Numbers and Number
Sense The learner demonstrates understanding of key concepts of sets and the real number system.
The learner is able to formulate intellectually challenging
situations involving sets and real numbers and solve these in a variety of strategies
The learner…
describes and illustrates well-defined sets, subsets, universal sets and the null set and cardinality of sets
describes the union and intersection of sets and the complement of a set.
uses Venn Diagrams to represent sets, subsets and set operations
solves problems involving sets.
describes and illustrates the absolute value of a number on a numberline as the distance of a number from 0.
performs fundamental operations on integers.
states and illustrates the different properties of operations on the sets of integers.
illustrates rational numbers and arrange them on a numberline.
expresses rational numbers (both repeating and terminating/repeating and non-terminating) from fraction form to decimal form and vice versa.
performs operations on rational numbers and illustrate their properties.
describes principal roots and tells whether they are rational or irrational.
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 53
determines between what two integers the square root of a number is.
estimates the square root of a number to the nearest tenth.
illustrates and graphs irrational numbers (square roots) on a numberline with and without technology.***
describes, represents and compares the different subsets of real numbers.
finds the union, intersection and complement of the set of real numbers and its subsets.
arranges real numbers in increasing or decreasing order.
determines the significant digits in a given situation.
writes very large and very small numbers in scientific notation.
describes and represents real-life situations which involve integers, rational numbers, square root of rational and irrational numbers.
solves problems involving real numbers.
Algebra The learner demonstrates understanding of key concepts algebraic
expressions, the properties of real numbers as applied in linear equations and
inequalities in one variable.
The learner is able to model situations using oral, written, graphical and algebraic methods in solving problems involving algebraic expressions, linear equations and inequalities in one variable.
The learner …
o translates verbal phrases to mathematical phrases and vice versa.
o differentiates between constants and variables in a given algebraic expression.
o evaluates algebraic expressions for given values of the variables.
o gives examples of algebraic expressions which are polynomials and classifies these
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 54 as to number of terms.
o Interprets the meaning of an where n is a positive integer.
o adds and subtracts polynomials
o interprets the meaning of an where n is a positive integer.
o multiplies and divides polynomials o derives the laws of exponent.
o finds inductively using models the (a) product of two binomials; (b) product of the sum and difference of two terms; (c) square of a binomial; (d) cube of a binomial; (e) product of a binomial and a trinomial.
o finds algebraically the (a) product of two binomials; (b) product of the sum and difference of two terms; (c) square of a binomial; (d) cube of a binomial; (e) product of a binomial and a trinomial. ***
o differentiates between mathematical expressions and mathematical equations.
o translates English sentences to
mathematical sentences and vice versa.
o differentiates between equations and inequalities.
o illustrates the meaning of absolute value.
o finds the solution of an equation or inequality involving one variable, including one that involves absolute value (a) from a given replacement; (b) intuitively by guess and check; (c) by algebraic procedures (applying properties of equalities and inequalities); (d)
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 55 graphing.
o solves problems that use equations and inequalities.
Measurement The learner demonstrates understanding of the key concepts of measurement
The learner is able to formulate real-life problems and solve these using a variety of strategies.
The learner …
o describes what it means to measure.
o describes the development of measurement from the primitive to the present international system of units.
o estimates or approximates the measures of quantities particularly length , weight/mass, volume, time, angle and temperature.
o uses appropriate instruments to measure quantities such as length, weight/mass, volume, time, angle and temperature.***
o converts measurements from one unit to another for each type of measurement including the English system.***
o solves problems involving measurements such as perimeter, area, weight/mass, time, speed, temperature, volume/capacity and utilities usage.***
Geometry The learner demonstrates understanding of key concepts of geometry of shapes and sizes, and geometric relationships.
The learner is able to create artistically models of plane figures, formulate and solve with accuracy authentic problems involving geometric shapes and sizes.
The learner ..
o presents point, line and plane using concrete and pictorial models.
o names the subsets of a line.
o illustrate and classify the different kinds of angles.
o derives relationships of geometric figures using measurements and by inductive reasoning; supplementary angles, complementary angles, equal angles,
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 56 vertical angles, adjacent angles, linear pairs, perpendicular lines and parallel lines.***
o derives relationships between vertical angles and among angles formed by parallel lines cut by a transversal using measurement and by inductive reasoning.
o uses a compass and straight edge to bisect line segments and angles and construct perpendiculars and parallels.
o classifies triangles according to their angles and according to their sides.
o illustrates and names different kinds of triangles.
o derives relationships among the sides and angles of a triangle using measurement and inductive reasoning.
o illustrates and names different kinds of quadrilaterals.
o derives relationships among the sides and angles of a quadrilateral using
measurement and inductive reasoning.
o illustrates convex polygons.
o derives the relationship of exterior and interior angles of any convex polygon using measurement and inductive reasoning.
o illustrates a circle and the terms related to it; radius, diameter, center, arc and central angle.
