Matrices

Types of Matrices

There are various types of matrices, depending on their structure. Let's explore the most common types:

Null Matrix

A matrix that has all 0 elements is called a null matrix. It can be of any order. For example, we could have a null matrix of the order 2 X 3. It's also a singular matrix, since it does not have an inverse and its determinant is 0. 

Any matrix that does have an inverse can be called a regular matrix.

Row Matrix

A row matrix is a matrix with only one row. Its order would be 1 X C, where C is the number of columns. For example, here's a row matrix of the order 1 X 5: 

Column Matrix

A column matrix is a matrix with only one column. It is represented by an order of R X 1, where R is the number of rows. Here's a column matrix of the order 3 X 1: 

Square Matrix

A matrix where the number of rows is equal to the number of columns is called a square matrix. Here's a square matrix of the order 2 X 2: 

Triangle Properties

Properties of Triangles

Triangles are three-sided closed figures. Depending on the measurement of sides and angles triangles are of following types:

• Equilateral Triangles: An equilateral triangle has all the sides and angles of equal measurement. This type of triangle is also called an acute triangle as all its sides measure 60° in measurement.
• Isosceles triangle: An isosceles triangle is the one with two sides equal and two equal angles.
• Scalene triangle: In a scalene triangle, no sides and angles are equal to each other.

Depending on angles, triangles are of following types:

• Acute Triangle: Triangles, where all sides are acute-angled to each other, are called acute triangles. The best example of this kind of triangle is the equilateral triangle.
• Obtuse Triangle: The obtuse angled triangle is the one with one obtuse angled side. Isosceles triangles and scalene triangles come under this category of triangles.
• Right Angled triangle: A triangle with one angle equal to 90° is called right-angled triangle.

Triangles Types

Types of Triangles

Equilateral Triangle

This type of triangle consists of three equal sides and equal angles. Every side of the triangle is of the same length and every angle will be of the same measure of 60°. The following figure is an equilateral triangle –

Isosceles triangle

The triangle with only two equal sides is known as the isosceles triangle. Not only two equal sides, the isosceles triangle also consists of two equal angles. The following figure is of the isosceles triangle-

Scalene Triangle

The triangle with no equal sides is the scalene triangle. Each line of this triangle is of different length. Following is the figure of the scalene triangle:

Geometry

Geometry is all about shapes and their properties.

If you like playing with objects, or like drawing, then geometry is for you!

Geometry can be divided into:

Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper

Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres.

Point, Line, Plane and Solid

A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)

Angles

Types of Angles

Coordinates

• Cartesian Coordinates
• Interactive Cartesian Coordinates
• Hit the Coordinate Game

Trigonometry

Trigonometry is a special subject of its own, so you might like to visit:

• Introduction to Trigonometry
• Trigonometry Index

Number Patterns

Types of Number Patterns in Math

Arithmetic Sequence

A sequence is group of numbers that follow a pattern based on a specific rule. An arithmetic sequence involves a sequence of numbers to which the same amount has been added or subtracted. The amount that is added or subtracted is known as the common difference. For example, in the sequence “1, 4, 7, 10, 13…” each number has been added to 3 in order to derive the succeeding number. The common difference for this sequence is 3.

Geometric Sequence

A geometric sequence is a list of numbers that are multiplied (or divided) by the same amount. The amount by which the numbers are multiplied is known as the common ratio. For example, in the sequence “2, 4, 8, 16, 32...” each number is multiplied by 2. The number 2 is the common ratio for this geometric sequence.

Square Numbers

In a square number sequence, the terms are the squares of their position in the sequence. A square sequence would begin with “1, 4, 9, 16, 25…”

Cube Numbers

In a cube number sequence, the terms are the cubes of their position in the sequence. Therefore, a cube sequence starts with “1, 8, 27, 64, 125…”

Triangular Numbers

The numbers in a sequence are referred to as terms. The terms of a triangular sequence are related to the number of dots needed to create a triangle. You would begin forming a triangle with three dots; one on top and two on bottom. The next row would have three dots, making a total of six dots. The next row in the triangle would have four dots, making a total of 10 dots. The following row would have five dots, for a total of 15 dots. Therefore, a triangular sequence begins: “1, 3, 6, 10, 15…”)

Techniques

How to Solve Math Problems Faster: 4 Techniques to Show Students

1. Squaring a Two-Digit Numbers that Ends with 5

Squaring numbers ending in 5 is easier, as there are only two parts of the process.

First, students will always make 25 the product’s last digits.

Second, to determine the product’s first digits, students must multiply the number’s first digit — 9, for example — by the integer that’s one higher — 10, in this case.

So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the result is 9,025.

Just like that, a hard problem becomes easy multiplication for many students

2. Calculating Percentages

Cross-multiplication is an important skill to develop, but there’s an easier way to calculate percentages.

For example, if students want to know what 65% of 175 is, they can multiply the numbers together and move the decimal place two digits to the left.

The result is 113.75, which is indeed the correct answer.

This shortcut is a useful timesaver on tests and quizzes.

3. Multiplying Odd Numbers by 5

This is another time-saving tactic that works well when teaching students the 5 times table.

This one has three steps, which 5 x 7 exemplifies.

First, subtract 1 from the number being multiplied by 5, making it an even number. Second, cut that number in half — from 6 to 3 in this instance. Third, add 5 to the right of the number.

The answer is 35.

Who needs a calculator?

4. Subtracting from 1,000

You can give students confidence to handle four-digit integers with this fast technique.

To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.

Let’s say students must solve 1,000 – 438. Here are the steps:

• 9 – 4 = 5
• 9 – 3 = 6
• 10 – 8 = 2
• 562

This also applies to 10,000, 100,000 and other integers that follow this pattern.

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