Bec Mathematics, PSSLC Mathematics

CONTENTS

INTRODUCTION 3

DESCRIPTION 5

UNIT CREDIT 6

TIME ALLOTMENT 6

EXPECTANCIES 7

SCOPE AND SEQUENCE 8

SUGGESTED STRATEGIES AND MATERIALS 9

GRADING SYSTEM 10

LEARNING COMPETENCIES 11

SAMPLE LESSON PLANS 30

INTRODUCTIONThis Handbook aims to provide the general public – parents, students, researchers, and other

stakeholders – an overview of the Mathematics program at the secondary level. Those in education, however, may use it as a reference for implementing the 2002 secondary education curriculum, or as a source document to inform policy and guide practice.

For quick reference, the Handbook is outlined as follows:

** The description defi nes the focus and the emphasis of the learning area as well as the language of instruction used.

** The unit credit indicates the number of units assigned to a learning area computed on a 40-minute per unit credit basis and which shall be used to evaluate a student’s promotion to the next year level.

** The time allotment specifi es the number of minutes allocated to a learning area on a daily (or weekly, as the case may be) basis.

** The expectancies refer to the general competencies that the learners are expected to demonstrate at the end of each year level.

** The scope and sequence outlines the content, or the coverage of the learning area in terms of concepts or themes, as the case may be.

** The suggested strategies are those that are typically employed to develop the content, build skills, and integrate learning.

** The materials include those that have been approved for classroom use. The application of information and communication technology is encouraged, where available.

** The grading system specifi es how learning outcomes shall be evaluated and the aspects of student performance which shall be rated.

** The learning competencies are the knowledge, skills, attitudes and values that the students are expected to develop or acquire during the teaching-learning situations.

** Lastly, sample lesson plans are provided to illustrate the mode of integration, where appropriate, the application of life skills and higher order thinking skills, the valuing process and the differentiated activities to address the learning needs of students.

The Handbook is designed as a practical guide and is not intended to structure the operationalization of the curriculum or impose restrictions on how the curriculum shall be implemented. Decisions on how best to teach and how learning outcomes can be achieved most successfully rest with the school principals and teachers. They know the direction they need to take and how best to get there.

DESCRIPTIONFirst Year is Elementary Algebra. It deals with life situations and problems involving measurement, real number system, algebraic expressions, fi rst degree equations and inequalities in one variable, linear equations in two variables, special products and factoring.

Second Year is Intermediate Algebra. It deals with systems of linear equations and inequalities, quadratic equations, rational algebraic expressions, variation, integral exponents, radical expressions, and searching for patterns in sequences (arithmetic, geometric, etc) as applied in real-life situations.

Third Year is Geometry. It deals with the practical application to life of the geometry of shape and size, geometric relations, triangle congruence, properties of quadrilaterals, similarity, circles, and plane coordinate geometry.

Fourth Year is still the existing integrated ( algebra, geometry, statistics and a unit of trigonometry) spiral mathematics but in school year 2003-2004 the graduating students have the option to take up either Business Mathematics and Statistics or Trigonometry and Advanced Algebra.

UNIT CREDITMathematics in each year level shall be given 1.5 units each.

TIME ALLOTMENTThe daily time allotment for Mathematics in all year levels is 60 minutes or 300 minutes weekly

EXPECTANCIES IN MATHEMATICSThe student will be able to compute and measure accurately, come up with reasonable estimate,

gather, analyze and interpret data, visualize abstract mathematical ideas, present alternative solutions to problems using technology, among others, and apply them in real-life situations.

ññAt the end of Third Year, the student is expected to demonstrate understanding and skills

in geometric relations, proving and applying theorems on congruence and similarity of triangles, quadrilaterals, circles and basic concepts on plane coordinate geometry.

ññAt the end of Second Year, the student is expected to demonstrate understanding of concepts

and skills related to systems of linear equations and inequalities, quadratic equations, rational algebraic expressions, variation, integral exponents, radical expressions and searching for patterns in sequences: arithmetic, geometric and others and apply them in solving problems.

ññAt the end of First Year, the student is expected to demonstrate understanding and skills in

measurement and use of measuring devices, performing operations on real numbers and algebraic expressions, solving fi rst degree equations and inequalities in one variable, linear equations in two variables and special products and factoring and apply them in solving problems.

SCOPE AND SEQUENCEELEMENTARY ALGEBRA (FIRST YEAR)

1. Measurement2. Real Number system3. Algebraic Expressions4. First Degree Equations and Inequalities in One Variable5. Linear Equations in Two Variables6. Special Products and Factors

INTERMEDIATE ALGEBRA (SECOND YEAR)

1. Systems of Linear Equations and Inequalities2. Quadratic Equations3. Rational Algebraic Expressions4. Variation5. Integral Exponents6. Radical Expressions7. Searching for Patterns in Sequences: Arithmetic, Geometry,etc.

GEOMETRY (THIRD YEAR)

1. Geometry of Shape and Size2. Geometric Relations3. Triangle Congruence4. Properties of Quadrilaterals5. Similarity6. Circles7. Plane Coordinate Geometry

SUGGESTED STRATEGIES AND MATERIALSStrategies in mathematics teaching include discussion, practical work, practice and consolidation, problem solving, mathematical investigation and cooperative learning.

DISCUSSION• It is more than the short question and answer which arise during exposition • It takes place between teacher and students or between students themselves.

PRACTICAL WORK• More student-centered activities• Teacher acts as facilitator• Concretizes abstract concepts• Develops students’ confi dence to discover solutions to problems

PRACTICE AND CONSOLIDATION• Develops mastery of a particular concept which is needed in problem solving and

investigation

PROBLEM SOLVING• Process of applying mathematics in the real world• Involves the exploration of the solution to a given situation

MATHEMATICAL INVESTIGATION• An open-ended problem solving• It involves the exploration of a mathematical situation, making conjectures and reason

logically

COOPERATIVE LEARNING• Members are encouraged to work as a team in exchanging ideas, successes and failures.

