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NZC: Glossary of Mathematics Terms

Absolute value

For all real numbers a, |a| (pronounced ‘absolute value of a’), is the positive magnitude of a.

So |6|=6, |100|=100, |-6|=6, |-100|=100
So |a|=a if a is positive, |a|=-a if a is negative, |a|=0 if a=0.
So |3|=3 and |-3|=3

An intuitive way of looking at absolute value is to consider that the absolute value of a real number is its distance from zero on the number line. For example:

Addition of:

  • Whole numbers: Addition is an operation of composition. On whole numbers, addition may be described as the joining of disjoint sets. In a physical model it is represented by the bringing together of objects. Initially it involves counting the objects in the joined set to determine the sum, or result, of the operation. When the basic addition facts are known, more complex addition problems can be answered using additive strategies.
    Addition is a binary operation, that is, it is an operation on two numbers.
    Addition is commutative, that is, the order of the numbers does not change the answer. For example, 4+5 = 5+4.
    Addition is associative, that is, the grouping of the numbers does not affect the answer. For example, (2+3)+5 = 2+(3+5).
    Zero is the identity element for addition, because the addition of zero to a number does not change it.
  • Fractions: Fractions may be added. If their denominators are the same then we can simply add the numerators to obtain the sum. For example, 2/9 + 4/9 = 6/9. If their denominators are not the same then we must choose equivalent fractions so that their denominators are the same. For example, to add 1/6 and 1/4 we must find equivalent fractions for 1/6 and 1/4 that have a common denominator. We could multiply the two denominators, 6 and 4, and that process would always give us a common denominator. However, we might observe that the least common multiple of 6 and 4 is actually 12.
    1/6 = 2/12, 1/4 = 3/12, so 1/6 + 1/4 = 5/12.
  • Decimals: Decimal fractions (commonly called decimals) may be added in the same way that whole numbers are, with care being taken to consider the position of the digits in the decimal. So, for example, tenths are added to tenths, hundredths to hundredths, etc. We can add the decimals because they represent fractions whose denominators are the powers of 10. For example, 2.4 = 2 + 4/10 and 3.56 = 3 + 5/10 + 6/100 so the 4/10 can be added to the 5/10 because they already have the common denominator of 10.
  • Percentages: Percentages may be added as if they were whole numbers or decimals. For example, 8% + 23% = 31%, 2.6% + 3.1 % + 120% = 125.7%. At an abstract level, a percentage is a numeral representing a number and therefore, just like decimals, they may be added. Care must be taken however because of the way that society uses percentages. One often refers to a percentage of something and that can lead to difficulties. For example, although it is true that 5% of 80 plus 10% of 80 is 15% of 80, it is not true that 5% of 80 plus 10% of 60 is 15% of either 80 or 60.
  • Integers: Integers may be added by observing the following rule:
    a+-b=a–b. For example, 4+-3 = 4–3 = 1.
    This rule, and similar rules for subtraction are best discovered using models, such as a black-and-white counters model, in which a white counter represents one, and a black counter represents -1. The first thing to establish is that opposites cancel. So, for example, 1+-1=0.
    The properties of addition outlined for whole number also apply to the addition of fractions, decimals, percentages and integers.

Additive strategies

Additive strategies are techniques used to solve addition problems from known facts. For example, we can change 9+6 into 10+5, so 9+6=15. Similarly, since most children learn the ‘doubles’ early on, 8+7 can be thought of as one more than 7+7. More advanced additive strategies would be such as the following: To find 47+38; shift 2 from the 47 to the 38 (i.e. partition 47 as 45+2). The problem then becomes 45+40, which can more easily be solved.
So the term ‘additive strategies’ involves the partitioning of numbers, that is the understanding that numbers can be ‘broken up’ and recombined as in the calculation of 47+38 above. It also involves methods of finding answers to subtractions such as 63-29. One strategy would be to subtract 30 from 63 to obtain 33 and then to add 1 because we have subtracted 1 too much. Another strategy would be to add 1 to each number and make the subtraction 64-30. That is effectively shifting both numbers along the number line one position, and hence the difference between them remains the same.

Angle

An angle is the figure formed by two rays (or line segments) meeting at a point. The rays are the sides of the angle, while the point is its vertex. The size (or measure) of the angle is usually measured in degrees and is determined by the amount of rotation (or turn) about the vertex that would be required to move one side of the angle onto the other side. The size of the angle is often loosely referred to as the angle itself, e.g. "an angle of 60o."

Angle properties of intersecting lines

  • Adjacent angles are angles that have a common vertex and a common side. So (with reference to the diagram) 1 and 4 are adjacent angles, 1 and 2 are adjacent angles etc.
  • Vertical angles (or vertically opposite angles) are non-adjacent angles formed by two intersecting lines. Thus, in the diagram, 2 and 4 are vertical angles and 1 and 3 are vertical angles. These angles are called vertical because each side of one is an extension through the vertex of a side of the other. Vertical angles are equal. For example, in the diagram, angle 1 equals angle 3 etc.
  • Supplementary angles are two angles whose sum equals 180o. So, in the diagram, 1 and 4 are supplementary angles as are 1 and 2, 2 and 3, and 3 and 4. We can see that adjacent angles are supplementary if their exterior sides lie on the same straight line.
  • Complementary angles are two angles whose sum equals 90o. Adjacent angles are complementary if their exterior sides are perpendicular to each other.

    a and b are adjacent with exterior sides perpendicular and are therefore complementary.

Angle properties of parallel lines

In Euclidean geometry, parallel lines are lines that lie in the same plane and do not intersect no matter how far they are extended. A transversal of two or more lines is a line that cuts across those lines. In the above diagram we have a transversal intersecting a pair of parallel lines. Properties and terminology of the angles created are:

  • 1,2,7, and 8 are exterior angles. They are the angles outside the two parallel lines.
  • 3,4,5 and 6 are interior angles. They are the angles between the two parallel lines.
  • Corresponding angles are angles on the same side of the transversal and on the same side of the parallel lines. So 1 and 5 are corresponding angles, 2 and 6 are corresponding angles, 3 and 7 are corresponding angles, and 4 and 8 are corresponding angles. Corresponding angles are equal.
  • Alternate interior angles are interior angles that are on opposite sides of the transversal. Thus 4 and 5 are alternate interior angles, and 3 and 6 are alternate interior angles. Alternate interior angles are equal.
  • Alternate exterior angles are exterior angles that are on opposite sides of the transversal. Thus 2 and 7 are alternate exterior angles, and 1 and 8 are alternate exterior angles. Alternate exterior angles are equal.
  • Opposite angles (or vertically opposite angles) are equal. 1 and 4 are opposite angles, 2 and 3 are opposite angles, 5 and 8 are opposite angles, and 6 and 7 are opposite angles.

 

Angle properties of polygons

A polygon is a portion of a plane bounded by straight lines. If n is the number of sides of the polygon then the smallest value for n is 3, which is the triangle. The interior angles of a triangle add to 180o a fact that can be easily shown by drawing a triangle on paper, tearing off the corners and putting the vertex angles together. For n=4 we have a quadrilateral which can be divided into two triangles, and so we see that the sum of the interior angles of a quadrilateral is 2 x 180o, or 360o. For n = 5 we have a pentagon which can be divided into three triangles so the sum of the interior angles of a pentagon is 3 x 180o, or 540o. Continuing in this manner we see that the sum of the interior angles of any n-sided polygon is (n-2)x 180o. This can also be written as (180n-360)o or (2n-4) right angles.
A regular polygon is a polygon whose sides are all congruent and whose angles are all equal. Hence for a regular polygon with n sides we can find the size of each interior angle by dividing the sum of the interior angles by n. A list of the regular polygons for n = 3,4,5,…,12 with their names and interior angle size is given below.

No of SidesName of regular polygonSize of each angle (in degrees)
3Equilateral triangle60 (180÷3)
4Square90 (360÷4)
5Regular pentagon108 (540÷5)
6Regular hexagon120 (720÷6)
7Regular heptagon128.57 (900÷7)
8Regular octagon135 (1080÷8)
9Regular nonagon (or Enneagon)140 (1260÷9)
10Regular decagon144 (1440÷10)
11Regular hendecagon147.27 (1620÷11)
12Regular dodecagon150 (1800÷12)

 

Note: If students have difficulty with the names of the polygons they could refer to them by their number of sides. For example, a pentagon could be referred to as a 5-gon, a hexagon as a 6-gon etc.

Angle properties relating to circles

The following are angle properties relating to circles:

  1. The angles on the same arc (or chord) of a circle are equal.

    Angle A equals angle B because they are supported by the same arc.
     
  2. The angle at the centre is twice the angle at the circumference.

    α is twice the size of β
     
  3. The angle supported by a diameter is a right angle:
  4. Opposite angles of a quadrilateral inscribed in a circle are supplementary, that is, they add to 180o.

    So angles A and C add to 180o and angles B and D add to 180o.

Antidifferentiation

Antidifferentiation is the reverse process of differentiation. So given a function, say f(x), it is the process of finding a function that, when differentiated, is equal to f(x). So it is the task of finding a function whose derivative is known. So, for example, if f(x) = 3x2 then an anti-derivative of f is x3. Another antiderivative of 3x2 is x3+5.
Anti-differentiation is also referred to as integration. The function to be integrated is referred to as the integrand, and the result of an integration is referred to as an integral. The indefinite integral of the function f is represented by ∫f(x)dx and is the set of all antiderivatives of f. So the integral of f(x) with respect to x is x3+C where C is a real number. That is, ∫f(x)dx = x3+C

Appropriate statistical variables

In statistics variables refer to measurable or countable attributes such as height, number of children in a family etc. The variables being measured in a statistical survey need to be clearly defined and measurable. Furthermore, the nature of the variable needs to be considered. For example, if the nature of the variable (such as the heights of children) were such that it led to the collection of measurement data then that would not be suitable for the curriculum levels 1 to 3.

Area

Area is a measure of the size of a surface. It is a measure of a two dimensional surface, measuring the size of a portion of a plane. It can also be used to measure the size of a curved surface. The basic SI unit of measurement of area is the square metre (m2), with square millimetre (mm2), square centimetre (cm2), and hectare (ha) also being used. (See SI measurement units)

Areas of polygons

Areas of polygons can be explored using squared paper. A good sequence is to start with a square (diagram 1 below), then move to a rectangle (diagram 2 below), observing that the areas of those are simply the product of two non-parallel sides. The area of the non-rectangular parallelogram (diagram 3 below) is easily discovered by cutting a triangle from one side and joining it to the opposite side to create a rectangle. This shows that the area of a parallelogram is the product of the base and the vertical height. Cutting a parallelogram on one diagonal creates two congruent shapes and shows that the area of a triangle (diagram 4 below) is a half of the area of the associated parallelogram, that is, a half of the product of the base and the vertical height. Next, the area of a trapezium (diagram 5 below) can be found by cutting the non-parallel sides through the midpoints and rotating them to make a rectangle. This shows that the area of the trapezium is the product of the average length of the two parallel sides and the distance between them.
Areas of other polygons might be found by seeing them as combinations of the polygons mentioned above, or might require a trigonometric approach.

Areas of rectangles, triangles, parallelograms etc.

See rectangles, triangles, parallelograms etc.

Arithmetic sequence

An arithmetic sequence is a sequence that is such that there is a common difference between successive terms of the sequence. For example, 1, 4, 7, 10, 13, is an arithmetic sequence because there is a common difference of 3 between the terms of the sequence.
If the common difference is d then the terms of the sequence can be written as a, a+d, a+2d, a+3d, … We can see that the nth term will be a + (n-1)d The sum of the n terms of the arithmetic series is:
a + (a+d) + (a+2d) + (a+3d) +… a + (n-1)d is 1/2n[2a + (n-1)d]
This may be observed by realising that the sum of the coefficients of d is 1 + 2 + 3 + ... + (n-2) + (n-1). This sum may be found by pairing the first term with the last term which gives a sum of n, the second term with the second-to-last term which gives a sum of n etc and by realising that there are 1/2 (n-1) such terms.
Hence the sum of the first n terms is na + 1/2(n-1)nd which can be written as 1/2n[2a + (n-1)d]

Backwards Counting Sequences

A counting sequence is an ordering of the counting numbers such that the difference between any two successive numbers is constant. The basic backwards counting sequence is …, 5, 4, 3, 2, 1. An example of a backwards skip counting sequence is …, 50, 40, 30, 20,10 , as is … 10, 8, 6, 4, 2 etc.

Base ten numeration system

Our numeration system (the Hindu-Arabic system) is a code in which the value of a digit is determined not only by its face value but also by its place value, or in other words the position that it is in. The base of this number system is 10. That means that the value of a digit in each place in a numeral is ten times greater than the value the same digit would have were it in the place to the right of it. For example, in the numeral 333, the left-hand 3 is worth 300, the middle 3 is worth 30, and the right hand 3 is worth just 3. It is an additive numeration system so the whole numeral is worth 300+30+3.

Basic addition and subtraction facts

The basic facts of addition are those equations in which two single-digit numbers are combined by addition to give a sum Hence they range from 0+0=0 to 9+9=18. For each basic addition fact there is a related basic subtraction fact, for example, 18-9=9. An understanding of the commutative property of addition halves the number of facts that need to be learned since 3+7 = 7+3 etc.
Addition and subtraction facts can be grouped into ‘families’ of facts, e.g., 5+4=9 so 4+5=9, 9-5=4 and 9-4=5

Basic multiplication and division facts

The basic facts of multiplication are those equations in which two single-digit numbers are combined by multiplication to give a product Hence they range from 0x0=0 to 9x9=81. For each basic multiplication fact there is a related basic division fact, for example, 81÷9=9. An understanding of the commutative property of multiplication halves the number of facts that need to be learned since 3x7 = 7x3 etc.
Multiplication and division facts can be grouped into ‘families’ of facts, e.g., 5x4=20 so 4x5=20, 20÷5=4 and 20÷4=5.

Binomial distribution

If p is the probability that an event will happen in any single trial (called the probability of a success) and q=1-p is the probability that it will fail to happen in any single trial (called the probability of a failure) then the probability that the event will happen exactly x times in n trials is given by P(x) = nCx pxqn-x where nCx = n!/[x!(n-x)!]

Capacity

Capacity is a measure of the interior volume of a container. Hence it is a measure of how much a container can hold. It is measured in units of volume. (See SI measurement units)

Cartesian plane

See Coordinate systems.

