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

Measuring Angles

An angle can be measured in three ways:

1. In degrees: The circumference is divided into 360 equal parts. Each part represents an angle, with the apex at the center of the circle (sexagesimal measure), and is indicated by placing 1°.
2. In grads: The circumference is divided into 400 equal parts. Each part represents an angle, with the apex at the center of the circle, measuring a grad, and is indicated by placing 1^g.
3. In radians: An angle of one radian is an angle whose arc on the circumference has the same length as the radius of the circle, and is indicated by placing 1 rad.

Definition of Sine, Cosine, and Tangent

For an acute angle β in a right triangle:

• Sine (sin β): Defined as the ratio of the opposite leg to the hypotenuse.
• Cosine

Correlation Diagram: Analyzing Variable Relationships

Understanding Correlation Diagrams

A correlation diagram is a powerful tool that studies the relationship between two variables. It is used to understand if two variables are linked and to what extent they are.

Types of Variable Relationships

The relationship between variables can be:

• A direct cause-and-effect relationship.
• Multiple causes leading to one effect, or one cause leading to several effects.

It is represented by a graph with two axes, where the values of each variable are plotted on a respective axis. For the chart to be reliable, it requires at least 20 pairs of data. Once a sufficient number of points are plotted, you can determine whether a relationship exists between the variables.... Continue reading "Quality Management Tools: Correlation, Control Charts, and Poka-Yoke" »

Process Control Elements

Raw Materials Management

Raw materials required for the process must be as specified in the transaction records, with checks on their source, condition, quantity, and production/receipt date. All details should be properly registered.

Labor Force and Training

Every process, regardless of its degree of automation, involves a quantitative and qualitative workforce component, which may be more or less specialized. The workforce employed in a process must be educated and trained in operations. They must be able to distinguish a good product from one that is not, know how to react to non-conformities, and maintain relevant operational records. Personnel can range from trainees requiring strong supervision to highly experienced... Continue reading "Essential Elements of Process Control and Quality Assurance" »

Identification of Financial Statements

The financial statements shall be clearly identified and distinguished from any other information published in the same document.

Characteristics of Financial Statements

• Accuracy: Displaying information in a clear, precise, and true manner.
• Opportunity: Must be submitted on time.
• Relevance: Only present information unique to the company's operations.
• Integrity: The information submitted must be complete and not incomplete or unreliable.
• Frequency: Financial statements should be made with a fixed periodicity.

Notes to Financial Statements

Clarifications or explanations of events or situations, quantifiable or not, are an integral part of every financial statement. They must be read together for correct interpretation.... Continue reading "Financial Statements: Identification, Characteristics, and Inventory Accounting" »

Teaching and Learning Concepts

Concrete Bases (Teaching)

Convert mathematics into something concrete. Create recognizable contexts to which children can assign their own meanings.

Construction (Learning)

Learning is a constructive activity, contrary to the idea that children simply absorb knowledge presented to them.

Models (Teaching)

To achieve progress, students must have tools at their disposal that enable them to establish a link between informal and formal mathematics. Models and materials are important.

Raising the Level (Learning)

The learning process occurs at different levels of formality. Changes can be sudden and create discontinuity in the learning process.

Time for Reflection (Teaching)

The teacher has to find time to include reflection... Continue reading "Principles of Math Teaching and Learning" »

Secondary Data Analysis: Sources and Elaboration

Secondary data analysis considers both qualitative and quantitative perspectives. It uses references like coding sheets in quantitative designs and identifies interviews or focus groups for qualitative data. Official statistics and public sources are used to verify field data.

Desrosieres argues that statistical indicators from official sources reflect social reality in institutions. These indicators are presented without analysis, as tables of results. This means the original data has been elaborated, which is typical of secondary sources. Secondary sources have developed, summarized, and synthesized the data. The degree of elaboration is the key difference between secondary data analysis and... Continue reading "Secondary Data Analysis: Sources, Types, and Elaboration" »

Hurwicz Criterion: Coefficient of Optimism

In this approach, a decision-maker may have an intermediate attitude. Since no probabilities are associated with each state of nature, this author suggests using a coefficient of optimism, simultaneously called C, and a coefficient of pessimism, (1 - C), where 0 ≤ C ≤ 1.

The coefficient C indicates the decision-maker's stance toward risk. When it is closer to 1, the decision-maker will be more optimistic, while closer to 0, the decision-maker is more pessimistic.

This criterion focuses only on the extreme scores of each alternative, weighing the result of maximum optimism with the coefficient C and the result of minimal value with the coefficient of pessimism (1 - C). The sum of these two products... Continue reading "Decision-Making Under Uncertainty: Criteria and Tables" »

Fundamental Data Mining Tasks and Concepts

Prediction: Forecasting Time Series Data

Prediction involves estimating the future value of a variable that is subject to random changes over time. This technique is strictly applied to time series, which are datasets whose domain is time.

Regression Analysis: Modeling Relationships

Regression is a generalization of classification (when the domain involves continuous classes) and prediction. The resulting model (classification or prediction) depends on the significance of the dependent and independent variables.

The goal is to find a mathematical or statistical model that properly relates the dependent variable with the independent variables. Geometrically, regression seeks a function that passes as close... Continue reading "Fundamental Data Mining Tasks and Algorithmic Approaches" »

The Concept of Materiality in Auditing

The auditor assesses the amount of errors or omissions that could affect the decisions of a reasonable user. This concept involves:

• Flexible guidelines.
• Both quantitative and qualitative elements.

Stages in the Application of Materiality

1. Establish a Preliminary Judgment on Materiality (Planning)

   This is the maximum amount of misstatement the auditor believes can exist in the financial statements without affecting the decisions of reasonable users. It is affected by the relative size of the company being audited.

   First, the auditor establishes a base (e.g., total assets, profits, turnover), which is multiplied by a percentage factor to determine an initial quantitative judgment. This initial judgment is then adjusted

Cutting speed depends on mining velocity. The cutter's speed must remain constant throughout its journey through the ore flow.

Sampling Methods

Sampling can be performed using manual or mechanical methods. Manual methods include cone and quartering, sampling with scoops or shovels, and using sampling tubes.

Manual Sampling Techniques

Cone and Quartering

1. Stack material into a conical pile by shoveling.
2. Form a cake or 'qeqe'.
3. Divide the cake into four equal fractions.
4. Take two opposite quarters as the sample; reject the other two.
5. If the sample size is adequate, combine the two retained quarters and reduce their size.
6. Repeat steps 1-4 as necessary until the desired sample size is achieved.

Sampling Scoop or Shovel

This procedure is used when ore is moved... Continue reading "Mining Ore Sampling Techniques and Procedures" »