MATERIALS INCLUDE DEPED APPROVED TEXTBOOKS AND LESSON PLANS. FEATURES OF THE LESSON PLANS ARE:

• Application of higher order thinking skills• Integration of values education• Provision of teaching-learning activities that address multiple intelligences• Use of cooperative learning strategies

GRADING SYSTEMThe grade will be based on certain criteria weighted accordingly as follows:

PERIODICAL TEST 25%UNIT TEST 25%QUIZZES 20%PARTICIPATION 15%HOMEWORK HOMEWORK HOMEWORK 15% ______TOTAL 100%

DETAILED LISTING OF LEARNING COMPETENCIESHIGH SCHOOL MATHEMATICS

ELEMENTARY ALGEBRA (1ST YEAR HIGH SCHOOL)

A. Measurement1. Illustrate the development of measurement from the primitive to the present

international system of units

2. Use instruments to measure length, weight, volume, temperature, time, angle

3. Express relationships between two quantities using ratios

4. Convert measurements from one unit to another

5. Round off measurements; round off numbers to a given place (e.g. nearest ten, nearest tenth)

6. Solve problems involving measurement

B. Real Number System1. Describe the real number system: natural numbers, whole numbers, integers, rational

numbers, irrational numbers, real numbers1.1 Review operations on whole numbers1.2 Describe opposite quantities in real life; illustrate integers on the number line;

use integers to describe positive or negative quantities 1.3 Visualize integers and their order on a number line; represent movement along

the number line using integers1.4 Arrange integers in increasing/decreasing order1.5 Defi ne the absolute value of a number on a number line as distance from the

origin.1.6 Determine the absolute value of a number; solve simple absolute value equations

using the number line 1.7 Perform fundamental operations on integers: addition, subtraction, multiplication,

division; state and illustrate the different properties (commutative, associative,

distributive, identity, inverse)1.8 Defi ne rational numbers; translate rational numbers (both terminating and

repeating/non-terminating) from fraction form to decimal form and vice versa 1.9 Arrange rational numbers in increasing/decreasing order1.10 Review simplifi cation of and operations on fractions1.11 Review operations on decimals

2. Square roots of positive rational numbers2.1 Defi ne the square root of a rational number; approximate the square root of a

positive rational number2.2 Identify square roots which are rational and which are not rational (irrational

numbers)2.3 If the square root of a number is not rational, determine two integers or rational

numbers between which it lies2.4 Give examples of other irrational numbers2.5 Use knowledge related to signed numbers and square roots in problem-solving

C. Algebraic Expressions1. Defi ne constants, variables, algebraic expressions

2. Simplify numerical expressions involving exponents and grouping symbols

3. Translate verbal phrases to mathematical expressions and vice versa

4. Evaluate mathematical expressions for given values for the variable(s) involved

5. Defi ne monomials, binomials, trinomials and multinomials and illustrate these

6. Simplify monomials using the laws on exponents6.1 Identify monomials; identify the base, coeffi cient and exponent in a monomial6.2 Laws on exponents

•

•

•

•

•

where m – n is a positive number if m > n.

m – n is a negative number if m < n.

6.3 Simplify and perform operations on monomials6.4 Express numbers in scientifi c notation

7. Defi ne polynomials; classify algebraic expressions as polynomials and non-polynomials

8. Perform operations on polynomials8.1 Addition and subtraction8.2 Multiplication : polynomial by a monomial8.3 Multiplication : polynomial by another polynomial8.4 Division : polynomial by a monomial

Division : polynomial by a polynomial

9. Problem solving involving polynomials

D. First Degree Equations and Inequalities In One Variable1. Introduce fi rst degree equations and inequalities in one variable

1.1 Distinguish between mathematical phrases and sentences1.2 Distinguish between expressions and equations1.3 Distinguish between equations and inequalities

2. Translate verbal statements involving general or unknown quantities to equations and inequalities and vice versa

3. Defi ne fi rst degree equations and inequalities in one variable1.1 Defi ne the solution set of a fi rst degree equation or inequality1.2 Illustrate the solution set of equations and inequalities in one variable on the

number line1.3 Find the solution set of simple equations and inequalities in one variable from a

given replacement set1.4 Find the solution set of simple equations and inequalities in one variable by

inspection

4. Review the basic properties of real numbers; state and illustrate the different properties of equality

5. Determine the solution set of fi rst degree equations in one variable by applying the properties of equality

6. Determine the solution set of fi rst degree inequalities in one variable by applying the properties of inequality; visualize solutions of simple mathematical inequalities on a number line

7. Solve problems using fi rst degree equations and inequalities in one variable (e.g. relations among numbers, geometry, business, uniform motion, money problems,etc.)

E. Linear Equations in Two Variables1. Describe the Cartesian Coordinate Plane (x-axis, y-axis, quadrant, origin)

2. Describe points plotted on the Cartesian Coordinate Plane; plot points on the Cartesian Coordinate Plane2.1 Given a point on the coordinate plane, give its coordinates2.2 Given a pair of coordinates, plot the point2.3 Given the coordinates of a point, determine the quadrant where it is located

3. Defi ne a linear equation in two variables: Ax+By=C.3.1 construct a table of values for x and y given a linear equation in two variables,

Ax+By=C2.2 Draw the graph of Ax+By=C based on a table of values for x and y3.3 Defi ne x and y intercepts, slope, domain, range3.4 Determine the following properties of the graph of a linear equation

Ax + By = C : • Intercepts• Trend (increasing or decreasing)• Domain • Range• Slope

4. Given a linear equation Ax + By = C, rewrite in the form y = mx + b, and vice versa 4.1 draw the graph of a linear equation in two variables described by an equation

using

• the intercepts • any two points• the slope and a given point

4.2 determine whether the graph of Ax+ By = C is increasing or decreasing4.3 obtain the equation of a line given the following:

• the intercepts • any two points• the slope and a point

4.4 use linear equations in two variables to solve problems

ENRICHMENT FOR LINEAR EQUATIONS IN TWO VARIABLES:

5. Defi ne an absolute value equation

5.1 Review the meaning of the absolute value of a number5.2 Construct a table of ordered pairs and draw the graphs of the following:

