Lesson 1: Mathematics in Our World

Question 1

• It is defined in many ways.
• Study of numbers.
• Set of problem-solving tools, a language, a process of thinking, and a study of pattern.

Answer 1

Mathematics

Question 2

• A branch of science, which deals with numbers and their operations. It involves calculation, computation, solving of problems etc.
• Its dictionary meaning states that it is the science of numbers and space or the science of measurement, quantity and magnitude.
• It is exact, precise, systematic and a logical subject.

Answer 2

Mathematics

Question 3

It is an arrangement which helps observers anticipate what they might see or what happens next. It also shows what may have come before and organizes information so that it becomes more useful.

Answer 3

Pattern

Question 4

The ___ is programmed to make sense of data or to bring order where there is disorder. It seeks to discover relationships and connections between seemingly unrelated bits of information. In doing so, it sees patterns.

Answer 4

human mind

Question 5

According to the __ the nature of mathematics is what follows: Mathematics is a study of patterns and relationship, a way of thinking, an art, a language, and a tool. It is about patterns and relationships.

Answer 5

National Council of Teachers of Mathematics (1991)

Question 6

___ are just a way to express those patterns and relationships in Mathematics.

Answer 6

Numbers

Question 7

It is the fundamental language of patterns. Occurs when there is congruence in dimensions, due proportions and arrangement. It provides a sense of harmony and balance.

Answer 7

Symmetry

Question 8

___ is one of the foremost predominant themes in arts, design and architecture all over the world and throughout human history.

Answer 8

Symmetry

Question 9

__ is the simplest kind of symmetry. It is one of the most common kinds of symmetry that we see in the natural world. It can also be called mirror symmetry because an object with this symmetry looks unchanged if a mirror passes through its middle.

Answer 9

Bilateral or reflection symmetry

Question 10

If a shape can be folded in half so that one half fits exactly on top of the other, then we say that the shapes are ___.

Answer 10

symmetric

Question 11

The fold in a bilateral or reflection symmetry is called a ___because it divides the shape into two equal parts. Bilateral-symmetric objects have at least one line or axis of symmetry.

Answer 11

line of symmetry

Question 12

__ is rotational symmetry around a fixed point known as the center. Images with more than one lines of symmetry meeting at a common point exhibits this symmetry.

Answer 12

Radial symmetry

Question 12

___ consist of taking motif or an element and rotating and/or reflecting that element. It repeats in no direction.

Answer 12

Rosette patterns

Question 13

A rosette pattern is ___ if it
only admits rotational symmetries.

Answer 13

cyclic

Question 14

A rosette pattern is __ if it admits both rotational symmetries and bilateral or reflectional symmetries.

Answer 14

dihedral

Question 15

This frieze pattern only admits a translational symmetry

Answer 15

Hop

Question 16

What are the two types of rosette patterns?

Answer 16

Cyclic and dihedral

Question 17

A __ or border pattern is a pattern in which a basic motif repeats itself over and over in one
direction. It extends to the left and right in a way that the pattern can be mapped onto itself by a horizontal translation. We can usually find these patterns in unique places like on the walls of buildings, fabrics, borders of rugs and tiled floor.

Answer 17

frieze

Question 18

Enumerate the 7 types of frieze patterns

Answer 18

1. Hop
2. Step
3. Sidle
4. Spinning Hop
5. Spinnng Siddle
6. Jump
7. Spinning Jump

Question 19

This frieze pattern only admits a translational and glide symmetries

Answer 19

Step

Question 20

The frieze pattern only admits translations and vertical reflections

Answer 20

Sidle

Question 21

The frieze pattern only admits translations and 180◦
rotations (half-turns).

Answer 21

Spinning Hop

Question 22

The frieze pattern only admits translations, vertical reflections, rotations, and glide reflections.

Answer 22

Spinning Siddle

Question 23

The frieze pattern only admits translations, a horizontal reflection, and glide reflection.

Answer 23

Jump

Question 24

The frieze pattern admits *translations, vertical reflections, horizontal reflections, rotations, and glide reflections.*

Answer 24

Spinning Jump

Question 25

Who is the mathematician who invented the names of the seven frieze patterns?

Answer 25

John B. Conway

Question 26

A __ is a *pattern with translation symmetry in two directions*. It is, therefore, essentially an *arrangement of friezes stacked upon one another* to fill the entire plane.

