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UNIT 1: SET OF NUMBERS
Key Unit competence: Classify numbers into naturals, integers, rational and irrationals.
1.0. Introductory Activity 1
1.1. Natural numbers
1.0.1. Definition
Activity 1.1.1
For any school or organization, they register their different assert for good
management.
For example
1. The number of desks may be 152
2. The number of Mathematics textbooks may be 2000.
3. The number of classrooms is 15
4. The number of kitchen may be 1
5. The number of car may be 0
6. And so on
All of these numbers are elements of a set. Which set do you think, they can
belong to?
Can you give other example of elements of that set?
CONTENT SUMMARY
Usually, when counting, people begin by one, followed by two, then three and so
on. The numbers we use in counting including zero, are called Natural numbers.
The set of natural numbers is denoted by = … 0, 1, 2, 3, 4, . { }
On a number line, natural numbers are represented as follows:
Application activity 1.1.1
1. Write down, first ten elements of natural numbers starting from zero.
2. Apart from recording assets, give two examples of where natural
numbers can be used in daily life
2 Mathematics ECLPE | TTC Year 1
1.1.2. Sub sets of natural numbers
Activity 1.1.2
Carry out the following tasks.
(a) Use a dictionary or internet to define the terms: even, odd and prime numbers.
(b) You are given the set of natural numbers between 0 and 20,
(i) make a set of odd numbers.
(ii) make a set of even numbers.
(iii) make a set of prime numbers.
(iv) identify even numbers which are prime numbers?
(v) How many odd numbers are prime numbers?
(c) Represent the information from (b) in a Venn diagram.
CONTENT SUMMARY
There are several subsets of natural numbers:
(a)Even numbers
Even numbers are numbers which are divisible by 2 or numbers which are multiples of 2.
Even numbers from 0 to 20 are 0, 2, 4, 6, 8, 10,12, 14, 16, 18 and 20.
The set of even numbers is E = {2,4,6,8,} and it is a subset of natural numbers.
(b) Odd numbers
Odd numbers are numbers which leave a remainder of 1 when divided by 2. Odd
numbers between 0 and 20 are 1,3,5,7,9,11,13,15,17 and 19. The set of odd numbers
is O = {1,3,5,7,} and it is a subset of natural numbers.
(c) Prime numbers
Prime number is a number that has only two divisors 1 and itself. Prime numbers
between 0 and 20 are 2, 3, 5, 7, 11, 13, 17 and 19. The set of prime numbers is
P = {2,3,5,7,11,13,19,} and it is a subset of natural numbers.
Application activity 1.1.2
Given E = {1, 4, 8, 11, 16, 25, 49, 53, 75}, list the elements of the following subsets
(a) Even numbers
(b) Odd numbers
(c) Prime numbers
Represent the above information on a Venn diagram.
1.1.3 Operations and properties on natural numbers
Activity 1.1.3
1. From the given any three natural numbers a ,b and c , investigate the following operations
a) is a b + always a natural number? Use example to justify your answer.
b) is a b + and b a + always giving the same answer? Use example to justify your
answer.
c) is a bc + +
and (ab c + +) always giving the same answer? Use example to
justify your answer.
2. Given any three natural numbers a ,b and c , investigate the following operations:
a b − and b a − , a bc − −
What do you notice? Is always the answer an element of natural numbers?
Justify your answer.
3. Given any three natural numbers a ,b and c , investigate the following
operations : a b × , a b × and b a × , a bc × ×
ab ac + , a bc × −
4. Given any two natural numbers a ,b different from zero, investigate the
following operations a b ÷ , a b ÷ and b a ÷ ,
CONTENT SUMMARY
Addition
(i) Closure property: The sum of any two natural numbers is always a natural
number. This is called ‘Closure property of addition’ of natural numbers. Thus, is
closed under addition. If a and b are any two natural numbers, then (a + b) is also a
natural number.
Example: 2 + 4 = 6 is a natural number.
(ii) Commutative property: If a and b are any two natural numbers, then, a+b=b+a.
Addition of two natural numbers is commutative.
Example: 2+ 4 = 6 and 4 + 2 = 6. Hence, 2 + 4 = 4 + 2
(iii) Associative property:
If a, b and c are any three natural numbers, then a + (b + c) = (a + b) + c. Addition
of natural numbers is associative.
Example: 2 + (4 + 1) = 2 + (5) = 7 and (2 + 4) + 1 = (6) + 1 = 7. Hence, 2+(4+1)=(2+4)+1
Subtraction
(i) Closure property: The difference between any two natural numbers need not be
a natural number. Hence is not closed under subtraction.
Example: 2 - 5 = -3 is a not natural number.
(ii) Commutative property: If a and b are any two natural numbers, then
(a-b) ≠ (b-a). Subtraction of two natural numbers is not commutative.
Example: 5 - 2 = 3 and 2 - 5 = -3. Hence, 5 - 2 ≠ 2 – 5. Therefore, Commutative
property is not true for subtraction.
(iii) Associative property: If a, b, c and d are any three natural numbers, then
a - (b - d) ≠ (a - b) – d. Subtraction of natural numbers is not associative.
