Introduction

In previous lessons on quadratic equations and the discriminant, we learned how to find roots by factoring, completing the square, and using the quadratic formula. The roots of a quadratic equation may be different, equal, real, or imaginary.

Consider this problem. Suppose there are two consecutive even numbers whose product is 224. If the first number is $x$, then the second is $x+2$. The mathematical model is:

Introduction

A sphere is one of the most familiar curved-surface solids in everyday life. Footballs, marbles, and many round containers are good examples. This article covers the elements of a sphere, its surface area, its volume, and how its volume changes when the radius changes.

A. Basic Competencies

3.7 Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.
4.7 Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.

Introduction

In a quadratic equation, there is an important component called the discriminant. The discriminant is used to determine the type of roots of a quadratic equation or quadratic function.

It is denoted by $D$ and defined as:

where $a$, $b$, and $c$ are the real constants in the quadratic equation:

Introduction

Consider the following problem.

A bathroom wall has a rectangular shape and will be covered with square tiles. The length of the wall is 5 tiles longer than its width. If 300 tiles are needed to cover the wall, determine the length of the wall. Assume the area of one tile is 1 square unit.

Introduction

Irrational numbers are closely related to radical forms. Numbers such as $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ are examples of irrational numbers written in radical form.

Introduction

This page summarizes Grade 9 mathematics materials for junior secondary school based on the revised 2013 curriculum. In general, the material is divided into 6 main chapters.

Main Chapters

1. Exponents and Radical Forms
2. Quadratic Equations
3. Quadratic Functions
4. Geometric Transformations
5. Similarity and Congruence
6. Curved Surface Solids

Detailed Topics

Chapter 1. Exponents and Radical Forms

1. Integers raised to integer powers
2. Fractions raised to integer powers
3. Radical forms and fractional exponents
4. Algebraic operations on radical forms
5. Rationalizing denominators

Chapter 2. Quadratic Equations

1. Definition of quadratic equations
2. Roots of quadratic equations
3. Constructing quadratic equations

Chapter 3. Quadratic Functions

1. Quadratic functions
2. Axis of symmetry and discriminant
3. Graphing quadratic functions
4. Applications of quadratic functions

Chapter 4. Geometric Transformations

1. Translation
2. Reflection
3. Rotation
4. Dilation

Chapter 5. Similarity and Congruence

1. Similarity of plane figures
2. Similar triangles
3. Congruent triangles
4. Finding side lengths and angle measures in congruent triangles

Chapter 6. Curved Surface Solids

1. Surface area of cylinders, cones, and spheres
2. Volume of cylinders, cones, and spheres
3. Changes in the volume of cylinders, cones, and spheres

Note

The materials on this blog are presented as summaries, illustrations, example problems, GeoGebra simulations, and interactive evaluations. They are intended to support understanding, not replace the main textbook.

Introduction

This lesson discusses two important geometry ideas: similarity and congruence. Both are used to compare figures based on shape, size, side relationships, and angle relationships.

Basic Competencies

1. Explain and determine similarity and congruence among plane figures.
2. Solve problems related to similarity and congruence among plane figures.

Learning Objectives

After studying this material, students are expected to be able to:

Introduction

A cone is a solid with one circular base and one apex. This lesson covers the elements of a cone, its slant height, its net, surface area, volume, and several common example problems.

A. Basic Competencies

3.7 Generalize the surface area and volume of curved solids, especially cylinders, cones, and spheres.
4.7 Solve contextual problems involving the surface area and volume of curved solids and combinations of those solids.

Introduction

Curved-surface solids include cylinders, cones, and spheres. This article focuses on the cylinder, a solid formed by two parallel congruent circles and one curved lateral surface.

A. Basic Competencies

3.7 Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.
4.7 Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.

Introduction

Learning Objectives

After studying dilation, students are expected to be able to:

1. Explain dilation as a geometric transformation in contextual problems.
2. Solve contextual problems involving dilation.

Material

Digital cameras often provide a zoom feature that enlarges or reduces an image. This idea is closely related to dilation in geometry.