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 57 Statistics and Probability The learner demonstrates
understanding of key concept, uses and
importance of Statistics, data collection/gathering and the different forms of data representation.
The learner is able to collect and organize data
systematically and make the necessary representation to solve problems in research, business, education, technology, science, economics, etc.
The learner …
o explains the basic concepts, uses and importance of Statistics.
o poses questions and problems that may be answered using Statistics.
o collects or gathers statistical data and organizes the data in a frequency table according to systematic consideration.
o uses appropriate graphs to represent organized data: pie chart, bar graph, line graph and a histogram.***
o analyzes, interprets accurately and draws conclusions from graphic and tabular data.
*** Suggestion for ICT enhanced lesson when available and where appropriate
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 58 GRADE 8
CONTENT CONTENT STANDARDS PERFORMANCE
STANDARDS
LEARNING COMPETENCIES
Algebra The learner demonstrates understanding of key concepts of special products and factors of polynomials, rational algebraic expressions, algebraic expressions with integral exponents, linear functions, and systems of linear equations and inequalities in two variables.
The learner is able to formulate real-life problems involving special products and factors, rational algebraic expressions, algebraic expressions with integral exponents, linear functions, and systems of linear equations and inequalities in two variables and solve these with utmost accuracy using a variety of strategies.
The learner…
identifies polynomials which are special products: polynomials with common monomial factors, trinomials that are product of two binomials, trinomials that are product of squares of a binomial, and products of sum and difference of two terms.***
finds special products and factors of certain polynomials: product of two binomials, product of sum and difference of two terms, square of a binomial, cube of a binomial, and product of special case of multiplying a binomial with a trinomial.
factors completely different types of polynomials (polynomials with common monomial factors, a difference of two squares, sum and difference of two cubes, perfect square trinomials, general trinomials) using special formulas, grouping and other techniques.
solves problems involving polynomials and their products and factors.
describes and illustrates rational algebraic expressions.
simplifies rational algebraic expressions
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 59
performs operations on rational algebraic expressions.
simplifies complex fractions.
solves problems involving rational algebraic expressions.
interprets zero and negative exponents
evaluates and simplifies algebraic
expressions involving integral exponents.
describes and illustrates the rectangular coordinate system and its uses.***
describes and plots positions on the
coordinate plane using coordinate axes. ***
Illustrates a relation and a function.
determines if a given relation is a function using ordered pairs, graphs and equations.
differentiates between dependent and independent variables.
describes the domain and range of a function.
describes a linear function using its points, equation and graph.
identifies the domain and range of a linear function.
illustrates the meaning of a slope of a line.
finds the slope of a line given two points, equation and a graph.
writes the linear equation ax + by = c in the form y = mx + b and vice versa.
graphs a linear equation given (a) any two
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 60 points; (b) the x – and y – intercepts; (c) the slope and a point on the line.***
describes the graph of a linear equation in terms of its intercepts and slope.***
finds the equation of a line given (a) two points; (b) the slope and a point; (c) the slope and its intercepts.
solves problems involving linear functions and linear patterns.
differentiates between mathematical expressions and mathematical equations.
differentiates between mathematical equations and inequalities.
illustrates linear inequalities in two variables.
graphs linear inequalities in two variables on the coordinate plane.
describes systems of linear equations and inequalities using practical situations and mathematical expressions.
identifies which given systems of linear equations have graphs that are parallel, intersect and coincide.
graphs systems of linear equations in two variables.***
solves systems of linear equations by (a) graphing; (b) elimination; (c) substitution.***
graphs system of linear inequalities in two variables.***
solves a system of linear inequalities in two
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 61 variables by graphing.***
solves problems involving systems of linear equations and inequalities in two variables.
Geometry The learner demonstrates understanding of key concepts of axiomatic development of geometry, triangle congruence, inequalities in triangles and parallel and perpendicular lines.