Example: 2 - (4 - 1) = 2 - 3 = -1 and (2 - 4) - 1 = -2 - 1 = -3
Hence, 2 - (4 - 1) ≠ (2 - 4) – 1. Therefore, Associative property is not true for
subtraction.
Multiplication
(i) Closure property: If a and b are any two natural numbers, then a x b = ab is also
a natural number. The product of two natural numbers is always a natural number.
Hence is closed under multiplication.
Example: 5 x 2 = 10 is a natural number.
(ii) Commutative property: If a and b are any two natural numbers, then
a x b = b x a. Multiplication of natural numbers is commutative.
Example: 5 x 9 = 45 and 9 x 5 = 45. Hence, 5 x 9 = 9 x 5. Therefore, Commutative
property is true for multiplication.
(iii) Associative property: If a, b and c are any three natural numbers,
then a x (b x c) = (a x b) x c. Multiplication of natural numbers is associative.
Example: 2 x (4 x 5) = 2 x 20 = 40 and (2 x 4) x 5 = 8 x 5 = 40 , Hence,
2 x (4 x 5) = (2 x 4) x 5. Therefore, Associative property is true for multiplication.
(iv) Multiplicative identity: a is any natural number, then a x 1 = 1 x a = a. The
product of any natural number and 1 is the whole number itself. ‘One’ is the
multiplicative identity for natural numbers.
Example: 5 x 1 = 1 x 5 = 5
Division
(i) Closure property: When we divide of a natural number by another natural number,
the result does not need to be a natural number. Hence, is not closed under
multiplication.
Example: When we divide the natural number 3 by another natural number 2, we
get 1.5 which is not a natural number.
(ii) Commutative property: If a and b are two natural then a ÷ b ≠ b ÷ a. Division
of natural numbers is not commutative.
Example: 2 ÷ 1 = 2 and 1 ÷ 2 = 1.5. Hence, 2 ÷ 1 ≠ 1 ÷ 2
Therefore, Commutative property is not true for division.
(iii) Associative property: If a, b and c are any three natural numbers,
then a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c. Division of natural numbers is not associative.
Example : 3 ÷ (4 ÷ 2) = 3 ÷ 2 = 1.5 and (3 ÷ 4) ÷ 2 = 0.75 ÷ 2 = 0.375
Hence, 3 ÷ (4 ÷ 2) ≠ (3 ÷ 4) ÷ 2. Therefore, Associative property is not true for
division.
Distributive Property
(i) Distributive property of multiplication over addition :
If a, b and c are any three natural numbers, then a x (b + c) = ab + ac. Multiplication
of natural numbers is distributive over addition.
Example : 2 x (3 + 4) = 2 x 3 + 2 x 4 = 6 + 8 = 14
2 x (3 + 4) = 2 x (7) = 14. Hence, 2 x (3 + 4) = 2 x3 + 2 x 4
Therefore, Multiplication is distributive over addition.
(ii) Distributive property of multiplication over subtraction:
If a, b and c are any three natural numbers, then a x (b - c) = ab – ac. Multiplication
of natural numbers is distributive over subtraction.
Example: 2 x (4 - 1) = (2x4) - (2x1) = 8 - 2 = 6
2 x (4 - 1) = 2 x (3) = 6. Hence, 2 x (4 - 1) = (2x4) - (2x1)
Therefore, multiplication is distributive over subtraction.
Application activity 1.1.3
1.2 Integers
1.2.1 Definition
Activity 1.2.1
Carry out the following activities
1. Using a Maths dictionary, define what an integer is.
2. What integer would you give to each of the following situation?
(a) A fish which is 50 m below the water level.
(b) Temperature of the room which is 42ºC.
(c) A boy who is 2 m below the ground level in a hole.
(d) A bird which is 3 m high on a tree
CONTENT SUMMARY
Integers are whole numbers which have either negative or positive sign and include
zero. The set of integers is represented by . Iintegers can be negative {-1,-2,-3,-4,-5,... },
positive {1, 2, 3, 4, 5, ... }, or zero {0}
The set of integers is represented using Carly brackets as follow :
{ ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }
Example 1:
Taking 4 steps forward can be considered as +4 (positive 4). This means it is 4 steps
ahead of the starting point.
Taking 4 steps backward is considered as -4 (negative 4). This means it is 4 steps
behind the starting point.
Example 2:
Application activity 1.2.1
John borrowed $3 to pay for his lunch.
Alex borrowed $5 to pay for her lunch.
Virginia had enough money for lunch and has $3 left over.
Place these people on the number line to find who is poorest and who is richest
given that they will pay the same amount.
1.2.2 Sub sets of integers
Activity 1.2.2
Discuss the following:
From integers between 12 and +20, form small sets which contain;
a) odd numbers
b) even numbers
c) factors of 6
d) multiples of 3.
CONTENT SUMMARY
Integers have several subsets such as set of natural numbers, set of even numbers,
set of odd numbers, set of prime numbers, set of negative numbers and so on.
Some of special subset of integers are the following:
Application activity 1.2.2
Among them,
• which are negative integers?
• Which are positive integers?
• Which are neither positive nor negative integers?
• Which are not integers?