Category data

Category data is data that can be organized into distinct categories. For example, it could be colours of cars, types of food preferred, brand of shoes etc. Category data does not lend itself to the calculation of statistics such as mean and standard deviation since it is not numeric. Category data might best be displayed by pictograms or bar graphs, or for younger students, block graphs, where, for example, a square of sticky paper could represent an element in a category.

Central Limit theorem

Suppose we take a sample of size n from a population. The Central Limit Theorem states that if n is large then the distribution of the means of the samples (called the sampling distribution of the mean) can be approximated closely with a normal distribution. In fact as n increases, the sampling distribution of the mean asymptotically approaches a normal distribution with mean μ and standard deviation ϑ / √n.

Chance

Elements of chance occur when the outcome of an experiment or a trial is not predetermined or fixed. (See Probability). Students can compare experimental distributions (See Statistical distributions) with expectations from models of the possible outcomes. For example, children in a class might collectively roll a dice 600 times and establish a frequency distribution for the outcomes. They could then compare that distribution with the expectation of each possible outcome having a frequency of 100.

Charts

A chart is a table containing data. The ability to read charts is a social requirement. Many charts, such as tide charts or weather charts, contain information that can be transferred to other forms of representation such as a time series graph.

Circle

A circle of radius r units and centre P is the set of points in a plane whose distance from P is r units. The length of the circumference of a circle is π x d, where d is the diameter of the circle and π is the ratio of the circumference to the diameter. (See Perimeters of circles) The area of a circle of radius r is πr2 (i.e. π x r x r) For example, suppose a circle has a radius of 3 cm then its diameter is 6 cm, its area is 9π cm2 and the length of its circumference is 6π cm. The area of a circle of radius r can be approximated by cutting the circle into (say) 16 congruent sectors and rearranging them to approximate a rectangle with sides of length πr and r, and hence area of πr2.

Circumference

The perimeter (or boundary) of a circle or an ellipse is called its circumference. The length of the circumference of a circle is πd where d is the diameter of the circle. See also Perimeters of circles.

Combinations

A combination of n different objects taken r at a time is a selection of r out of the n objects with no attention given to the order of arrangement. The number of combinations of n objects taken r at a time is denoted by nCr or (nr) and is equal to n!/(r!(n - r)!) where n! (pronounced n factorial) is equal to the product of the natural numbers from 1 to n. So for example, 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720.

Common factor

An integer a is a common factor of two integers b and c if it is a factor of both b and c. Take, for example, the factors of 18 and the factors of 24. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The common factors of 18 and 24 are 1, 2, 3, and 6 since those are the numbers that are factors of both 18 and 24. The greatest common factor of 18 and 24 is 6 since that is the greatest number that is a factor of both 18 and 24. The greatest common factor (also called the greatest common divisor) is often abbreviated to g.c.f. (or g.c.d.).

Common multiple

An integer a is a common multiple of two integers b and c if it is a multiple of both b and c. For example, the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …The common multiples of 4 and 6 are therefore the numbers 12, 24, 36, 48, …Unlike a set of common factors, this is an infinite set, the set of multiples of 12. The prime factorisations of 4 and 6 are 2 x 2 and 2 x 3 respectively. So the common multiples of 4 and 6 are multiples of 2 x 2 x 3, which equals 12.
So the least common multiple (often abbreviated to l.c.m.) of 4 and 6 is 12. The least common multiple of any two integers can also be found as the product of the numbers divided by their greatest common factor, that is, for integers a and b, l.c.m. of a and b = (a x b)÷(g.c.f. of a and b) In the example above, the l.c.m. of 6 and 4 = (6 x 4)÷2 = 12.

Comparing

The process of determining relative size or number. For example, if sticks were being compared by their length that could be done by measuring them against a standard (such as a metre rule) and comparing the measurement obtained, or by direct comparison by putting them alongside each other.

Comparing experimental results with expectations from models of outcomes

We can compare actual results of an experiment with the expectations of a probability model to test the model or to determine the significance of the outcome of the experiment. For example, suppose our experiment is to toss a fair coin three times and count the number of heads obtained. We could repeat this experiment a large number of times and record the number of times that we obtained no heads, one head, two heads, or three heads. We could then compare the number of times that each of these events occurred with the expected number based on our probability model.

Compass directions

Directions on planet Earth can be given in terms of a compass (or magnetic compass) bearing. The magnetised needle of a magnetic compass is attracted to Earth’s so-called magnetic pole, a position of strong magnetic attraction in the northern hemisphere. True north is the direction from any point on the Earth to the North Pole. The North Pole is one of the two points of intersection of Earth’s surface with its axis of rotation, the other point being the South Pole. Magnetic north differs considerably from true north and changes every year as Earth’s magnetic pole changes position. In New Zealand, the magnetic deviation in 2006 was approximately 18o east of north at Kaitaia and 26o east of north at Stewart Island. The variation is roughly proportional to these figures for the length of the country. For example, the figure for Wellington is approximately 22o. So in Wellington the needle of a magnetic compass would point to a true bearing of 22o. These figures are increasing by approximately 0.5o every six years.

Complex Numbers

The solution of some equations, such as x2 + 1 = 0, cannot be found within the set of real numbers since it requires that we find a number x such that x2 = -1 The equation requires a value of x which, when multiplied by itself, is -1. No such number exists within the set of real numbers for the product of any two real numbers that have the same sign is always positive or zero. Hence a new number i is defined for which i2 = -1.
The set of numbers of the form a + bi where a and b are real numbers and i2 = -1 is called the set of complex numbers. If z = a + bi then we have the following possibilities:

  1. b = 0 in which case z is a real number
  2. a = 0 in which case z is a pure imaginary number, such as 6i, -5i etc
  3. Neither a nor b is zero in which case z is called a complex number.

The set of complex numbers gives completeness to the number system since the roots of all polynomials can be found within the set of complex numbers and the nth root of any complex number is a complex number. This was obviously not true for the set of real numbers since, as above, the square root of -1 cannot be found within the set of real numbers.
Complex numbers can be represented graphically on the complex plane, a modified Cartesian plane in which the horizontal axis is called the real axis and represents the real part of the complex number, and the vertical axis is called the imaginary axis and represents the imaginary part of the complex number. The complex plane is also referred to as the Argand diagram.

 

Compounding

Quantities may change over time in applications involving growth or decay. In particular the concept of compound interest is one in which the quantity changes over time and the change is proportional to the size of the quantity. Suppose capital (the principal, P) is invested at a growth rate of r (r could be 10% say). If the amount of the investment plus the interest in year n is An then,
A0 = P
A1 = P + r P = P (1+r)
A2 = P (1+r) + r P (1+r) = P (1+r)(1+r) = P (1+r)2
A3 = P (1+r)2 + r P (1+r)2 = P (1+r)2 (1+r) = P (1+r) 3

Hence the amount An after n compoundings is given by
An = PRn where R = 1+r.
A function of the form A(n) = PRn , or y = 3x is called an exponential function.

Cone

A circular cone is a solid whose base is a circle and whose lateral surface comes to a point. A line from the vertex of the cone to the centre of its base is called the axis. In a right cone the base is perpendicular to the axis. If the base is not perpendicular to the axis it is an oblique cone. A cone is effectively a pyramid with an infinite number of lateral faces and therefore it should not be surprising that, like a pyramid, its volume is one -third of the product of the base area and the vertical height. So for a cone whose base is a circle of radius r and whose vertical height is h, the volume (V) is given by: V = 1/3 πr2h.

Confidence interval

A confidence interval, given with an associated level of probability, is an interval within which a population parameter will lie with that degree of probability.

Congruent

Two figures are congruent if they are related so that for very point on one there is a corresponding point on the other and that the distance between any two points on one is equal to the distance between the corresponding points on the other. So two figures (shapes, lines, angles etc.) are congruent if they are identical in shape and could be made to fit exactly on to each other. Fitting one figure on to the other may require turning it over. (See Direct and indirect transformations)

Conic sections

The conic sections are so called because they can all be obtained as the outline of the intersection of a plane with a cone. Their equations are described by the general quadratic polynomial in two variables, x and y, which is of the form: f (x,y) = ax2 +bxy +cy2 +dx + ey +f.
The conic sections and their standard forms are:

  1. The circle of radius r and centre (0,0), which has equation x2 + y2 = r2 A circle is the locus of a point P (x, y) that is a fixed distance from a given point. The fixed distance is the radius and the given point is the centre.
    A circle may be obtained as the outline of the intersection of a right circular cone with a plane that is parallel to the base of the cone.
  2. The ellipse with centre at (0, 0), which has equation x2/a2 + y2/b2 = 1. An ellipse is the locus of a point P (x, y), the sum of whose distances from two fixed points (called the foci – plural of focus) is constant.
    An ellipse may be obtained as the outline of the intersection of a right circular cone with a plane that does not cut the base of the cone. In the case where the plane is parallel to the base we obtain a circle.
  3. The hyperbola with centre (0, 0), which has equation x2/a2 - y2/b2 = 1.
    An hyperbola is the locus of a point P (x, y), the absolute value of the difference of whose distances from two fixed points (the foci) is constant. A branch of the hyperbola may be obtained as the outline of the intersection of a right circular cone with a plane that cuts the base of the cone but is not parallel to a side (or generator) of the cone.
  4. The parabola with centre (0, 0) and focus on the x-axis, and whose directrix is the y-axis, which has equation y2 = 4cx.
    A parabola is the locus of a point P (x, y), whose distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix).
    The parabola may be obtained as the outline of the intersection of a right circular cone with a plane that cuts the base of the cone and is parallel to a side (or generator) of the cone.

 

Continuity of functions

A function f (x) is continuous at x = a if the following conditions hold:

  1. f (a) is defined
  2.  
    lim
    x → a
    f(x) exists and
  3.  
    lim
    x → a
    f(x) = f(a)

A function is discontinuous at a if one or more of the conditions for continuity fails there.

Conversions between fractions, decimals and percentages

See Fraction, decimal and percentage conversions.

Coordinate geometry techniques applied to graphs and lines

 

  • Distance The distance between two points in the plane is found by the use of Pythagoras’ theorem. If the points are P1 (x1, y1) and P2 (x2, y2) then the distance between them is:
    |P1 P2| = √ [(x2-x1)2 + (y2-y1)2] So the distance between (2,3) and (5,7) is √ (32+42) = 5
  • Gradient The gradient (or slope) of a line containing two points P1 (x1, y1) and P2 (x2, y2) is:
    change in y/change in x = (y2-y1)/ (x2-x1)
    So the slope of the line containing (-2,3) and (5, -1) is -1-3/5--2 = - 4/7
  • The equation of the line through two points P1 (x1, y1) and P2 (x2, y2) is: given by: y-y1/ x-x1 = y2-y1/ x2-x1
    So the line through (2,3) and (4,7) has equation y-3/x-2 = 7-3/4-2
    This can be rearranged to: y = 2x -1

Coordinate systems

There are many different possible types of coordinate system. They are designed to define the position of a point on the plane or in space. In the Cartesian coordinate system, a point in the plane can be uniquely represented by an ordered pair of numbers, each of which represents a distance along an axis, measured from the origin. An illustration of the Cartesian coordinate system, in which the axes are perpendicular, is shown below. The coordinates of point P are the ordered pair (2,3).

A simple coordinate system for younger students could be a system in which the spaces are labelled rather than points in the plane.

Cosine rule

For any triangle ABC the following law of cosines holds:
a2 = b2 + c2 - 2bc Cos A
b2 = a2 + c2 - 2ac Cos B
c2 = a2 + b2 - 2ab Cos C where a is the length of the side opposite angle A, b is the length of the side opposite angle B and c is the length of the side opposite angle C.
This rule is necessary for solving a triangle when given either two sides and the included angle, or three sides.

Counting

Counting is the process of establishing a one-to-one correspondence between the set of objects being counted and the set of natural numbers in order. The number of objects in the set is the last number named.

Counting Numbers

The set of counting numbers is the same as the set of natural numbers, i.e. 1, 2, 3, 4,… (See Base ten numeration system)

Critical path

An activity digraph is a directed network showing the time taken to complete a certain sequence of activities. A longest path (in units of time) in an activity digraph D is called a critical path of D. It shows the minimum time necessary to complete the project.

Cube root

See Roots.

Cuboid

A cuboid is a solid figure bounded by six rectangular faces. Hence it is like a box that has all sides rectangular. A special case of the cuboid is the square cuboid, which has two (or more) opposite faces squares. A special case of the square cuboid is the cube, which has all faces as squares.
The square cuboid could also be classified as a right square prism.
The volume of a cuboid is the product of the length of three of its sides, none of which are parallel to each other, expressed in appropriate units. (This can easily be discovered by making cuboids with blocks). For example, if a cuboid has edge lengths of 6cm, 8cm and 10cm then its volume is 480 cm3.

Curve fitting

Very often a relationship is found to exist between two or more variables. For example, weight depends to some degree on height. We may wish to express this relationship in mathematical form by determining an equation connecting the variables. ‘The method of least squares’ is a method that finds a function which best fits the data points.

Cylinder

A right circular cylinder (usually just called a cylinder) is a solid with three faces, whose bases are parallel circles that are perpendicular to the third face. Further, its cross-sections parallel to the bases are also circles. In common terms it is the shape of a spaghetti tin. In similarity to the volume of a cuboid, or in fact any prism, the volume of a cylinder is the product of the area of its circular base and its height. So if a cylinder has base circles of radius r and a height h then its volume is πr2h. For example, if a cylinder had a height of 9cm and a radius of 4cm then its volume would be π x 16 x 9 which is 452cm3 (to 3 dp).

Data

A set of known facts, numbers, or information used as a basis for reasoning. Raw data is data that has been collected but has not been organised.

Data collection methods

Data collection methods need to be appropriate to the variable being considered. For example, if the variable being considered by children in Room 5 at Kiwi School is the amount of pocket money received by children at Kiwi School then the data must be representative of the whole school, not just of Room 5. Sampling must be random and children can consider how that could be achieved.

Data displays

Data can be displayed in a variety of ways, but whatever way is chosen the key intention of using some form of data display is to make the data more readily accessible or more understandable to the viewer. Category data might best be displayed by pictograms or bar graphs, or for younger students, block graphs, where, for example, a square of sticky paper could represent an element in a category. Whole number data can be displayed by block graphs, pictographs, tally charts, bar graphs, pie graphs and stem and leaf graphs. Also possible are dot plots, strip graphs and time series graphs. Measurement data can be displayed by histograms.

