Elementary mathematics

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Elementary mathematics consists of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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 * ...
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 '' secondary education, lower secondary education'' (ages 11 to 14) ...
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 and 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
such as
divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
and the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only wa ...
s, are studied in basic
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
, another part of elementary mathematics. Elementary Focus *
Abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since Ancient history, ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the ado ...
* LCM and GCD * Fractions and
Decimals The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
* Place Value &
Face Value The face value, sometimes called nominal value, is the value of a coin A coin is a small, flat (usually depending on the country or value), round piece of metal A metal (from ancient Greek, Greek μέταλλον ''métallon'', "min ...
*
Addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
and
subtraction Subtraction is an Arithmetic#Arithmetic operations, arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches ...
*
Multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
and
Division Division or divider may refer to: Mathematics *Division (mathematics) Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and ...
*
Counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
*Counting
Money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic context. The primary functions which distinguish money are as ...
*
Algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
*Representing and ordering numbers * Estimating * Approximating *
Problem Solving Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
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 A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...
* telling
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
using 12 hour clock and 24 hour clock * comparing objects using measurable attributes * measuring
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
, length, width * centimetres and
metres The metre (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
*
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
and capacity * temperature change * days, months, weeks, years * distances using kilometres * measuring kilograms and litres * determining
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
and
perimeter A perimeter is a closed path (geometry), path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length (mathematics), length. The perimeter of a circle or an ellipse is called its circumference. Calc ...
* determining grams and millilitre * determining measurements using
shapes A shape or figure is a graphics, graphical representation 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 pl ...
such as a
triangular prism In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field o ...
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 measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
requires a significant amount of
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
or its geometrical analogue, the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by Inscribed figure, inscribing inside it a sequence of polygons whose areas limit (mathematics), converge to the area of the containing shape. If the sequence is correctly c ...
; 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 (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
(the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
for example), mathematicians have produced a formula to describe the volume. : An equation is a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the terminology, term ''formula'' in science refers to the Commensurability (philosophy o ...
of the form ''A'' = ''B'', where ''A'' and ''B'' are expressions that may contain one or several variables called unknowns, and "=" denotes the equality
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
. Although written in the form of
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same me ...
, an equation is not a statement 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 of qualitative or quantitative variables; restated, pieces of data are individual pieces of
information Information is an Abstraction, abstract concept that refers to that which has the power to Communication, inform. At the most fundamental level information pertains to the Interpretation (logic), interpretation of that which may be sensed. ...
. Data in
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hardware and software. ...
(or
data processing Data processing is the data collection, collection and manipulation of digital data to produce meaningful information. Data processing is a form of ''information processing'', which is the modification (processing) of information in any manner ...
) is represented in a
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
that is often tabular (represented by rows and columns), a
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondar ...
(a set of
node In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex). Node may refer to: In mathematics *Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two ...
s with
parent A parent is a caregiver of the offspring in their own species. In humans, a parent is the caretaker of a child (where "child" refers to offspring, not necessarily age). A ''biological parent'' is a person whose gamete resulted in a child, a male t ...
-
children A child (plural, : children) is a human being between the stages of childbirth, birth and puberty, or between the Development of the human body, developmental period of infancy and puberty. The legal definition of ''child'' generally refers ...
relationship), or a graph (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. In other words, measurement is a process of determi ...
s and can be visualized using graphs or
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
s. Data as an abstract
concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by sev ...
can be viewed as the lowest level of
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. T ...
, from which
information Information is an Abstraction, abstract concept that refers to that which has the power to Communication, inform. At the most fundamental level information pertains to the Interpretation (logic), interpretation of that which may be sensed. ...
and then
knowledge Knowledge can be defined as Descriptive knowledge, awareness of facts or as Procedural knowledge, practical skills, and may also refer to Knowledge by acquaintance, familiarity with objects or situations. Knowledge of facts, also called pro ...
are derived.