1.2.3 Operations and properties on integers
Activity 1.2.3
1. Work out the following on a number line:
(a) (+3) + (+2) (b) –(5) + - (3) (c) (+4) + (-3)
(d) Which side of the number line did you move when adding a negative
number to a positive number?
(e) Which side of the number line did you move when adding the given
numbers which are all negative?
2. Work out the following and show your solutions on a number line.
(a) (-4)-(+3) (b) (+5) – (+3) (c) (–6) – (-6)
(d) On your number line, which direction do you move when subtracting two
negative numbers?
(e) In case you have two positive numbers that you are finding the difference,
which side of the number line would you move?
3. Work out the following:
(a) (+5) × (-6) (b) (+5) × (+6) (c) (-5) × (+6) (d) (-5) × (-6)
(e) Did you obtain the same results in all the four tasks?
4. Work out the following and show your solutions on a number line.
(a) (-4) ÷(+4) (b) (+4) ÷ (+4) (c) (–4) ÷ (-4) (d) (+4) ÷ (-4)
(e) Did you obtain the same results in all the four questions above?
CONTENT SUMMARY
The table below shows how addition, subtraction, multiplication and division of
integers are performed.
Application activity 1.2.3
A common example of negative integer usage is the thermometer.
Thermometers are similar to number lines, but vertical. They have positive
integers above zero and negative integers below zero. Commonly, people recognize a temperature of -25°C as cold. People use this number system to measure and represent the temperature of the air. Also, if it is -23°C outside, and
the temperature drops 3 degrees, what is temperature now? If we picture the thermometer, we know that as the temperature drops, we look downwards on the thermometer
1.3 Rational and irrational numbers
1.3.1 Definition of rational numbers
Activity 1.3.1
CONTENT SUMMARY
From any two integers a and b , we deduce fractions expressed in the form a/b , where b is a non-zero integer. A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers; the denominator in a rational number cannot be zero. As fractions are rational numbers, thus set of fractions is known as a set of rational numbers.
Application activity 1.3.1
From research on internet or using reference books, identify different type of fractions and give example for each type.
1.3.2 Sub sets of rational numbers
Activity 1.3.2
Knowing that a rational number is a number that can be in the form of p q, where p and q are integers and q is not zero. Tell whether the given statements are true or false. Explain your choice.
1. All integers are rational numbers.
2. No rational numbers are whole numbers.
3. All rational numbers are integers.
4. All whole numbers are rational numbers
CONTENT SUMMARY
From the concept of subset and definition of a set of rational numbers we can establish some subsets of rational numbers; among them we have:
• Integers and its subsets: 0 , , , ,... +−+ Z Z;Z Z..........
• Natural numbers and its subsets: , , + N N prime numbers, odd numbers, even numbers,…
• Counting numbers and its subsets: ,+ N square numbers, prime numbers, odd numbers, even numbers …
Application activity 1.3.2
Using Venn diagram, establish the relationship between the following sets:
Natural numbers; Integers and rational numbers.
1.3.3 Operations and properties on rational numbers
Activity 1.3.3
CONTENT SUMMARY
The following table shows how addition, subtraction, multiplication and division of rational numbers are performed
Application activity 1.3.3
Read carefully the following text and make a research to find out two more examples where fractions or rational numbers are used in real life and then present your findings in written and oral forms.“Imagine you are shopping with your $100 in birthday money. You really want a few items you have had your eye on for a while, but they are all very expensive. You are waiting for the items to go on sale, and when they do, you rush down to the store. Instead of being marked with a new price, though, the store has a large sign that reads: All items are currently 75% off. This sounds like great news, but without doing some Math, there is no way to know if you have enough money. Knowing that 75% is ¾ off the cost of each item is the best way to get started”.
1.3.4 Definition of irrational numbers
Activity 1.3.4
CONTENT SUMMARY
Application activity 1.3.4
Using internet or reference books, from your own choice of appropriate numbers, verify and discuss if the given statement is always true, sometimes true or never true.
1. The sum of rational number and irrational number is irrational.
2. The product of rational number and irrational number is irrational.
3. The sum of two irrational numbers is irrational.
4. The product of two irrational numbers is irrational.
5. Between two rational numbers, there is an irrational number.
6. If you divide an irrational number by another, the result is always an irrational number
1.4 Real numbers
1.4.1 Definition
Activity 1.4.1
From the following Venn diagram, representing the set of counting numbers, natural numbers, integers, decimals, rational numbers and irrational numbers, verify and discuss if the given statement is whether true or false.
1. The set of counting numbers is a subset of natural numbers.
2. The intersection of set of integers and counting numbers is the set of natural numbers.
3. The intersection of set of integers and natural numbers is the set of counting numbers.
4. The union of set of natural numbers and counting numbers is the set of natural numbers.
5. The intersection of set of rational numbers and irrational numbers is the set of irrational numbers.
6. The union of set of rational numbers and irrational numbers is a set of irrational numbers.
CONTENT SUMMARY
The set of rational numbers and the set of irrational numbers combined together, form the set of real numbers. The set of real numbers is denoted by R . Real numbers are represented on a number line as infinite points or they are set of decimal numbers found on a number line. This is illustrated on the number line below
Application activity 1.4.1
Some examples of applications of real numbers in our daily life are identified below. From research activity; find out at least 3 examples of other applications of real numbers in our daily life.
