The Practical Use of My Debt Ratio to My Income

Reasoning

Reasoning is fundamental to knowing and doing mathematics. Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw local conclusions, develop and describe solutions, and recognize how those solutions can be applied. So we can ask, when is reasoning needed? Reasoning is needed when first encountering a new challenge. When faced with a mathematical challenge, reasoning helps us to make use of relevant prior knowledge such as how to tackle this type of a problem. The reasoning involved is complex and unique to the individual as each of us has a different past of mathematical experiences. Mathematical reasoning can be inductive or deductive. Inductive reasoning involves looking for patterns and making generalizations. Deductive reasoning involves making a logical argument, drawing conclusions, and applying generalizations to specific situations. This kind of reasoning sometimes involves eliminating unreasonable possibilities and justifying answers. As students grow older, the deductive reasoning ability improves. As teachers and students of mathematics, it is essential for us to value and promote reasoning explicitly, persistently, consistently, and frequently. This will help to develop complete chains of reasoning. Mathematical reasoning is the glue that binds together all other mathematical skills.

Proportions

Real-world decisions are based on subjective probabilities arising from a complex and diverse mixture of our perceptions, memories, and reasoning processes. Should we bring an umbrella, wear a coat, or stock up on bread and milk? All of these questions can be answered by looking at our previous knowledge of weather and what the weather report is saying, the time of year, and what has happened in the past. Much research has been done on these situations to formulate an estimate of subjective probability. At the heart of probability, estimation is the concept of relative frequency or proportion. When looking at proportion judgments, an investigation occurs between the judged proportion and actual proportion. When looking at proportions for mathematics, we are using the same principles. Proportions are mathematical comparisons between two numbers. Often, these numbers can represent a comparison between things or people. A good example would be if you visit a new city, and you want to look at the number of fast food restaurants as compared to all of the restaurants. The number of fast food restaurants would be proportional to the total number of restaurants. Proportions are helpful when you are comparing or finding equal amounts. They are useful when reading maps and blueprints. Suppose you are looking at a distance between two cities on a map. On the map, the distance may appear to be 3 inches. However, according to the map legend, 1 inch equals 100 miles. Therefore, when setting up the proportion, we would see that 1 inch : 100 miles as 3 inches : x miles. Notice how we used the variable x as we do not know that value until we calculate the proportion. Our calculations will show that 1 inch is to 100 miles as 3 inches is to 300 miles. To be sure our fractions are proportional, we can check them with cross multiplication. Proportions are also useful when reading a blueprint for a house. If on a blueprint, the living room is 10 inches by 12 inches, a key most likely equates inches and feet. If 1 inch equals 1 foot in the key, then the living room would actually be 10 feet by 12 feet.

Variability

Imagine you are deciding on the types of materials to build a house. You want to examine the costs of various products: wood, brick, steel, and plastic. You will notice that there is a price-per-load difference between all four items. Variability refers to how spread out a group of data is. In other words, variability measures how much the costs of the materials differ from one another. Variability can also be referred to as dispersion or spread. Data sets with similar values are said to have little variability, while data sets that have values spread out have high variability. There are various measures of variability. The range is the simplest measure of variability. The range represents the largest amount minus the smallest amount. The range is very sensitive to outliers or values that are significantly higher or lower than the rest of the data set. If there are outliers, using the range may not be the best measurement. However, if you do have outliers, you can use the interquartile range. The interquartile range is also called the middle 50. It is the range for the middle 50% of the data, so it will only consider the middle values and not the outliers. Another measure of variability is variance. The variance is the measure of how close the values of the data set are to the mean. The variance is mainly used to calculate the standard deviation and other statistics. When we look back at our materials, we can look at the mathematical variability in different ways: cost, size, usefulness, durability, and availability. Taking all of this into account, we can determine the optimal amount of materials at the desired cost.

Historical Origins of Algebra

The diverse history of algebra began in ancient Egypt and Babylon. It was there that people learned to solve linear, quadratic, and indeterminate equations. In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic expose of the basic theory of equations including both examples and proofs. As these ancient civilizations wrote out algebraic equations using only occasional abbreviations, in medieval times, Islamic mathematicians were able to talk about arbitrarily high powers of the unknown, x, and work

out the basic algebra of polynomials. In the 16th century, an important development in algebra was the introduction of symbols for the unknown and for algebraic powers and operations. Rene Descartes wrote a book, La geometrie, which looks much like a modern algebra text. His most significant contribution to mathematics, however, was his discovery of analytic geometry, which reduces the solution of geometric problems to the solutions of algebraic ones. The 17th century, also known as the period of the scientific revolution, was a period of intense activity and innovation in mathematics. Advances in numerical calculation, the development of symbolic algebra and analytic geometry, and the invention of differential and integral calculus resulted in a major expansion of subject areas in mathematics. By the end of the 17th century, a program of research based in analysis had replaced classical Greek geometry at the center of advanced mathematics. Throughout the next century, this growth in mathematics would continue due to its close association with physics, or more specifically, mechanics and theoretical astronomy. Until the mid-17th century, mathematicians worked alone or in small groups. However, in 1660, the Royal Society of London was founded, and in 1666, the Academy of Sciences was founded by the French. In 1700, more academies of science emerged in Berlin, and in 1724, St Petersburg. By working as universities, these academies were able to open communications about research findings. We cannot talk about algebra without talking about Sir Isaac Newton and Gottfried Leibniz. Throughout the late 17th century, these two scientists/mathematicians had many accomplishments in the field of calculus. During this 200-year period, significant advances occurred in the theory of equations, foundations of Euclidean geometry, number theory, projective geometry, and probability theory. These subjects became mature branches of mathematics in the 19th century.