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Questions and Answers
What is an example of bilateral symmetry?
What is an example of bilateral symmetry?
Which pattern is characterized by arrangements around a central point?
Which pattern is characterized by arrangements around a central point?
What do fractals exhibit?
What do fractals exhibit?
What is a characteristic of a mosaic pattern?
What is a characteristic of a mosaic pattern?
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Which of these patterns resembles the branching of a tree?
Which of these patterns resembles the branching of a tree?
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What is a defining feature of tessellations?
What is a defining feature of tessellations?
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Which pattern is identified as having lines or bands differing in color or tone?
Which pattern is identified as having lines or bands differing in color or tone?
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Which type of symmetry is primarily found in starfish?
Which type of symmetry is primarily found in starfish?
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What is denoted by the symbol ∅?
What is denoted by the symbol ∅?
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Which method involves listing out the elements of a set inside braces?
Which method involves listing out the elements of a set inside braces?
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How many elements are in the power set of a set containing n elements?
How many elements are in the power set of a set containing n elements?
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What does the symbol ⊂ represent?
What does the symbol ⊂ represent?
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What is the primary focus of D'Arcy Thompson's studies?
What is the primary focus of D'Arcy Thompson's studies?
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Which of the following describes equal sets?
Which of the following describes equal sets?
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Which mathematician predicted mechanisms of morphogenesis?
Which mathematician predicted mechanisms of morphogenesis?
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What does the term engineering mathematics refer to?
What does the term engineering mathematics refer to?
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What is true about a null set in relation to other sets?
What is true about a null set in relation to other sets?
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What is mathematical language primarily used for?
What is mathematical language primarily used for?
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Which of the following best describes a finite set?
Which of the following best describes a finite set?
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What do equivalent sets have in common?
What do equivalent sets have in common?
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Which of the following actions is NOT one of the four main actions attributed to problem-solving according to various authors?
Which of the following actions is NOT one of the four main actions attributed to problem-solving according to various authors?
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What do Venn Diagrams represent?
What do Venn Diagrams represent?
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Which individual is credited with popularizing the Fibonacci Sequence in Europe?
Which individual is credited with popularizing the Fibonacci Sequence in Europe?
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What distinguishes mathematical language from ordinary language?
What distinguishes mathematical language from ordinary language?
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What does the Golden Ratio represent numerically?
What does the Golden Ratio represent numerically?
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Which of the following terms describes an expression with three terms?
Which of the following terms describes an expression with three terms?
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What is the term for a mathematical expression that combines expressions using comparative operators?
What is the term for a mathematical expression that combines expressions using comparative operators?
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In the expression 3x + 5 - 7y, how many terms are present?
In the expression 3x + 5 - 7y, how many terms are present?
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Which option defines a monomial?
Which option defines a monomial?
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What does an open sentence in mathematics indicate?
What does an open sentence in mathematics indicate?
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Which mathematical expression represents a binomial?
Which mathematical expression represents a binomial?
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What is the characteristic of a closed sentence in mathematics?
What is the characteristic of a closed sentence in mathematics?
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What does the union of sets A and B represent?
What does the union of sets A and B represent?
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Which of the following accurately describes the intersection of sets A and B?
Which of the following accurately describes the intersection of sets A and B?
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What characterizes the difference of sets A and B?
What characterizes the difference of sets A and B?
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What does the complement of Set A signify?
What does the complement of Set A signify?
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How is the cross product of sets A and B defined?
How is the cross product of sets A and B defined?
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What distinguishes inductive reasoning from deductive reasoning?
What distinguishes inductive reasoning from deductive reasoning?
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Which statement correctly defines a premise?
Which statement correctly defines a premise?
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What is the purpose of a mathematical proof?
What is the purpose of a mathematical proof?
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Study Notes
Mathematics
- The study of relationships between numbers, quantities, and shapes.
- Includes arithmetic, algebra, trigonometry, geometry, statistics, and calculus.
- Gary Smith's landscapes incorporate eight patterns: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial, and dendritic.
- These patterns are found in nature, such as plants, animals, rock formations, and human creations.