Real numbers help us to count and to measure out quantities of different items. For instance, in catering you may have to ask the client how many sandwiches they need for the event. Certainly, those working in accounts and other financial related jobs may use real numbers mostly. Even when relaxing at the end of the day in front of the television flicking from one channel to the next you are using real numbers.
1.4.2 Subsets of real numbers and intervals
Activity 1.4.2
Carry out research on sets of real numbers to determine its subsets and present your finds using a number line.
Application activity 1.4.2
Note that, in writing intervals, it is also possible to include only one endpoint in an interval. Included point is geometrically represented by the solid dot. From this notice, complete the following table.
1.4.3 Operations and properties on real numbers
CONTENT SUMMARY
The following table shows how addition, subtraction, multiplication and division of real numbers are performed
Application activity 1.4.3
1. Discuss whether Closure property under division for real numbers is satisfied.
2. Ngoma District wants to sell two fields on the same price. One has width of 60m out 160m of length. Another one has the width of 100m as it is its length. Among these fields, what is biggest? Interpret your result.
1.4. END UNIT ASSESSMENT 1
1. List three rational numbers between 0 and 1
2. Identify the sets to which each of the following numbers belong by ticking (˅) in the appropriate boxes (cells).
UNIT 2: SET THEORY
Key Unit competence: Solve problems that involve Sets operations, using Venn diagram.
2.0. Introductory Activity 2
At a TTC school of 500 students-teachers, there are 125 students enrolled in Mathematics club, 257 students who play sports and 52 students that are enrolled in Mathematics club and play sports.
If M stands for the set of students in Mathematics club, S stands for the set of students in Sports and U stands for all students at the school or universal set,
1. Complete the following table
2.1 Sets and Venn diagrams
Activity 2.1
Given set A = { 1, 3, 5, 7, 9 } and B = { 1, 2, 3, 4, 5 },
• Identify common elements for both sets.
• Determine elements of set A which are not in set B
• Determine elements of set B which are not in set A
• Determine all elements in set A and set B
• Represent elements of set A in a Venn diagram
• Represent elements of set B in a Venn diagram
• Represent elements of sets A and B using one Venn diagram
CONTENT SUMMARY
A well-defined collection of objects is called a set. Each member of a set is called an element. All elements of a set follow a certain rule and share a common property amongst them.
A set that contains all the elements and sets in a given scenario is called a Universal Set (U).
Venn Diagrams consist of closed shapes, generally circles, which represent sets. The capital letter outside the circle denotes the name of the set while the small letters inside the circle denote the elements of the set.
The various operations of sets are represented by partial or complete overlap of these closed figures. Regions of overlap represent elements that are shared by sets.
In practice, sets are generally represented by circles. The universal set is represented by a rectangle that encloses all other sets.
The given figure is a representation of a Venn diagram. Here each of the
circles A, B and C represents a set of elements.
• Set A has the elements a, d, e and g.
• Set B has the elements b, d, g and f.
• Set C has the elements e, g, f and c.
• Both A and B have the elements d and g.
• Both B and C have the elements g and f.
• Both C and A have the elements e and g.
• A, B and C all have the element g.
The circular pattern used to represent a set and its elements is called a Venn diagram.Venn diagrams are an efficient way of representing and analysing sets and performing set operations. As such, the usage of Venn diagrams is just the elaboration of a solving technique. Problems that are solved using Venn diagrams are essentially problems based on sets and set operations.
Example
1. Given set A= {2, 4, 5, 7, 8}, represent set A on a Venn diagram.
Solution
First, express the data in terms of set notation and then fill the data in the Venn diagram for easy solution.
When drawing Venn diagrams, some important facts like “intersection”, “union” and “complement” should be well considered and represented.
Example: Consider the Venn diagram below.
List the elements of set M and N.
Solution
Application activity 2.1
Consider these two sets A = {2,4,6,8,10 ] and B = { 2,3,5,7} . Represent them in a Venn diagram
2.2 Operations of Sets
Activity 2.2
Consider a class of students that form the universal set. Set A is the set of all students who were present in the English class, while Set B is the set of all the students who were present in the History class. It is obvious that there were students who were present in both classes as well as those who were not present in either of the two classes. The shaded part shows the elements which are considered in the diagram.
Observe the diagrams and identify which one to represent the following:
• All students who were absent in the English class
• All students who were present in at least one of the two classes.
• All the students who were present for both English as well as History classes.
• All the students who have attended only the English class and not the History class
• All the students who have attended just the History class and not the English class.
CONTENT SUMMARY
From the activity, we realize that two or more sets can be represented using one Venn diagram and from the representations, different sets can be determined. To determine those sets, one may perform different operations on sets such as:
• Intersection of sets
• Union of sets
• Universal sets
• Simple difference of sets
• Symmetric difference of sets
• Complement of sets
a. The intersection of sets
The common elements which appear in two or more sets form the intersection of sets. The symbol used to denote the intersection of sets is∩ . The intersection of sets A and B is denoted by A ∩ B and consist of those elements
a) Universal set
A set that contains all the subsets under consideration is known as a universal set. A Universal set is denoted by the symbolU .