Decimals

A number may have many numerals and one commonly used form of numeral is the ‘decimal’, or more correctly, ‘decimal fraction’. This is a system which extends the base ten numeration system to have place values less than 1. For example, whereas 324 is a compact numeral which can be written in expanded form as (3x100) + (2x10) + (4x1), we can extend this system to include fractional parts. For example, 324 15/100 could be written as 324.15 which means (3x100) + (2x10) + (4x1) + (1x1/10) + (5x1/100). So decimals are another way of recording fractional parts and are an extension of the base 10 numeration system. Obviously there is no restriction on the length of the decimal part, the part to the right of the decimal point. Other ways of recording parts of a whole are common fractions and percentages.
The full benefit of having a positional notation numeration system such as the system of decimal fractions is in having a system of units of measurement that is in harmony with it. Hence we see the importance of the metric system to industry and to society in general. (See SI measurement units)

Decimal place value

See Rounding

Denominator

When a rational number is written as a fraction, that is in the form a/b then b is called the denominator of the fraction.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative of a function is a new function whose domain consists of those points where the former function is differentiable, and whose values are the slopes of tangent lines at the corresponding points. Thus the value of the derivative of a function at any given point in the domain is a measure of the rate of change of the function at that point.
The derivative of a function f(x) is denoted by f ‘(x); its value at the point x is the slope of the line tangent to f at the point x. The term ‘derivative’ is based on the fact that the new function is derived from the original one.
The derivative of a function f(x) with respect to x is defined as follows:

 

f‘(x) =limf (x + h) - f(x)provided the limit exists.
hh → 0 0h


It can be shown that if f(x) = axn then f‘(x) = naxn-1 and that (f (x) + g (x))‘ = f‘(x) + g‘(x). These results enable the differentiation of polynomials.
Note: An alternative notation for the derivative of y with respect to x is dy/dx.
Approximate values of derivatives of functions may be found by using numerical differentiation techniques and formulas such as
f‘(x) = [f (x+h) - f (x)]/h and f‘(x) = [f (x+h) - f (x - h)]/2h

 

Differential equations

A differential equation is an equation that involves derivatives. For example: y’ = x+5, y’’+3y’+2y = 0 etc.
If there is a single independent variable such as above, the equation is called an ordinary differential equation.
The order of a differential equation is the order of the highest derivative that occurs.

Direct and indirect relationships with linear proportions

A quantity a is said to vary directly as another quantity b if a = kb where k is a constant. (This is often said as "a is directly proportional to b"). k is called the constant of proportionality.
For example, the length of the circumference of a circle (c) varies directly as the radius (r) and is expressed by the relationship c = 2πr. So if the radius doubles then so does the circumference.
A quantity a is said to vary inversely as another quantity b if a = k x 1/b where k is a constant of proportionality.
(This is often said as "a is inversely proportional to b"). For example, suppose the distance between two towns A and B is 30 km. The time (t) taken to travel from A to B is inversely proportional to the speed (s) of travel.
So t=30/s. If s=30 kph then t=1 (hour). If s doubles then t is halved.

Direct and indirect transformations

Transformations can be classified as direct or indirect. The direct isometries are rotation and translation. They are called direct because they do not flip (or turn over) the shape being transformed. Reflection and glide reflection are indirect isometries because they do flip the shape being transformed.

Direct comparison

The process of comparing directly rather than through an independent standard. For example, we can compare the number of elements in two given sets by counting the elements of each set and determining which is the greater number. In direct comparison we would match the elements against each other physically. Similarly, to determine which of two sticks is the longer we could use direct comparison by laying the sticks side-by-side and determining the longer one by observation, rather than measuring the two sticks against a standard measure.

Direction

The direction between two points A and B is the description of the path an object (or person) travelling from A to B would take. This could be in simple social terms such as forwards or backwards, left or right; in terms of compass directions such as north, south, south-east etc.; in terms of an angle of turn from an origin; or as a direction vector.

Discontinuities of functions

See Continuity of functions.

Distance

The distance between any two points on a line, on a plane or in space is the length of the straight line between them. This could be expressed in non-standard units such as steps, handspans etc., or in standard units such as meters, kilometres etc. (See length)

Distributions

See Statistical distributions.

Division of:

  • Whole numbers: Division is the inverse operation of multiplication. It arises from contexts that involve sharing or measurement. Both sharing and measurement can be viewed as repeated subtraction. For each basic fact of multiplication there is a family of facts that includes two basic facts of division. For example, 5 x 7 = 35 so 7 x 5 = 35, 35÷7 = 5 and 35÷5 = 7.
    Division is a binary operation, that is, it is an operation on two numbers.
    Division is not commutative, that is, the order of the numbers does matter. For example, 12÷4 ≠ 4÷12.
    Division is not associative, that is, the grouping of the numbers does affect the answer. For example, (12÷6) ÷2 = 1, whereas 12÷(6 ÷2) = 4.
  • Fractions: The operation of division may be performed on fractions. Division by a number is essentially multiplication by the reciprocal of the number. For example, 12÷4 = 12 x 1/4. Similarly, 2/3÷4/5 = 2/3 x 5/4 = 10/12
  • Decimals: Decimal fractions (commonly called decimals) may be divided in the same way that whole numbers are, with care being taken to consider the position of the decimal point. It is often best to leave the decimal point out while operating on the numbers and put the decimal point in the answer by estimation.
  • Percentages: Percentages may be divided but the answer needs to be interpreted carefully. For example, 50%÷40% = 50/100÷40/100 = 50/100 x 100/40 = 5/4. The answer is a fraction and if it is necessary to express the answer as a percentage then it would be necessary to convert it to a percentage, namely 125%
  • Integers: Integers may be divided in the following way: a÷-b = -(a÷b) e.g. 6÷-3 = -2
    -a÷b = - (a÷b) e.g. -6÷3 = -2
    --b = a÷b e.g. --3 = 2
    These results are probably best discovered from their associated multiplication results. (See Multiplication of integers). The properties of division outlined for whole number also apply to the division of fractions, decimals, percentages and integers.

Domain

See Function.

Drawings and models

Objects may be represented by drawings or models. The drawing or model and the object it represents may be similar in that the drawing or model may be a scale representation of the object. However drawings in particular can often be used to good effect in mathematics to represent relationships between elements, and the drawings might have no scale representation to the relationships they model.
Isometric plan views and nets may be used to effect. Drawing a picture is a useful problem solving strategy and such a picture might be approximately to scale or might have no scale relationship.

Elements of chance

See Chance.

Enlargement

An enlargement in the plane or in space is a mapping of a set of points such that for each point the distance of its image from a fixed point (the centre of enlargement) is a given multiple of the distance from the point to the centre of enlargement. For example, if the fixed multiple were 2 then every point of the image of the transformation would be twice the distance from the centre of the enlargement that the original point was. If it were a figure being transformed then all the length measurements of the figure would be doubled under that specific transformation. . If the multiple were 3 then all the length measurements of the image would be three times the length measurements of the original figure. The centre of enlargement is the only invariant point under the transformation of enlargement. Enlargement is not an isometric transformation since although the shape is similar it is not preserved because the dimensions have been changed.

Equal and different likelihoods

Events can have equal likelihoods or they can have different likelihoods. If a fair coin is tossed, the likelihood of getting a ‘head’ is equal to the likelihood of getting a ‘tail’. However if we draw a card from a standard pack of cards, the likelihood of getting a king is not equal to the likelihood of getting a red card.

Equality

The equality relation is fundamental for numbers and is usually taken as understood. It is understood that things that are equal, have all and only the same properties. Of importance are the symmetric and transitive properties of the relation ‘equals’, namely that if a=b then b=a (symmetric property), and if a=b and b=c then a=c (transitive property) for all numbers a, b, and c, The symbol "=" often evokes a meaning of "the answer follows" which, while being one interpretation of the symbol, is unhelpful in understanding the symmetric and transitive properties of the relation.

Equal-sharing

Equal sharing is a division concept based on the action of distributing the elements of a set evenly amongst a given number of subsets. E.g. Grandma shares $24-00 evenly amongst her four grandchildren. How much will each grandchild receive? Strategies for solving this problem will depend on a student’s level of numeracy understanding and could include dealing (physically sharing out), finding four equal addends, and inverse multiplication (i.e. what do I multiply by 4 to get 24?).

Equivalent decimal and percentage forms for everyday fractions

Decimals, percentages, and fractions are the three main numeral systems used to represent parts of a whole. For example, the proportion that is one part out of two parts can be represented as 1/2, or 0.5, or 50%. Children can use two- or three- dimensional models such as 100s blocks and place value rods to explore the relationships between these three numeral systems. Older students can use the implied division operation of a fraction to convert fractions to decimals and percentages.

Equivalent fractions

Two different fractions that represent the same number are referred to as equivalent fractions. For example, 1/2, 2/4, 3/6, and 4/8 are equivalent fractions because they represent the same number.
In the example given, 1/2 is said to be in irreducible form because the numerator and denominator have no common factor. The others are all reducible.

Experiment

See Statistical experiment.

Exponent

See Powers.

Exponential equations

An exponential equation is an equation in which the variable appears in an exponent (See Roots and Powers). So y = 3x is an exponential equation because y is a function of x and the variable x appears as an exponent. Equations of the form ax = an and ax = bx can be solved easily, while other equations may be solved by the use of logarithms. Example; 3x = 9x-2 Hence 3x = 32x-4 so x = 2x – 4. therefore x = 4.

Extrapolation

Extrapolation is the process of finding estimates of function or data values that lie outside the set of known values.

Factor

An integer a is a factor of an integer b if a divides b. It is often helpful to state the number set that we are working in. For instance, if we operate within the set of natural numbers, then the factors of 6 are 1, 2, 3, and 6 since those are the natural numbers that divide 6. Similarly, the factors of 28 are 1, 2, 4, 7, 14, and 28. (See also Common factor)

Factorial

n! (pronounced n factorial) is equal to the product of the natural numbers from 1 to n. So for example, 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720

Features of simple data displays

At an early stage children can observe and identify features of simple data displays – features such as greatest frequency, least frequency, how spread out the data is, mode, ‘middle’, and unusual values such as outliers.

Forward Counting Sequences

A counting sequence is an ordering of the counting numbers such that the difference between any two successive numbers is constant. The basic forward counting sequence is 1, 2, 3, 4, 5,…
An example of a forward skip counting sequence is 10, 20, 30, 40, 50, …, as is 2, 4, 6, 8, 10, … etc.

Fraction

A fraction is a numeral of the form a/b where a and b are both integers and b≠0. If the fraction lies between –1 and 1 then the fraction is called a proper fraction. (e.g. 1/2, 3/5, -2/7 etc), otherwise it is called an improper fraction (e.g. 11/5, 257/17, -3/2 etc). In the example 2/7, the two is called the numerator, and the 7 is called the denominator. If the fraction has arisen from the part-whole concept then the whole has been divided into seven equal parts and the 2 represents the number of the parts. Fractions are useful for representing parts of a whole, that is, they are numerals that can be used when whole numbers cannot describe a certain number.

Fractions, decimals and percentages of numbers

  • Whole numbers: Students can use multiplication properties to answer simple problems involving fractions of whole numbers. For example, 1/4 of 28 can be answered by realising that 4 x 7 = 28. Therefore, 1/4 of 28 is 7.
    Decimal fractions of whole numbers involve problems such as finding 0.5 of 8. Students can do this by knowing that 0.5 = 1/2 and by taking 1/2 of 8, or by multiplying 5 by 8 and realising that the answer is 4, not 40. Similarly, 0.8 of 9 must be 0.72 because 8 x 9 = 72 and the answer must be a bit less than 9. It can be confirmed by realising that 0.8 of 9 equals 8/10 of 9 and that 8/10 x 9 = 72/10.
    Percentages of whole numbers may be found in the same way as decimals of whole numbers, using appropriate multiplicative strategies For example, 80% of 9 is 7.2 because 9 x 80 = 720 and the answer must be a bit less than 9. Knowing that 10% = 1/10 can be helpful. For example, 35% of 40 is 4+4+4+2 since 10% of 40 is 4 and 5% is half of 10%. So 35% of 40 is 14.
    Again, results can be confirmed by converting to fractions and using fraction multiplication.
  • Fractions: Students can find fractions of simple fractions by using models such as a grid. For example, to find 1/4 0f 3/5, a 4 by 5 grid can be used with the quarter being shaded in one direction and the three=fifths in the other direction. It is then obvious that 1/4 of 3/5 is 3/20. Folding a rectangular sheet of paper and shading in the fractions in each direction is also a good approach.
    Decimal fractions of simple fractions involves finding answers to questions such as: "What is 0.25 of 1/3?" Students can answer this by using their knowledge of decimals. Then this problem becomes: "What is 1/4 of 1/3?" and can be answered using a grid (See Fractions above).
    Percentages of simple fractions involve finding answers to questions such as: "What is 25% of 1/3?" Again students can use their knowledge of percentages and convert the problem to a fraction of a fraction as above.
  • Percentages: Finding percentages of decimals involves finding answers to questions such as: "What is 40% of 0.4?" Students might reason this as 10% of 0.4 is 0.04 so 40% of 0.4 must be 0.16. Or they might reason that 40% = 0.4 and since 4 x 4 = 16 the answer must be 0.16 because it is a little less than half of 0.4. Alternatively they might reason that 40% of 0.4 is equal to 4/10 x 4/10 and by fraction multiplication arrive at 16/100, which is 0.16.

Fraction, decimal and percentage conversions

Fractions, decimals and percentages can be used to represent numbers that are not integers, that is, they include parts of a whole. Consequently for each number written as a numeral in one of these three forms there is a corresponding numeral written in each of the other two forms. Initially, connections between these numerals should be made by exploration with equipment such as a linear model for decimals and fractions, a closed abacus or a hundreds grid for percentages and sets and regional models for fractions.
After the concepts of the conversions have been fully explored with equipment, converting a fraction to a decimal may be done abstractly by using the division property of a fraction, namely that a/b = a÷b. So, for example, 3/4 = 3÷4=0.75 etc.
Since a percentage represents a proportion out of 100, the decimal for 3/4 may be multiplied by 100 to obtain the percentage for 3/4. (Students should have already observed this from the use of equipment.) So 3/4 = 75% Converting decimals or percentages to fractions involves using the definitions of the numerals. For example, 1/8 = 0.125, so converting 0.125 to a fraction simply involves writing the number as 125/1000. This can be reduced to 1/8 as they are equivalent fractions. Fractions that should be commonly known as decimals and percentages include halves, thirds, quarters, fifths, eighths, tenths, twentieths, twenty-fifths, and fiftieths.

