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Elementary mathematics consists of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
topics frequently taught at the
primary Primary or primaries may refer to: Arts, entertainment, and media Music Groups and labels * Primary (band), from Australia * Primary (musician), hip hop musician and record producer from South Korea * Primary Music, Israeli record label Works * ...

primary
or
secondary school A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both lower secondary education (ages 11 to 14) and upper secondary educat ...
levels. In the Canadian curriculum, there are six basic strands in Elementary Mathematics: Number, Algebra, Data, Spatial Sense, Financial Literacy, and Social emotional learning skills and math processes. These six strands are the focus of Mathematics education from grade 1 through grade 8. In secondary school, the main topics in elementary mathematics from grade nine until grade ten are: Number Sense and algebra, Linear Relations, Measurement and Geometry. Once students enter grade eleven and twelve students begin university and college preparation classes, which include: Functions, Calculus & Vectors, Advanced Functions, and Data Management.


Strands of Elementary Mathematics


Number Sense & Numeration

Number Sense is an understanding of numbers and operations. In the 'Number Sense and Numeration' strand students develop an understanding of numbers by being taught various ways of representing numbers, as well as the relationships among numbers. Properties of the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
such as
divisibility In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
and the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, are studied in basic
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, another part of elementary mathematics. Elementary Focus *Abacus *LCM *Fractions and Decimals *Place Value & Face Value *Addition and subtraction *Multiplication and Division *Counting Money *Counting *Algebra *Representing and ordering numbers *Estimating *Problem Solving To have a strong foundation in mathematics and to be able to succeed in the other strands students need to have a fundamental understanding of number sense and numeration.


Spatial Sense

'Measurement skills and concepts' or 'Spatial Sense' are directly related to the world in which students live. Many of the concepts that students are taught in this strand are also used in other subjects such as science, social studies, and physical education In the measurement strand students learn about the measurable attributes of objects, in addition to the basic metric system. Elementary Focus * Standard and non-standard units of measurement * telling time using 12 hour clock and 24 hour clock * comparing objects using measurable attributes * measuring height, length, width * centimetres and metres * mass and capacity * temperature change * days, months, weeks, years * distances using kilometres * measuring kilograms and litres * determining area and perimeter * determining grams and millilitre * determining measurements using shapes such as a triangular prism The measurement strand consists of multiple forms of measurement, as Marian Small states: "Measurement is the process of assigning a qualitative or quantitative description of size to an object based on a particular attribute."


Equations and formulas

A formula is an entity constructed using the symbols and formation rules of a given logical language. For example, determining the
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
of a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
requires a significant amount of
integral calculus In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
or its geometrical analogue, the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
; but, having done this once in terms of some
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

parameter
(the
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

radius
for example), mathematicians have produced a formula to describe the volume: This particular formula is: : An equation is a
formula In science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well a ...

formula
of the form ''A'' = ''B'', where ''A'' and ''B'' are
expressions Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphor#Common types, Metaphorical expression, a parti ...
that may contain one or several variables called unknowns, and "=" denotes the equality
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
. Although written in the form of
proposition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
, an equation is not a
statement Statement or statements may refer to: Common uses *Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a Sentence_(lin ...
that is either true or false, but a problem consisting of finding the values, called solutions, that, when substituted for the unknowns, yield equal values of the expressions ''A'' and ''B''. For example, 2 is the unique ''solution'' of the ''equation'' ''x'' + 2 = 4, in which the ''unknown'' is ''x''.


