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 such as
divisibility and the distribution of
prime numbers, are studied in basic
number theory, another part of elementary mathematics. Elementary Focus: •
Abacus •
LCM and
GCD •
Fractions and
Decimals •
Place Value &
Face Value •
Addition and
Subtraction •
Multiplication and
Division •
Counting • Counting
Money •
Factorial •
Combination without repetition •
Combination with repetition •
Permutation without repetition •
Permutation with repetition •
Algebra •
Equation •
Inequality • Representing and ordering numbers •
Estimating •
Approximating •
Problem Solving 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. : 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 of the heights of 31
Black Cherry trees. Histograms are a common tool used to represent 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
measurements and can be
visualized using
graphs or
images. 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 areas. A
polygon is a shape that is bounded by a finite chain of straight
line segments 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
formulas for the areas of simple shapes such as
triangles,
rectangles, and
circles.
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
equations for
planes,
straight lines, and
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
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{b \times \cdots \times b}_n Roots are the opposite of exponents. The
nth root of a
number x (written \sqrt[n]{x}) is a number
r which when raised to the power
n yields
x. That is, :\sqrt[n]{x} = 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
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 the
first three postulates of
Euclid.
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
measurements of
volumes of various
solid figures (
three-dimensional figures) including
pyramids,
cylinders,
cones,
truncated cones,
spheres, 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{Q}).
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 pattern. A
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 A2 = . 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
x2. 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
angles of
triangles. The field emerged during the 3rd century BC from applications of
geometry to astronomical studies. ==United States==