Nature of Mathematics Lesson 1 Patterns and Sequences

This set of vocabulary flashcards covers concepts from Lesson 1 including types of mathematical sequences, symmetry in nature, and population growth models.

Patterns

Regular, repeated, or recurring forms or designs, such as floor tile layouts, building designs, or the way we tie shoelaces.

Sequence

An ordered list of numbers or other elements like geometric objects that often follow a specific pattern or function, and can be finite or infinite.

Arithmetic Sequence

An ordered set of numbers with a common difference between each consecutive term, generated by adding multiples of that common difference ($d$) to previous terms.

Common Difference ($d$)

The constant value added to each term in an arithmetic sequence, calculated using the formula $D = A_2 - A_1$.

Arithmetic Sequence $n^{th}$-term Formula

The formula used to find any term in an arithmetic sequence: $A_n = a_1 + (n-1)d$.

Geometric Sequence

Ordered sets of numbers that progress by multiplying and dividing each term by the same amount called the common ratio.

Common Ratio ($r$)

The factor by which terms in a geometric sequence are multiplied, calculated using the formula $R = \frac{a_2}{a_1}$.

Geometric Sequence $n^{th}$-term Formula

The formula used to find the specific term in a geometric sequence: $A_n = a_1 r^{n-1}$.

Harmonic Sequence

A progression formed by taking the reciprocals of an arithmetic sequence.

Symmetry

Indicates that an imaginary line can be drawn across an object such that the resulting parts are mirror images of each other.

Bilateral Symmetry

Also known as line symmetry, it occurs when the left and right portions of an object are exactly the same, such as in a butterfly.

Rotational Symmetry

A property where a figure, like a starfish, can be rotated by a specific angle and still achieve the same appearance as the original position.

Angle of Rotation

The smallest angle that a figure can be rotated while still preserving its original formation.

Order of Rotation ($n$)

A figure has rotational symmetry of order $n$ ($n$-fold rotational symmetry) if $\frac{1}{n}$ of a complete turn leaves the figure unchanged.

Angle of Rotation Formula

The formula used to determine the degrees needed for a figure's rotation to look unchanged: $\text{Angle of rotation} = \frac{360^\text{degrees}}{n}$.

Packing Problem

The challenge of finding the optimum method of filling up a given space, such as finding the most efficient way to store honey in a honeycomb using the smallest amount of wax.

Alan Turing

A British mathematician who proposed that the formation of biological patterns like hyena spots and tiger stripes is governed by a system of reaction-diffusion equations.

Morphogens

Chemicals proposed by Alan Turing that react with each other and diffuse through an embryo to form patterns.

Exponential Growth Model

The formula used to model population growth over time: $A = Pe^{rt}$.

Euler's Constant ($e$)

A mathematical constant used in modeling population growth with an approximate value of $2.718$.