15152907 Topic 7 Mathematical Concepts

7/28/2019 15152907 Topic 7 Mathematical Concepts

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X INTRODUCTION

In mathematics, concepts refer to the basic mathematical knowledge needed forsolving mathematical problems. A concept is a group of related mathematical facts.You have to learn related mathematical facts before you can understand amathematical concept. For example, the concept of a quadrilateral involves theknowledge of the parallelogram, rectangle, rhombus and trapezium, as shown inFigure 7.1.

Figure 7.1 : Example of a mathematical concept

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77 X Mathematical

Concepts

LEARNING OUTCOMES

By the end of this topic, you should be able to:1. Explain what is meant by mathematical concepts;

2. Differentiate between mathematical facts and concepts; and

3. List the mathematical facts for each mathematical concept.


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X TOPIC 7 MATHEMATICAL CONCEPTS90

A mathematical fact is a true mathematical statement. This topic will discussvarious concepts in mathematics, namely numerical concepts, geometrical concepts,statistical concepts and algebraic concepts.

NUMERICAL CONCEPTSIn this sub-topic, you will be exposed to numerical concepts.

7.1.1 Concept of Real Numbers

The concept of real numbers involve the knowledge of rational and irrationalnumbers.

(a) Rational number : Any number that can be expressed as an exact fraction.

Example of rational numbers:31

,43

, 0.4,13

,2

13,

(b) Irrational number : Any number that cannot be expressed in the form of afraction.

Example of irrational numbers: 2 , 3 , 5 , S

Look at the number system in Figure 7.2. As you can see, all the numbers areinterrelated.

Figure 7.2 : Concept of real numbers

7.1


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TOPIC 7 MATHEMATICAL CONCEPTS W 91

7.1.2 Concept of Fraction G

Generally, a fraction is part of a whole number. For instance, when a cake is cut intotwo equal slices, each slice is a fraction of the cake. A fraction consists of anumerator and a denominator. An example of this is shown below:

The concept of fraction involves the knowledge of proper fraction, improperfraction and mixed numbers/fractions.

(a) Proper fraction : The numerator is smaller than the denominator.

Examples :53

;61

(b) Improper fraction : The numerator is greater than, or equal to the denominator.

Examples :37

;66

(c) Mixed numbers/fractions : Comprises an integer and a proper fraction.

Examples :311 and

57

9

List two mathematical facts in the real number system.

ACTIVITY 7.1

What are the mathematical facts behind the concept of fraction?Discuss.

SELF-CHECK 7.1


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Do the following exercise to test your understanding.

GEOMETRICAL CONCEPTThe geometrical concept comprises various concepts pertaining to angles andquadrilaterals.

7.2.1 Concept of Angles

The concept of angles involves the knowledge of straight angles, right angles, acuteangles, obtuse angles, reflex angles, complementary angles, supplementary anglesand vertically opposite angles as well as others. See Figure 7.3.

Figure 7.3 : Types of angles

(a) A full circle has a measure of 360.

(b) A straight angle is equal to 180.

(c) A right angle is equal to 90.

(d) An acute angle is less than 90.

7.2

Can you think of other mathematical facts involved in the concept of fraction? Discuss with your course mates in myLMS.

ACTIVITY 7.2


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(e) An obtuse angle is between 90 and 180.

(f) A reflex angle is between 180 and 360.

(g) Complementary angles add up to 90 (see Figure 7.4).

Figure 7.4 : Complementary angles

(h) Supplementary angles add up to 180 (see Figure 7.5).

Figure 7.5 : Supplementary angles

(i) Vertically opposite angles are equal (see Figure 7.6).

Figure 7.6 : Vertically opposite angles

When a transversal cuts two parallel lines (see Figure 7.7):

Corresponding angles are equal;

Alternate angles are equal; and

Co-interior angles are supplementary.

Figure 7.7 : A transversal cutting through two parallel lines and its related angles


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Try the following exercise to test your understanding.

7.2.2 Concept of Triangles

The concept of triangles involves the knowledge of acute-angled triangles, right-angled triangles, obtuse-angled triangles, equilateral triangles, isosceles triangles

and scalene triangles as well as the sum of angles in a triangle.(a) All sides of a scalene triangle are of different lengths (see Figure 7.8).

Figure 7.8 : Scalene triangle

(b) An isosceles triangle has two sides of equal length (see Figure 7.9).