## Basic two-dimensional geometry

Two-dimensional geometry is a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the tw ...
is a shape that is bounded by a finite chain of straight
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
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, a polytope is a geometric object with Flat (geometry), flat sides (''Face (geometry), faces''). Polytopes are the generalization of three-dimensional polyhedron, polyhedra to any number of dimensions. Polytopes may exist ...
in any number of dimensions. A
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
is a simple
shape A shape or figure is a graphics, graphical representation 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 pl ...
of two-dimensional geometry that is the set of all points in a plane that are at a given distance from a given point, the center.The distance between any of the points and the center is called the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''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 (geometry), path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length (mathematics), length. The perimeter of a circle or an ellipse is called its circumference. Calc ...
is a path that surrounds a
two-dimensional In mathematics, a plane is a Euclidean space, Euclidean (flatness (mathematics), flat), two-dimensional surface (mathematics), surface that extends indefinitely. A plane is the two-dimensional analogue of a point (geometry), point (zero dimensi ...
shape A shape or figure is a graphics, graphical representation 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 pl ...
. 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 consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
or
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
is called its
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
.
Area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
is the
quantity Quantity or amount 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 multiple of a u ...
that expresses the extent of a
two-dimensional In mathematics, a plane is a Euclidean space, Euclidean (flatness (mathematics), flat), two-dimensional surface (mathematics), surface that extends indefinitely. A plane is the two-dimensional analogue of a point (geometry), point (zero dimensi ...
figure or
shape A shape or figure is a graphics, graphical representation 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 pl ...
. There are several well-known
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the terminology, term ''formula'' in science refers to the Commensurability (philosophy o ...
s for the areas of simple shapes such as
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
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 par ...
s, and
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
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, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
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 shows 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 the ...
$\tfrac yx$ 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 using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
is the study of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
using a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space ...
. This contrasts with
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Coordinate-free, without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, ...
. Usually the Cartesian coordinate system is applied to manipulate equations for Plane (mathematics), planes, Line (geometry), straight lines, and Square (geometry), squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (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 is a real number that is inequality (mathematics), less than 0 (number), zero. Such numbers are often used to represent the amount of a loss or absence. For example, a 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 scales for temperature.

Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as ''b''''n'', involving two numbers, the Base (exponentiation), base ''b'' and the exponent (or power) ''n''. When ''n'' is a natural number (i.e., a positive integer), exponentiation corresponds to repeated multiplication of the base: that is, ''bn'' is the product (mathematics), product of multiplying ''n'' bases: :$b^n = \underbrace_n$ Roots are the opposite of exponents. The nth root of a number ''x'' (written $\sqrt\left[n\right]$) is a number ''r'' which when raised to the power ''n'' yields ''x''. That is, :$\sqrt\left[n\right] = r \iff r^n = x,$ where ''n'' is the ''degree'' of the root. A root of degree 2 is called a ''square root'' and a root of degree 3, a ''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, angles, and other geometric figures using only an Idealization (science philosophy), idealized ruler and Compass (drafting), compass. The idealized ruler, known as a 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 theorem.) More formally, the only permissible constructions are those granted by 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 a graphics, graphical representation 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 pl ...
and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation (geometry), translation, a rotation, and a reflection (mathematics), reflection. 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 similarity (geometry), similar if they both have the same
shape A shape or figure is a graphics, graphical representation 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 pl ...
, 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 (geometry), scaling (enlarging or shrinking), possibly with additional Translation (geometry), translation, Rotation (mathematics), rotation and Reflection (mathematics), reflection. 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 congruence (geometry), congruent to the result of a uniform scaling of the other.

## Three-dimensional geometry

Solid geometry was the traditional name for the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
of three-dimensional Euclidean space. 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. In other words, measurement is a process of determi ...
s of
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
s of various solid figures (three-dimensional figures) including Pyramid (geometry), pyramids, Cylinder (geometry), cylinders, cone (geometry), cones, Frustum, truncated cones,
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s, and Prism (geometry), prisms.

## Rational numbers

Rational number is any number that can be expressed as the quotient or fraction ''p''/''q'' of two integers, with the denominator ''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 $\mathbb$).

## Patterns, relations and functions

A 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. A binary relation, relation on a set ''A'' is a collection of ordered pairs of elements of ''A''. In other words, it is a subset of the Cartesian product ''A''2 = . Common relations include divisibility between two numbers and inequalities. A function is a binary relation, 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 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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
that studies relationships involving lengths and angles of
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
s. The field emerged during the 3rd century BC from applications of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
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,