Example
Consider a school in which one can find various categories of people such as pupils, teachers and other workers or staff.
1. Use S to represent the set of people in a school.
2. Write all the subsets of S.
Solution
1. We can present the set of all people in the school with sets as follows:
Set S pupils teachers workers = { ,, . }
Thus, set S contains all the subsets of the various categories of people in the school.
Let us use sets P, T and W to represent the subsets of set S.
2.3 Analysis, interpretation and presentation of a problem using Venn diagram
Activity 2.3
1. A survey was carried out in a shop to find the number of customers who bought bread or milk or both or neither. Out of a total of 79 customers for the day, 52 bought milk, 32 bought bread and 15 bought neither.
a. Without using a Venn diagram, find the number of customers who:
i. bought bread and milk
ii. bought bread only
iii. bought milk only
b. With the aid of a Venn diagram, work out questions (i), (ii) and (iii) in (a) above.
c. Which of the methods in (a) and (b) above is easier to work with? Give reasons for your answer.
2. The Venn diagram below shows the number of senior one students in a school who like Mathematics (M), Physics (P) and Kinyarwanda (K). Some like more than one subject in total 55 students like Mathematics.
CONTENT SUMMARY
1. Venn diagrams are great for comparing things in a visual manner and to quickly identify overlaps. They are diagrams containing circles that show the logical relations between a collection of sets or groups
Venn diagrams are used in many areas of life where people need to categorize or group items, as well as compare and contrast different items. Although Venn diagrams are primarily a thinking tool, they can also be used for assessment. However, students must already be familiar with them.
Example: In a room, there are 5 people a, b, c, d, e. Out of them, a, b and c are Males while d and e are Females. Also, a and e study science while b, c and d study English. The set of males is M = {a, b, c} and the set of females is F = {d, e}
2.4. Modelling and solving problems involve Set operations using Venn diagram
Activity 2.4
A survey was carried out in Kigali. 50 people were asked about their preferred hotel for taking lunch among Hilltop, Serena and Lemigo hotels. It was found out that 15 people ate at Hilltop, 30 people ate at Serena, 19 people ate at Lemigo, 8 people ate at Hilltop and Serena, 12 people ate at Hilltop and Lemigo, 7 people ate at Serena and Lemigo. 5 people ate at Hilltop, Serena, and Lemigo.
a. Model the problem using variables and represent the information on a Venn diagram.
b. How many people ate at Hilltop?
c. How many ate at Hilltop and Serena but not at Lemigo?
d. How many people did not eat from any of these three hotels?
CONTENT SUMMARY
Pictorial representations of sets represented by closed figures are called set diagrams or Venn diagrams and they are used to illustrate various operations like union and intersection. A Mathematician John Venn introduced the concept of representing the sets
pictorially by means of closed geometrical figures called Venn diagrams. In Venn diagrams, the Universal Set U is represented by a rectangle and all other sets under consideration by circles within the rectangle. Venn diagrams are useful in solving simple logical problems.
A Venn Diagram is an illustration that shows logical relationships between two or more sets (grouping items). Venn diagram uses circles (both overlapping and non-overlapping) or other shapes. Commonly, Venn diagrams show how given items are similar and different.
Despite Venn diagram with 2 or 3 circles are the most common type, there are also many diagrams with a larger number of circles (5,6,7,8,10…)theoretically, they can have unlimited circles.
Venn Diagram in case of two elements
Where w is the number of elements that belong to none of the sets A, B or C
Tip: Always start filling values in the Venn diagram from the innermost value or intersection part
Example:
150 TTC student-teachers were interviewed. 85 were were registered for a Math class, 70 were were registered for an English class while 50 were registered for both Math and English.
Model this problem using variables and Venn diagram and find out the following:
• How many student-teachers signed up only for Math class?
• How many student-teachers signed up only for English class?
• How many student-teachers signed up for Math or English?
• How many student-teachers signed up for neither Math or English?
Solution
• Let x be the number of student- teachers who signed up for both Math and English.
• The number of student- teachers who signed up only for Math class is 85 – x. Knowing that x = 50, student- teachers who signed up only for Math is 35.
• The number of student- teachers who signed up only for English class is 70 – x. Knowing that x = 50, student- teachers who signed up only for English is 20.
• The number of student- teachers who signed up for Math or English is given by the total number of all student teachers in both sets. This is 35 +50 + 20 = 105
• The number of student- teachers who signed up for neither Math or English is given by the total number of all TTC student- teachers who were interviewed minus the total number of all student teachers in both sets.
This is 150 - 105 = 45
Application activity 2.4
The Venn diagram below shows the number of year two student-teachers in SME who like Mathematics (M), Physics (P) and Kinyarwanda (K). Some like more than one subject in total 55 students like Mathematics.
a. How many student- teachers who like the three subjects?
b. Find the total number of year two student- teachers in SME.
c. How many student-teachers who like Physics and Kinyarwanda only?