Data

Data is a set of
values In ethics Ethics or moral philosophy is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philoso ...
of qualitative or quantitative variables; restated, pieces of data are individual pieces of
information Information is processed, organised and structured data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to c ...

information
. Data in
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and software. It has sci ...

computing
(or
data processing Data processing is, generally, "the collection Collection or Collections may refer to: * Cash collection, the function of an accounts receivable department * Collection agency, agency to collect cash * Collections management (museum) ** Colle ...

data processing
) is represented in a
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...

structure
that is often tabular (represented by rows and
columns A column or pillar in architecture upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Par ...
), a
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with se ...

tree
(a set of
node In general, a node is a localized swelling (a "knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splic ...
s with
parent A parent is a caregiver of the offspring In biology, offspring are the young creation of living organisms, produced either by a Asexual reproduction, single organism or, in the case of sexual reproduction, two organisms. Collective offspring ...

parent
-
children Biologically, a child (plural children) is a human Humans (''Homo sapiens'') are the most abundant and widespread species In biology Biology is the natural science that studies life and living organisms, including their anat ...

children
relationship), or a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
(a set of connected nodes). Data is typically the result of
measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ...

measurement
s and can be visualized using
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
s or
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

image
s. Data as an
abstract
abstract
concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the bo ...

concept
can be viewed as the lowest level of
abstraction Abstraction in its main sense is a conceptual process where general rules Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, mak ...

abstraction
, from which
information Information is processed, organised and structured data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to c ...

information
and then
knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to e ...
are derived.


Basic two-dimensional geometry

Two-dimensional geometry is a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
concerned with questions of shape, size, and relative position of two-dimensional figures. Basic topics in elementary mathematics include polygons, circles, perimeter and area. A
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
is a shape that is bounded by a finite chain of straight
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

line segment
s closing in a loop to form a closed chain or ''circuit''. These segments are called its ''edges'' or ''sides'', and the points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of the polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides. A polygon is a 2-dimensional example of the more general
polytope In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...
in any number of dimensions. A
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
is a simple
shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...

shape
of
two-dimensional geometry 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρ ...
that is the set of all
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
in a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
that are at a given distance from a given point, the
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
.The distance between any of the points and the center is called the
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

radius
. It can also be defined as the locus of a point equidistant from a fixed point. A
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which t ...

perimeter
is a path that surrounds a
two-dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic ...
shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...

shape
. The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
or
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
is called its
circumference In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
.
Area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

Area
is the
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ...
that expresses the extent of a
two-dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic ...
figure or
shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...

shape
. There are several well-known
formula In science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well a ...

formula
s for the areas of simple shapes such as
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
s,
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
s, and
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
s.


Proportions

Two quantities are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called the
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of proportionality or proportionality constant. *If one quantity is always the product of the other and a constant, the two are said to be ''directly proportional''. are directly proportional if the
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
\tfrac yx is constant. *If the product of the two quantities is always equal to a constant, the two are said to be ''inversely proportional''. are inversely proportional if the product xy is constant.


Analytic geometry

Analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
is the study of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
using a
coordinate system In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

coordinate system
. This contrasts with
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
. Usually the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
is applied to manipulate
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
s for
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
s,
straight line In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...

straight line
s, and
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

square
s, often in two and sometimes in three dimensions. Geometrically, one studies the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
(2 dimensions) and
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
(3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. Transformations are ways of shifting and scaling functions using different algebraic formulas.


Negative numbers

A
negative number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
that is
less than In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different not ...
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
. Such numbers are often used to represent the amount of a loss or absence. For example, a
debt Debt is an obligation that requires one party, the debtor A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to ...

debt
that is owed may be thought of as a negative asset, or a decrease in some quantity may be thought of as a negative increase. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and
Fahrenheit The Fahrenheit scale ( or ) is a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to convenient and stable parameters, such a ...

Fahrenheit
scales for temperature.


Exponents and radicals

Exponentiation is a
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
operation, written as ''b''''n'', involving two numbers, the
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, ...
''b'' and the exponent (or power) ''n''. When ''n'' is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
(i.e., a positive
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
), exponentiation corresponds to repeated
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
of the base: that is, ''bn'' is the product of multiplying ''n'' bases: :b^n = \underbrace_n Roots are the opposite of exponents. The
nth root In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
''x'' (written \sqrt /math>) is a number ''r'' which when raised to the power ''n'' yields ''x''. That is, :\sqrt = r \iff r^n = x, where ''n'' is the ''degree'' of the root. A root of degree 2 is called a ''
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
'' and a root of degree 3, a ''
cube root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cube root
''. Roots of higher degree are referred to by using ordinal numbers, as in ''fourth root'', ''twentieth root'', etc. For example: * 2 is a square root of 4, since 22 = 4. * −2 is also a square root of 4, since (−2)2 = 4.


