Lesson 1.1: POLYGON
Lesson 1.2 Triangles
Lesson 1.3 Quadrilaterals
Week 1 and Week 2
Math 13
Solid Mensuration
A polygon is a closed plane figure that is
joined by line segments.
A polygon may also be defined as a union
of line segments such that: i) each
APPLICATIONS
OF
DIFFERENTTIATI
ON
EXTREMA ON AN
INTERVAL
OBJECTIVE
Understand the definition of extrema of a function
on an
interval.
Understand the definition of relative extrema of a
function on an open interval.
Find extrema on a closed interval.
EXTRE
SYSTEMS OF EQUATIONS
MATH10
ALGEBRA
Systems of Equations(Algebra and Trigonometry, Young 2nd Edition, page 874904)
GENERAL
OBJECTIVE
Week 6 Day
1
At the end of the chapter the students are expected to:
Solve systems of equations in two variables with the
FUNCTIONS
OPERATIONS ON FUNCTIONS
OBJECTIVES:
perform operations on functions;
determine the domain of the given functions;
determine the domain of the resulting
functions
after performing the operations on functions;
and
define a composite function a
Triangles
Example
The sides of the triangle are 12 cm, 16 cm and 21 cm. Find the area of the triangle.
Given : a = 16 cm, b = 21 cm, c = 12 cm
Required: Area of the triangle
s=
12 +16 + 21
2
s = 24.5
12
16
A = s(s a)(s b)(s c)
A = 24.5(24.5 12)(24.5 16)(2
Presentation on Figures with
Volume = Bh
CHOOSE WHAT TYPE OF PROBLEM
DO YOU WANT TO EXPLORE
Problem on Ratio of
Two Solids
Problem on Lateral
Area of a Prism
Problem on Volume
of a Regular Prism
Problem on Volume
of Cylinder
Problem on Total
Surface of a
Lesson 3 POLYHEDRONS
Week 5
Math 13
Solid Mensuration
Dihedral Angles
The dihedral angle is the angle
formed between two intersecting planes.
In the figure shown, the two planes are
called faces of the dihedral angle, and the
line of intersection between
Lesson 2.1: CIRCLES
Lesson 2.2
MISCELLANEOUS
WeekPLANES
3 and Week 4
Math 13
Solid Mensuration
2.1 CIRCLES
A circle is a set of points, each of which
is equidistant from a fixed point called the
center.
The line joining the center of a circle to
any point
Lesson 4: PRISMS AND
CYLINDERS
Week 6
Solids for which V=Bh
Math 13
Solid Mensuration
A prism is defined as a polyhedron with
two congruent bases that lie in parallel
planes, and whose every section that is
parallel to a base has the same area as
that of
Chapter 8
Prism & Cylinder
I.
Prism
A. Definition
B. Parts
a. Base
b. Lateral Base
c. Lateral Edge
d. Height/Latitude
C. Example
a. Special Type (Square & Rectangular)
b. Right Prism
c. Regular
d. Oblique
D. Properties
E. Surface Area
a. General LSA=Pe
b.
CHAPTER 5
CIRCLE
AReminderaboutpartsoftheCircle
Circumference
radius
diameter
Definition: Circle is a set of points each of which is
equidistant from a fixed point called the center.
Circumference is the distance around the outer edge.
Parts
AReminderabou
Polyhedra
Solid is any limited portion of space
bounded by surfaces or plane figures.
Vertex
Lateral face
l
na
o
iag
Lateral edge
Miscellaneous Planes
Types of solid:
(1) Right Circular Cylinder
(2) Pyramid
(3) Cone
(4) Sphere
Miscellaneous Planes
2 Main
Miscellaneous Planes
Star Polygon a starlike figure which
generally consists of a polygon with
triangles on its sides.
5Pointed Star
6Pointed Star
Miscellaneous Planes
Example#1.
Determine the area of a regular 6pointed
star if the inner regular hexag
HIGHER ORDER
SQUARE MATRICES
MATH 15  Linear Algebra
PROPERTIES OF DETERMINANTS
For all square matrices, the following properties
hold:
1.
If a row or a column of a given matrix is a
multiple or equal to another row or column, then
the determinant is equ
APPLICATIONS OF
REF/RREF
MATH 15  Linear Algebra
HOMOGENEOUS SYSTEM
HOMOGENEOUS SYSTEM
EXAMPLES
GAUSSIAN ELIMINATION