Function

A function consists of two things:

  1. A set of elements called the domain, and another set of elements called the range.
  2. A rule for associating each element of the domain with exactly one element of the range.

The domain is often called the set of values of the independent variable, and the range is the set of values of the dependent variable. A typical value in the domain is usually denoted x, and is called the independent variable; and a typical value in the range is denoted y and is called the dependent variable.

 

Generalising

Mathematics is considered to be a deductive science, working from proven general results (theorems etc.) to the specific. However the best practice in teaching new concepts to children is to work inductively from specific discoveries and observations of patterns to generalised conclusions. This is referred to as generalising, or using inductive reasoning.
For example, 1+3=4, 3+5=8, 7+9=16, 13+5=18. From these results we might generalise that the sum of two odd numbers is an even number.

Geometric properties

Geometric properties by which children can sort geometric shapes and objects include number of vertices, number of edges, number of faces, types of face, symmetry, curvature, thickness, dimension, size of vertex angle etc.

Geometric sequences

A geometric sequence is a sequence of the form: a, ar, ar2, ar3, ar4, … arn-1, …
For example, 1, 2, 4, 8, 16, 32, 64, … is a geometric sequence in which r, the common ratio, is 2, and a, the first term, is 1.
The sum (Sn) of the finite geometric series a+ar+ar2+ar3+ar4+ …+ arn-1 consisting of n terms is:
S n = [a(1-rn)]/(1-r) where a is the first term and r is the common ratio.
If |r| < 1 then the sum to infinity of the geometric series a+ar+ar2+ar3+ar4+ …+ arn-1 + … is S=a/(1-r) where |r| is the absolute value of r.
For example 1/2 + 1/4 + 1/8 + 1/16 + … is a geometric series with a=1/2 and r=1/2.
So S = 1/2 ÷ 1/2 = 1

Glide

A glide (also called a glide reflection) is a transformation in the plane that is the composition of a reflection in a mirror line followed by a translation parallel to the mirror line. It is a necessary addition to the isometries if we wish to explain all isometric symmetry movements in terms of one isometry.

Gradient

See Rate of change and graphs.

Graphs, tables and rules

A graph is a visual representation of data or of a relationship of some kind. In its simplest form a graph is any set of points in a plane. Different types of graph best serve different purposes. For comments on statistical data graphs see Data displays>. Linear (See Linear equations) and non-linear relationships found in number patterns are often displayed on the Cartesian plane (See Coordinate systems).
Some relationships might also be shown quite well by way of a table, and relationships can be described by a rule.


For example, the diagram above shows a tiling pattern of concrete slabs laid around central grassed areas. The number of slabs needed is given by the sequence 8, 12, 16,…
This could be shown by way of a table:

 

 

Side length of grass square123456 
No. of slabs81216202428  


Or a graph:

Or a rule:
The number of slabs needed is four times the number of the term plus 4. This could be expressed algebraically as s=4n+4.

 

 

Graphs, tables and rules for simple quadratic relationships

As shown above for linear relationships, graphs, tables and rules can be developed for simple quadratic equations. Consider the pattern of balls set up in the triangular arrangements shown below:

We could draw up the following table for the number of balls in each term:

 

 

n12345n
Term n: T(n)1361015  

 


Looking at the table, students will see that to get the nth term, you add n to the (n-1)th term. So T(n) = T(n-1)+n.
That is helpful if you already know the (n-1)th term. Closer observation should lead to the discovery that the number of balls in each term is half the product of the term number and the next term number. This leads to the relation:
T(n) = n(n+1)/2
= 1/2 n2 + 1/2 n
This is a polynomial equation of degree two since the highest power to which the variable n is raised is two. Its graph will be a part of a parabola. Polynomial equations of degree two are also referred to as quadratic equations.

 

 

 

Greatest common factor

See Common factor.

Grid references

(See also Coordinate systems). Grid references on a map are similar to coordinates in the Cartesian plane. For the major New Zealand topographical map series (NZMS 260) the grid references are given as a six-digit number, the first three digits being the distance east (in kilometres to one decimal place) and the second three digits being the distance north (in kilometres to one decimal place) from a starting origin. Simple grid systems can be added to maps for children by labelling (with letters or numbers) the bottom horizontal border of the map and the left-hand vertical border. Children can then read the coordinates of the map by reading the horizontal reference first followed by the vertical reference, thus simulating the coordinate system of the Cartesian plane. Activities can then include things such as a coordinate journey through the map describing the view according to the features shown on the map.

Grouping

Putting sets of objects into groups, usually according to some attribute e.g. colour, size, shape etc.

Half turn

A rotation through 180o, or half of a complete rotation.

Hectare

See SI measurement units – Area.

Inferences

See Statistical inference.

Integers

The set of integers is an extension of the set of whole numbers to include the negatives of the whole numbers. Thus it is {…, -3, -2, -1, 0, 1, 2, 3, …} The integers give us answers to questions such as :"What can I add to 5 to get 3?".

Integration

Integration is the reverse process of differentiation. The task with integration of a function f (x) is to find a function F(x) such that the derivative of F(x) is f(x) for all x in the domain of f. That is, F(x) is the integral of f(x) if and only if F’ (x) = f (x) for all x in the domain of f. Then we write F(x) = ∫f (x) dx which reads as F (x) is the integral of f (x) with respect to x. For example,
If f (x) = 3x2+2x+5 then F (x) = ∫f (x) dx = x 3+x2+5x +C, where C is an arbitrary constant.
Such an integral is known as an
indefinite integral, or an antiderivative. Differentiation of F(x) gives f(x).
An integral F(x) which is defined between certain limit values a and b of x is a definite integral.
Geometrically, a definite integral can be thought of as the area contained between the graph of a function and the x-axis and between the lines x=a and x=b. Areas located above the x-axis count as positive in integration, areas below the x-axis as negative.
If the indefinite integral cannot be found it may still be possible to find the numerical value of the definite integral using numerical integration methods such as the use of the trapezoidal rule and Simpson’s rule. This is also referred to as approximate integration.

Interpolation

Interpolation is the process of finding estimates of function or data values that lie between two known values.

Invariant

Unaltered or unchanged.

Invariant properties of transformations

A point is invariant under a transformation in the plane or in space if the transformation leaves it unaltered. A rotation in the plane has one invariant point (the centre of rotation) and a rotation in space has a line of invariant points (the axis of rotation). A reflection in the plane has a line of invariant points (the mirror line or line of reflective symmetry) and a reflection in space has a plane of invariant points (the plane of reflective symmetry). A translation in the plane or in space has no invariant points, since by the definition of translation all points move the same distance. An enlargement has one invariant point (the centre of enlargement).

Inverse of a function

Suppose we have a function y = f(x) which maps values of x to values of y. If there is a function g(y) that maps the values of y back to values of x (i.e. g(y) = x) then g is said to be the inverse of f. The inverse of f is written as f-1
So g = f-1
So we have f-1 [f (x)] = x
To have an inverse, f must be one-to-one, that is each value in the domain of f must be associated with just one value in the range of f. The domain of f-1 is the range of f, and the range of f-1 is the domain of f.

Irrational numbers

Real numbers that have no fractional form (that is, cannot be written in the form a/b where a and b are integers) are called irrational numbers. For example, √2 is an irrational number and therefore, like all irrational numbers:

  1. Has no fractional representation, and
  2. Its decimal form is non-repeating and has no pattern.

To 50 decimal places:
√2 = 1.41421356237309504880168872420969807856967187537694

 

Isometric transformation

An isometric transformation (or isometry) is a shape-preserving transformation (movement) in the plane or in space. The isometric transformations are reflection, rotation and translation and combinations of them such as the glide, which is the combination of a translation and a reflection.

Language of direction and distance

See Direction, and Distance.

Least common multiple

See Common multiple.

Length

Length is the concept of distance in a straight line between two points on a line, in a plane, or in space. It is a measure of one dimension, measuring the size of a line. The basic SI unit of measurement of length is the metre, with millimetre, centimetre and kilometre also being commonly used units. (See SI measurement units)

Limit of a function

Consider the function f(x) = x2. As x tends towards 2, x2 tends towards 4. It is a case of closeness of x to 2 forces closeness of x2 to 4. So the limit of f(x) as x tends to 2 is 4. This is written as:

 

lim f(x)= 4
x → a


As x tends to 2 from the left, (that is, x is approaching 2 and is less than 2), f (x) tends to 4, and as x tends to 2 from the right, (that is, x is approaching 2 and is greater than 2), f(x) tends to 4. The limit as x tends to a of f(x) exists if and only if the limit as x tends to a from the left exists and the limit as x tends to a from the right exists and both of these limits are equal.
The explanation of the limit as implying closeness is illustrated by the function f(x) = (x2-1)/(x-1) x ≠ 1 So f(x) = x+1 for all values of x other than x=1 and its graph is the graph of f(x) = x+1 with a hole at x = 1. Although f (x) is undefined at x=1 we can still say that the limit of f(x) as x tends to 1 is 2 since closeness of x to 1 forces closeness of f(x) to 2. Put another way, f (x) can be made as close to 2 as we wish simply by choosing a value of x sufficiently close to 1.

 

Linear equations

The simplest linear equation is one which expresses a direct linear relationship between two variables. For example, if sticky bars cost $3-00 each then the cost (c) in dollars of buying n of them is given by the equation c=3n. This is called a first degree or linear equation. If the associated values of c and n are graphed on a coordinate plane the points that satisfy this equation will lie on a straight line. Suppose there is a packaging fee of $2-00. Then c=3n+2. We could now ask questions such as:" If the cost was $23-00, how many sticky bars did we buy?" Note that in a linear equation the variables (such as c and n above) are not raised to higher powers such as squares or cubes.
So a linear equation is an equation of the form y=ax+b where a and b are real numbers. The solution, or root, of the equation ax+b=0 is x= -b/a.

Linear inequalities

A linear inequality is an expression of the form
ax + b > c The relations >,<, ≥, and ≤ give rise to inequalities.
2x + 3 > 5 is a linear inequality because x is raised only to the power of 1. To solve this inequality we can use the rules of algebra, with a little caution.
2x + 3 > 5 Adding -3 to both sides we obtain
2x > 2 Dividing both sides by 2 we obtain
x > 1 So the solution is the set of all real numbers greater than 1.
Caution has to be exercised in handling inequalities. Two things in particular are worth mentioning:

  1. Changing the unknown to the other side of the inequality changes the sign. For example: If 22
  2. Multiplying both sides of the inequality by a negative number also changes the sign. For example: If -x>3 then x<-3.

An examination of the number line helps with an understanding of these two properties.

 

Linear programming

A problem of linear programming is that of finding nonnegative values of a number of variables for which a certain linear function of those variables assumes the greatest (or the least) possible value while subject to certain linear constraints.
Graphing techniques can often be used to find the maximum or minimum values.

Linear scales

Drawings, maps and models are scale representations and that scale is a linear (adjective of line) scale if the linear dimensions of the scale representation are in direct proportion to the linear measurements of the region being represented.
For example, a map of a playground might be drawn to a scale of one centimetre to one metre (that is, 1cm on the map represents 1m on the ground). As a ratio, this is a scale of 1:100, so every distance on the map is one one-hundredth of the distance it represents on the ground.

Linear proportion

Any situation that can be modelled using the equation a/b = c/d involves a linear proportion. Reasoning with linear proportions involves part-whole relationships (equivalent fractions), operating on fractions, measurement, rates and ratios, and division with remainders. It also involves competence with connecting fractions, decimals and percentages and using graphs to solve problems. Linear proportions apply in a wide range of contexts including trigonometry, probability, metric measurement conversions, calculating best deals, and physical rates such as speed. (See also rate and ratio)

Linear regression of bivariate data

Very often in practice a relationship is found to exist between two variables (for example, height and weight). It is often desirable to express this relationship in mathematical form by finding an equation connecting the variables. If the data appears to be approximated well by a straight line we say that a linear relationship exists between the variables. This is called a regression line.

Location

Location refers to the position of an object (or a point) on a line, on a plane or in space. The location of a point on a line can be defined by its distance from a fixed origin. The location of a point in a plane can be described by an ordered pair, and the location of a point in space can be described by an ordered triple. (See Coordinate system) Location can also be described by direction and distance, such as a compass bearing and a distance from a fixed point.

Locus (plural loci)

A locus is a geometric figure for which all points satisfy a given condition. It is the set of points and only those points that satisfy the condition. So the locus of a point that is three centimetres from a given point P is a circle of radius 3cm whose centre is at P. The locus of a point that is equidistant from two given intersecting lines is the bisector of the angles formed by the lines. The locus of a point equidistant from two given parallel lines is a line parallel to the two lines and midway between them.