2.5. END UNIT ASSESSMENT 2
1. In a class, 15 students play Volleyball, 11play Basketball, 6 play both games and everyone plays at least one of the games. Find the total number of students in the class.
2. Out of 17 teachers in a school, 10 teach Economics and 9 teach Mathematics. The number of teachers who teach both subjects is twice that of those who teach none of the subjects. With the aid of a Venn diagram, find the number that teach:
(a) Both subjects
(b) None of the subject
(c) Only one subject
3. In a class the students are required to take part in at least two sports chosen from football, gymnastics and tennis. 9 students play football and gymnastics; 19 play football and tennis; 6 play all the three sports. If there were 30 students in the class, draw a Venn diagram to show this information. With the help of a Venn diagram, find out how many students did not participate in any of the sports.
4. At a certain school, 100 students were interviewed about the subject they like. 28 students took Physical Education (PE), 31 took Biology (BIO), 42 took English (ENG), 9 took PE and BIO, 10 took PE and ENG, 6 took BIO and ENG, while 4 students took all three subjects.
• Represent the situation in the following Venn diagram.
• Model the problem using variables and find out the following:
a. How many students took none of the three subjects?
b. How many students took PE, but not BIO or ENG?
c. How many students took BIO and PE but not ENG?
UNIT 3: PROBLEMS ON RATIOS AND PROPORTIONS
Key Unit competence: Apply ratios, proportions and multiplier proportion change to solve real life related problems
3.0. Introductory Activity 3
In daily life people compare quantities, share proportionally things or objects for different reasons. Do you ever wonder why ratio and proportion are necessary in daily life situations?
For example, look at your classmates:
a. Find out the number of boys, then the number of girls. Can you express the number of boys or girls in terms of fraction or ratio?
b. Can you equally share a certain number of Mathematics textbooks to different groups in your classroom and then figure out the ratio of Mathematics textbooks per learner?
c. Give examples where the concept of ratio and proportion are used in daily life situations
3.1 Equal and unequal share, Ratio, Direct and indirect proportions.
3.1.1 Equal and unequal share, Ratio and proportion
Activity 3.1.1
1. Suppose that two brothers from your village received 7 000 Frw from their child who lives in the city. The condition to share this amount of money is that for every 2 Frw that the young brother gets, the other one gets 3 Frw. The two brothers have come to you for help after they disagreed on how to share such money.
(i) In what ratio would you share the money between them?
(ii) Tell your partner how you would share the money and how much each would get.
2. Read carefully the following word problem and express the given data into fraction or ratio.
a. John and Lucy partnered to save money for x time and later buy a taxi. For every 800 Frw that John saved, Lucy saved 120 Frw. In what fraction or ratio were their contributions? What other simplest fraction or ratio is same as this?
b. Jane and David sold milk to a vendor in the morning. Jane sold 4 500 ml while David sold 7.5 litres. In what fraction or ratio are their milk sales?
CONTENT SUMMARY
• Equal and unequal share
There are many cases in real life where people or organizations need to share items or resources equally or unequally. For example, a father may want to share 24 acre of land among his two sons. One of them who is disabled gets double of what the other son gets. In such case, the land is unequally shared.
Ratios:
The mathematical term ‘ratio’ defines the relationship between two numbers of the same kind. The relationship between these numbers is expressed in the form “a to b” or more commonly in the form a b
A ratio is used to represent how much of one object or value there is in relation to another object or value.
Example: If there are 10 apples and 5 oranges in a bowl, then the ratio of apples to oranges would be 10 to 5 or 10: 5. This is equivalent to 2:1. In contrast, the ratio of oranges to apples would be 1:2.
Ratios occur in many situations such as in business where people compare profit to loss, in sports where compare wins to losses etc.
Application activity 3.1.1
Ingabire, Mugenzi and Gahima have jointly invested in buying and selling of shares in the Rwanda stock exchange market. In one sale as they invested different amount of money, they realised a gain of 1 080 000 Frw and intend to uniquely share it in the ratio 2:3:4 respectively. How much did Mugenzi get?
3.1.2 Direct and indirect proportions
CONTENT SUMMARY
• Direct proportion
The mathematical term ‘proportional’ describes two quantities which always have the same relative size or ‘ratio’.
Example: An object weighs 2kg on the 1st day. If weight is proportional to age, then the object will weigh 4kg on the 2nd day, 6kg on the 3rd day, 8kg on the 4th day, 10kg on the 5th day and so on.
There are many phenomena in real life that involve a squared relationship.
For example: • The area (A) of a circle varies directly with the square of the radius r
• The value (V) of a diamond varies directly with the square of its mass (M) make the following comparison:
3.2 Calculation of proportional and compound proportional change
Activity 3.2
1. Discuss with your classmate what you understand by the word multiplier.
• Consider a shirt that is sold at a 20% discount.
• What is the percentage of the selling price?
• Convert this percentage you have gotten into fraction. What do you notice?
2. Consider a shirt with a marked price of 500 Frw. After negotiating with the customer, the shirt is sold at a 10% lower. Discuss with your classmate the change in price and the new price (selling price) of the shirt in Frw.
