. When a car reaches the highest
point on the ferriswheel, its altitude from ground level is 31 m. How far away
from the center, horizontally, is the car when it is at an altitude of 25 m?
AnswerHint
REAL-LIFE CONICS

37. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
3. The cable of a suspension bridge hangs in the shape of a parabola. The
towers supporting the cable are 400 ftapart and 150 fthigh. If the cable, at
its lowest, is 30 ftabove the bridge at its midpoint, how high is the cable 50 ft
away (horizontally) from either tower?
97.5 ftAnswer
REAL-LIFE CONICS

38. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
4. A light reflector is in the shape of a paraboloid so that its cross-section is
parabolic. If a light source is placed on the location of the focus of the
reflector, the light rays will reflect off the surface, forming a beam of light
parallel to the axis. Suppose the light reflector is 12 inches in diameter. If the
light source is located 1.5 inches from the vertex, determine the depth of the
reflector so that the beam of the reflected light is parallel to the axis.
6 inchesAnswer
REAL-LIFE CONICS

39. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
5. The arch of a bridge is in the shape of a semiellipse, with its major axis at the
water level. Suppose the arch is 20 ft high in the middle, and 120 ft across its
major axis. How high above the water level is the arch, at a point 20 ft from
the center (horizontally)? Round off to 2 decimal places.
18.86 ftAnswer
REAL-LIFE CONICS

40. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
6. Two control towers are located at points Q(−500,0) and R(500,0), on a
straight shore where the x-axis runs through (all distances are in meters). At
the same moment, both towers sent a radio signal to a ship out at sea, each
traveling at 300 m/µs. The ship received the signal from Q 3 µs
(microseconds) before the message from R.
a. Find the equation of the curve containing the possible location of the ship.
b. Find the coordinates (rounded off to two decimal places) of the ship if it is
200 m from the shore (y = 200).
(−610.76, 200)
????????????
????????????
????????????????????????????????????????????????????????????????????????
−
????????????
????????????
????????????????????????????????????????????????????????????
= 1Answer
REAL-LIFE CONICS

41. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
Use the substitution method to solve the system, and
sketch the graphs in one Cartesian plane showing the point
of intersection.
46
534
xy
xy
+=
+=

42. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations

43. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
Solve the following system, and sketch the graphs in one
Cartesian plane.
2
20
1
xy
yx
−+=
−=

44. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations

45. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations

46. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
Solve the system, and sketch the graphs in one Cartesian
plane showing the point(s) of intersection.
22
16
4
xy
xy
+=
−=

47. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations

48. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations

49. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
The screen size of television sets is given in inches. This
indicates the length of the diagonal. Screens of the same
size can come in different shapes . Wide-screen TV's
usually have screens with aspect ratio 16:9, indicating
the ratio of the width to the height. Older TV models often
have aspect ratio 4:3. A 40-inch LED TV has screen aspect
ratio 16:9. Find the length and the width of the screen.
Answer: 34.86 inches wide and 19.61 inches high

50. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
Solve the system.
22
21
7
xy
xy
−=
+=
()5,2

51. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
Solve the system.
22
650
10
xyxy
xy
+−++=
++=
()
57
0, 1 ,
22
and

−−



52. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Systems of Nonlinear Equations
The difference of two numbers is 12, and the sum of their
squares is 144. Find the numbers.
()( )12,0 0, 12and−

53. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sequences and Series
A sequence is a function whose domain is the set of
positive integers or the set 1,2,3,…,????????????
A series represents the sum of the terms of a sequence.

54. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sequences and Series

55. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

56. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

57. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

58. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

59. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

60. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

61. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

62. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Sigma Notation

63. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction

64. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
In a proof by mathematical induction, it is not assumed
that P(k) is true for all positive integers! It is only shown
that if it is assumed that P(k) is true, then P(k + 1) is also
true. Thus, a proof by mathematical induction is not a case
of begging the question, or circular reasoning.

65. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that
()1
1 2 3 ...
2
nn
n
+
++++=

66. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()
()
()
1
1 2 3 ...
2
111
1
2
12
1
2
11
nn
n
+
++++=
+
=
=
=
????????????=1

67. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()1
1 2 3 ...
2
kk
k
+
++++=
????????????=????????????≥1

68. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()
()()1 11
1 2 3 ... 1
2
kk
kk
+ ++

+++++ +=
????????????=????????????+1

69. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()
()
()
1
1 2 3 ... 1 1
2
kk
kk k
+
++++++= ++
??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=????????????
??????????????????????????????????????????????????????????????????????????????????????????+1??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

70. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()
()()12 1
1 2 3 ... 1
2
kk k
kk
++ +
+++++ +=
()()12
2
kk++
=
()()1 11
2
kk+ ++

=

71. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that
()( )222 2
12 1
123
6
nn n
n
++
++++=

72. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Conjecture a formula for the sum of the first n positive
odd integers. Then prove your conjecture using
mathematical induction.

73. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
1+2+2
2
+⋯+2
????????????
=2
????????????+1
−1
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

74. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
�
????????????=0
????????????
????????????????????????
????????????
=????????????+????????????????????????+????????????????????????
2
+⋯+????????????????????????
????????????
=
????????????????????????
????????????+1
−????????????
????????????−1
????????????ℎ????????????????????????????????????≠1????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

75. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that for every
nonnegative integer n, ????????????
3
−????????????+3is divisible by 3.

76. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()
3
0 0 3 3 31−+=⇒
()
3
33 3kk a a−+= ⇒
()()
3 32
1 13 3 23k k kkk+ −++= + + +

77. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()()
3 32
1 13 3 23k k kkk+ −++= + + +
()() ( )
3 32
1 13 33 3k k kk k k+ −++= −++ +
()()
3 2
1 133 3 3k k ak k+ −++= + +
()() ( )
3 2
1 133k k ak k+ −++= + +

78. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that for every
nonnegative integer n, 7
????????????+2
+8
2????????????+1
is divisible by 57.

79. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that 2 divides ????????????
2
+????????????
whenever n is a positive integer.

80. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that ????????????
2
−1is
divisible by 8 whenever n is an odd positive integer.

81. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
Using mathematical induction, prove that 2
????????????
>????????????
2
if n is
an integer greater than 4

82. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
2
52
2
25
32 25
n
n>
>
>
????????????=5

83. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
2
2
k
k>
????????????=????????????≥5

84. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
()
21
21
k
k
+
>+
????????????=????????????+1

85. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
1
2 22
kk+
= ⋅
22
22
k
kk⋅>+
()
22
21 1kk k+ += +
222
4kkk k+>+
22
4 21kkkk+≥++

86. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
2
????????????
<????????????!
??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????≥4

87. DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT
Mathematical Induction
????????????<2
????????????
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

88. THANK YOU!