Elementary mathematics

A collection of geometric shapes. All shapes of a given color are similar to each other. Shapes and basic geometry are important topics in elementary mathematics.

An illustration using apples to show that 2+3 and 3+2 are both equal to 5. Apples are frequently used to explain arithmetic in textbooks for children.^[1]

Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.

The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.^[2]

In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.^[3]

Topics

According to a survey of the math curriculum of countries participating in the TIMSS exam, the following topics were considered important to the elementary curriculum (years 1-8) by at least two-thirds of the highest-performing countries:^[4]

Whole numbers

The whole numbers^[5][6][7][8] are those 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"). In common language, these purposes are distinguished by the use of cardinal and ordinal numbers, respectively. A third use of natural numbers is as nominal numbers, such as the model number of a product, where the natural number is used only for naming (as distinct from a serial number where the order properties of the natural numbers distinguish later uses from earlier uses).

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 mathematics focuses on the (+) and (×) operations and their properties:

• Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.
• Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
• Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.
• Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
• Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).
• No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0.

Measurement units

A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same physical quantity.^[9] Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement.

For example, length is a physical quantity. The metre is a unit of length that represents a definite predetermined length. When we say 10 metres (or 10 m), we actually mean 10 times the definite predetermined length called "metre".

The definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common. Now there is a global standard, the International System of Units (SI), the modern form of the metric system.

Fractions

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: $\tfrac{1}{2}$ and 17/3) consists of an integer numerator, displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero.

Equations and formulas

A formula is an entity constructed using the symbols and formation rules of a given logical language.^[10] For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion;^[11] 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:

V = 4/3 π r³

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.^[12]

Data representation and analysis

An example histogram 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 area.

A polygon 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.

Rounding and significant figures

Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing £23.4476 with £23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414. Rounding is often done on purpose to obtain a value that is easier to write and handle than the original. It may be done also to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is better stated as "about 123,500."

The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:^[13]

• All leading zeros;
• Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures); and
• Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

Estimation

Estimation is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available.^[14]

An informal estimate when little information is available is called a guesstimate, because the inquiry becomes closer to purely guessing the answer.

Decimals

A decimal fraction is a fraction the denominator of which is a power of ten.^[15]

Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed) at the position from the right corresponding to the power of ten of the denominator; e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as 0.8, 0.83, 0.083, and 0.0008. In English-speaking, some Latin American and many Asian countries, a period (.) or raised period (·) is used as the decimal separator; in many other countries, particularly in Europe, a comma (,) is used.

Percentages

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviation "pct." A percentage is a dimensionless number (pure number).

For example, 45% (read as "forty-five percent") is equal to 45/100, or 0.45. Percentages are used to express how large or small one quantity is relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity. For example, an increase of $0.15 on a price of $2.50 is an increase by a fraction of 0.15/2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase. While percentage values are often between 0 and 100 there is no restriction and one may, for example, refer to 111% or −35%.

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. x and y 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. x and y are inversely proportional if the product $xy$ is constant.

Analytic geometry

Cartesian coordinates

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 bⁿ, 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 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.^[16] 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.^[17] 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.

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 = A × A. Common relations include divisibility between two numbers and inequalities.

A function^[18] 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.^[19] Slope is often denoted by the letter m.^[20]

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.^[21]

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.^[22] The No Child Left Behind program was one attempt to address this deficiency, requiring that all American students be tested in elementary mathematics.^[23]

References

1. Enderton, Herbert (1977). Elements of set theory. Academic Press. p. 138. ISBN 0-12-238440-7.: "...select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."
2. Gary L. Musser, Blake E. Peterson, and William F. Burger, Mathematics for Elementary Teachers: A Contemporary Approach, Wiley, 2008, ISBN 978-0-470-10583-2.
3. Timothy J. McNamara, Key Concepts in Mathematics: Strengthening Standards Practice in Grades 6-12, Corwin Prss, 2006, ISBN 978-1-4129-3842-6
4. Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum. American educator, 26(2), 1-18.
5. Weisstein, Eric W., "Whole Number", MathWorld.
6. Clapham & Nicholson (2014): "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers."
7. James & James (1992) give definitions of "whole number" under several headwords:
   INTEGER … Syn. whole number.
   NUMBER … whole number. A nonnegative integer.
   WHOLE … whole number.
       (1) One of the integers 0, 1, 2, 3, … .
       (2) A positive integer; i.e., a natural number.
       (3) An integer, positive, negative, or zero.
8. The Common Core State Standards for Mathematics say: "Whole numbers. The numbers 0, 1, 2, 3, ...." (Glossary, p. 87) (PDF)
   Definitions from The Ontario Curriculum, Grades 1-8: Mathematics, Ontario Ministry of Education (2005) (PDF)
       "natural numbers. The counting numbers 1, 2, 3, 4, ...." (Glossary, p. 128)
       "whole number. Any one of the numbers 0, 1, 2, 3, 4, ...." (Glossary, p. 134)
   These pre-algebra books define the whole numbers:
   Szczepanski & Kositsky (2008): "Another important collection of numbers is the whole numbers, the natural numbers together with zero." (Chapter 1: The Whole Story, p. 4). On the inside front cover, the authors say: "We based this book on the state standards for pre-algebra in California, Florida, New York, and Texas, ..."
   Bluman (2010): "When 0 is added to the set of natural numbers, the set is called the whole numbers." (Chapter 1: Whole Numbers, p. 1)
   Both books define the natural numbers to be: "1, 2, 3, …".
9. "measurement unit", in International Vocabulary of Metrology – Basic and General Concepts and Associated Terms (VIM) (PDF) (3rd ed.), Joint Committee for Guides in Metrology, 2008, pp. 6–7.
10. Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York, NY: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6
11. Smith, David E (1958). History of Mathematics. New York: Dover Publications. ISBN 0-486-20430-8.
12. "Equation". Dictionary.com. Dictionary.com, LLC. Retrieved 2009-11-24.
13. Chemistry in the Community; Kendall-Hunt:Dubuque, IA 1988
14. C. Lon Enloe, Elizabeth Garnett, Jonathan Miles, Physical Science: What the Technology Professional Needs to Know (2000), p. 47.
15. "Decimal Fraction". Encyclopedia of Mathematics. Retrieved 2013-06-18.
16. Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Congruent Figures" (PDF). Addison-Wesley. p. 167. Retrieved September 2013.
17. Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. ISBN 978-0-07-288008-3.
18. The words map or mapping, transformation, correspondence, and operator are often used synonymously. Halmos 1970, p. 30.
19. Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Gradient" (PDF). Addison-Wesley. p. 348. Retrieved September 2013.
20. Weisstein, Eric W. "Slope". MathWorld--A Wolfram Web Resource. Retrieved September 2013.
21. R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002)
22. Liping Ma, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning.), Lawrence Erlbaum, 1999, ISBN 978-0-8058-2909-9.
23. Frederick M. Hess and Michael J. Petrilli, No Child Left Behind, Peter Lang Publishing, 2006, ISBN 978-0-8204-7844-9.