3. Consider that 3 people working at the same rate can cultivate 2 acres of land in 3 days. What do you think will happen if the working days are increased to five and people are still working at the same rate? Discuss.
2. A farmer gets 80 litres of milk from his cow. The amount of milk from the cow reduced by 5% after illness. What is the new amount of milk produced by the cow?
Repeated proportional change is an extremely useful mathematical process because it can be used to calculate real world financial problems such as compound interest.
‘Compound interest’ refers to the interest added to a deposit or loan. The added interest will also earn interest as time passes. To calculate the compound interest of a loan, we may use repeated proportional change.
Example
(a) £500 is borrowed for 6 years at 5 % compound interest. Calculate the amount of compound interest which will be paid
Solution
(a) From the question, we know that 5% compound interest is added each year. This means that there will be 105% of the original amount borrowed at the end of the first year or 1.05.
With 1.05 as your multiplier, you can calculate the total amount of money borrowed after 6 years. The money was borrowed for 6 years, so you must raise 1.05 to the power of 6. Therefore:
Total amount of money borrowed is 500 x (1.05) 6 = 670.047= £670
The question has asked you to calculate the amount of compound interest. To do so, you must subtract the original amount borrowed (£500) from the value you have just generated:
670 - 500 = 170. As a result, the amount of compound interest which will be paid is £170
To calculate compound proportionality problems, one can use the unitary method or compound rule of three
Example: 9 men working in a factory produce a certain number of pans in 6 working days. How long will it take 12 men to produce the same number of pans if they work at the same rate?
Application activity 3.2
1. What is the multiplier of 45% decrease?
2. Deborah’s salary last year was 15 000 Frw. This year it was increased by 20%. What is her salary this year?
3. In 2004 a company processed 800 tonnes of maize. In 2005, the company decreased production by 30 %. How many tonnes did the company process in 2005?
4. Four men working at the same rate can dig a piece of land in ten days. How long would it take five men to do the same job?
3.3 Problems involving direct and indirect proportions
Activity 3.3
Read carefully the given problems and discuss how can you solve problems involving direct and indirect proportions.
1. F is directly proportional to x. When F is 6, x is 4. Find the value of F when x is 5.
2. A is directly proportional to the square of B. When A is 10, B is 2. Find the value of A when B is 3.
3. A is inversely proportional to B. When A is 10, B is 2. Find the value of A when B is 8
4. F is inversely proportional to the square of x. When F is 20, X is 3. Find the value of F when x is 5.
CONTENT SUMMARY
Two values x and y are directly proportional to each other when the ratio x : y or x y ∝ is a constant (i.e. always remains the same). This would mean that x and y will either increase together or decrease together by an amount that would not change the ratio.
Knowing that the ratio does not change allows you to form an equation to find the value of an unknown variable.
Example:
If two pencils cost $1.50, how many pencils can you buy with $9.00?
Problem solving plan
1. Understand the problem: Think what information is given and what information is required
2. Decide on a strategy: List the strategies with which you think the solution can be found
3. Apply the strategy: Find the solution using the strategy you have chosen
4. Look back:
• Have you verified your solution?
• Are there other solutions?
• Can you solve a simpler problem?
• Have you answered the question as it was initially stated?
Example
1. The voltage V(volts) across an electrical circuit is directly proportional to the current I (Amperes) flowing through the circuit. When I=1.2 A, V=78V
a) Express V in terms of I
b) Find V when I=2A
c) Find I when V=162.5V
UNIT 4 :PROPOSITIONAL AND PREDICATE LOGIC
Key Unit competence: Use Mathematical logic as a tool of reasoning and argumentation in daily situation
4.1 Definition
4.1.1 Simple statement and compound statements
true or false, but never both, and to always have the same truth value. The two truth values of proposition are true and false and are denoted by the symbols T and F respectively. Occasionally they are also denoted by the symbols 1 and 0 respectively.
Application activity 4.1.1
1. Find out which of the following sentences are statements and which are not. Justify your answer.
a) Uganda is a member of East African Community.
b) The sun is shining.
c) Come to class!
d) The sum of two prime numbers is even.
e) It is not true that China is in Europe.
f) May God bless you!
2. Write down the truth value (T or F) of the following statements
a) Paris is in Italy.
b) 13 is a prime number.
c) Kigeri IV Rwabugiri was the king of the Kingdom of Rwanda
d) Lesotho is a state of South Africa.
e) Tanzania is in east of Rwanda and is in SADC (Southern African Development Community)
4.1.2 Truth tables
Activity 4.1.2
Suppose we are given two simple statements, named p and q to get a compound statement C pq ( , ).
a) What are the possibilities for the truth-values of p and of q?
b) Using a table,
i) How many possibilities are there, for their pairs of truth-values?
ii) How many possibilities are there, for the triples of truth-values of three statements p, q and r?
The way we will define compound statements is to list all the possible combinations of the truth-values (abbreviated T and F) of the simple statements (that are being combined into a compound statement) in a table, called a truth table. The name of each statement is at the top of a column of the table. The truth values used to define the truth-value of the compound statement appear in the last column.