The learner is able to communicate mathematical thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems involving parallelism and perpendicularity of lines and congruent triangles using appropriate and accurate representations
The learner ..
o identifies the hypothesis and conclusions of if-then and other types of statements.
o formulates the inverse, converse and contrapositive of an implication.
o distinguishes between inductive and deductive reasoning.
o provides formal arguments that explain results of a phenomenon or a situation.
o writes formal arguments as a series of statements that make up a proof (both direct and indirect).
o illustrates the undefined terms in geometry.
o explains the need for defined terms previously introduced.
o differentiates between postulate and theory and give the importance of each.
o iIllustrates triangle congruence.***
o states and illustrates the SAS, ASA and SSS congruence postulates.***
o applies the postulates and theorems on triangle congruence to prove statements on congruence, including right triangles.
o applies triangle congruence to geometric constructions to perpendicular and angle bisector.
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 62 o states and illustrates the theorems on
triangle inequalities, e.g. exterior angle inequality theorem, triangle inequality theorem, hinge theorem.***
o applies theorems on triangle inequalities to:
(a) determine possible measures for the angles and sides of triangles, and (b) justify claims about the unequal relationships between side and angle measures.
o applies the theorems on triangle inequalities to prove results involving triangle inequalities.
o illustrates parallel and perpendicular lines.***
o illustrates and proves properties of parallel lines cut by a transversal.***
o names the different kinds of quadrilaterals and identify those that are parallelograms.
o determines the conditions that make a quadrilateral a parallelogram,
o determines and proves the conditions under which lines and segments are parallel or perpendicular.
o uses properties to find measures of angles, sides, and other quantities involving
parallelograms.
Statistics and Probability The learner demonstrates understanding of key concepts of measures of central tendency and measures of variability of data and basic concepts of probability
The learner is able to ..
o compute accurately measures of central tendency and measures of variability and apply these appropriately to data
The learner …
o finds the mean, median and mode of statistical data
o describes and illustrates the mean, median and mode of ungrouped and grouped data.
o discusses the meaning of variability.
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 63 .
analysis and interpretation in fields such as research, business, education, science, technology, economics, etc.
o calculates the different measures of
variability of a given set of data: (a) range; (b) average deviation; (c) variance; (d) standard deviation.***
o describes a set of data using measures of central tendency and measures of variability.
o use precisely counting techniques and probability in solving problems related to different fields of
endeavor.
o describes an experiment, outcome, sample space and event.***
o explains and interprets the probability of an event.
o differentiates between an experimental probability and a theoretical probability.
o counts the number of occurrences of an outcome in an experiment and organize them using a table, tree diagram, systematic listing and the fundamental counting
principle.***
o solves simple problems involving probabilities of events.
*** Suggestion for ICT enhanced lesson when available and where appropriate
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 64 GRADE 9
CONTENT CONTENT STANDARDS PERFORMANCE
STANDARDS
LEARNING COMPETENCIES
Algebra The learner demonstrates understanding of key concepts of quadratic functions and equations, rational algebraic equations and variations.
The learner is able to…
investigate thoroughly mathematical relationships in various situations and formulate real-life problems involving quadratic functions and equations, rational algebraic equations and variations and solve these using a variety of strategies.
The learner…
o makes mathematical models to represent real-life situations using quadratic functions.
o recognizes quadratic functions and quadratic equations.
o differentiates a quadratic function from a quadratic equation.
o represents and describes a quadratic function using its: (a) tables of values; (b) graph; (c) equation.
o finds the domain and range of a quadratic function.
o graphs quadratic functions and determines or describes its: (a) intercepts; (b) zeroes; (c) axis of symmetry; (d) maximum or minimum point; (e) shape.***
o writes the equation of a quadratic function given: (a) a table of values; (b) graph; (c) zeros.
o transforms the quadratic function y = ax2 + bx + c to y = a(x – h)2 + k.
o analyzes the effects of changing the values of a, h and k in the equation y = a(x – h)2 + k of a quadratic function on its graph.***
o solves quadratic equations by: (a) extracting square roots; (b) factoring; (c) completing the
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 65 square; (d) the quadratic formula.
o characterizes the roots of a quadratic equation using the discriminant.
o describes the relationship between the coefficients and the roots of a quadratic equation.
o solves equations transformable to quadratic equations.
o solves problems involving quadratic functions and quadratic equations.
o recognizes equations that involve rational algebraic expressions.
o solves equations involving rational algebraic expressions.
o solves problems involving equations with rational algebraic expressions
o recognizes relationships between two
quantities that involve the following variations:
(a) direct; (b) inverse; (c) joint; (d) combination of any of these.
o translates statements that involve variations to: (a) a table of values; (b) a mathematical equation; (c) a graph and vice versa.
o identifies situations that involve quantities that vary (a) directly; (b) inversely; (c) jointly; (d) according to a combination of any of these.
o solves problems involving variation.
create real-life problems involving expressions with rational exponents and
applies the laws involving positive integer exponents to zero and negative exponents.
simplifies expressions with rational
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 66 radicals and solve these
correctly.
exponents.
solves problems involving expressions with rational exponents.
identifies radical expressions.
relates expressions with fractional exponents to radical expressions.***
writes expressions with rational exponent as radicals and vice versa.
proves the laws of radicals.
simplifies radical expressions using the laws of radicals.
performs operations on radical expressions.***
solves equations and problems involving radical expressions.***
Geometry The learner demonstrates understanding of key concepts of triangle similarities and quadrilaterals.
