Mathematics in Our World – Key Concepts & Problem-Solving

These Question-and-Answer cards review definitions, examples, formulas, and problem-solving methods from Modules 1 & 2 on Mathematics in Our World, covering patterns, Fibonacci/Golden Ratio, mathematical language, and Polya’s strategy.

What is mathematics primarily the study of?

Patterns and structure.

Name three broad fields that rely fundamentally on mathematics.

Physical and biological sciences, engineering & information technology, and economics.

Give two reasons why mathematics is considered a useful way to think about nature.

It helps quantify and organize the world and predicts phenomena to make life easier.

List three roles that mathematics plays in our world.

Organizes patterns, predicts natural behavior, and helps control phenomena for human purposes.

Which visible regularities in the natural world provide clues to natural laws?

Patterns in nature (e.g., snowflake symmetry, star motion, animal markings).

What daily astronomical pattern illustrates circular motion across the sky?

Stars appear to move in circles each day.

Give two animal groups whose skins display stripe patterns illustrating mathematical regularity.

Zebras and tigers (cats and snakes also acceptable).

Which symmetry divides an organism into mirror-image left and right halves?

Bilateral symmetry.

Which symmetry involves repeated arrangement around a central point?

Radial (rotational) symmetry.

Define a fractal in one sentence.

A never-ending, self-similar pattern that repeats at different scales.

What geometric shape describes a curved pattern revolving around a center?

A spiral.

State the recursive rule for the Fibonacci sequence.

Xₙ = Xₙ₋₁ + Xₙ₋₂.

Write the first seven non-zero Fibonacci numbers.

1, 1, 2, 3, 5, 8, 13.

What special rectangle is constructed using successive Fibonacci squares?

The Golden Rectangle.

What spiral results from drawing arcs in adjoining Fibonacci squares?

The Fibonacci (logarithmic) spiral.

How many petals does a black-eyed Susan typically have, according to Fibonacci patterns?

13 petals.

Which paired spiral counts in a sunflower usually match consecutive Fibonacci numbers?

34 clockwise and 55 counter-clockwise spirals (or vice-versa).

Provide the algebraic definition of the Golden Ratio (φ).

(a + b)/a = a/b = 1.618033987… where a > b.

What happens to the ratio of successive Fibonacci numbers as they increase?

It approaches the Golden Ratio (≈1.618).

Name three human body features that often exhibit golden-ratio proportions.

Relative positions of mouth and nose, overall facial layout, limb sections (e.g., forearm to hand).

Identify two plant structures that display logarithmic spirals related to φ.

Pinecones and pineapples (also shells or seed heads).

Which galaxy type commonly follows a logarithmic (Golden Ratio) spiral?

Spiral galaxies such as the Milky Way.

Give one weather phenomenon whose swirl approximates a Fibonacci spiral.

A hurricane.

Name two famous artworks that employ the Golden Ratio in composition.

Da Vinci's Mona Lisa and Michelangelo's The Creation of Adam (others acceptable).

Which ancient Greek building’s façade embodies golden-ratio proportions?

The Parthenon in Athens.

List three world landmarks whose designs incorporate the Golden Ratio.

Great Pyramid of Giza, Taj Mahal, Eiffel Tower (Notre Dame, CN Tower also valid).

Explain why bees use hexagons in honeycombs from a mathematical standpoint.

Hexagons minimize wax while maximizing storage strength and leaving no gaps.

What scientific method uses tree-ring patterns to determine age?

Dendrochronology.

How can turtle age estimation illustrate natural geometry?

Counting growth rings (scutes) that form hexagonal patterns on the shell.

Name two natural phenomena that produce fractal patterns besides lightning.

River networks and snowflakes (foam bubbles also acceptable).

State three practical fields that apply mathematics daily.

Forensics, medical drug design, information technology (others: fluid dynamics, cryptography).

Give an example of mathematics in social sciences.

Economists use matrices, probability, and statistics to model markets.

What three adjectives describe the nature of mathematical language?

Precise, concise, and powerful.

Define a mathematical expression.

A finite, well-defined combination of symbols representing an object; it lacks a complete thought.

Define a mathematical sentence.

A correct symbol arrangement stating a complete thought that can be true or false.

Give one example of a technical term whose meaning differs in math and everyday English.

‘Group’ (everyday: collection; math: a specific algebraic structure).

What name is given to short mathematical phrases like “if and only if” or “without loss of generality”?

Mathematical jargon.

Which alphabet is traditionally used for simple variables in mathematical notation?

The Latin alphabet.

What does the symbol ∀ mean?

‘For all’ (universal quantifier).

Write the four steps in Polya’s problem-solving framework.

1) Understand the problem 2) Devise a plan 3) Carry out the plan 4) Look back.

During Polya’s Step 1, name two guiding questions you might ask.

Do I understand all the words? What is given and what is the goal?

Give three common strategies listed in Polya’s Step 2 (Devise a Plan).

Guess and test, draw a picture, work backwards (others: use a variable, make a list, etc.).

What is the key task in Polya’s Step 3?

Carry out the devised plan and check each step for correctness.

What is the main purpose of Polya’s Step 4?

Look back to verify the solution and consider alternative methods or applications.

Translate this English statement into a mathematical equation: “Two times the sum of a number and 3 equals thrice the number plus 4.”

2(x + 3) = 3x + 4.

In a tour problem, if 10 tourists paid a total of ₱170,000 for options costing ₱15,000 and ₱20,000, what variable choice simplifies solving the number who chose the higher-priced option?

Let x = number of tourists who paid ₱20,000 (side trip to HK).

What property of mathematical language allows lengthy ideas to be expressed briefly?

Conciseness.

Why is precision crucial in mathematical language?

Because even small symbol changes can alter meaning, precise distinctions avoid ambiguity.

What makes mathematical language ‘powerful’?

Its ability to convey complex concepts succinctly and unambiguously.

Provide an everyday example of cryptography’s mathematical application.

Securing ATM card transactions via encryption algorithms.

Which mathematical fields underpin computer science development?

Logic, set theory, combinatorics, graph theory, discrete probability, and number theory.

Name two symbolism rules (conventions) mathematicians follow to ensure universal understanding.

Using established notation (e.g., ‘=’ for equality) and punctuating symbolic sentences as clauses.

What term describes a statement proved from axioms and earlier results?

Theorem (related: lemma, corollary).

Which pattern category includes the branching of rivers and human lungs?

Fractals.

What type of pattern is observed in V-formations of geese or fish schooling?

Spatial organization patterns illustrating optimization and aerodynamics (also example of symmetry/formation).

How does mathematics assist archaeologists in carbon dating artifacts?

By applying exponential decay models and statistical analysis to radiocarbon measurements.