Compass-and-straightedge

Compass-and-straightedge, also known as ruler-and-compass construction, is the construction of lengths,
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s, and other geometric figures using only an idealized
ruler A ruler, sometimes called a rule or line gauge, is a device used in geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of spac ...

ruler
and
compass A compass is a device that shows the cardinal direction The four cardinal directions, or cardinal points, are the directions north, east, south, and west, commonly denoted by their initials N, E, S, and W. East and west are perpendicular ( ...
. The idealized ruler, known as a
straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and mac ...

straightedge
, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass, see
compass equivalence theoremThe compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a ''divider'' or ''collapsing compass'', that is, a Compass (drafting), compass that "collapses ...
.) More formally, the only permissible constructions are those granted by
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
's first three postulates.


Congruence and similarity

Two figures or objects are congruent if they have the same
shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...

shape
and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
are called congruent if, and only if, one can be transformed into the other by an
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, i.e., a combination of rigid motions, namely a
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
, a
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

rotation
, and a
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. Two geometrical objects are called similar if they both have the same
shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A plane shape, two-dimensional s ...

shape
, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energy ...
(enlarging or shrinking), possibly with additional
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
,
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
and
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
to the result of a uniform scaling of the other.


Three-dimensional geometry

Solid geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
was the traditional name for the
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
of three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. Stereometry deals with the
measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ...

measurement
s of
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
s of various solid figures (
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (), generally, is any characteristic that can help in defining or classifying a ...
figures) including
pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilater ...

pyramids
,
cylinders A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

cylinders
,
cones A cone is a three-dimensional space, three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the Apex (geometry), apex or vertex (geometry), vertex. A cone is fo ...

cones
,
truncated cones
truncated cones
,
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
s, and
prisms An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape o ...
.


Rational numbers

Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
is any
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
that can be expressed as the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
or fraction ''p''/''q'' of two
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, with the
denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
''q'' not equal to zero. Since ''q'' may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or
blackboard bold Image:Blackboard bold.svg, 250px, An example of blackboard bold letters Blackboard bold is a typeface style that is often used for certain symbols in mathematics, mathematical texts, in which certain lines of the symbol (usually vertical or near-v ...

blackboard bold
\mathbb).


Patterns, relations and functions

A
pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric Geometry (from the grc, ...

pattern
is a discernible regularity in the world or in a manmade design. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeating like a
wallpaper Wallpaper is a material used in interior decoration Interior design is the art and science of enhancing the interior of a building to achieve a healthier and more aesthetically pleasing environment for the people using the space. An inte ...

wallpaper
. A relation on a set ''A'' is a collection of
ordered pair In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

ordered pair
s of elements of ''A''. In other words, it is a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
of the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''A''2 = . Common relations include divisibility between two numbers and inequalities. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number ''x'' to its square ''x''2. The output of a function ''f'' corresponding to an input ''x'' is denoted by ''f''(''x'') (read "''f'' of ''x''"). In this example, if the input is −3, then the output is 9, and we may write ''f''(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function.


Slopes and trigonometry

The
slope of a line
slope of a line
is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''. Trigonometry is a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
that studies relationships involving lengths and
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s of
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
s. The field emerged during the 3rd century BC from applications of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
to astronomical studies. The slope is studied in grade 8.


United States

In the United States, there has been considerable concern about the low level of elementary mathematics skills on the part of many students, as compared to students in other developed countries. The No Child Left Behind program was one attempt to address this deficiency, requiring that all American students be tested in elementary mathematics.Frederick M. Hess and Michael J. Petrilli, ''No Child Left Behind'', Peter Lang Publishing, 2006, .


References

{{DEFAULTSORT:Elementary Mathematics Elementary mathematics,