• y =

• y = + b

• y = - b

• y =

• y =

• y = + c

F. Special Products and Factoring 1. Review multiplication of polynomials

1.1 monomial by polynomial – using the distributive property1.2 binomial by binomial – using the distributive property, using the FOIL method

2. Identify special products• Polynomials whose terms have a common monomial factor

• Trinomials which are products of two binomials• Trinomials which are squares of a binomial• Products of the sum and difference of two quantities

3. Factor polynomials• Polynomials whose terms have a common monomial factor• Trinomials which are products of two binomials• Trinomials which are squares of a binomial• Products of the sum and difference of two quantities

4. Given a polynomial, factor completely

ENRICHMENT FOR SPECIAL PRODUCTS AND FACTORING

5. Use special products and factoring to solve problems

INTERMEDIATE ALGEBRA (2ND YEAR HIGH SCHOOL)

A. Systems of Linear Equations and Inequalities1. Review the Cartesian Coordinate System

2. Review graphs of linear equations in two variables

3. Defi ne a system of linear equations in two variables

4. Solve systems of linear equations in two variables4.1 Given a pair of linear equations in two variables, identify those whose graphs

are parallel, those that intersect, and those that coincide4.2 Given a system of linear equations in two variables fi nd the solution of the

system graphically (i.e. by drawing the graphs and obtaining the coordinates of the intersection point)

4.3 Given a system of linear equations in two variables, determine whether or not their graphs intersect, and if they do, fi nd the solution of the system algebraically• By elimination• By substitution

5. Use systems of linear equations to solve problems (e.g. number relations, uniform motion, geometric relations, mixture, investment, work)

6. Review the defi nition of inequalities; defi ne a system of linear inequalities

6.1 Translate certain situations in real life to linear inequalities

6.2 Draw the graph of a linear inequality in two variables

6.3 Represent the solution set of a system of linear inequalities by graphing

B. Quadratic Equations

1. Defi ne a quadratic equation ; distinguish a quadratic equation from a linear equation

2. Find the solution set of a quadratic equation

1.1 Review the defi nition of solution set of an equation; defi ne “root of an equation” 1.2 Determine the solution set of a quadratic equation by algebraic

methods:• Factoring• Quadratic formula• Completing the square

2.3 derive the quadratic formula

3. Solve rational equations which can be reduced to quadratic equations

4. Use quadratic equations to solve problems

C. Rational Algebraic Expressions1. Review simplifi cation of fractions including complex fractions; review operations on

fractions

2. Defi ne a rational algebraic expression; domain of a rational algebraic expression; identify rational algebraic expressions; translate verbal expressions into rational algebraic expressions

3. Simplify rational algebraic expressions(reduce to lowest terms)

4. Add and subtract rational algebraic expressions4.1 Find the least common denominator4.2 Change two or more rational expressions with unlike denominators to those with

like denominators4.3 Simplify results

5. Multiply and divide rational algebraic expressions

6. Simplify complex fractions

7. Solve rational equations 7.1 Check for extraneous solutions

8. Solve problems involving rational algebraic expressions

D. Variation1. Defi ne the following:

• Direct variation• Direct square variation• Inverse variation• Joint variation

2. Identify relationships between two quantities in real life that are direct variations, direct square variations, inverse square variations or joint variations

3. Translate statements that describe relationships between two quantities using the following expressions to a table of values, a mathematical equation, or a graph, and vice versa• “_____ is directly proportional to _____”• “_____ is inversely proportional to _____”• “_____ varies directly as _____”• “_____ varies directly as the square of _____”• “_____ varies inversely as _____”

4. Solve problems on direct variation, direct square variation, inverse variation and joint variation

E. Integral Exponents1. Review concepts related to positive integer exponents

• The meaning of ax when x is a positive integer• Laws on exponents• Multiplying and dividing expressions with positive integral exponents

2. Demonstrate understanding of expressions with zero and negative exponents

1.1 Give the meaning of ax when x is 0 or a negative integer

1.2 Evaluate numerical expressions involving negative and zero exponents

1.3 Rewrite algebraic expressions with zero and negative exponents

1.4 Use laws of exponents to simplify algebraic expressions containing integral exponents

3. Review the use of scientifi c notation

4. Solve problems involving expressions with exponents

F. Radical Expressions1. Review roots of numbers

1.1 Identify expressions which are perfect squares or perfect cubes, and fi nd their square root or cube root respectively

1.2 Given a number in the form where x is not a perfect nth root, name two rational numbers between which it lies

2. Demonstrate understanding of expressions with rational exponents2.1 Use laws of exponents to simplify expressions containing rational exponents2.2 Rewrite expressions with rational exponents as radical expressions and vice

versa

3. Simplify radical expressions3.1 Identify the radicand and index in a radical expression3.2 Simplify the radical expression in such a way that the radicand contains no

perfect nth root3.3 Rationalize a fraction whose denominator contains square roots

4. Add and subtract radical expressions

5. Multiply and divide radical expressions

6. Solve radical equations

7. Solve problems involving radical equations

G. Searching for Patterns in Sequences, Arithmetic, Geometric and Others1. Demonstrate understanding of a sequence

1.1 List the next few terms of a sequence given several consecutive terms1.2 Derive, by pattern-searching, a mathematical expression (rule) for generating the

sequence

2. Demonstrate understanding of an arithmetic sequence2.1 Defi ne and give examples of an arithmetic sequence

2.2 Describe an arithmetic sequence by any of the following ways:• Giving the fi rst few terms• Giving the formula for the nth term• Drawing the graph

2.3 Derive the formula for the nth term of an arithmetic sequence2.3.1 Given the fi rst few terms of an arithmetic sequence, fi nd the common

difference and the nth term for a specifi ed n2.3.2 Given two terms of an arithmetic sequence, fi nd: the fi rst term; the

common difference or a specifi ed nth term2.4 Derive the formula for the sum of the n terms of an arithmetic sequence2.5 Defi ne an arithmetic mean; solve problems involving arithmetic means2.6 Solve problems involving arithmetic sequences

3. Demonstrate understanding of a geometric sequence 3.1 Defi ne and give examples of a geometric sequence3.2 Describe a geometric sequence in any of the following ways:

