1. Prepared by: Jimmy M. Romero
Licensure Examination for Teachers (Refresher Course) Focus: Contemporary Mathematics (Gen Ed Math2)
Prepared by:
Jimmy M. Romero

2. LET Competencies: 3. Plane and Solid Geometry
3.1. Show mastery of basic terms and concepts in plane and solid geometry
lines and curves, perpendicular and parallel lines
angles, angle properties
special triangles and quadrilaterals
polygons and solids
3.2. Solve problems involving basic terms and concepts in plane and solid geometry
   

3. LET Competencies: 4. Statistics and Probability
4.1. Show mastery and knowledge of basic terms and concepts in statistics and probability
counting techniques
probability of an event
measure of central tendency
measures of variability
4.2. Solve, evaluate, manipulate symbolic and numerical problems in statistics and probability by applying fundamental rules, principles and processes
   

4. Concepts and Terminologies

5. Plane and Solid Geometry

6. A point names a location. • A Point A
Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures.
A point names a location.
• A
Point A
A line is perfectly straight and extends forever in both directions. B C l
line l, or BC

7. A segment, or line segment, is the part of a line between two points.
A plane is a perfectly flat surface that extends forever in all directions. E
plane P, or plane DEF F D
A segment, or line segment, is the part of a line between two points. H GH G
A ray is a part of a line that starts at one point and extends forever in one direction. J KJ K

8. Collinear. Three or more points are said to be collinear if a single line contains all of them. Otherwise they are said to be non collinear. (Figure 1.2)
Coplanar . Points and lines which lie in the same plane are called coplanar otherwise they are called non-coplanar.
Figure 1.2

9. Line Segment
A line segment is a straight line segment which is part of the straight line between two points. To identify a line segment, one can write AB. The points on each side of the line segment are referred to as the end points.

10. An angle () is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees.
One degree, or 1°, is 1/360 of a circle. m1 means the measure of 1. The angle can be named XYZ, ZYX, 1, or Y. The vertex must be the middle letter. X Y Z 1
m1 = 50°

11. The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°. F K J G H

12. Pairs of Angles: Complementary Angles
Complementary Angles
Two angles adding up to 90° are called
complementary angles.
Supplementary Angles
Two angles adding up to 180° are called
supplementary angles. 

13. Additional Example 2D: Classifying Angles
D. Name a pair of complementary angles.
TQP, RQS
mTQP + m RQS = 47° + 43° = 90°

14. Additional Example 2E: Classifying Angles
E. Name two pairs of supplementary angles.
TQP, RQT
mTQP + mRQT = 47° + 133° = 180°
SQP, RQS
mSQP + mRQS = 137° + 43° = 180°

15. The measures of angles that fit together to form a complete circle, such as MRN, NRP, PRQ, and QRM, add to 360°. P R Q M N

16. A right angle measures 90°.
An acute angle measures less than 90°.
An obtuse angle measures greater than 90° and less than 180°.
A reflex angle measures greater than 180° and less than 270°.

17. Additional Example 2A & 2B: Classifying Angles
A. Name a right angle in the figure.
TQS
B. Name two acute angles in the figure.
TQP, RQS

18. Additional Example 2C: Classifying Angles
C. Name two obtuse angles in the figure.
SQP, RQT

19. Congruent figures have the same size and shape.
Segments that have the same length are congruent.
Angles that have the same measure are congruent.
The symbol for congruence is , which is read “is congruent to.”
Intersecting lines form two pairs of vertical angles.
Vertical angles are always congruent

20. Bisectors
Bisectors refers to the line, ray or line segment that passes through the midpoint. The bisector divides a segment into two congruent segments as demonstrated here:

21. Transversal
A line which intersects two or more given coplanar lines in distinct points is called a transversal of the given lines. In the figure the line l is the transversal of lines a and b.

22. Parallel Lines
Parallel lines are defined as those lines which are coplanar and do not intersect

23. Transversal across two parallel lines
If however a transversal (line n) is drawn across lines l and m in the figure below, all the eight angles formed can be measured.

24. The Interior and Exterior Angles

25. Sum of the Interior Angle
Sum of the Interior and Exterior Angles
Sum of the Exterior Angle

26. Each Interior Angle
n-gon n
(n-2) × 180° / n

27. Areas and Perimeters Circumference, Sector and Chord

29. Area of triangle with three sides given: (Heron’s Formula)

31. Median =

34. A = bh

35. A = bh

37. The chord of a circle is a line joining any two points on the circle.
The product of the segments of two chords intersecting each other are equal, i.e.