Answer 26

wallpaper pattern

Question 27

Any particular wallpaper pattern is made up of a combination of the following symmetries; ___

Answer 27

reflection, rotation and glide reflection

Question 27

A __ or tiling is a *repeating pattern of figures that covers a plane with no gaps or overlaps*. It is just like a wallpaper group in which patterns are created by repeating a shape to fill the plane.

Answer 27

tessellation

Question 27

According to Nocon (2016), in order for a plane figure to be considered a wallpaper pattern, it must have at least the basic unit, ___, and a ___. There must be at least two rows, each one of at least two units long.

Answer 27

one copy by translation, copy of these two by translation in the second direction

Question 27

It is shown that there are only ___ distinct types of wallpaper patterns.

Answer 27

17

Question 28

Tessellations can be created with __, __, and __.

Answer 28

translations, rotations, reflections

Question 29

A __ is a perfect example of *natural tessellations.*

Answer 29

honeycomb

Question 30

The Fibonacci sequence was invented by the Italian ___.

Answer 30

Leonardo Pisano Bigollo (1180-1250)

Question 30

Leonardo Pisano Bigollo is known in mathematical history by several names:

Answer 30

Leonardo of Pisa (Pisano means “from Pisa”) and Fibonacci (which means “son of Bonacci”)

Question 31

The sequence F1; F2; F3; : : : is then the Fibonacci sequence. Such a definition is called a __ because it *starts by defining some initial values and defines the next term as a function of the previous terms*

Answer 31

recursive definition

Question 32

If we take the ratio of Fn to Fn−1 for n ≥ 1, we see that n gets larger and larger, the ratio gets closer and closer to a value dentoted by, this is called the ___

Answer 32

golden ratio

Question 33

___ emphasized that the Fibonacci sequence has captivated mathematicians, scientists, artists and designers for centuries. It is a sequence with many interesting prop

Answer 33

George Dvorsky (2013)

Question 34

Spiral patterns in sunflower seeds, pinecone seed pods, cauliflower, and pineapples can be deciphered as Fibonacci sequences. Divide spirals into left and right directions to get two consecutive Fibonacci numbers. Spiral patterns also appear in cauliflower and pineapples.

Answer 34

Pinecones, Speed Heads, Vegetables and Fruits

Question 35

Most flowers express the Fibonacci sequence if you count the number of petals on these flowers. Some plants also exhibit the Fibonacci sequence in their growth points, on the places where tree branches form or split

Answer 35

Flowers and branches

Question 36

The family tree of a honey bee perfectly resembles the Fibonacci sequence.

Answer 36

Honeybees

Question 37

The human body has many elements that show the Fibonacci numbers and the golden ratio. Most of your body parts follow the Fibonacci sequence and the proportions and measurements of the human body can also be divided up in terms of the golden ratio.

Answer 37

Human body

Question 38

Fibonacci numbers and the relationships between these numbers are evident in spiral galaxies, sea wave curves and in the patterns of stream and drainages

Answer 38

Geography, Weather and Galaxies

Question 38

The Great Pyramid of Giza built around 2560 BC is one of the earliest examples of the use of the golden ratio

Answer 38

Architecture

Question 39

An Old man by Leonardo Da Vinci: Leonardo Da Vinci explored the human body involving in the ratios of the lengths of various body parts. He called this ratio the "___" and featured it in many of his paintings.

Answer 39

divine proportion

Question 39

It is believed that Leonardo, as a mathematician tried to incorporate of mathematics into art. This painting seems to be made purposefully line up with the *golden rectangle.*

Answer 39

Arts

Question 39

It is a well defined *collection of objects* called *elements.*

Answer 39

Sets

Question 40

A collection is __ if for any given object, we can objectively decide *whether it is or is not in the collecition.*

Answer 40

well-defined

Question 41

Object which *belongs to a given set.*

Answer 41

Element/member of a set

Question 42

How many ways can sets be described?

Answer 42

3

Question 43

Described sets by *listing* all elements between braces and separated by commas.

Answer 43

Listing/Roster Method

Question 44

Uses a *variable* (a symbol, usually a letter, that can represent different elements of a sets,) *braces,* and a *vertical bar* | (such that).

Answer 44

Set-builder notation

Question 45

Uses a short verbal *statement* to describe the set.

Answer 45

Descriptive method

Question 46

N

Answer 46

Natural numbers

Question 47

Z

Answer 47

Integers

Question 48

Q

Answer 48

Rational numbers

Question 48

*Possible* to list down all the elements in the list.

Answer 48

Finite set

Question 49

R

Answer 49

Real numbers

Question 50

*Impossible* to list down all the elements in the list.

Answer 50

Infinite set