• Giving the fi rst few terms of the sequence• Giving the formula for the nth term• Drawing the graph

3.3 Derive the formula for the nth term of a geometric sequence3.3.1 Given the fi rst few terms of a geometric sequence, fi nd the common ratio

and the nth term for a specifi ed n3.3.2 Given two specifi ed terms of a geometric sequence, fi nd: the fi rst term;

the common ratio or a specifi ed nth term3.4 Derive the formula for the sum of the terms of a geometric sequence3.5 Derive the formula for an infi nite geometric series3.6 Defi ne a geometric mean; solve problems involving geometric means3.7 Solve problems involving geometric sequences

4. Defi ne a harmonic sequence, harmonic series, and harmonic mean4.1 Illustrate a harmonic sequence and determine the sum of the fi rst n terms4.2 Determine the harmonic mean of two numbers4.3 Solve problems involving harmonic sequences

5. Introduce the Fibonacci sequence; defi ne and illustrate the Fibonacci sequence

6. Introduce the Binomial Theorem6.1 State and illustrate the Binomial Theorem6.2 State and apply the formula for determining the coeffi cients of the terms in the

expansion of .

GEOMETRY (3RD YEAR HIGH SCHOOL)

A. Geometry of Shape and Size 1. Undefi ned Terms

1.1 Describe the ideas of point, line, and plane1.2 Defi ne, identify, and name the subsets of a line

• Segment• Ray

2. Angles1.1 Illustrate, name, identify and defi ne an angle1.2 Name and identify the parts of an angle1.3 Read or determine the measure of an angle using a protractor1.4 Illustrate, name, identify and defi ne different kinds of angles

• Acute• Right• Obtuse

3. Polygons1.1 Illustrate, identify, and defi ne different kinds of polygons according to the

number of sides• Illustrate and identify convex and non-convex polygons• Identify the parts of a regular polygon (vertex angle, central angle, exterior

angle)1.2 Illustrate, name and identify a triangle and its basic and secondary parts (e.g.,

vertices, sides, angles, median, angle bisector, altitude)1.3 Illustrate, name and identify different kinds of triangles and their parts (e.g.,

legs, base, hypotenuse)• classify triangles according to their angles and according to their sides

1.4 Illustrate, name and defi ne a quadrilateral and its parts1.5 Illustrate, name and identify the different kinds of quadrilaterals1.6 Determine the sum of the measures of the interior and exterior angles of a

polygon• Sum of the measures of the angles of a triangle is 180• Sum of the measures of the exterior angles of a quadrilateral is 360• Sum of the measures of the interior angles of a quadrilateral is• (n – 2)180

4. Circle4.1 Defi ne a circle4.2 Illustrate, name, identify, and defi ne the terms related to the circle (radius,

diameter and chord)

5. Measurements5.1 Identify the following common solids and their parts: cone, pyramid, sphere,

cylinder, rectangular prism)5.2 state and apply the formulas for the measurements of plane and solid fi gures

• Perimeter of a triangle, square, and rectangle• Circumference of a circle• Area of a triangle, square, parallelogram, trapezoid, and circle• Surface area of a cube, rectangular prism, square pyramid, cylinder, cone,

and a sphere• Volume of a rectangular prism, triangular prism, pyramid, cylinder, cone,

and a sphere

5.3 Solve problems involving plane and solid fi gures

B. Geometric Relations1. Relations involving Segments and Angles

1.1 Illustrate and defi ne betweeness and collinearity of points1.2 Illustrate, identify and defi ne congruent segments1.3 Illustrate, identify and defi ne the midpoint of a segment1.4 Illustrate, identify and defi ne the bisector of an angle1.5 Illustrate, identify and defi ne the different kinds of angle pairs

• Supplementary• Complementary• Congruent• Adjacent• Linear pair• Vertical angles

1.6 Illustrate, identify and defi ne perpendicularity1.7 Illustrate and identify the perpendicular bisector of a segment

2. Angles and Sides of a Triangle2.1 Derive/apply relationships among the sides and angles of a triangle

• Exterior and corresponding remote interior angles of a triangle• Triangle inequality

3. Angles formed by Parallel Lines cut by a Transversal3.1 Illustrate and defi ne Parallel Lines 3.2 Illustrate and defi ne a Transversal3.3 Identify the angles formed by parallel lines cut by a transversal3.4 Determine the relationship between pairs of angles formed by parallel lines cut

by a transversal• Alternate interior angles• Alternate exterior angles• Corresponding angles• Angles on the same side of the transversal

4. Problem Solving involving the Relationships between Segments and between Angles 4.1 Solve problems using the defi nitions and properties involving relationships

between segments and between angles

C. Triangle Congruence1. Conditions for Triangle Congruence

1.1 Defi ne and illustrate congruent triangles1.2 State and apply the Properties of Congruence

• Refl exive Property• Symmetric Property• Transitive Property

1.3 Use inductive skills to establish the conditions or correspondence suffi cient to guarantee congruence between triangles

1.4 Apply deductive skills to show congruence between triangles• SSS Congruence• SAS Congruence• ASA Congruence• SAA Congruence

2. Applying the Conditions for Triangle Congruence2.1 Prove congruence and inequality properties in an isosceles triangle using the

congruence conditions in 1.3• Congruent sides in a triangle imply that the angles opposite them are

congruent• Congruent angles in a triangle imply that the sides opposite them are

congruent• Non-congruent sides in a triangle imply that the angles opposite them are not

congruent

• Non-congruent angles in a triangle imply that the sides opposite them are not congruent

2.2 Use the defi nition of congruent triangles and the conditions for triangle congruence to prove congruent segments and congruent angles between two triangles

2.3 Solve routine and non-routine problems

EnrichmentApply inductive and deductive skills to derive other conditions for congruence between two

right triangles• LL Congruence• LA Congruence• HyL Congruence• HyA Congruence

D. Properties of Quadrilaterals1. Different type of Quadrilaterals and their Properties

1.1 Recall previous knowledge on the different kinds of quadrilaterals and their properties (square, rectangle, rhombus, trapezoid, parallelogram)

