Elementary mathematics

Elementary mathematics consists of mathematics topics frequently taught at the

primary

or secondary school 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 such as divisibility and the distribution of prime numbers, are studied in basic

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

of a

sphere

requires a significant amount of integral calculus or its geometrical analogue, the

method of exhaustion

; but, having done this once in terms of some

parameter

(the

radius

for example), mathematicians have produced a formula to describe the volume: This particular formula is:

:

An equation is a

formula

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. Although written in the form of proposition, 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

. Data in

computing

(or

data processing

) is represented in a

structure

that is often tabular (represented by rows and columns), a

tree

(a set of nodes with

parent

-

children

relationship), or a graph (a set of connected nodes). Data is typically the result of

measurement

s and can be visualized using graphs or

image

s.

Data as an

abstract

concept

can be viewed as the lowest level of

abstraction

, from which

information

and then knowledge are derived.

Basic two-dimensional geometry

Two-dimensional geometry is a branch of 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

is a shape that is bounded by a finite chain of straight

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 any number of dimensions.

A

circle

is a simple

shape

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

. It can also be defined as the locus of a point equidistant from a fixed point.

A

perimeter

is a path that surrounds a two-dimensional

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

or

ellipse

is called its circumference.

Area

is the quantity that expresses the extent of a two-dimensional figure or

shape

. There are several well-known

formula

s for the areas of simple shapes such as

triangle

s,

rectangle

s, and

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 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

$\tfrac yx$ 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 is the study of

geometry

using a

coordinate system

. This contrasts with synthetic geometry.

Usually the Cartesian coordinate system is applied to manipulate

equation

s for planes,

straight line

s, and

square

s, 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 less than

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.

Exponents and radicals

Exponentiation is a mathematical operation, written as ''b''^''n'', involving two numbers, the 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 of multiplying ''n'' bases:

:$b^n = \underbrace_n$

Roots are the opposite of exponents. The nth root of a

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

'' 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,

angle

s, and other geometric figures using only an idealized

ruler

and 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

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, a

rotation

, and a 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 similar if they both have the same

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 (enlarging or shrinking), possibly with additional translation, rotation and 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 congruent to the result of a uniform scaling of the other.

Three-dimensional geometry

Solid geometry was the traditional name for the

geometry

of three-dimensional Euclidean space. Stereometry deals with the

measurement

s of

volume

s of various solid figures (three-dimensional figures) including

pyramids

,

cylinders

,

cones

,

truncated cones

,

sphere

s, and 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 relation on a set ''A'' is a collection of

ordered pair

s 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 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 that studies relationships involving lengths and

angle

s of

triangle

s. The field emerged during the 3rd century BC from applications of

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

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