Figure 7.9 : Isosceles triangle

(c) All sides and angles of an equilateral triangle are equal (see Figure 7.10).

Figure 7.10 : Equilateral triangle

List three mathematical facts about the concept of angles.

SELF-CHECK 7.2


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(d) An obtuse-angled triangle has one angle greater than 90 (see Figure 7.11).

Figure 7.11 : Obtuse-angled triangle

(e) All angles of an acute-angled triangle are less than 90 (see Figure 7.12).

Figure 7.12 : Acute-angled triangle

(f) A right-angled triangle has one angle of 90 (see Figure 7.13).

Figure 7.13 : Right-angled triangle

(g) The sum of angles in a triangle is 180 (see Figure 7.14).

Figure 7.14 : Sum of angles in a triangle


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7.2.3 Concept of Quadrilaterals G

The concept of quadrilaterals involves the knowledge of parallelograms, rectangles,rhombus, squares, kites and trapeziums.

(a) The opposite sides of a parallelogram are parallel and equal. The oppositeangles are equal (see Figure 7.15).

Figure 7.15 : Opposites of a parallelogram

(b) The opposite sides of a rectangle are parallel and equal, and all internal anglesare 90 (see Figure 7.16).

Figure 7.16 : Opposite sides of a rectangle

(c) The opposite sides of a rhombus are parallel and all sides are equal (see Figure7.17).

Figure 7.17 : Opposite sides of a rhombus

List three mathematical facts related to the concept of triangles.SELF-CHECK 7.3


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(d) The opposite sides of a square are parallel, all sides are equal and all internalangles are 90 (see Figure 7.18).

Figure 7.18 : Opposite sides of a square

(e) A kite has two pairs of adjacent equal sides. Each pair of opposite angles areequal (see Figure 7.19).

Figure 7.19 : Angles of a kite

(f) A trapezium has one pair of opposite sides that are parallel (see Figure 7.20).

Figure 7.20 : Opposite sides of a trapezium


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Do the following exercise to test your understanding.

STATISTICAL CONCEPTSStatistical concepts include the concept of central tendency.

7.3.1 Concept of Central TendencyCentral tendency refers to a variables typical and most common values. Generally,these are the values around which most values tend to cluster or converge. Thethree primary measures of central tendency are mode, median and mean.

(a) Mode : This is the value with the greatest frequency. In other words, it isthe most commonly occurring value.

(b) Median : This is the value which has an equal number of values higher andlower than it, i.e., the middle value or the average of two middle

values.(c) Mean : This is the sum of the values of all divided by the number of cases.


ALGEBRAIC CONCEPTSBasic algebra consists of knowledge of the following term, coefficient, expression,equation and constant (see Figure 7.21).

7.4

7.3

List three mathematical facts related to the concept of quadrilaterals.

SELF-CHECK 7.4

List down some other mathematical facts found in statistics.

ACTIVITY 7.3


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Figure 7.21 : Algebraic expression

(a) Expressions which contain letters that are used to represent unknownnumbers are called a lgebraic expressions.

(b) A t erm of an algebraic expression is the product of numbers and/orunknowns.

(g) Example : 2a is a term.

(c) A term which has no unknown is called a c onstant term.

(d) An e quation contains an equal (=) sign.


x A mathematical fact is a true mathematical statement.

x The concept of real numbers involves the knowledge of rational and irrationalnumbers.

x Fraction involves the knowledge of proper fraction, improper fraction and mixednumbers/fractions.

x The geometrical concept involves knowledge about angles and quadrilaterals.

1. List three mathematical facts related to the concept of algebra.

2. Define the term mathematical fact and give an example.

ACTIVITY 7.4


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x The concept of angles involves knowledge of straight, right, acute, obtuse, reflex,complementary, supplementary and vertically opposite angles as well as others.

x

The quadrilateral concept involves knowledge of parallelograms, rectangles,rhombus, squares, kites and trapeziums.

x Central tendency refers to a variables typical and most common value(s).

x The three primary measures of central tendency are mode, median and mean.

x Basic algebra consists of knowledge of terms, coefficients, expressions, equations andconstants.

Acute angle

Coefficient

Constant

Denominator

Equilateral triangle

Irrational numbersIsosceles triangle

Mean

Median

Mode

Numerator

Obtuse angle

Parallelogram

Quadrilaterals

Reflex angle

RhombusScalene triangle

Trapezium

Unknown

Whole numbers

15152907 Topic 7 Mathematical Concepts

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