1.2 Apply inductive and deductive skills to derive certain properties of the trapezoid • Median of a trapezoid• Base angles and diagonals of an isosceles trapezoid

1.3 Apply inductive and deductive skills to derive the properties of a parallelogram• Each diagonal divides a parallelogram into two congruent triangles• Opposite angles are congruent• Non-opposite angles are supplementary• Opposite sides are congruent• Diagonals bisect each other

1.4 Apply inductive and deductive skills to derive the properties of the diagonals of special quadrilaterals• Diagonals of a rectangle• Diagonals of a square • Diagonals of a rhombus

2 Conditions that Guarantee that a Quadrilateral is a Parallelogram2.1 Verify sets of suffi cient conditions which guarantee that a quadrilateral is a

parallelogram2.2 Apply the conditions to prove that a quadrilateral is a parallelogram2.3 Apply the properties of quadrilaterals and the conditions for a parallelogram to

solve problems

EnrichmentApply inductive and deductive skills to discover certain properties of the Kite

E. Similarity1. Ratio and Proportion

1.1 State and apply the defi nition of a ratio1.2 Defi ne a proportion and identify its parts1.3 State and apply the fundamental law of proportion

• Product of the means is equal to the product of the extremes1.4 Defi ne and identify proportional segments1.5 Apply the defi nition of proportional segments to fi nd unknown lengths

2. Proportionality Theorems1.1 State and verify the Basic Proportionality Theorem and its Converse

3. Similarity between Triangles3.1 Defi ne similar fi gures3.2 Defi ne similar polygons3.3 Defi ne similar triangles 3.4 Apply the defi nition of similar triangles

• Determining if two triangles are similar • Finding the length of a side or measure of an angle of a triangle

3.5 State and verify the Similarity Theorems3.6 Apply the properties of similar triangles and the proportionality theorems to

calculate lengths of certain line segments, and to arrive at other properties

4. Similarities in a Right Triangle1.1 Apply the AA Similarity Theorem to determine similarities in a right triangle

• In a right triangle the altitude to the hypotenuse divides it into two right triangles which are similar to each other and to the given right triangle

1.2 Derive the relationships between the sides of an isosceles triangle and between the sides of a 30-60-90 triangle using the Pythagorean Theorem

EnrichmentState and verify consequences of the Basic Proportionality Theorem

• Parallel lines cut by two or more transversals make proportional segments• Bisector of an angle of a triangle separates the opposite side into segments whose

lengths are proportional to the lengths of the other 2 sides

State, verify and apply the ratio between the perimeters and areas of similar triangle

Apply the defi nition of similar triangles to derive the Pythagorean Theorem• If a triangle is a right triangle, then the square of the hypotenuse is equal to the sum

of the squares of the legs

5. Word Problems involving Similarity1.1 Apply knowledge and skills related to similar triangles to word problems

F. Circles1. The circle

1.1 Recall the defi nition of a circle and the terms related to it• Radius• Diameter• Chord• Secant• Tangent• Interior and exterior

2. Arcs and Angles2.1 Defi ne and identify a central angle2.2 Defi ne and identify a minor and major arc of a circle2.3 Determine the degree measure of an arc of a circle2.4 Defi ne and identify an inscribed angle2.5 Determine the measure of an inscribed angle

3. Tangent Lines and Tangent Circles3.1 State and apply the properties of a line tangent to a circle

• If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency

• If two segments from the same exterior point are tangent to a circle, then the two segments are congruent

4. Angles formed by Tangent and Secant Lines4.1 Determine the measure of the angle formed by the following:

• Two tangent lines• A tangent line and a secant line• Two secant lines

EnrichmentIllustrate and identify externally and internally tangent circlesIllustrate and identify a common internal tangent or a common external tangent

Geometric Constructions• Duplicate or copy a segment• Duplicate or copy an angle• Construct the perpendicular bisector and the midpoint of a segment

Derive the Perpendicular Bisector Theorem• Construct the perpendicular to a line

From a point on the lineFrom a point not on the line

• Construct the bisector of an angle• Construct parallel lines• Perform construction exercises using the constructions in 4.1 to 4.6• Use construction to derive some other geometric properties (e.g., shortest distance

from an external point to a line, points on the angle bisector are equidistant from the sides of the angle)

G. Plane Coordinate Geometry1. Review of the Cartesian Coordinate System, Linear Equations and Systems of Linear

Equations in 2 Variables1.1 Name the parts of a Cartesian Plane1.2 Represent ordered pairs on the Cartesian Plane and denote points on the Cartesian

Plane1.3 Defi ne the slope of a line and compute for the slope given the graph of a line1.4 Defi ne a Linear Equation1.5 Defi ne the y-intercept1.6 Derive the equation of a line given two points of the line1.7 Determine algebraically the point of intersection of two lines1.8 State and apply the defi nitions of Parallel and Perpendicular Lines

2. Coordinate Geometry2.1 Derive and state the Distance Formula using the Pythagorean Theorem2.2 Derive and state the Midpoint Formula 2.3 Apply the Distance and Midpoint Formulas to fi nd or verify the lengths of

segments and fi nd unknown vertices or points2.4 Verify properties of triangles and quadrilaterals using coordinate proof

3. Circles in the Coordinate Plane3.1 Derive/state the standard form of the equation of a circle with radius r and center

at (0,0) and at (h,k) 3.2 Given the equation of a circle, fi nd its center and radius3.3 Determine the equation of a circle given:

• Its center and radius• Its radius and the point of tangency with a given line

3.4 Solve routine and non-routine problems involving circles

MATH I : LINEAR EQUATIONS IN TWO VARIABLES

Competency E1. Describe the Cartesian Coordinate Plane (x-axis, y-axis, quadrant, origin)

Time Frame. 2 Sessions

Objectives:At the end of the sessions, the students must be able to:

1. Describe the Cartesian coordinate plane2. Given a point, describe its distance from the x or y axis3. Given a point on the coordinate plane, give its coordinates4. Given a pair of coordinates, plot the points5. Given the coordinates of a point, determine the quadrant where it is located

Development of the Lesson:A. Introduce the Cartesian coordinate plane using the number line. State that the rectangular

coordinate plane are also called Cartesian plane can be constructed by drawing a pair of perpendicular number lines to intersect at zero on each line.