38. Shapes
Formula
Rectangle
P = 2l + 2w
A = lw
Rectangular Prism
TSA = 2lw + 2wh +2hl
V = lwh
Square
P = 4s
A = S2
Cube
TSA = 6s2
V = s3
Parallelogram
P = 2a + 2b
A = bh
Circular Cylinder
TSA = 2πr2 + 2πrh
V = πr2h
Trapezoid
P = a1 + a + b1 + b2
A= (½)(b1 + b2)h
Pyramid
TSA = LSA + B
V = Bh
Triangle
P = S1 + S2 + S3
A = (½) bh
Cone
TSA = πr2 + πrs
V = (1/3)πr2h
Circle
C = 2πr
A = πr2
Sphere
TSA = 4πr2
V = πr3

39. Volumes Lateral and Surface Areas (Solid)

45. PROBABILITY and STATISTICS

46. Counting Techniques, Permutations and Combination Probability

47. Counting Techniques
An experiment is a situation involving chance or probability that leads to results called outcomes. (e.g. tossing a coin, throwing 2 dice or rolling a die)
An outcome is the result of a single trial of an experiment (e.g. possible outcome is a head or a tail appears).
An event is one or more outcomes of an experiment (e.g. pair of numbers that are the same).
The sample space of an experiment is the set of all possible outcomes of that experiment. Example: In tossing a coin, S = {H, T}. In rolling a die: S = {1, 2, 3, 4, 5, 6}.
The sample point is an element of the sample space. Example: In rolling a die, there are 6 sample points.

48. Fundamental Principle of Counting.
If a choice consists of k steps, of which the first can be performed by n1 ways, for each of these the second can be performed by n2 ways, for each of these the third can be performed by n3 ways, and so on, until the kth can be performed in nk ways, then the whole choice can be made in n1·n2 ·n3…nk ways.
Example: A deli has a lunch special which consists of a sandwich, soup, dessert and drink for P238. 
They offer the following choices:
Sandwich: chicken salad, ham, tuna, and roast beef
Soup:  tomato, chicken noodle, vegetable
Dessert: cookie and pie
Drink: tea, coffee, coke, diet coke and sprite
How many lunch specials are there?
Solution: · 3 · 2 · 5 = 120 ways

49. Permutation. It is an arrangement of objects in which the order is taken into consideration.
Linear Permutation
If n objects are to be arranged all at a time, then the number of distinct arrangements is given by the formula n! = (n – 1)(n – 2)(n – 3) ··· 3 ·2 · 1.
Example 1: In how many ways can 6 people be lined up in a straight line.
If n objects are to be arranged in r objects at a time, then the number of distinct arrangements is
nPr =
Example 2: In a qualifying race for an Olympic road cycling event, 15 men competed for 5 slots. How many different ways could the top 5 slots be arranged if the order in which the men finished mattered?

50. Permutation. It is an arrangement of objects in which the order is taken into consideration.
Linear Permutation
If n objects are to be arranged all at a time, then the number of distinct arrangements is given by the formula n! = n(n – 1)(n – 2)(n – 3) ··· 3 ·2 · 1.
Example 1: In how many ways can 6 people be lined up in a straight line.
If n objects are to be arranged in r objects at a time, then the number of distinct arrangements is
nPr = n!/(n-r)!
Example 2: In a qualifying race for an Olympic road cycling event, 15 men competed for 5 slots. How many different ways could the top 5 slots be arranged if the order in which the men finished mattered?

51. Permutations with Repetitions.
The number of different permutations of n objects, where there are n1 indistinguishable objects of the first kind, n2 indistinguishable objects of the 2nd kind, ..., and nk indistinguishable objects of the kth kind is Pn = n!/(n1!n2!n3!...nk!) , where n = n1 + n2 + n3 +… + nk.
Example 3: How many different six-digit numerals can be written using all of the following six digits: 4, 4, 5, 5, 5, 7.