The learner is able to … create, investigate, analyze
and solve mathematical situations involving similarity through appropriate and accurate representation.
The learner…
o describes a proportion.
o applies the fundamental theorems of proportionality to solve problems involving proportions.
o differentiates between triangle similarity and triangle congruence and relate one to the other.
o illustrates, explains, and proves the conditions for similarity of triangles. ***
a. SAS similarity theorem b. SSS similarity theorem c. AA similarity theorem
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 67 d. right triangle similarity theorem
e. special right triangle theorems o applies the theorems to show that given
triangles are similar.
o derives and proves the Pythagorean theorem.
o solves problems that involve: (a) triangle similarity; (b) Pythagorean theorem; (c) special right triangles.***
create various designs/
models of structures that utilizes the properties of quadrilaterals.
o describes the different quadrilaterals.
o determines the conditions that make a quadrilateral a parallelogram.
o uses properties of quadrilaterals to find measures of angles, sides and other quantities.
o proves theorems on the different kinds of parallelogram (rectangle, rhombus, square).
o proves theorems on trapezoids and kites.
o proves the midline and intercept theorems.
o applies the properties of quadrilaterals in real life.
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 68 Patterns and Algebra
(Trigonometry)
The learner demonstrates understanding of the basic concepts of trigonometry.
The learner is able to apply the concepts of trigonometric ratios to solve real-life problems with precision and accuracy.
The learner …
o defines the six trigonometric ratios: sine, cosine, tangent, secant, cosecant and cotangent.
o finds the trigonometric ratios of special angles.
o describes and illustrates angles of elevation and angles of depression.
o uses trigonometric ratios to solve real-life problems involving right triangles (e.g., navigation, surveying). ***
*** Suggestion for ICT enhanced lesson when available and where appropriate
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 69 GRADE 10
CONTENT CONTENT STANDARDS PERFORMANCE
STANDARDS
LEARNING COMPETENCIES
Patterns and Algebra The learner demonstrates understanding of key concepts of sequences and polynomial functions.
The learner is able to … create, analyze and solve
problems involving sequences and series in disciplines such as arts, music, science, business, agriculture, etc, through appropriate and accurate representations
The learner …
o finds and generates pattern.***
o describes and illustrates an arithmetic sequence.
o gives examples of arithmetic sequences.
o finds the terms of an arithmetic sequence including the general nth term of the sequence.***
o finds the sum of terms of a given arithmetic sequence.***
o describes and illustrates a geometric sequence.
o gives examples of geometric sequences.
o differentiates a finite geometric sequence from an infinite geometric sequence.
o differentiates an arithmetic sequence from a geometric sequence.
o finds the terms of a geometric sequence including the general nth term of the sequence.***
o finds the sum of terms of a given geometric sequence.***
o illustrates other types of sequences (e.g., harmonic, Fibonacci).
o solves problems involving sequences and series.
investigate thoroughly o describes polynomial functions.
K to 12 Curriculum Guide version as of December 2012 MATHEMATICS 70 mathematical relationships
involving polynomial functions in business, economics, science, etc. to maximize profits, minimize cost and as basis for decision making.
o identifies and gives examples of polynomial functions.
o differentiates polynomial functions from polynomial expressions.
o performs operations on polynomial expressions.
o uses synthetic division in dividing polynomial expressions.
o proves the remainder theorem and the factor theorem.
o factors polynomial expressions using
synthetic division, the remainder theorem and the factor theorem.
o finds the zeros of polynomial functions.
o sketches graphs of polynomial functions.***
o solves problems involving factors and zeros of polynomial functions.
Geometry The learner demonstrates understanding of key concepts of circles.
The learner is able to … create, analyze and find
solutions to intellectually challenging situations involving circles and other related terms in disciplines such as art, engineering, architecture, navigation, etc.
solutions to intellectually challenging situations involving circles and other related terms in disciplines such as art, engineering, architecture, navigation, etc.