Logarithmic algebraic expressions

The logarithm function is the inverse of the exponential function. The logarithm function is defined, when b is positive and b ≠ 1, as y = logbx. y is called the logarithm of x to the base b. If y is the logarithm of x to the base b, then by = x. For example:
102 = 100, so log10 100 = 2
103 = 1,000, so log10 1,000 = 3
105 = 100,000 so log10 100,000 = 5
From this we can see that the two fundamental properties of the logarithm can be derived from the corresponding laws of exponents (See Powers). They are: (i) The logarithm of a product is equal to the sum of the logarithms of the factors.
So logb xy = logb x + logb y (ii) The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.
So logb x/y = logb x - logb y From (i) above it follows that the logarithm of the power of a number is equal to the power times the logarithm of the number. That is:
logb xp = p logb x
The logarithm of 1 is 0 since b0 = 1
There is a number e called the base of the natural logarithm. Equations involving growth and decay are best written in terms of e, and logarithms to the base e are called natural logarithms. e is an irrational number, its decimal expansion to 10 places of decimals being 2.7182818285.
It can be seen how this number occurs naturally in situations of growth and decay. For example, the formula for compounding interest is An= P (1 + r)n. Suppose we set interest at 100%. Then r = 1. After one year, A1 = P (1+1) = 2P
If we decide to modify the system of accumulation to 50% paid twice a year we have A1 = P (1+1/2)2
If we decide to modify the system of accumulation to 25% paid four times a year we have A1 = P (1+1/4 )4
Continuing in this way we can see that as we move towards continuous growth, that is as n tends to infinity with r = 1/n, the formula for the amount after one year becomes:

 

A1 =limP (1+1/n)n
n→∞


The limit as n tends to infinity of (1+1)n/n is called e.
To ten decimal places e = 2.7182818285 Putting some values for n into the expression (1+1)n/n shows this to be a ‘reasonable’ value.
Growth (or decay) functions where the growth (or decay) is continuous can be written in the form y = A ekt where A is the initial size, t is the time and k is the nominal rate of growth. Compare this with the compound interest formula: Suppose we invest $100-00 for two years at a rate of 10% paid annually. Then A2 = 100 (1+0.1)2 = 121.
If we invested $100-00 for two years at a rate of 10% per annum paid continuously we would have: A2 = 100 e0.2 = 122.14.

 

Log modelling

In curve fitting processes it is helpful to obtain scatter diagrams of transformed variables in order to decide which type of curve should be used. For example, if a scatter diagram of log y against x shows a linear relationship then the equation has the form y = abx or log y = log a + (log b) x while if log y versus log x shows a linear relationship the equation has the form y = axb or log y = log a + b log x In order to see these relationships, special graph paper for which one or both scales are calibrated logarithmically can be used. These are referred to as semi-log or log-log graph paper respectively.

Maps

A map is a scale drawing, a projection of a portion of our three-dimensional world onto a plane surface such as a piece of paper. Maps can be simple, such as scale drawings of the classroom, or more complex, involving compass directions and coordinate grid systems.

Margins of error

The standard deviation of a sampling distribution of a statistic is often called its standard error.

Mass

The mass of an object is a measure of the amount of matter in it. It represents in a quantitative way that property of matter that is described qualitatively as inertia. While an object on the Moon would have about one-sixth of the weight that it would have on Earth, its mass would be unchanged. The common SI units of mass are the kilogram (kg), milligram (mg), gram (g) and tonne.

Measurement data

Statistical data falls into two main classes. Things that can be counted (for example, the number of children in a family) give rise to whole number data and things that can be measured (for example, the heights of a group of people) give rise to measurement data. A third class is category data, which is data of things that fall into categories such as colour, brand, model etc. So height, weight, distance, volume, time etc. give rise to measurement data.
Measurement data is considered to be continuous since (in theory) a measurement can take any value in a given interval. Whole number data is discrete since it can take only whole number values.

Measures of Central Tendency

Also referred to as measures of central location, the measures of central tendency are those statistics that describe the centre or the most typical value of a set of data. They might often be loosely described as averages in the sense that they are indicative of the middle of a set of data. Most common amongst these measures is the arithmetic mean, usually referred to as the mean. The arithmetic mean of a set of n numbers is found by taking the sum of the numbers and dividing that sum by n. Other measures of central tendency are the median (the middle value when the numbers are ordered), and the mode (the most commonly occurring value). The mode may not exit, and if it does it may not be unique.

Measures of spread

Measures of spread measure the degree of variability in a set of data. They are used as an indicator of the dispersion of a set of data. Some measures of spread are the range (the difference between the largest and smallest numbers in the set), the mean deviation (or mean absolute deviation), the semi-interquartile range, the standard deviation, and the variance.

Metric System

See SI.

Multiple

An integer a is a multiple of an integer b if and only if a = m x b for some integer m. So, for example, the positive multiples of 5 are 5, 10, 15, 20, 25, …Obviously, the set of multiples of any number other than zero is infinite. (See also Common multiple)

Multiple transformations

The term multiple transformations refers to the composition of two or more transformations. Suppose R is a transformation of rotation in the plane about the origin through 90o (anticlockwise); and M represents a reflection in the plane in the x-axis. What is the effect of the multiple transformation R followed by M? The multiple transformation R followed by M is written as MR. We can find the effect of MR by considering its effect on the two points (1,0) and (0,1). That is because those two points ‘represent’ the x-axis and the y-axis respectively and so every point in the plane can be written as a combination of those two points. MR maps (1,0) to (0, -1) and (0,1) to (-1, 0). So MR is the same as a reflection in the line y = - x. Note that transformation composition is not commutative since MR is not the same as RM.
It is helpful in considering the effect of multiple transformations to classify transformations as direct or indirect. The direct isometries are rotation and translation. They are called direct because they do not flip (or turn over) the shape being transformed. Reflection and glide reflection are indirect isometries because they do flip the shape being transformed. So, for example, the product of two reflections is a direct isometry and is therefore either a rotation or a translation. In fact the Fundamental Isometry Theorem assures us that every isometry in the plane can be expressed as a product of at most three reflections.

Multiplication of:

  • Whole numbers: Multiplication may be described as repeated addition of the same number. For example, 5x7 means 7+7+7+7+7 or alternatively, 5+5+5+5+5+5+5. When the basic multiplication facts are known, more complex multiplication problems can be answered using partitioning strategies. For example, 8 x 15 can be seen as 8x10 plus a half of 8x10 since 5 is a half of 10. So 8x15=120. Similarly, to find 7x19 take 7x20, which is 140, and subtract 7 to get 133.
    Multiplication is a binary operation, that is, it is an operation on two numbers.
    Multiplication is commutative, that is, the order of the numbers does not change the answer. For example, 4x5 = 5x4.
    Multiplication is associative, that is, the grouping of the numbers does not affect the answer. For example, (2x3) x 5 = 2 x (3x5).
    1 is the identity element for multiplication, because multiplication by one does not change a number.
    Multiplication is distributive over addition, that is,
    a x (b+c) = (a x b) + (a x c)
    For example, 3x(4+7) = (3x4)+(3x7) This property of multiplication is most useful when operating beyond the basic facts of multiplication. For example, 8x56 can be calculated as (8x50) + (8x6) which equals 400+48 or 448
  • Fractions: Fractions may be multiplied as follows: a/b x c/d = (a x c)/ (b x d). For example, 2/3 x 4/7 = 8/21. This rule is best discovered by students, using grids or folded paper, and then observing the pattern of results.
  • Decimals: Decimal fractions (commonly called decimals) may be multiplied in the same way that whole numbers are, with care being taken to consider the position of the digits in the decimal. So, for example, to find the result of 1.5 x 0.8, multiply 15 by 8 to obtain 120 and realise by estimation that the answer must be 1.2. Alternatively, one could multiply 15/10 by 8/10 to obtain 120/100, which is 1.2.
  • Percentages: There is a danger in looking for the answer to problems such as 20% x 40% in that it would be tempting to give the answer of 800% since 20 x 40 = 800. The easiest way to multiply the percentages is to change them to decimals. Then 20% x 40% = 0.2 x 0.4 = 0.08 = 8%. Alternatively, one could multiply 2/10 by 4/10 to obtain 8/100, which is 8%.
  • Integers: Integers may be multiplied in the following way:
    a x-b = -(a x b) e.g. 2 x -3 = -6
    -a x b = - (a x b) e.g. -2 x 3 = -6
    These results are best discovered using models, such as a black-and-white counters model, in which a white counter represents one, and a black counter represents -1.
    -a x -b = a x b e.g. -2 x -3 = 6
    This result is difficult to model but can be demonstrated as follows:
    (3 + -3) = 0 so -2 x (3 + -3) = -2 x 0 = 0
    Also, -2 x (3 + -3) = (-2 x 3) + (-2 x -3)
    So (-2 x 3) + (-2 x -3) = 0
    But (-2 x 3) = -6
    So -2 x -3 = 6
    Alternatively, one can take an inductive approach and look for a pattern. For example, consider some of the multiples of negative three.
    4 x -3 = -12
    3 x -3 = -9
    2 x -3 = -6
    1 x -3 = -3
    0 x -3 = 0
    -1 x -3 = 3
    -2 x -3 = 6
    -3 x -3 = 9 etc.
    From this, and other similar examples, it is apparent that the product of two negative numbers is a positive number.

    The properties of multiplication outlined for whole number also apply to the multiplication of fractions, decimals, percentages and integers.

Multiplicative strategies

Multiplicative strategies are techniques used to solve multiplication problems from known facts. A multiplication strategy involves one or more of the properties of multiplication, specifically, commutativity, associativity, distributivity and inverse properties (See Multiplication). For example, 3x20 could be seen as 3x10x2 or 30x2, which is 60. At a harder level, 19x3 could be seen as (20x3)-3 which is 60-3 or 57.
Similarly, 315÷45 = 630÷90 = 63÷9 = 7.

Multivariate data

Data that consists of measurements of three or more attributes or variables for each individual or object in a sample (for example age, height and gender). This contrasts with univariate data where measurements are made of only one attribute, and bivariate data where measurements are made of two attributes.

Natural numbers

The set of natural numbers is {1, 2, 3, 4, 5, …} They are often also referred to as the Counting numbers.

Nets

A net consists of a set of connected polygons that can be folded to form a polyhedron

Network

In graph theory a graph consists of a set of points, called vertices, along with edges connecting the vertices. The edges express a relationship between the vertices. A graph may have no edges (the null graph), one or more edges, an edge connecting every vertex to every other vertex (the complete graph) or more than one edge between two vertices (a multigraph). A graph may have numerical values attached to each edge (a network), an arrowed direction on each edge (a directed graph, or digraph) or both numerical values and direction arrows (a directed network). In directed graphs the edges are called directed edges or arcs.
The positioning of the vertices in the plane is not usually significant as it is the edges that define the relationship between the vertices.
The following directed network shows the cost (in $m) of stormwater piping in Watersville.

Non-linear function

A linear function is a function of the kind f(x)=ax b, where a and b are real numbers. The graph of such a function on the Cartesian plane is a straight line. Other functions, such as polynomials of higher degree, are non-linear functions. Their graphs are not straight lines. For the example, a quadratic function equation is of the form f(x)=ax2+bx+c where a, b, and c are real numbers. The graph of a quadratic function is a parabola.

Normal distribution

The normal distribution is a continuous probability distribution which is defined by the equation
y = [1/(ϑ √2π)]e - 0.5(x-μ)22
The total area bounded by the curve and the x-axis is 1, hence the area under the curve between the ordinates x = a and x = b, where a

Numeral

A name or symbol (or combination of symbols) that describes a number. So two, 2, 1+1, rua and 4/2 are all numerals for the same number.

Numeration system

See Base ten numeration system.

Numerator

When a rational number is written as a fraction, that is in the form a/b, then a is called the numerator of the fraction. (See also Fraction)

Operations

Addition, subtraction, multiplication, and division are examples of binary operations on numbers. They are called binary operations because they use a rule to map two numbers to one number (the answer). For example, if the two numbers are 4 and 5 and the binary operation is addition then the answer is 9. Note that the one restriction on these operations is that division by zero is not possible. For example, 7÷0 has no answer, since it asks the question: "What do I multiply zero by to get 7?" Other types of operations are possible, such as finding the square root of a number, which is a unary operation since it requires only one number to be operated on. (See also Addition, Subtraction, Multiplication, Division)

Optimal solutions using numerical approaches

Students who are not at the stage of using calculus techniques to find maxima and minima may still be able to solve simple practical problems using graphs, charts, estimation techniques etc. For example, to find the maximum area of a rectangle that has a perimeter of length 20 metres, students could set up a table as follows to find the answer. They could then draw a graph of that.

Height (m)012345678910
Width (m)109876543210
Area (m2)091621242524211690

Ordering

Arranging according to some chosen attribute. Ordering involves the relations ‘less than’ and ‘greater than’ as defined on numbers. For example, objects could be ordered by area where the measures of the areas, given as numbers, are used to order the areas.

Ordering fractions

To order two fractions, we can write them in equivalent fraction forms so that they have the same denominator and then compare the numerators. For example, to order the fractions 2/3 and 5/8 we could write them in equivalent fraction forms with the same denominator. The least common multiple of 3 and 8 is 24. 2/3 = 16/24 and 5/8 = 15/24. So 5/8 < 2/3.
We can use mental strategies to order some pairs of fractions. For example, 7/12 > 1/2 because we know that 1/2 = 6/12 and 7>6. However, this apparently intuitive approach to ordering fractions is underpinned by mentally considering equivalent fraction forms.

Ordinal position of members of sequential patterns

See Sequential patterns.

Parallelogram

A parallelogram is a four-sided plane figure (quadrilateral) with two pairs of parallel sides. Thus a parallelogram is of the shape shown below.

A parallelogram has opposite sides equal.
The opposite angles of a parallelogram are equal
Two angles not opposite add to 180o
Note that rectangles, rhombuses and squares are all parallelograms.
A rectangle is a parallelogram with all angles equal.
A rhombus is a parallelogram with all sides equal.
A square is a parallelogram with all sides equal and all angles equal.
Therefore a square is both a rectangle and a rhombus.
The area of a parallelogram is found by multiplying the base length by the vertical height.

For example, in the above parallelogram, the length of the base is 8cm and the vertical height is 5cm. So the area is 40 cm2.
This result may be explored and discovered by students using paper parallelograms and cutting off a triangle from one side of the parallelogram and joining it to the other side to form a rectangle.

Partitioning

Partitioning is the process of ‘breaking up’ numbers, usually to simplify operations. For example, to compute 28 +37 we might partition 28 into 25+3 and see the problem as 25+40.

Partitioning and combining like measures

Measures of, for example, length can be added or subtracted providing they are in the same units. So a length of 31 metres can be broken into lengths of 8 metres and 5 metres.

Path

Technically, a path is a sequence of connected vertices, none of which is repeated. Less specifically, the term ‘path’ may refer to a route given by bearing and direction or by a set of points such as a sequence of coordinates on a Cartesian plane

Pattern

A pattern can be described as any regularity that the mind can perceive. (See Sequential pattern and Repeating pattern.)

Patterns and trends within and between data sets

With time series data, patterns and trends may be observable both within a time series data set and between two or more time series data sets. For example, there might be a clear relationship between average monthly temperature and monthly power consumption.

