Module 4: Mathematical Patterns Flashcards
a set of numbers arranged in some order
sequence
ordered arrangement of a set of numbers
sequence
a sequence of values that follows a pattern of adding a fixed amount (always the same) from one term to the next
arithmetic sequence
the fixed amount in the arithmetic sequence is called
common difference
the formula in the arithmetic sequence
an=a1+(n-1)d
a sequence of values that follows a pattern of multiplying a fixed amount from one term to the next
geometric sequence
the fixed amount in the geometric sequence
common ration, r
the formula for finding the common ratio
r= an/an-1
the formula for geometric sequence
an=an-1r or an=a1r^n-1
a means of solving practical problems. sequence is formed by starting with 1,1 and adding the two preceding numbers to get the next number
Fibonacci Sequence
who developed the fibonacci sequence
Leonardo Fibonacci
examples of Fibonacci sequence in nature
nautilus shell, hurricane, galaxy
the ratio of 2 successive Fibonacci numbers approach the number called
Golden Ratio
the golden ratio is irrational and is noted by the ratio of…
(1+√5)/2 approx. 1.618
involves flexible thinking, creativity, judgment, and logical problem-solving.
abstract reasoning
refers to the ability to analyze information, detect patterns, and relationships and solve problems on a complex, intangible level.`
abstract reasoning
factors to Compare in abstract reasoning
size, location, color, shades, angles, movement
created when a shape or a combination of shapes are repeated over and over again covering a place without any gaps or overlaps using transformations; another word is
tessellation; tiling
who were the first people who used tessellations; where did they put it
Sumerians at about 4000BC; build wall decorations in patterns of clay tiles
some of the most famous tessellations
Moorish wall tiles of Islamic architecture
became the first person to complete the study of tessellations after he explored the structure of honeycombs and snowflakes
Johannes Kepler
300 years later, Russian crystallographer,_____, began the study of tessellations in mathematics.
Yvgraf Fyodorov
methods used in tesselations
translation, rotation, reflection
a tesselation made up of congruent regular polygons
regular tessellation
3 regular polygons that tessellate in the Euclidian plane
triangles, squares, hexagons
formed by regular polygons. the arrangement of polygons at every vertex is identical
semi-regular tessellations
2 method of producing irregular shapes to tessellate
- translating (or sliding) the midpoint of any side of the starting shape making some curved lines
- rotating the midpoint of any side of the starting shape
a world-famous graphic artist that attributed to the idea of transformation of shapes to create new, irregular, tessellating shapes
Mauritis Cornelis Escher (1898-1972)
a rough or fragmented geometric shape that can be split into parts, each of which is approximately a reduced-size copy of the whole.
fractal
amazingly complicated patterns often produced by very simple processes- reflection , rotation, and translation
fractals
- demonstrate a fourth time of symmetry
- possess self-similarity
fractals
a shape is _____ when it looks essentially the same from a distance as it does closer up.
self-similar
a characteristic where an object is composed of smaller copies of itself
scaling symmetry or scale invariance
fractals from mathematical constructions
Koch Curve; Sierpinski’s Triangle