52. When you have n things to choose from ... you have n choices each time!
When choosing r of them, the permutations are: n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilities for the second choice, and so on, multplying each time.) Which is easier to write down using an exponent of r:
n × n × ... (r times) = nr , where n is the number of things to choose from, and you choose  r  of them (Repetition allowed, order matters).
Example 4: In a lock, there are 10 numbers to choose from (0 to 9) and you choose 3 of them. How many arrangements will there be?
Solution: (10)3 = 10 ·10 · 10 = 1,000

53. Circular Permutation.
Circular permutation is the number of ordered arrangements that can be made of n objects in a circular manner is given by (n – 1)!.
Example 4: In how many ways can 5 students be arranged in a round table?
Solution: (5 – 1)! = 4! = 4 · 3 · 2 · 1 = 24 arrangements.

54. Combination.
Combination is the arrangement of objects into specified number of groups without regard to order. Thus, the order of arrangements should not be taken into consideration. So, if n objects are to be arranged r at a time, the number of distinct combinations is given by the formula
nCr = n!/(n – r)!r!
Example 5: From a squad of 16 members, find the total number of different arrangements of 9 players.

55. Probability

56. Probability of an Event E:
Probability is the measure of how likely an event is. It is the likelihood of occurrence of an event.
Probability of an Event E:
P(E) = no. of ways in an event occurs/total no. of ways
If E is any event that occurs, then the probability of that event denoted by P(E) has a value between o and 1, inclusively. That is, 0 ≤ P(E) ≤ 1.
Example 1. A standard deck has four suits: spades (), hearts (), diamonds (), and clubs (). It has thirteen cards in each suit: ace, 2, 3, . . ., 10, jack, queen, and king. Each of these cards is equally likely to be drawn. What is the probability of getting (a) an ace?, (b) a diamond?

57. If event E is an event that occurs surely, then P(E) = 1, but if it is impossible to happen then P(E) = 0.
Moreover, if event E’ will not happen, then P(E) + P(E’) = 1.
Example 2. A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner?
Sample Space: {yellow, blue, green, red}
P (not red) = 1 – P(Red)

58. Addition Rule
Mutually Exclusive
If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as P( A or B) = P(A  B). If two events are mutually exclusive then the probability of either occurring is P(A  B) = P(A) + P(B).
For example, the chance of rolling a 1 or 2 on a six-sided die is P(1  2) = P(1) + P(2) = 1/ /6 = 2/6 or 1/3.

59. Not mutually Exclusive
If the events are not mutually exclusive then P(A  B) = P(A) + P(B) – P(A  B).
For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is 13/ /52   – 3/52 = 22/52 or 11/26, because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but should only be counted once.

60. Statistics

61. Measures of Central Tendency
MEAN – The most widely used and familiar average. The most reliable and the most stable of all measures of central tendency.
Advantage: It is the best measure for regular distribution.
Disadvantage: It is affected by extreme values
What is the mean?
75,60,78,

62. Measures of Central Tendency
MEDIAN – The scores that divides the distribution into halves. It is sometimes called the counting average.
Advantage: It is the best measure when the distribution is irregular or skewed. It can be located in an open-ended distribution or when the data is incomplete (ex. 80% of the cases is reported)
Disadvantage: It necessitates arranging of items according to size before it can be computed
What is the median?
75,60,78,

63. Measures of Central Tendency
MODE – the crude or inspectional average measure. It is most frequently occurring score. It is the poorest measure of central tendency.
Advantage: Mode is always a real value since it does not fall on zero. It is simple to approximate by observation for small cases. It does not necessitate arrangement of values.
Disadvantage: It is not rigidly defined and is inapplicable to irregular distribution
What is the mode of these scores?
75,60,78,

64. Symmetrical or Normal Distribution

65. Asymmetrical or Skewed Distribution:
When low scores pull the mean toward the left tail, the skew is negative

66. Asymmetrical or Skewed Distribution:
Conversely, when high scores pull the mean toward the right tail, the skew is positive.

67. Measures of Variability or Scatter
1. RANGE
R = highest score – lowest score
2. Quartile Deviation
QD = ½ (Q3 – Q1)
It is known as semi inter quartile range
It is often paired with median

68. Measures of Variability or Scatter:
STANDARD DEVIATION
It is the most important and best measure of variability of test scores.
A small standard deviation means that the group has small variability or relatively homogeneous.
It is used with mean.

70. Z-Score or Standardized Score
tells the number of standard deviations equivalent to a given raw score

71. Thank you
and
GOD BLESS!