B. Ask the students to describe the two lines and their point of intersection, to develop the following

ideas:The two number lines, which are perpendicular lines, are called coordinate axes.The horizontal line is called the x-axis.The vertical line is called the y-axisy-axis. The point where the two lines intersect is called the originorigin and is labeled 0 on both axes.The two axes divide the plane into four regions called quadrants: the fi rst, second, third and fourth quadrants in a counterclockwise direction.

SAMPLE LESSON PLANS

C. State that each point in the coordinate plane has corresponding distance from the y-axis and from the x-axis, that a pair of numbers is needed to tell how many units to the right or left of the y-axis and how many units above or below of the x-axis the point is located. The pairs of numbers will be the name of the point. This pair of numbers is called ordered pairordered pair.

D. Present the following examples and ask students to describe the distance of each point from the y or x-axis

1. If x = -2 answer: the point is 2 units to the left of the y-axis

2. If x = 0 answer: the point is in the y-axis

3. If x = 2 answer: the point is 2 units to the right of y-axis

4. If y = -3 answer: the point is 3 units below the x-axis

5. If y = 3 answer: the point is 3 units above the x-axis

Hence, the ordered pair (-2, 3) is located 2 units to the left of the y-axis and 3 units above the x-axis.

E. Let the student observe what the signs are of the coordinates of the points in the different quadrants. (Both positive in quadrant 1, negative-positive in II, negative-negative in III, and positive-negative in IV.)

F. State that in the ordered pair (x, y), x and y are called coordinates of the point. x is called the x-coordinate or abscissa and y is called the y-coordinate or ordinate.

Ask students to give the coordinates of each point pictured in the graph. e.g. A (3,2)

1. B ans. (5,6)2. C (-7,4)3. D (-4,5)4. E (1,0)5. F (0,-2)6. G (8,-4)7. H (9,3)8. I (-9,-3)

9. J -2

10. K

G. Ask the students to what quadrant each point is located. To see whether the students understand the concept, go over the exercises on _________.

H. Then proceed to the plotting of points by asking the students to locate the points in the plane whose coordinates are (3,5). State that the process of marking a point in a plane is called plotting plotting the pointsthe points.

I. Present the following exampleLocate the points P(-1,2), Q(2,3), R(-3,-4), S(3,-5) in the plane.

J. State that when an entire set of ordered pairs is plotted, the corresponding set of points in the

plane represents the graph of the set. Sometimes the points in the graph form a recognizable pattern, just like the example that follows:

Plot the points on the graph provided. Connect each point with the next one by a line segment in the order given.

1. (2,0) 6. (-3, -3) 11. (2, -7) 2. (2,6) 7. (-3. -7) 12. (3, -7) 3. (0,10) 8. (-2, -7) 13. (3, -3) 4. (-2,6) 9. (-1, -6) 14. (2, -2) 5. (-2, -2) 10. (1, -6) 15. (2,0)

To see whether the students understand the concept of plotting points, go over exercises on ___________.

Suggested Teaching Strategies:1. Provision for Life Skills or Higher Order Thinking Skills

- In plotting points, help the students to realize through several examples that every point on a vertical line has the same x-coordinate and every point on the horizontal line has the same y-coordinate.

- Cite instances where the use of the Cartesian plane is found. Assign student to observe and fi nd other applications of the plane.

2. Provision for Multiple Intelligences- To tap the visual/spatial intelligence of students ask them to draw pictures on a

graph paper using only lines. The students will then give the coordinates of the points where the lines intersect.

- To tap the interpersonal intelligence of the students, prepare a game of treasure hunting. Indicate in the treasure map the reference point and the locations or position of buildings, places. The whole group will work for a common goal-to fi nd the treasure.

3. Provision for Cooperative Learning- Prepare a group game on plotting of points. Done outside the classroom, ask the

students to serve as markers in plotting the set of points given to them. The fi rst group to plot the points correctly in the coordinate plane wins.

MATH I : SPECIAL PRODUCTS AND FACTORING

Competency F2. identify special products


Objective:At the end of the sessions, the students should be able to:

1. Identify the following special products:a. Square of a binomial,b. Difference of two squares,c. Sum or difference of two cubes.

Development of the Lesson:

A. Give the students a review of products of polynomials by going over the following exercises in class and asking the students to recite.

1. Product of a polynomial and a monomialFind the following products:

a. 2x(3x+4)=6xb.c.d.e.

2. Product of two binomialsUse the FOIL method to fi nd the following products:

a. b. c. d. e.

B. Start the study of special products with a discussion of squares of binomials.1. Let the students do the following exercise by pairs:

Find the following products:

a. b. c. d. e.

Answer the following questions: a. How many terms are there in each product? b. What do you observe about the fi rst and last terms of each product?

c. Observe the middle terms of the products. What do you notice about the numerical coeffi cient of the middle term and the constant in each factor?

C. Process the activity by going over the answers to the questions. State that these answers suggests the characteristics of a special product called a Perfect Square Trinomial (PST). Based on the exercise they just did, the students should be able to see that a PST results from multiplying a binomial with itself. In other words, a PST is a square of a binomial. Repeat the characteristics of a PST.

D. Test if the students would be able to identify perfect square trinomials by asking them to answer the exercises on page _____. (Note: The teacher may give exercises of the suggested form below:

Practice Exercise:

Identify whether the given trinomial is a PST or NOT. Write PST or NOT PST.

_____1. _____4. _____2. _____5. _____3.

E. Introduce the next special product by asking the students to fi nd the following products using the FOIL method.

1. 2. 3. 4. 5.

F. Let the students observe the product in each case. (The products are all binomials; the operation in each one is subtraction; the terms are both perfect squares.) Ask them to describe what are the products of the outer terms and inner terms when they apply FOIL. (They are additive inverses of each other.) Present the special product called Difference of Two squares (DOTS). Summarize the characteristics of a difference of two squares and describe what factors result to DOTS.