Symmetry
- A harmonious and beautiful proportion of balance or an object is invariant to any various transformations (reflection, rotation, or scaling).
- Bilateral Symmetry - A symmetry in which the left and right sides of the organism can be divided into approximately mirror images of each other along the midline.
- Radial Symmetry - A symmetry in which the organism can be divided into equal parts by lines from its central axis.
- Animals have mainly bilateral or vertical symmetry, even leaves of plants and some flowers such as orchids.
- A five-fold symmetry is found in the echinoderms, which includes starfish (dihedral-D5 symmetry), sea urchins, and sea lilies.
Curved Patterns
- A pattern arranged around a central point is called a radiating pattern.
Fractals
- Never-ending patterns found in nature, with the same shape but different sizes.
Spirals
- A curved pattern focusing on a center point and a series of circular shapes revolving around it.
- Pinecones are examples of spirals.
Dendritic Pattern
- Resembles the branches of a tree and develops gently into a sloping basin with a uniform rock type.
Scattered Pattern
- A dispersed settlement pattern where buildings are not squished together.
Mosaic Pattern
- A piece of art or image made from assembling small pieces of colored materials.
Stripe
- A line or band that differs in color or tone.
Spots
- Usually consists of dots.
Tessellations
- The tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.
Important figures in the study of patterns:
- Joseph Plateau (19th century Belgian physicist) - Examined soap films and formulated the concept of minimal surfaces.
- Ernst Haeckel (19th century German biologist and artist) - Painted hundreds of marine organisms to emphasize their symmetry.
- D'Arcy Thompson (Scottish biologist) - Pioneered the study of growth patterns in both plants and animals, showing simple equations could explain spiral growth.
- Alan Turing (20th century British mathematician) - Predicted mechanisms of morphogenesis, which give rise to patterns of spots and stripes.
- Aristid Lindenmayer (Hungarian biologist) and Benoit Mandelbrot (French American mathematician) - Showed how the mathematics of fractals could create plant growth patterns.
Fibonacci
- Leonardo Pisano Bogollo (1170-1250) - Italian mathematician known as Fibonacci.
- Famous for the Fibonacci sequence and helping spread Hindu Arabic numerals through Europe.
- Fibonacci Day: November 23rd.
- Golden Ratio: 1.1618
Engineering Mathematics
- The art of applying mathematics to complex real-world problems.
- Combines mathematical theory, practical engineering, and scientific computing.
- Found in a wide range of careers, from designing cars to inventing robotics.
Mathematical Language
- The system used to communicate mathematical ideas.
- Numbers, measurements, shapes, spaces, functions, patterns, data, and arrangements are mathematical nouns or objects.
- Mathematical verbs are the four main actions attributed to problem-solving and reasoning.
Four Main Actions in Problem Solving
- Modeling and Formulating: Creating appropriate representations and relationships to mathematize the original problem.
- Transforming and Manipulating: Changing the mathematical form in which a problem is originally expressed to equivalent forms that represent solutions.
- Inferring: Applying derived results to the original problem situation, and interpreting and generalizing the results in that light.
- Communicating: Reporting what has been learned about a problem to a specified audience.
Venn Diagrams
- Pictorial representations of relationships and operations of sets.
- The universal set is represented by a rectangle, and circles within the rectangle represent its subsets.
Mathematical Language
- Differ from ordinary speech in three ways:
- Non-temporal: No past, present, or future.
- Devoid of emotional content.
- Precise.
Mathematical Expressions
- Consist of terms, separated by plus or minus signs.
- May contain expressions in parentheses or other grouping symbols.
- Variables or letters are used to represent numbers in algebra.
- Variable: Represents the unknown and makes use of letters. Also called the literal coefficient.
- Numerical Coefficient: The number with the variable.
- Constant: Any single number.
- Monomial: A mathematical expression with one term, e.g., 21(x - 8).
- Binomial: A mathematical expression with two terms, e.g., 5x + 12y.
- Trinomial: A mathematical expression with three terms, e.g., 3x + 2(x +y) - 36.
