Notes, abstracts, papers, exams and problems of Mathematics

4.1 The stages of learning as Dienes
The learning process is a process based on abstraction, generalization and communication. This process of abstraction is to accurately analyze and Dienes identifies six different stages in it:
Stage 1: introduces the individual in the middle => Game Free
2nd stage: review, manipulate, get rules => Structured Games
3rd stage: becoming aware of the common structure to games made
4th stage: representation of the common structure graphically or schematically => Stage representative
5th stage study of the properties of abstract structure, which implies the need to invent a language => Stage symbolic
6th stage: Construction of axioms and theorems => formal Stage
Her pedagogical approach

Qualitative Variables

Nominal Variables

Nominal variables are qualitative variables that cannot be ordered in an ascending or descending manner; that is, they cannot be ranked. For example, blood group.

Ordinal Variables

Ordinal variables are variables that can be ordered in an ascending or descending manner; that is, they can be ranked.

Quantitative Variables

Discrete Variables

Discrete variables are variables whose values are obtained by counting.

Continuous Variables

Continuous variables are variables whose values are obtained by measurement using a scale.

Mean

Advantages

• Has many good theoretical properties
• Used as the basis of many statistical tests
• Good summary statistic for symmetrical distribution
• Easy to calculate
• Possible for further algebraic treatment

Disadvantages

• Less

De Morgan's Law

De Morgan's Law: (Flip if the union is true)

, image of set: [min, max]; one-to-one: horizontal line test; Onto: Image must equal domain; Bijective: one-to-one and Onto

|| EV

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Possible Outcomes and Probability Calculations

• Repetition formula: n^k
  • Example: 5 awards (k) and 30 students (n), with no limit to awards per student.
• Permutation formula: P(n, k) = n! / (n - k)!
  • Example: Each student gets 1 award, so the number of students decreases by one each award.
• No overlap probability: P(n, k) / repetition formula
• Arrangements: a = slots → a! can be multiplied by arrangements within slots
• Die sum probability:
  • List combinations that lead to the sum for each die.
  • If a die is rolled multiple times, each combination has (rolls)! permutations.
  • Add

Map Generalization

Types of Symbols

• Line Symbols: Represent real-life objects with a linear path.
• Point Symbols: Represent objects occurring at a single point on Earth's surface using a dot.
• Area (Polygon) Symbols: Represent real-life objects spread over Earth's surface using geometric shapes.

Generalization Techniques

Reality contains too much information for a single 2D map. Generalized geometry and content make a map useful. A good map suppresses less important information to highlight what needs to be seen.

• Selection: Only relevant line, point, and area features are chosen.
• Classification: Grouping similar features and using a common symbol to represent them.
• Simplification: Reduction of unnecessary detail.
• Smoothing: Smoothing out abruptly joined

Chapter 2: Predetermined Overhead Rate

Predetermined Overhead Rate = Estimated Total Manufacturing Overhead (MOH) / Estimated Total MOH Driver (e.g., Direct Labor hours, Direct Labor costs, Machine Hours)

Prime Cost = Direct Materials + Direct Manufacturing Labor

Conversion Cost = Direct Manufacturing Labor + Indirect Manufacturing Overhead

Cost Accumulation: Data is collected in an organized way (also known as cost pools).

Cost Assignment: Systematically links an actual cost pool to a distinct cost object (e.g., Tires, engine, labor assigned to car cost).

Activity Base: Examples include kilometers driven in a car, units produced, units sold, machine hours.

Product Cost: Costs tied to creating a product (Direct Materials, Direct Labor, Manufacturing... Continue reading "Cost Accounting Essentials: Key Concepts and Calculations" »

System Types
The systems of equations can be classified by the number of solutions that can arise. According to that case may have the following cases:
· Incompatible system if it has no solution.
· Compatible system if you have any solution in this case can also distinguish between:
or compatible system determined when it has a finite number of solutions.
indeterminate or compatible system when it admits an infinite set of solutions.
Fitting and classification:

Calculating the rank of a matrix for determining

1. We can rule a line if:.
· All the coefficients are zeros.
· There are two equal lines.
• A line is proportional to another.
• A line is a linear combination of others.
Delete the third column because it is a linear... Continue reading "Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem" »

Part I: Complete the Following Table

Part II: Complete the Following Sentences

1. She is from the United States. She is American.
2. She is from Nigeria. She is Nigerian.
3. She is from Germany. She is German.
4. They are from Egypt. They are Egyptian.
5. We are from Canada. We are Canadian.

Part III: Select the Correct Sentence

1. Select the correct sentence.
   • a) She lives with Carlos, and she works on Saturdays.
2. Select the correct sentence.
   • c) They are running in the park.
3. Select the correct sentence.
   • b) She is 20 years old.
4. Select the correct sentence.
   • a) We always

Bond Characteristics

• Coupon: The interest payment made by the bond issuer, usually expressed as an annual percentage of the bond's face value.
• Par (Face Value): The amount the bondholder receives when the bond matures, typically $1,000.
• Term to Maturity: The time remaining until the bond's maturity date when the issuer must repay the bond's par value.
• Denomination: The face value of the bond, usually in increments of $1,000.
• Quotation: Bonds are quoted as a percentage of their face value (e.g., a bond quoted at 95 is selling for 95% of $1,000, or $950).

Bond Prices, Yield to Maturity (YTM), Current Yield, and Rate of Return (HPR)

• Bond Prices: The market price of a bond depends on interest rates. Prices and interest rates have an inverse relationship.

Discrete Random Variables

Discrete random variables are variables that can take on a finite number of distinct values. In simpler terms, a discrete random variable is a set of possible outcomes that is countable.

Continuous Random Variables

Continuous random variables are random variables that take an infinitely uncountable number of potential values, typically measurable amounts.

Example

1. List the sample space in the given experiment. How many outcomes are possible?

The sample space is: S = {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD}

2. Count the number of defective keyboards in each outcome in the sample space and assign this number to the outcome. For instance, if you list NND, then the number of defective keyboards is 1.

The possible values of X are 0,... Continue reading "Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques" »