G. For the development of the idea of a sum of two cubes or difference of two cubes, use the same strategy used to develop the idea of a difference of two squares. Let the students fi nd the products of pairs of factors which result to a sum of two cubes and factors which result to a difference of two cubes. Ask the students to observe the products and what are common to these products. Explain that these are special products because they can be easily obtained by inspecting the factors without having to do the multiplication process.

H. Assign the exercises on page ____.

Suggested Teaching Strategies:1. Provision for Integration of Content Areas in Language Teaching

- Go over the meaning of the following terms: polynomial, factor, product, and common factor.

2. Provision for Life Skills or Higher Order Thinking Skills- In introducing the special product PST, you may use a problem like, “What is the

area of a square whose side has a length of (x+6) meters?”

3. Provision for Cooperative Learning- Prepare a group puzzle on fi nding the products of binomials, including squares of

binomials and factors of DOTS. Let the students do the puzzle in groups of 5 or 6.

MATH I: SPECIAL PRODUCTS AND FACTORING

Competency F4. Given a polynomial, factor completely


Objective:At the end of the sessions, the students must be able to:

1. factor completely a given polynomial.

Development of the Lesson:A. Review factoring by giving 3 examples for each of the following cases: polynomials whose terms

have a common monomial factor, trinomials which are products of two binomials, perfect square trinomials, difference of two squares , and sum and difference of two cubes. Ask for volunteers to give the factors orally. After each case, state the technique used to determine the factors.

B. Present the following case:

Factor .

Ask the students to examine the polynomial and fi nd out what case it is. State that it is a trinomial but of 3rd degree so it is not the same as the trinomials we studied which are products of two binomials. Lead the students to see the common monomial factor.

Call the students’ attention to the trinomial factor. Ask them to examine it. They should realize that it is still factorable.

Present now the idea of a completely factored polynomial. Consider other examples.

1.

2. Let the students do the exercises on page _____.

C. Present a polynomial of the form ax+ay+bx+by

Challenge the students to factor completely. Let them investigate and discuss with a seatmate.

Discuss the technique of grouping the terms before factoring, using the given polynomial.

Ask the students to work on the following exercises.

1. 4xy+4x+3y+3 = (4xy+4x)+(3y+3) = 4x(y+1)+3(y+1) = (y+1)(4x+3)

2. ax+2a-bx-2b+cx+2c = (ax-bx+cx)+(2a-2b+2c) = x(a-b+c)+2(a-b+c) = (a-b+c)+(x+2)

Stress that in each case, the terms are grouped in such a way that a common factor appears in each group.

D. Consider other examples which involve factoring polynomials with more than two factors. Guide the students in factoring by asking them to examine each of the factors in every step of the solution.

1. Is still factorable? Do you see any common factor?

2.

Note: Ask the students to justify the following when the need comes up in the discussion.

a. Is equal to (x+y) ?b. Is equal to (x+y) ?

E. Give a practice set covering all cases of factoring polynomials.

Suggested Teaching Strategies:1. Provision for Cooperative Learning

- Prepare a group puzzle on factoring polynomials of different types. Ask the students to work on the puzzle in groups of 5 or 6, or in dyads.

2. Provision for Values Education and the Valuing Process- Try to bring out individual trials in life, then enumerate possible solutions on how to

overcome these trials in a gradual manner, then in an abrupt manner. Whichever way, these are possible solutions for the said trials.

MATH II: QUADRATIC EQUATIONS

Competency B1. Defi ne a quadratic equation ax2 + bx + c = 0; distinguish a quadratic equation from a linear equation.

Time Frame. 1 Session

Objectives :At the end of the session, the students must be able to :

1. defi ne, identify and give an example of a quadratic equation2. distinguish a quadratic equation from a linear equation

Development of the Lesson:A. Defi ne a quadratic equation as an equation of the form ax2 + bx + c = 0 where a,b and c are

constants and a 0.

Ask the students why the value of a should not be 0. Clearly, when a = 0, the equation is linear and not quadratic.

Cite some examples of quadratic equations like the following:

3x2 + 5x – 3 = 0 -9x2 = 10 (3x-7)(5+2x) = 0

B. Lead the students to distinguish between a linear equation and a quadratic equation by asking them to identify the linear equations and the quadratic equations from a given set of equations.

C. To check whether the students understood the lesson well, ask them to give examples of quadratic equations.

After the fi rst few examples, challenge them by asking for examples of quadratic equations where

a. b = 0

b. c = 0

c. b and c are both 0


- Prepare a group puzzle on distinguishing linear equations from quadratic equations. Then ask the students to work in groups of 4 or 5.


Competency B2. Review the defi nition of solution set of an equation; defi ne “root of an equation”

Time Frame. 1 Session

Objectives :At the end of the session, the students must be able to :

1. recall the defi nition of the solution set of an equation2. defi ne “root of an equation”


A. Ask the student to recall what the “solution set of an equation” means.Defi ne the solution set of an equation as the set of all values for the variable which will make the equation true.Below are examples :

Example 1 : The solution set of x+2= 0 is {-2} because –2 + 2 = 0.

Example 2 : The solution set of 3x = 1 is { } because 3( ) = 0.

Example 3 : The solution set of x - 49 = 0 is {7, -7} because(7) - 49 = 0 and (-7) - 49 = 0.

B. Then state that each element in the solution set of an equation is a root of the equation.Hence,

-2 is the root of the equation in Example 1.

is the root of the equation in Example 2. 7 and –7 are the roots of the equation in Example 3.

Ask the student to give the roots of some equations. Make sure that some equations and some are quadratic. Make sure that the quadratic equations you will give at this point can be solved by inspection.

C. Through the other examples in part B, proceed to lead the students to draw a conclusion about the number of roots a linear equation has and the number of roots a quadratic equation has.


- Give the student a puzzle which will allow them to practice how to fi nd the solution set of simple linear and quadratic equations.


Competency B2.3. Derive the quadratic formula


Objective : At the end of the sessions, the students must be able to derive the quadratic formula.

Development of the Lesson :A. To derive the quadratic formula ask the students to “solve” the general quadratic equation ax

+ bx + c = 0 by competing the square. It may help to guide them using the steps on the left side below so that they can come up with the derivation as outlined on the right side.