1.
Augmented Matrix. Write the augmented
matrix of the system.
A matrix that includes the entire linear system is called
an augmented mat
LINEAR ALGEBRA
VECTOR SPACE
Math 15: Linear Algebra
VECTOR SPACE
Math 15: Linear Algebra
VECTOR SPACE
Math 15: Linear Algebra
Theorem
Math 15: Linear Algebra
Subspace
Math 15: Linear Algebra
Theorem:
Math 15: Linear Algebra
ILLUSTRATIONS:
Math 15: Linear
ANALYTIC GEOMETRY
Math 14
Plane and Analytic Geometry
SPACE
COORDINATES
and
SURFACES
OBJECTIVES:
At the end of the lesson, the student is expected
to be able to:
define space coordinates.
plot points of space coordinates.
write and sketch the graphs of
INVERSE OF A MATRIX
Definition: An n x n matrix A is called
nonsingular (or invertible) if there exists
an an n x n matrix B such that
Math 15: Linear Algebra
INVERSE OF A MATRIX
Math 15: Linear Algebra
Theorems (Properties) OF INVERSE OF A
MATRIX
Math 15
DETERMINANTS
Math 15: Linear Algebra
DETERMINANT of a 2 x 2
matrix
Math 15: Linear Algebra
DETERMINANT of a 3 x 3
matrix
Math 15: Linear Algebra
DETERMINANTS
Math 15: Linear Algebra
DETERMINANTS
Math 15: Linear Algebra
COFACTOR EXPANSION
Math 15: Linear A
EIGENVALUES/EIGENVECTO
RS
Math 15: Linear Algebra
EIGENVALUES/EIGENVECTO
RS
Math 15: Linear Algebra
STEPS
FOR
EIGENVALUES/EIGENVECTORS
FINDING
Math 15: Linear Algebra
ILLUSTRATION:
Math 15: Linear Algebra
EIGENSPACES
Math 15: Linear Algebra
ILLUSTRATION:
Vectors in Space
Math 15: Linear Algebra
Vector Operations
Math 15: Linear Algebra
Properties
Operations
on
Vector
Math 15: Linear Algebra
Properties
Operations
on
Vector
Math 15: Linear Algebra
Math 15: Linear Algebra
Inner/Dot Product
Angle Between Two
Linear Combination
Math 15: Linear Algebra
Linear Combination
Math 15: Linear Algebra
Linear Combination
Math 15: Linear Algebra
Practice:
Math 15: Linear Algebra
Spanning
Math 15: Linear Algebra
Practice:
Math 15: Linear Algebra
Linear Independence
Math
SPHERICAL
TRIGONOMETRY
DEFINITION OF
TERMS:
Spherical Trigonometry is a branch of
trigonometry that concerns with triangles extracted
from the surface of the sphere.
Great Circle is a circle obtained by passing a
section through the center of the sphere
SOLUTIONS
OF
LINEAR
SYSTEM OF EQUATIONS
Consider the following linear system of
three equations in three unknowns:
Math 15: Linear Algebra
SOLUTIONS
OF
LINEAR
SYSTEM OF EQUATIONS
We shall define the following matrices
Math 15: Linear Algebra
SOLUTIONS
OF
Math 15: Linear Algebra
Column Vectors of A
Math 15: Linear Algebra
Row Space and Column Space
Math 15: Linear Algebra
Theorems on Row Space:
Math 15: Linear Algebra
Illustration: Basis for a row
space
Math 15: Linear Algebra
Illustration: Basis for a col
LINEAR EQUATIONS
MATH10
ALGEBRA
Week 1 Day 1 Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 9099)
GENERAL
OBJECTIVE
Week 1 Day 1
At the end of the lesson the students are
expected to:
Classify equations as linear, fractional, or rat
Rules of Differentiation
1. Constant Rule:

If , where c is a constant, then
Example:
2. Power Rule:

If , where n is a rational number, then
Example:

GGT (gawagawang theorem)
a.i.1.
Examples:
2.
Examples:
Shortcut:
If, where n is the index, m is the
ANALYTIC GEOMETRY
(Lesson 14)
Math 14
Plane and Analytic Geometry
PARAMETRIC
EQUATIONS
OBJECTIVES:
At the end of the lesson, the student is expected
to be able to:
define parametric equation.
sketch the graphs of parametric equations by point
plotting.
ALGEBRAIC EXPRESSIONS
An algebraic expression is the result
of associating constants and variables
by addition, subtraction, multiplication,
division, taking roots and raising to a
power.
Math 101: College Algebra
ALGEBRAIC EXPRESSIONS
A constant is any