- Polynomial: A mathematical expression with more than two terms. A trinomial is also a polynomial.
Mathematical Sentences
- Combine two mathematical expressions using a comparison operator.
- Use numbers, variables, or both.
- Comparison operators include equal, not equal, greater than, greater than or equal to, less than, and less than or equal to.
- Equation: A mathematical sentence containing the equal sign.
- Inequality: A mathematical sentence containing the inequality sign.
- Open sentence: Uses variables and is not known whether it is true or false.
- Closed sentence: Is known to be either true or false.
Context and Convention in Mathematical Symbols
- Context: Refers to the particular topics being studied.
- Convention: Techniques used by mathematicians, engineers, and scientists where each symbol has a particular meaning
Sets
- A well-defined collection of distinct objects.
- Elements or members can be numbers, people, letters, or other sets.
- Sets are conventionally named with capital letters.
- Braces are used to specify that the objects written between them belong to a set.
- Elements/Members inside the braces are separated by a comma.
Ways to Describe a Set:
- Roster/Tabular Method: Elements are listed or enumerated, separated by commas, inside a pair of braces.
- Rule/Descriptive Method: Common characteristics of the elements are defined. Uses set builder notation where x represents any element of the set.
Types of Sets:
- Empty/Null/Void set: Has no elements (∅ or {}).
- Finite set: Has a countable number of elements.
- Infinite set: Has an uncountable number of elements.
- Universal set: The totality of all the elements of the sets under consideration (denoted by U).
Set Relationships:
- Equal sets: Have the same elements.
- Equivalent sets: Have the same number (cardinality) of elements.
- Joint sets: Have at least one common element.
- Disjoint sets: Have no common element.
- Equal sets are equivalent sets, but not all equivalent sets are equal.
Subsets
- A set every element of which can be found in a bigger set.
- Symbol ⊂ means "a subset of", ⊄ means "not a subset of".
- If the first set equals the second set, then it is called an improper subset.
- Symbol ⊆ is used to mean an improper subset.
- A null set is always a subset of any given set and is considered an improper subset.
- All other subsets are considered proper subsets.
- Power set: The set containing all the subsets of the given set with n number of elements, has 2n number of elements.
Set Operations
- Union of sets: (A ∪ B) is a set whose elements are found in A or B or in both. Symbol: A ∪ B = {x | x ∈ A or x ∈ B}.
- Intersection of sets: (A ∩ B) is a set whose elements are common to both sets. Symbol: A ∩ B = {x | x ∈ A and x ∈ B}.
- Difference of sets: (A -- B) is a set whose elements are found in Set A but not in Set B. Symbol: {x | x ∈ A and x ∉ B}.
- Complement of a set: (A') is a set whose elements are found in the universal set but not in Set A. Symbol: A' = {x | x ∈ U and x ∉ A}.
- Cross Product of sets: (A × B) is the set of all ordered pairs (x, y) such that x belongs to A and y belongs to B.
Functions
- Mathematical entities that give a unique output to particular inputs.
- They have three important parts: input, relationship, and output.
Inductive Reasoning
- Making generalized decisions after observing repeated specific instances of something.
Deductive Reasoning
- Taking information from general observations and making specific decisions based on that information.
Reasoning Differences:
- Inductive reasoning: Leads to conclusions based on a series of observations, may or may not be valid.
- Deductive reasoning: Leads to conclusions based on previously known facts, conclusions are correct and valid.
- Inductive reasoning is used to form hypotheses, while deductive reasoning is used to prove ideas..
Premise:
- A previous statement or proposition from which another is inferred or follows as a conclusion.
Intuition
- The ability to understand something instinctively, without conscious reasoning.
Mathematical Proof
- An argument that convinces others that something is true.
- Conclusive evidence or an argument establishing a fact or truth.
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Description
Explore the intricate relationships within mathematics and discover the principles of symmetry. This quiz covers essential topics such as arithmetic, algebra, and the different types of symmetry evident in nature. Test your understanding of patterns and their significance in mathematics and the natural world.