1. Write the general form of a quadratic equation 1. ax + bx + c = 0

2. Multiply both sides of the equation by 4a 2. 4a x + 4abx + 4ac = 0

3. Subtract 4ac from both sides of the equation 3. 4a x + 4abx = - 4ac

4. Add to both sides of the equation a term which makes the left side a perfect square trinomial 4. 4a x + 4abx + b = b - 4ac

5. Express the left side as a square of a binomial 5. (2ax + b) = b - 4ac

6. Extract the left side as a square of a binomial 6 2ax + b =

7. Add -b to both sides of the equation 7. 2ax = - b =

8. Divide both sides by 2a 8. x = 2a

B. Ask the student to multiply both sides of the equation by a or 9a instead of 4a in the second step and carry out the derivation process. Find out if they are getting the same results. Draw a conclusion about the term which may be used to multiply the equation with, in the second step to carry out the derivation of the quadratic formula.

C. Ask the students to rewrite the quadratic formula as

x = 2a and to memorize this. Tell the students that b - 4ac is called the “discriminant”. Show how

they may use the discriminant to determine whether a given quadratic equation has:a. equal or unequal rootsb. real or imaginary rootsc. rational or irrational roots

Suggested Teaching Strategies:1. Provision for Multiple Intelligence

- To tap the interpersonal intelligence of the students, ask them to explain at the end of the lesson, to a partner, the process of deriving the quadratic formula.

MATH III : POLYGONS

Competency 3.1. Illustrate, identify and defi ne different kinds of polygons according to the number of sides

• illustrate and identify convex and non-convex polygons• identify the parts of a regular polygon (vertex angle,

central angle, exterior angle)

Time Frame: 1 Session

Objectives:At the end of the session, the students must be able to:

1. Defi ne and identify different kinds of polygons. 2. Illustrate and identify convex and non-convex polygon.3. Identify the parts of a regular polygon.


A. Show illustrations of different kinds of polygons. Let the students study the fi gures then ask them how these were formed. Lead them to the concept that polygons are made of segments intersecting at its endpoints. Also, no two of its segments with common endpoint are collinear.

B. Ask the students to count the number of vertices, sides and angles. Supply a name for each example. Clarify that polygons are named according to the number of sides.

Number of Sides Polygons3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon12 dodecagon

n-sides n-gon

C. Show illustrations of two kinds of polygons like the ones below.

Ask students to extend the sides. Focus on lines FE and ED. Below will be the result

Ask students like- What happens to the polygon when the line was formed?- Are all the other vertices of the polygon located on one side of the half-plane?

Answers will lead to the defi nition of convex and non-convex polygons. Be sure that the students will be able to distinguish that a polygon is a convex if no two points of a polygon lie on the opposite sides of a line containing any side of the polygon.

D. Show to the students the following fi gures in order to come up with the defi nition of a regular polygon.

Help the students defi ne what a regular polygon is.

E. After defi ning a regular polygon, discuss and identify the parts of the regular polygon

Attempt to defi ne these parts with students.

F. Tell the students that such kind of polygon is regular, let them formalize the defi nition.

G. Let the students identify the parts of regular polygon. Sides can be extended to name the exterior angles.

H. Provide practice exercises.

Suggested Teaching Strategies:1. Provision for Multiple Intelligence

- to tap the verbal/linguistic intelligence of the students, encourage them to cite their observations in the discussion in Part D.

- to tap the interpersonal intelligence, allow them to discuss their observations with the discussion in Part D with a seat mate.

MATH III: POLYGONS

Competency 3.3. Illustrate, name, and identify different kinds of triangles and their parts (e.g. legs, base, hypotenuse)


Objectives:At the end of the sessions, the students must be able to:

1. name and identify different kinds of triangle. 2. classify triangles according to sides and according to angles. 3. name and identify parts of a right triangle.

Development of the Lesson:A. Distribute 3 pieces of cut out triangles to the students. Let them measure the sides of the triangle.

Ask students the following questions:• What have you noticed about the sides of triangle A?• How will you distinguish triangle A from triangle B and C?• What is the difference between triangle B and C?• What are the properties of triangle A? triangle B? triangle C?

State the following:An equilateral triangle is a triangle with all sides congruent.An isosceles triangle is a triangle with exactly two sides congruent.A scalene triangle is a triangle with no sides congruent

Further ask the student the following questions.What kind of triangle is triangle A? triangle B? triangle C?What is the basis of classifi cation for these triangles?To see whether the student understand the classifi cation of triangles according to sides, let them

answer the exercises on _____________________.

B. Review the kinds of angles; acute, right and obtuse. Let them identify the kinds of angles from the chart.

Distribute cut out triangles to the students. (Prepare 3 triangles : acute, right and obtuse triangles)

Let them measure the angles of the triangles.

Ask the student the following questions:• What have you noticed about the measure of the angles of the triangle?• If you will group the triangles; how will you do it? Explain your answer.• What is your basis of classifi cation of the triangles?

State the following: Triangles can be classifi ed according to the measure of their angles?1.An acute triangle is a triangle with all three angles acute. 2.A right triangle is a triangle with one right angle.3.An obtuse triangle is a triangle with an obtuse angle.

To see whether the student understand the classifi cation of triangles according to sides, let them answer the exercises on _____________________.

C. Let the student observe the fi gures on the chart.

Let the student identify the kinds of triangles in the chart and describe the characteristics of the two triangles.

Explain to the students that in an isosceles triangle:

The two congruent sides are called LEGS, the third side is called the BASE, the angles on the base are called BASE ANGLES.

Explain further that in a Right Triangle the sides that are perpendicular are the legs and the side opposite the right angle is the hypotenuse.

Show the illustration to help the students visualize these parts.

Give some more fi gures then ask the students to identify the legs, hypotenuse, base and the base angles.

Suggested Teaching Strategies:

1. Provision for Higher Order Thinking Skills- In classifying triangles, conduct an activity where the students can compare and

identify the different kinds of triangles.

2. Provision for Cooperative Learning - Prepare cut out triangles which students can classify as well as discuss their basis for

classifi cation. This can be done in groups.