Percentage

A percentage is another way (as well as fractions and decimals) of representing a part of a whole. The term literally means ‘for each 100’. The percentage is therefore the numerator when the denominator is 100. For example, 1/2 = 50/100. So one half equals 50%. Similarly, 3/4 =75/100 so three quarters equals 75%. 29% = 29/100 Percentages can represent numbers greater than one. Just as 2/1 =2, so 200% = 2.

Perimeter

The boundary of a figure in a plane is called the perimeter The length of the perimeter of a polygon is the sum of the lengths of its sides.

Perimeters of circles

The perimeter of a circle is called its circumference. The length of the perimeter of a circle is π x d, where d is the length of the diameter of the circle and π is an irrational number whose value is a little less than 3 1/7. Because π is an irrational number it has no exact representation as a fraction or decimal fraction. To five decimal places, π = 3.14159.
Students may discover this relationship between the length of the diameter of a circle and the length of its circumference by winding string around circles with a variety of diameters and thereby measuring the lengths of the circumferences. This is most easily gone using jars, tins, saucepans, etc. Dividing the length of the circumference by the length of the diameter shows that the ratio of circumference to diameter is constant and is independent of the size of the diameter.

Permutation

A permutation of n different objects is an arrangement of the objects with attention given to the order of the arrangement. So, for example, the letters a, b, and c have six permutations. They are abc, acb, bac, bca, cab, and cba. The number of permutations of n objects taken n at a time is n factorial, which is written n! and is defined as follows:
n! = 1 x 2 x 3 x …x (n - 2)(n - 1)n
So 3! = 1 x 2 x 3 = 6

Perpendicular

Two lines are perpendicular if the angle between them is 90o, that is, they meet at right angles. Two planes, P1 and P2 are perpendicular if a line in P1 perpendicular to the line of intersection of the planes P1 and P2 is perpendicular to every line in P2.

Plane

A plane is a flat surface that is considered to extend indefinitely. It is a two-dimensional figure that has area but not volume. Hence a flat tabletop is a portion of a plane. The whole plane is the set of points defined by extending the tabletop indefinitely.

Plane shape (or plane figure)

A shape that can lie wholly in a plane. A plane shape is therefore a flat, two-dimensional shape and is imagined as having no volume.

Point estimate

A point estimate is a single number estimate of a population parameter based on collected data.

Poisson distribution

The discrete probability distribution p(x) = (λ x e)/ x! is called the Poisson distribution. The values of p(x) are computed from tables. Both the mean and the distribution of this distribution have the value of λ. As λ increases indefinitely the Poisson distribution approaches a normal distribution.

Polygon

A polygon is a portion of a plane bounded by straight lines. (See also Areas of polygons)

Polyhedron

A polyhedron (plural polyhedra) is a solid whose faces are all plane polygons. The faces need not be regular polygons. (A regular polygon is a polygon whose edges are all congruent and whose angles are all equal) If the faces of a polyhedron are all identical regular polygons then the polyhedron is referred to as a regular polyhedron, or a Platonic solid.
If the faces of a polyhedron are all regular polygons but are not identical, then the polyhedron is referred to as a semi-regular polyhedron, or an Archimedean solid.

Polyhedron nets

See Nets.

Polynomial

A polynomial equation of degree n is an equation of the form an xn + an-1 xn-1 + ... + a1 x + a0 = 0 where a0, a1, … an are all real numbers and all the powers of the variable x have nonnegative integer exponents.
For example, 3x+2=0 is a polynomial equation as is 2x5+√3x2=0, but 5x3+6x+3x-1=0 is not a polynomial because one of the exponents is negative.

Population

The source of the observations of a sample is called a population. It is the ‘whole’ of which a sample is a part, and consists of all the possible elements of the set from which the sample is taken.

Population parameters

Descriptions of populations (such as their mean and standard deviation) are called parameters. This compares with descriptions of samples, which are called statistics. We use statistics to reach decisions about parameters.

Position

Position may be described relative to many things. For example, latitude gives position relative to the equator. At Level One children might give their position relative to a nearby person or object using terms such as ‘behind’, ‘in front of’, ‘to the left of’, ‘to the right of’, ‘five steps away’ etc. At a more advanced level position could be described by coordinates on a map or on the Cartesian plane, or by latitude and longitude.

Possible outcomes

When we perform a nondeterministic, or random, experiment we can consider the possible outcomes. (sometimes called the sample space). For example, when we roll a normal dice (or die) once, the possible outcomes are 1, 2, 3, 4, 5, or 6.

Powers

Consider the number 4 x 4 x 4 x 4 x 4 We can write this in the abbreviated form of 45. Just as 4+4+4+4+4 can be written as 5 x 4, so 4 x 4 x 4 x 4 x 4 can be written as 45. 45 is called a power. It is the fifth power of 4. It is usually read as "four to the fifth". For the power ap, a is called the base and p the exponent. So 4 is the base and 5 is the exponent of the power 45. The exponent is often loosely referred to as the power.
Rules for calculating with powers: (Note that these are all derivable from the definition of a power.)
am x an = am+n
From the definition of am and an we can see that
am x an = (a x a x …..x a)[m times] x (a x a x …..x a)[n times]
= (a x a x a x a …..x a)[m+n times]
= am+n     (Example: 35 x 37 = 312)
Similar reasoning gives the following results:
am÷an = am-n     (Example: 25÷22 = 23)
(am) n = amn (that is, am x n)     (Example: (32)3 = 36)
If a ≠ 0 then a0 = 1    (Example: 100 =1 Note that100 equals, for example, 102÷102 which obviously equals 1.)
(ab)n = an bn    (Example: (3 x 2)4 = 3 x 2 x 3 x 2 x 3 x 2 x 3 x 2 = 34 x 24)
a-n = 1÷an (= 1/an)    (Example: 2-3 = 1/23 = 1/8 because, for example, 22÷25 = (2 x 2)÷(2 x 2 x 2 x 2 x 2) = 1÷(2 x 2 x 2) = 1/8)
But from above, 22÷25 = 22-5 = 2-3 So 2-3 = 1/23 = 1/8

Powers with fractional exponents

See Roots.

Prime numbers

A prime number is a natural number that has exactly two positive divisors, namely 1 and the number itself. So 2 is a prime because its only divisors are 1 and 2; 3 and 5 are primes for the same reason, that their only divisors are 1 and themselves. But 4 is not a prime because 1, 2, and 4 are all divisors of 4, so 4 has three divisors and is therefore not a prime. The first nine primes are 2, 3, 5, 7, 11, 13, 17, 19, and 23. There are infinitely many primes, that is, there is no ‘last’ prime number. Note that zero is not a prime number since every natural number divides zero, and one is not a prime number since it has only one divisor, namely itself.
The simplest means of sifting out the positive primes numbers from the natural numbers is to use the sieve of Eratosthenes.
Integers that are not 0, ±1, or prime, are called composite.

Prism

A prism is a polyhedron with two congruent and parallel faces (called the bases) whose remaining faces (called the lateral faces) are parallelograms. So a prism is a portion of space enclosed by polygons, with specific properties. Prisms are named after the shape of their base faces. For example, if the bases are pentagons then the prism is a pentagonal prism.
If the lateral faces are all perpendicular to the base then the prism is called a right prism. Hence a cuboid is a right rectangular prism, since the base faces are both rectangular. The volume of a prism is the product of the area of a base polygon and the altitude (or vertical height) of the prism.

Product

The product of two numbers is the result obtained when the numbers are multiplied together. For example, the product of 4 and 5 is 20 because 4 x 5 = 20.

Probability of an event

Probability is the study of random events – events in which the outcome is not fixed. For example, if an experiment is to toss a fair coin twice and count the number of heads obtained, then an event could be that two heads occur. We can then ask, "What is the probability of that event occurring"? The probability of any event is a number between 0 and 1 inclusive. If the probability of an event is zero then the event is impossible and if the probability of an event is one then the event is certain. Hence the probability of an event can be described as a number that tells us how likely it is that the event will occur.
Some properties are:

  1. Conditional probability. The conditional probability of A given B, written P(A|B), is the probability that A occurs given that B occurs and is equal to the probability of A and B both occurring divided by the probability of B. This can be written as P(A|B) = P(A∩B) / P(B)
  2. This can also be written as P(A∩B) = P(B). P(A|B) and is referred to as the general multiplication rule.
  3. If A and B are independent events, that is P(A|B) = P(A), then the above rule becomes P(A∩B) = P(A). P(B) and is referred to as the special multiplication rule.

 

Probability activity

A probability activity is a game, an experiment or some other activity that involves the concept of probability. For example, we might calculate or estimate the probability of drawing two successive hearts from a well shuffled ordinary pack of 52 cards. A probability activity would be to do this as an experiment many times over and check the degree of agreement between the theoretical outcome and the experimental result.

Properties of addition and subtraction with whole numbers

See Addition and Subtraction.

Properties of multiplication and division with whole numbers

See Multiplication and Division.

Proportionality

See Direct and indirect relationships with linear proportions.

Pyramid

A pyramid is a polyhedron whose base is a polygon and whose other faces are triangles with a common vertex. Pyramids are described in terms of the base polygon, for example, a triangular pyramid, hexagonal pyramid etc. A regular pyramid is a pyramid that has a regular polygon for a base and whose altitude meets the base at its centre. The volume of a pyramid is one third of the product of the base area and the vertical height. So for a pyramid, V = 1/3 x b x h where V is the volume, b is the area of the base and h is the vertical height.

Pythagoras’ Theorem

The Theorem of Pythagoras states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two other sides.
Hence we can state the theorem as:
In a right triangle, the area of the square constructed on the hypotenuse is equal to the total area of the squares constructed on each of the other two sides.
This suggests a discovery approach to Pythagoras’ Theorem by drawing a right triangle and constructing squares on the sides of the triangle. The two smaller squares may then be cut out and shown to cover the area of the largest square.
There are many different proofs of Pythagoras Theorem. One of the simplest is Bhaskara’s proof.

Quadratic equations

Quadratic equations are equations of the form y=ax2+bx+c where a, b and c are real numbers. This is a polynomial equation of the second degree, because the greatest exponent of its powers is two. ax2 is the quadratic term, bx the linear term and c the constant term. When graphed on the Cartesian plane its graph forms a parabola. The simplest quadratic equations are the pure quadratic equations such as x2-4=0.
If x2-4=0 then x2=4 so x ∈ {2, -2}
Quadratic equations arise in many situations in the real world, such as in the following measurement problem:
A carpet is 6 metres longer than it is wide and has an area of 27 square metres. What are the dimensions of the carpet?
If the length of the carpet is x metres then the width is x-6 metres and we have the quadratic equation x(x-6)=27, which rearranges to x2-6x-27=0. There are three ways to solve this equation:

  1. By factorising it:
    x2-6x-27 = 0
    So (x-9)(x+3) = 0 If the product of two numbers is zero then at least one of them is zero. So x-9 = 0 or x+3 = 0. So x = 9 or x = -3.
    The carpet is 9 metres long by three metres wide.
  2. By completing the square:
    x2-6x-27 = 0
    x2-6x = 27
    We see that (x-3)2 = x2-6x+9
    So x2-6x+9 = 27 +9
    (x-3)2 = 36
    x-3 = ± 6
    x ∈ {9,-3}
  3. By using the general solution, which is that if ax2 + bx +c = 0 then
    x = [-b ± √(b2 - 4ac)]/2a
    So for x2-6x-27 = 0 we have:
    x = [6 ± √(36 +108)]/2
    = (6 ± 12)/ 2
    = 9, -3

 

Quadrilateral

A quadrilateral is a four-sided plane figure bounded by straight lines. Some specific quadrilaterals are the parallelogram, rectangle, square, rhombus and trapezium.

Quarter turn

A rotation through 90o. A quarter of a complete rotation.

Random sample

A statistical sample is random if each element of the population has an equal chance of being selected.

Range

See Function.

Rate

A rate is a comparison of two different types of quantity or attribute. For example, 6 km / hour is a rate because it involves a comparison of distance with time. $5-00 per metre is a rate because it involves the comparison of money and length. This is close to the concept of ratio, which usually is used to make comparisons within the same attribute. A unit rate is a rate that is simplified so that it gives a measure of the first attribute for each unit of the second attribute. For example, if New Zealand’s 4,000,000 people are represented by 120 members of parliament, then that is a unit rate of approximately 33,333 people per MP.
$5-00 per metre is usually written as $5-00/m which emphasises the fact that a rate is a fraction.

Rate of change and graphs

The gradient of a straight-line graph is a measure of the slope of the line. Usually given the letter m, it is found by taking any two points on the line and putting

m=vertical distance between the points
horizontal distance between the points


m can also be defined as the tangent of the angle θ where θ is the smallest non-negative angle that the line makes with the positive end of the x-axis.

The gradient of a curve at a point P is the gradient of the tangent to the curve at P. The gradient at a point P is a measure of the rate of change of the variable on the vertical axis (usually the y-axis) with respect to the variable on the horizontal axis (usually the x-axis). For example, if the gradient is 3 then the vertical variable is increasing by 3 units for every 1 unit that the horizontal variable increases.

Ratio

A ratio is an expression of the relationship between two measures of the same attribute. Usually written as a:b it expresses how much of a and b are consistently combined in a whole, e.g. the ratio of weedkiller to water is 1:100, or the relationship of a to b, e.g. the scale on a map is 1 cm:100m. Common use distinguishes rates, which have two measurements of different units, e.g. kilometres per hour, and ratios, which have two or more measures of the same attribute. Both imply a division, although a ratio is not usually expressed in the form of a decimal fraction. As an example, suppose 15 lollies are shared between Molly and Dolly in the ratio 2:3. That means that for every 2 lollies that Molly receives, Dolly receives 3. So Molly gets 2 lollies out of every 5 (or 2/5 of the 15 lollies) and Dolly gets 3 lollies out of every 5 (or 3/5 of the 15 lollies).
In situations in which the ratio describes composition of a whole four fractional relationships exist. For example, in the ratio a:b, a/(a+b) gives the fraction of the whole made up by a. Similarly, b/(a+b) gives the fraction of the whole made up by b. So for a bag containing jelly beans in the ratio 3 black:5 red, 3/8 of the jellybeans are black and 5/8 are red.
In the ratio a:b, a/b describes the multiplicative relationship between the amount of a and the amount of b. Similarly this is b/a for the multiplicative relationship b to a. So for the jellybean example above, there are 3/5 as many black jellybeans as red, and there are 5/3 as many red jellybeans as black.

