Notes, summaries, assignments, exams, and problems for Mathematics

Levels of Number Understanding

Level 1

At this level, number figures are symbols linked to real-world objects.

Level 2

Children try to find correspondence between written figures and quantitative aspects of objects.

Level 3

Numbers, especially single-digit numbers, represent counts of objects. Two-digit numbers are not yet seen as composed of constituent figures.

Level 4

Two-digit numbers systematically represent the total quantity, but each digit is treated as a separate number without understanding place value. For example, in 16, '6' might represent six objects and '1' might represent one object. Alternatively, '6' might represent sets of 6, and '1' might relate to a single unit or group.

Level 5

Each digit in a two-digit number represents an amount... Continue reading "Key Concepts in Math Education" »

Fundamental Geometric Concepts

Basic Geometric Elements

• Line: A straight path without beginning or end, extending infinitely in both directions.
• Ray: A straight path that has a beginning point but no end, extending infinitely in one direction.
• Segment: A straight path delimited by two distinct endpoints.
• Angle: The opening formed by two rays that share a common starting point (vertex).

Relationships Between Lines

• Intersecting Lines (Secant): Lines that cross each other at a single point.
• Parallel Lines: Lines that lie in the same plane and never intersect, having nothing in common.
• Coincident Lines: Lines that occupy the exact same position, sharing all points in common.
• Perpendicular Lines: Lines that intersect to form four equal (90-degree) angles,

Exercise 1: Calculating Monthly Interest Rate

A person invested $12,000 in an institution and received $13,008.00 after seven months. What is the equivalent monthly interest rate that the investor earned?

Solution:

• Initial Capital (C) = $12,000
• Time (n) = 7 months
• Future Value (FV) = $13,008.00

Using the formula: FV / C = (1 + i * n)

13,008.00 / 12,000.00 = (1 + i * 7)

1. 0840 = 1 + 7i

2. 0840 = 7i

i = 0.0840 / 7

i = 0.012 or 1.2% per month

Exercise 2: Equivalent Financial Return

An investment of $15,000 is made for a period of three months at a simple interest rate of 26% per annum. What amount must be invested for two months at a linear rate of 18% per year to achieve the same financial return?

Solution:

Investment 1:

• C = $15,000
• n = 3 months
• i (annual)

Direct Methods

1a) Gauss Elimination Method

The Gauss Elimination method consists of converting a matrix into an equivalent matrix with zeros below the main diagonal of A. The first equation will have n variables, the second n-1, and so on until the final equation, which will have only one variable. Once the zeros are achieved, the values of the variables are found by starting with the last variable and substituting back up to the first. This solves the system.

1b) Gauss-Jordan Method

The Gauss-Jordan method consists of obtaining a diagonal matrix instead of just an upper triangular matrix. Variables are obtained directly from the resulting system, without substitutions. This saving in the final step comes at the cost of slightly increasing the... Continue reading "Direct and Indirect Methods: Gauss and Jacobi" »

Essential Language and Literary Concepts

Vocabulary Definitions

• Ratification: To confirm something is true or valid.
• Sullen: Bad-tempered and sulky; gloomy.
• Persuade: To induce someone to do something through reasoning or argument.
• Obey: To comply with a command, instruction, or law.
• Prescribed: To lay down as a rule or course of action to be followed; to determine.
• Literal: Taking words in their most basic sense without metaphor or allegory.
• Prolific: Producing much fruit, foliage, or many offspring; producing many works.
• Clause: A distinct article or provision in a document, such as a contract or law.
• Maize: A cereal grain, especially corn, that is already mature.
• Diligence: Careful and persistent work or effort.
• Ninety: The cardinal number equivalent

Parties Involved in a Bill of Exchange

The issuance and circulation of a bill of exchange involves the following parties:

• The Drawer: The creditor who issues the bill of exchange, instructing the debtor to pay.
• The Drawee: The debtor who is obligated to pay the bill of exchange at maturity. The drawee may accept or reject the payment order. If accepted, the drawee becomes the acceptor.
• The Payee/Holder/Beneficiary: The person who holds the bill of exchange and is entitled to receive payment.

Form of Bills of Exchange

A bill of exchange must be issued on official forms or stamped paper issued by the State. The amount should be proportional to the value stated on the bill. Improper formatting may cause difficulties in pursuing action against the debtor... Continue reading "Understanding Bills of Exchange: Parties, Form, and Accounting" »

Item 12

What Does the Acronym ICT Stand For?

ICT stands for Installation of Telephone and Common Infrastructure.

ICT Networks in a Building

• Food Network
• Distribution Network
• Dispersion Network
• Interior User Network

Elements Inside a RITI or RITS

The interior of a RITI or RITS contains the following elements:

• Point of Interconnection: Consists mainly of an input terminal strip where multi-pair cables from each operator arrive, and output strips where they connect the cables that go toward secondary multi-pair registers.
• End-point Distribution: A set of TLCA distributors, and if required, an amplifier. It is responsible for carrying the TLCA signal to the building's vertical risers and amplifying the signal to reach adequate quality levels.
• Elements of electrical

Time Study in Work Measurement

The study of time as one of the major work measurement techniques was formalized by Taylor.

Technical Insolvency

A firm is in a state of technical insolvency when the economic profit generated by its assets is not sufficient to meet the interest on its debts.

Quota Sampling

Quota sampling falls under the category of non-probability sampling.

Commercial Experimentation Technique

Analysis of Variance

The technique used in most commercial experimentation is Analysis of Variance.

Strategic Behavior and Social Groups

The set of shared values of a social group determines your preference for a particular type of strategic behavior.

Investment Risk

If a certain level of risk exists between two alternative investments, preference... Continue reading "Time Study, Sampling, and Business Management Concepts" »

Quadrilaterals

A quadrilateral is a closed plane figure, a polygon, consisting of four sides and four angles.

A diagonal is a straight line segment joining two non-consecutive vertices.

Types of Quadrilaterals

Square

A square has four equal sides and four 90-degree angles. Its diagonals are equal in length and perpendicular to each other.

Rectangle

A rectangle has opposite sides that are equal and parallel. All its angles are 90 degrees. Its diagonals are equal in length and are oblique (not perpendicular).

Rhombus

A rhombus has four equal sides, with opposite sides parallel. Its diagonals are not equal in length but are perpendicular to each other.

Parallelogram (Rhomboid)

A rhomboid (or parallelogram) has opposite sides that are equal and parallel.... Continue reading "Geometric Shapes: Quadrilaterals, Polygons, and Proportions" »