Rational algebraic expressions

A rational function is a function of the form:
f(x)/g(x) where f(x) and g(x) are algebraic expressions.
x/(x-1) is a rational algebraic expression.
x/(x-1)=2 is a rational algebraic equation. Multiplying both sides of the equation by x-1 gives x=2x-2 so x=2.
Rational algebraic expressions should be manipulated in the same way that rational numbers are manipulated.
For example, when adding or subtracting, a common denominator needs to be found as shown:

 

   5    +     3    =     5(x-2)    +     3(x+1)   
x+1x-2(x+1)(x-2)(x+1)(x-2)

 

 =      8x-7   
x2-x-2


Care is needed to ensure that the denominator is not zero, since if it is zero the expression becomes meaningless.

Rational numbers

The rational numbers, often given the label Q, are the numbers that can be written as fractions, that is in the form a/b where a and b are integers and b ≠ 0. (b cannot be equal to zero because division by zero is meaningless). The decimal form of all rational numbers is a repeating or a terminating decimal and all repeating or terminating decimals are rational numbers and can be written in the form a/b.

Real numbers

The set of real numbers is the union of the set of rational numbers and the set of irrational numbers. Hence it is all the numbers on the number line.

Reciprocal

The reciprocal of a number a is its multiplicative inverse, that is, the number which when multiplied by a gives 1 as the answer. (1 is the multiplicative identity element because multiplication of any number by 1 leaves the number unchanged.) So the reciprocal of 2 is 1/2, since 2 x 1/2 = 1
The reciprocal of 2/3 is 3/2 since 2/3 x 3/2 = 1

Rectangle

A rectangle is a four-sided polygon with opposite sides equal in length and all interior angles right angles (that is 90o). The area of a rectangle is the product of two adjacent sides. For example, if the sides of a rectangle are 8 cm and 6 cm then the area is 48 cm2. The area of a rectangle can be effectively explored using grid paper.

 

Reflection

A reflection in the plane has the effect of transforming an object in the plane onto its mirror image. Thus under reflection in the plane a figure in the plane is effectively flipped over a fixed line in the plane. The points on this line (called the mirror line or line of reflection) are the only fixed (invariant) points of the transformation. A reflection in space has a plane of reflection, that is, a plane of points that are invariant under the transformation of reflection. A reflection is a shape-preserving (isometric) transformation.

Relationships between successive elements of number patterns

Using a table and looking at the differences between successive terms can be an effective way of finding a rule that will generate all the terms of a sequential pattern.
Example: Consider the pattern: 1, 4, 7, 10, 13, 16, 19,…

 

n1234567
nth term14710131619
Differences 333333

Compare the pattern with the multiples of three: 3, 6, 9, 12, 15, 18, 21, …
It is apparent form the table that the nth term, Tn, is given by the expression:
Tn = (3xn)-2
Similarly, a graph might also display the relationship.

 

Relating three-dimensional models to two-dimensional representations and vice-versa

A three dimensional model or shape can be represented on a two dimensional surface (such as a piece of paper) by drawing cross-sectional views of the intersection of the shape with three planes that are perpendicular to each other. For example, for a building, we could take a front cross-sectional view, a side cross-sectional view and a so-called ‘bird’s-eye view’.
A model can also be represented by an isometric drawing or, in the case of a polyhedron, a net.

Relative size of positive and negative integers and decimals

Understanding the relative size of numbers in a place value system involves understanding the place value concept, and the face values of the numeral symbols (See. Base ten numeration system). Examples: 99 < 103 because 103 has a 1 in the hundreds column and 99 has no numeral in the hundreds column. 346 < 348 because although they have the same numerals in the hundreds and the tens columns, in the ones column they have a 6 and an 8 and 6 < 8. Some caution is needed with negative numbers as in the following example. -28 < -17 because although 28 > 17, -28 is to the left of -17 on the number line. A number line can help children with the order relation. Decimals are ordered in the same way that whole numbers are but some caution is needed as children will often see, for example, 0.32 to be greater than 0.8 since 32 is greater than 8. Using the number line or modelling decimals with materials will help to overcome this misconception.

Repeating pattern

A repeating pattern is a pattern that consists of a core that is repeated. The core is the shortest string of elements that repeats. E.g. 1, 2, 3, 1, 2, 3, 1, 2, 3, … (The core is 1, 2, 3 ) or a,b,a,b,a,b,a,b,…(The core is ab).

Risk and relative risk

In statistics, risk is often mapped to the probability of some event which is seen as undesirable. The risk function is defined as the expectation value of the loss function.
Relative risk is the risk of an event relative to exposure. It is a ratio of the probability of the event occurring in the set exposed to the risk to the probability of the event occurring in the population.
A relative risk of 1 means that there is n difference beteween the two groups. A relative risk <1 means that the event is less likely to occur in the exposed group.
A relative risk >1 means that the event is more likely to occur in the exposed group.

Roots

a1/n where n is a positive integer, is defined as the nth root of a.
Consider the meaning of a1/2 x a1/2.
By the laws of powers, a1/2 x a1/2 = a1/2+1/2 = a1 = a
So a1/2 is the number which when multiplied by itself gives a. We usually refer to this as the square root of a and write it as √a. So a1/2 = √a. So √4 = 2, since 2x2=4. Two is referred to as the square root of 4. It is the principal square root. There is another number which when multiplied by itself gives an answer of 4 and that is -2 since -2 x -2 = 4. (See Multiplication: Integers) -2 is referred to as a secondary root.
a1/3 x a1/3 x a1/3 = a1/3+1/3+1/3 = a1 = a, so a1/3 is the number which when multiplied by itself three times gives an answer of a. We usually refer to a1/3 as the cube root> of a and write it as 3√a. So a1/3 = 3√a
3√8 = 2 since 2 x 2 x 2 = 8. Two is the principal cube root of 8. There are no real secondary roots.
3-8 = -2 because -2 x -2 x -2 = -8. -2 is called the principal root of -8 because there are no other real roots.
-4 has no principal root since there are no real numbers which when multiplied by themselves give a negative number.
The fourth root of a is a1/ which can be written as 4√a etc.
Note the connection with geometry for square root and cube root. The square root of 9 is 3, which is the length of the side of a square of area 9 square units. The cube root of 27 is 3, which is the length of an edge of a cube of volume 27 cubic units.
The square roots of numbers that are not themselves squares of natural numbers (that is they do not belong to the sequence 1, 4, 9, 16, 25, 36, …)are irrational numbers and have neither an exact fractional representation nor a terminating or repeating decimal representation. For example, √2 as a decimal to 20 places is 1.414213562373095048801688724209.

Rotation

A rotation in the plane is a movement in a circular motion (a turn) through some angle that leaves shape unchanged and in which exactly one point (the centre of rotation) does not move i.e. a rotation in the plane has one invariant point. A rotation of a three-dimensional figure in space has a line of invariant points called the axis of rotation.
Rotation, like translation and reflection, is called an isometric transformation since it does not change the shape of the figure being rotated.

 

Rounding

Rounding of a number means replacing the numeral by another numeral that has fewer significant figures. The decision whether to round up or down depends on the value of the leading digit being rounded off. The convention is that 0, 1, 2, 3, and 4 are effectively just chopped off (truncated) while 5, 6, 7, 8, and 9 are truncated but the first digit not truncated is increased by a value of one.
Suppose that, measuring to the nearest millimetre we recorded some measurements as (in metres) 52.365, 12.764, 4.986, 2.031, and 5.699. If we decided to change them to measurements to the nearest centimet
re (that is, to two decimal places (2dp)) we could record them as 52.37, 12.76, 4.99, 2.03, and 5.70.
Using a metre ruler graduated in millimetres can help with an understanding of these roundings. The numbers 52.365 etc. above are expressed to three decimal places (3dp) while the numbers 52.37 etc above are expressed to two decimal places (2dp).

Sample

A sample is a part of a population. If it has been randomly selected then important conclusions about the population can be drawn. For example, the population could be the students in a certain school. A sample of a given size, say 50, could be selected by numbering the students and using a table of random numbers to choose 50 students.

 

Sample distributions

See Statistical distributions.

Sample variation

If two samples are selected from the same population they are unlikely to be the same or to have the same mean and standard deviation. Students need to be aware that a sample is not necessarily an exact representation of the population.

Scales

See Linear scales.

Sequence

A sequence is an ordered set (usually of numbers) arranged in such a way that the next element (or term) of the sequence is completely specified. E.g. 4, 7, 10, 13, … (The nth term (or general term) is given by the rule (3xn)+1 For example, the 5th term is (3x5)+1).
A finite sequence has a first and last term e.g. 2, 4, 6 , 8, …, 20
An infinite sequence continues indefinitely, e.g. 2, 4, 6, 8, …

Sequential pattern

A sequential pattern is a pattern whose terms change in an identifiable and consistent way. Examples are 2, 5, 8, 11, 14, 17, … and 1, 3, 6, 10,15, 21, … or a, b, d, g, k, p, …
The ordinal position of a term in a sequential pattern refers to its position in the pattern, that is, which term it is in the pattern. For example, in the pattern 2, 5, 8, 11, 14, 17, … the first term, T1 is 2; the second term T2 is 5 etc.

Series

A series is the sum of the terms of a sequence. For example, suppose we have the sequence 1/2, 1/4, 1/8, 1/16...
1/2 + 1/4 + 1/8 + 1/16 …is a series since it is the sum of the above sequence.
This is an infinite series since it continues indefinitely, but it is also a convergent series since it has a finite sum. The sum of this series is 1, which can be easily seen by observing the difference between 1 and the sum after one term, two terms, three terms etc.

Set

A set is a collection of objects or ideas

SI

SI stands for Systeme Internationale d’Unites (International System of Units). We usually refer to this as the metric system. It is a system of weights and measures based on powers of ten and the weight of water. (See SI measurement units)

SI measurement units

Most of the commonly used metric units and their abbreviations are as follows:

  • Length: metre (m), kilometre (km) (1000 metres), centimetre (cm) (1/100 metre), millimetre (mm) (1/1000 metre)
  • Volume: cubic metre (m3), cubic centimetre (cm3, or c.c.),
  • Capacity: Units of volume may appropriately be used as units of capacity. A commonly used unit (especially with fluids) is the litre (l) (1000 cm3), and the millilitre (ml) (1 cm3). These units may also be used as units of volume.
  • Weight: (See mass and weight). The basic unit of weight in common use is the gram (g). It is the weight of 1 cm3 of water. Other units are the kilogram (kg) (1000g, the weight of a litre of water), the milligram (mg) (1/1000 g), the tonne (1000 kg, the weight of a cubic metre of water).
  • Area: : The basic unit of area is the square metre (m2). Other units in use are the square millimetre (mm2), the square centimetre (cm2), and the hectare (10,000 m2). So a hectare is the area of a square of sides 100 m.

Significant figures

Numerals are often expressed as approximate values The concept of significant figures (or significant digits) is an assessment of the accuracy of the numeral as an expression of the real value of the number that it represents. When a number is given in decimal notation, the error should not exceed a half unit of the last digit retained. Suppose a number has been rounded to 1400. We should be able to expect that before it was rounded to the nearest whole number it was greater than or equal to 1399.5 and less than 1400.5 Hence it should be no more than 0.5 different due to the rounding. Hence we can say that the four figures are reliable, and that the number has been expressed to four significant figures.
However, suppose we had been rounding to the nearest 100, which we might do in scientific or other practical situations. Then 1379 would also be written as 1400 but only the first two digits would be significant. Then we would know that the true value was greater than or equal to 1350 and less than 1449.
When numerals are written in standard form, it is easy to see how many significant figures they have. In the examples above, the first example of 1400 which was to four significant figures should be written as 1.400 x 103, while the second example of 1400 which was to two significant figures should be written as 1.4 x 103.
The numeral 2.57000 x 102 is intended to indicate that it has six significant figures.
Similarly 0.000378 can be written as 3.78 x 10-4 and has three significant figures.

Similarity

Similar polygons are polygons whose corresponding angles are equal and whose corresponding sides are in proportion. For example, the two triangles below are similar because their corresponding angles are equal but they are not congruent.

Since the two triangles are similar their sides are in proportion and, since the vertical side of the triangle on the right is twice the length of the corresponding side of the triangle on the left, it follows that a = 6cm and b = 10 cm.

Simple additive strategies

See additive strategies.

Simultaneous equations

A system of two linear equations in two unknowns is solved simultaneously when all ordered pairs are found which satisfy both equations. In the case of three linear equations in three unknowns the solution is the set of ordered triples that satisfy all three equations. The system itself is often referred to as a system of simultaneous equations. There are three different types of solution:

  1. The system may have a unique solution. For example:
    3x+4y=17
    x-y=1 This system has the unique solution (3,2)
  2. The system may have an infinite solution. This occurs when one equation is a multiple of the other, or in the case of three equations in three unknowns, one equation is a combination of the other two. For example:
    2x+y=6
    4x+2y=12 The solution is {(t, -2t +6), t any real number}
    This is an infinite solution.
  3. The system may have no solution. The system is contradictory and is said to be inconsistent. The solution is the empty set ∅. For example:
    2x+y=6
    2x+y=8

Note that we cannot get two or three solutions. The possibilities are none, one or infinitely many solutions.
One equation non-linear A system of simultaneous equations may consist of only one linear equation and one other equation such as a quadratic equation. For example:
2x-y=1 and 3x2-xy+2y2=24
The linear equation can be used to express y in terms of x (or x in terms of y) and the resulting solutions back substituted into the linear equation.

 

Sine rule

In any acute triangle ABC the sine of an angle is proportional to the length of the side opposite the angle.
So a/sinA = b/sinB = c/sinC where a is the length of the side opposite angle A, b is the length of the side opposite angle B and c is the length of the side opposite angle C.
Care has to be taken when dealing with an obtuse triangle, because although the same relationship still holds, it is possible to become confused since, for example, sin 20o = sin 160o
(See also Trigonometric ratios)

Situations with elements of chance

Situations with elements of chance occur when we are involved with nondeterministic, or random, events. So whenever an event is not strictly determined as to its outcome there is an element of chance. (See Probability of an event). Situations such as tossing a coin and determining whether it comes up ‘heads’ or ‘tails’, or rolling a dice and observing the outcome are situations with elements of chance.

Skip counting

A counting sequence is an ordering of the counting numbers such that the difference between any two successive numbers is constant. We refer to a sequence in which the difference between any two successive numbers is greater than 1 as skip counting. An example of a backwards counting sequence is …, 50, 40, 30, 20, 10 , as is … 10, 8, 6, 4, 2 etc. An example of a forward counting sequence is 1,3,5,7,9,… etc.

Sorting

Putting objects into sets according to some chosen attribute, for example, the red ones, the square ones etc.

Spatial features

See Geometric properties.

Sphere

A sphere with centre P is a solid such that every point on its surface is at an equal distance from P. A sphere may be considered as a solid that is generated by a circle that revolves about its diameter. A sphere has only one surface. The volume (V) of a sphere of radius r is given by V = 4/3 πr3 The surface area of a sphere is 4πr2

Square

See rectangle.

Square root

See Roots.

Standard form

A number in decimal form may be written as the product of a number greater than 1 and less than 10, and a power of 10. A number written that way is said to be in standard form Standard form is also referred to as scientific notation. So, for example,

Statistical distributions

When summarising large masses of raw data it is often useful to distribute the data into classes, or categories, and to determine the number of data points belonging to each class, called the class frequency. A tabular arrangement of data by classes together with the corresponding class frequencies is called a frequency distribution or frequency table.

Statistical enquiry cycle

This refers to the sequence of actions that should occur during a statistical enquiry, namely, posing a question that invites statistical investigation; determining and gathering relevant data; sorting and organizing the data; displaying and interpreting the data with reference to the question posed; and discussing the results.

Statistical inference

Statistical inference is the part of statistics concerned with drawing conclusions about a population based on sample data.

Statistical experiment

A statistical experiment is a random or nondeterministic experiment. Its features are that:

  1. each experiment is capable of being repeated indefinitely under essentially unchanged conditions.
  2. Although we are in general not able to state what a particular outcome will be, we are able to describe the set of all possible outcomes of the experiment
  3. As the experiment is performed repeatedly, the individual

outcomes seem to occur in a haphazard manner. However as the experiment is repeated a large number of times, a definite pattern or regularity appears.

 

Subtraction of:

  • Whole numbers: Subtraction is an operation of decomposition. On whole numbers, subtraction may be described as the separating of a set into two disjoint sets. In a physical model it is represented by the separating of objects. Subtraction is the inverse operation of addition Consequently, the basic addition facts give rise to families of facts that also involve subtraction. For example, 4+5 = 9 has three other family members, namely, 5+4 = 9, 9-5 = 4 and 9-4 = 5.
    Subtraction is a binary operation, that is, it is an operation on two numbers.
    Subtraction is not commutative, that is, the order of the numbers does matter. For example, 7-3 ≠3-7.
    Subtraction is not associative. The grouping of the numbers does affect the answer. For example, (10-5) -2 ≠ 10- (5-2)
  • Fractions: Fractions may be subtracted. If their denominators are the same then we can simply subtract the numerators to obtain the difference. For example, 5/9 - 2/9 = 3/9. If their denominators are not the same then we must choose equivalent fractions so that their denominators are the same. For example, to subtract 1/6 from 1/4 we must find equivalent fractions for 1/6 and 1/4 that have a common denominator. We could multiply the two denominators, 6 and 4, and that process would always give us a common denominator. However, we might observe that the least common multiple of 6 and 4 is actually 12. 1/6 = 2/12, 1/4 = 3/12, so 1/4 - 1/6 = 1/12.
  • Decimals: Decimal fractions (commonly called decimals) may be subtracted in the same way that whole numbers are, with care being taken to consider the position of the decimal point. So, for example, tenths are subtracted from tenths, hundredths are subtracted from hundredths, etc.
  • Percentages: Percentages may be subtracted as if they were whole numbers or decimals. For example, 15%-7%=8%, 2.4%-1.2%=1.2.%, At an abstract level, a percentage is a numeral representing a real number and therefore, just like decimals, they may be subtracted. Care must be taken however because of the way that society uses percentages. One often refers to a percentage of something and that can lead to difficulties. For example, although it is true that 10% of 80 minus 5% of 80 is 5% of 80, it is not true that 10% of 80 minus 5% of 60 is 5% of either 80 or 60.
  • Integers: Integers may be subtracted by observing the following rule:
    a--b=a+b. For example, 4--3=4+3=7 This rule is best discovered using models, such as a black-and-white counters model, in which a white counter represents one, and a black counter represents -1. The number line can also be helpful.
    The properties of subtraction outlined for whole numbers also apply to the subtraction of fractions, decimals, percentages, and integers.

Symbols

Mathematics has developed from its early rhetorical phase where all words relating to operations were written, to its present symbolic phase, where various marks have meanings describing number, operations and relations. Some common symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, -, x, ÷, <, >, =, π, etc.

Symmetric Patterns

The essential feature of a symmetric geometric pattern is that it can be divided into two or more identical parts, and furthermore that these parts are systematically disposed to one another. In addition, some patterns, such as frieze patterns, will have repetitive elements.
Symmetry means that the parts of a figure are not only congruent but related by an isometry in such a way that the whole figure is self-coincident under that isometry. That is, the whole figure maps onto itself under that isometry. Symmetry of reflection and rotation can be found in many objects and patterns. Kowhaiwhai are an excellent example of frieze patterns and all seven different types of frieze pattern are found in them. They can be analysed in terms of a fundamental region (the smallest region that can generate the pattern) and the isometric transformations acting on the fundamental region to generate the whole pattern. All four isometries are found in the Kowhaiwhai.

Temperature

Temperature is a physical property of a system that underlies the common notions of hot and cold – something that is hotter has the greater temperature. The temperature of a body is a measure of its relative hotness or coldness. Temperature is usually measured in degrees Celsius (oC), a measurement system in which water freezes at 0oC and boils at 100oC.

Three-dimensional

A shape is three dimensional if it occupies a portion of space. If a shape has volume then it is three-dimensional. It is called three-dimensional because any point in space (often called 3-space) can be described by distances in three independent directions from a fixed point. (See Relating three-dimensional models to two-dimensional representations and vice-versa)

Time

Time is a fundamental property of physics and can be described only in terms of its measurement. It is defined in terms of the length of a mean solar day, i.e. the average duration of one rotation of Earth with respect to the sun. The basic unit of time is the second, which is defined as 1/86,400 of a day. Other units in common use are the minute (60 seconds) and the hour (60 minutes).

Timetables

Timetables are tables of data that involve time as one of the data measures. Timetables can contain much mathematical information. Tables such as bus timetables usually contain patterns that can be explored as a part of mathematics study. Interpretation of the timetable and its patterns develops the ability to read such tables.

Time series

When data is recorded at regularly spaced intervals of time (for example, every month or every year) it is referred to as time series data. Examples could be monthly rainfall, annual sales of consumer goods etc. Transformation

A transformation of the plane or of space is a function (or mapping) that maps the points of the plane or of space to points of the plane or of space. Common transformations are translation, reflection and rotation. See also Multiple transformations.

Translation

The movement of a figure in the plane (or in space) such that every point moves the same distance and in the same direction. Hence a translation is a shape-preserving (isometric) transformation that involves no rotation (turning) or reflection.

Trapezium

A trapezium is a quadrilateral (four-sided polygon) having exactly one pair of parallel sides. The area of a trapezium is the mean of the lengths of the parallel sides multiplied by the distance between them.

Treediagram

A tree diagram can be an effective way to show a schematic representation of the possible outcomes of an experiment. Each branch of the tree displays an outcome. Probabilities of that outcome can be written against the branch. For example, suppose an experiment is to toss a coin twice and observe each toss as a head (H) or a tail (T). The following tree diagram displays the possible outcomes and their associated probabilities.



From this we can see that there are four possible outcomes, each of whose probability is printed at the bottom of the branch. These probabilities are obtained by multiplying the probabilities on each branch of the path leading to the outcome.

 

Triangle

A triangle is a polygon with three sides, that is, a portion of the plane bounded by three straight lines. The interior angles of a triangle add to 180o a fact that can be easily shown by drawing a triangle on paper, cutting the triangle out, tearing off the corners and putting the vertex angles together. A vertex of a triangle is the point where two of the sides meet.
Triangles can be classified by their sides as follows:
Scalene triangle – no sides are equal
Isosceles triangle – at least two sides are equal
Equilateral triangle – all three sides are equal. The equilateral triangle is therefore a special case of the isosceles triangle.

They can also be classified according to the kind of angles they have:
Right triangle – one angle a right angle
Obtuse triangle – has an obtuse angle, that is, an angle greater than 90o but less than 180o.
Acute triangle – a triangle with three acute angles, that is, angles that are less than 90o.

The area of a triangle is half of the length of the base multiplied by the vertical height. This can be discovered by finding the area of a parallelogram as described (See parallelogram) and realising that every triangle can be obtained by bisecting a parallelogram. Hence the area of a triangle is half the area of the associated parallelogram.

Trigonometric equations

In trigonometric equations the unknown appears in the form of a trigonometric function, or functions. Simple or basic trigonometric equations are trigonometric equations in which only one kind of trigonometric function is present. For example, cos 3x/2 = 1
Care must be taken to allow for all possible solutions.
If cos 3x = 1/2 then 3x = arccos 1/2
So x = ± π/9 + 2nπ/3, n an integer.

Trigonometric ratios

If two triangles have equal angles then the lengths of corresponding sides will be in proportion. Triangles that have equal angles are called similar triangles. The lengths of any two corresponding sides in two similar triangles will be in proportion even though they may not be equal. So if one side in the larger triangle is double the length of the corresponding side in the smaller triangle then the other two sides in the larger triangle will also be double the length of the corresponding sides in the smaller triangle. This property is used in defining the fundamental trigonometric functions of angles. The two right triangles above are similar triangles. AC and DF are the two hypotenuses. CB is the side opposite angle A and FE is the side opposite angle D. AB is the side adjacent to angle A and DE is the side adjacent to angle D. The fundamental trigonometric functions are defined as follows:
The sine of A: sin A = length of CB divided by length of AC = sin D
The cosine of A: cos A = length of AB divided by length of AC = cos D
The tangent of A: tan A = length of CB divided by length of AB = tan D
The cosecant of A: cosec A = length of AC divided by length of CB
The secant of A: sec A = length of AC divided by length of AB
The cotangent of A cot A = length of AB divided by length of CB

The numerical value of these trigonometric functions depends only on the size of the acute angles in the right triangle and not on the ‘size’ of the triangle.

These functions may be summarised as follows:
sin = opposite/hypotenuse
cos = adjacent/hypotenuse
tan = opposite/adjacent
cosec = hypotenuse/opposite
sec = hypotenuse/adjacent
cot = adjacent/opposite

So cosec = 1/sin
sec = 1/cos
cot = 1/tan

The inverse functions are arcsin, arccos, arctan etc.

Turn

See rotation.

Two-dimensional

A shape is two dimensional if it can be made to lie wholly in a plane. Hence a two dimensional shape has area but no volume. It is called two-dimensional because any point in a plane can be described by distances from a fixed point in two independent directions, such as in the Cartesian plane (See Coordinate systems).

Two-dimensional representations of three-dimensional solids

See Relating three-dimensional models to two-dimensional representations and vice-versa.

Two-way tables

Two-way tables, often called contingency tables, are two-dimensional grids in which frequencies observed in a survey can be displayed. A table with r rows and k columns is referred to as an r x k table. For example, suppose three teachers, A, B, and C, looked at the pass rate of their students. This could be set out in a 2 x 3 table as follows.

 ABCTotal
Passed508372205
Failed105823
Total608880228

Uncertainty

If the probability of an event is neither 0 (impossible) nor 1 (certain) then there is an element of uncertainty involved. The event may or may not occur. So if today is Wednesday then we can be certain that tomorrow will be Thursday. But if it hasn’t rained in Alice Springs for three years we cannot be certain that it will not rain tomorrow. Children often confuse certainty with events that are highly likely to occur. For example, if the All Blacks are playing Botswana then a large proportion of children will claim that an All Black win is certain even though there is obviously an element of uncertainty in the outcome.

Units of measurement

A unit of measurement is an item or quantity that has the attribute being measured and which can be compared with the object being measured. For example, the length of a desk could be measured in handspans or in centimetres. Both have the attribute of length; the handspan is a nonstandard unit of measurement, the centimetre a standard unit of measurement. (For a list of the commonly used SI units see SI measurement units)

Variables

The basic concept of a variable is of a quantity that can take several (perhaps even infinitely many) values. In an equation it will often be represented by a letter such as an x or a y. Suppose, for example, that chocolate bars bought on the Internet cost $2-00 each plus $3-00 for postage. Then the cost (in dollars) of buying x bars will be 2x+3. We could write c=2x+3. c and x are both variables. Once we have chosen a value for x (say 10) then c will be automatically determined. So x is called the independent variable and c the dependent variable. (See also Appropriate statistical variables)

Views and pathways from locations on a map

Children can be asked questions regarding a map such as: "If you were standing at the corner of Smith and Jones Streets looking towards the water tower, what would you see on your left?" For more advanced learners, questions such as: "If you started at position (3,5) and you travelled northeast for 6 km what would you find?" etc.

Volume

Volume is a measure of the amount of space that a three-dimensional object occupies. Lines and planes are considered to have no volume. The basic unit of measurement of volume is the cubic metre (m3), with cubic centimetre (cm3 or c.c.) also in common use. Units used more commonly as units of capacity may also be used, in particular the litre (l) (1000 cm3) and the millilitre (ml) (1cm3), (See SI measurement units)

Volumes of cuboids, prisms etc.

See cuboids, prisms etc.

Weight

The measure of the heaviness of an object. It is the force that results from the action of gravity on matter. The term ‘weight’ is often used when strictly speaking ‘mass’ is meant. The distinction between mass and weight is unimportant for most practical purposes and we commonly use units of mass (kilogram, gram, tonne etc.) as units of weight. (See SI measurement units)

Whole Numbers

{0,1,2,3,…} is the set of whole numbers. Hence the whole numbers are all the natural numbers as well as the number zero. (See Base ten numeration system)

Whole number data

Statistical data falls into two main classes. Things that can be counted (for example, the number of children in a family) give rise to whole number data and things that can be measured (for example, the heights of a group of people) give rise to measurement data. A third class is category data, which is data of things that fall into categories such as colour, brand, model etc.