Application activity 4.1.2
Write down the truth table for three propositions p, q and r
4.2 Logical connectives
4.2.1 Negation
Activity 4.2.1
Let p, q, r, s be the propositions
1. It is raining
2. Uganda is African country
3. London is in France
4. Kagabo reads Newsweek
Give a verbal sentence which describes the opposite proposition
Application activity 4.2.1
1. Write the negation of each of the following statements.
a) Today is raining.
b) Sky is blue
c) My native country is Rwanda.
d) Benimana is smart and healthy.
4.2.2 Conjunction
4.2.3 Disjunction
Determine the truth value of each of the following statements
1. Paris is in France or 4 + 4= 8
2. The sun is a planet or the Jupiter is a star.
3. Paris is in Japan or 3 +4 =7
4. The first president of United States of America is George Washington or was inaugurated in 1879.
5. Nairobi is the capital of Tanzania or1+1 =2
6. The French revolution started in 1789 or ended in 1799
4.2.4 Conditional statement
Activity 4.2.4
Rephrase the following statements without changing the meaning
1. If you buy me a pen, I will go to school
2. If the earth is flat, then mars is flat
3. If you are tall, then you will be a member of our volleyball team
4. If you do not buy these shoes, then I will not go with you
5. If you do not pay school fees, then you will not get you school report
Application activity 4.2.4
1. Using the statements p :Mico is fat and p :Mico is happy
Assuming that “not fat” is thin, write the following statements in symbolic form
a) If Mico is fat then she is happy
b) Mico is unhappy implies that Mico is thin
2. Write the following statements in symbolic form and their truth table
a) If n is prime, then n is odd or n is 2.
b) If x is nonnegative, then x is positive or x is 0.
c) If Tom is Ann’s father, then Jim is her uncle and Sue is her aunt.
4.2.5 Bi-conditional statements
4.2.6 Tautologies and Contradictions
CONTENT SUMMARY
A tautology is a compound statement that is always true regardless of the truth values of the individual statements substituted for its statement variables.
Example
• The statement “The main idea behind data compression is to compress data” is a tautology since it is always true.
• The statement “I will either get paid or not get paid” is a tautology since it is always true
If you are given a statement and want to determine if it is a tautology, then all you need to do is construct a truth table for the statement and look at the truth values in the final column. If all of the values are T (for true), then the statement is a tautology.
The statement “I will either get paid or not get paid” is a tautology since it is always true. We can use p to represent the statement “I will get paid” and not p (written ¬p) to represent “I will not get paid.”
4.3 Quantifiers and their negations: Existential and Universal quantifiers
4.3.1 Predicates
4.3.2 Quantifiers
4.3.3 Negation of quantifiers
Activity 4.3.3
Negate the following statements
1. All grapefruit are pink.
2. Some celebrities are modest.
3. No one weighs more than two thousand pounds.
4. Some people are more than ten feet tall.
5. All snakes are poisonous.
6. Some whales can stay under water for two days without surfacing for air.
7. All birds can fly
Application activity 4.3.3
Negate each of the following statements and write the answer in symbolic form
1. Some students are math majors
2. Every real number is positive, negative or zero
3. Every good boy does fine
4. There is a broken desk in our classroom
5. Lockers must be turned in by the last day of class
6. Haste makes waste
4.4 END UNIT ASSESSMENT 4
UNIT 5: OPERATION ON POLYNOMIALS
Key Unit competence: Perform operations, on polynomials and solve related problems.
5.1 Defining and comparing polynomials
5.2. Operations on polynomials
5.3 Factorization of polynomials
5.4 Expansion of polynomials
UNIT 6:LINEAR AND QUADRATIC EQUATION AND INEQUALITIES
Key Unit competence: Solve algebraically or graphically linear, quadratic equations or inequalities
6.1 Linear and quadratic equations
6.1.1. linear equations
6.1.2. Quadratic equations
When completing the square to solve quadratic equation, remember that you must preserve the equality. When you add a constant to one side of the equation, be sure to add the same constant to the other side of equation.
6.2 Equations reducible to quadratic
6.3.Linear and quadratic inequalities
6.3.1. Linear inequalities
6.3.3. Quadratic inequalities
6.4 Solving word problems involving linear or quadratic equations
Activity 6.4
1. Make a research on internet and by means of examples prepare a short presentation on how linear equations and quadratic equations can be used in daily life.
2. Read carefully the following word problems and model or rewrite them using a mathematical statement.
• Six less than two times a number is equal to nine
• Jane paid 22100 Frw for shoes and clothes. She paid 2100 Frw more for clothes than she did for shoes. How much did Jane pay for shoes?
3. A manufacturer develops a formula to determine the demand for its product depending on the price in Rwandan franc. The formula is , where is the price per unit, and is the number of units in demand. At what price will the demand drop to 1000 units?
CONTENT SUMMARY
It is not always easy to tell what kind of equation a word problem involves, until you start translating it to math symbols.
Words like “together,” “altogether,” or “combined” often indicate that the problem involves addition. The word “left,” as in “he had xx amount left,» often indicates subtraction.
The following table explains the keywords used when writing equations from verbal models.
6.5 Solving and discussing parametric equations
Application activity 6.5
UNIT 7:PROBLEMS ON POWERS, INDICES, RADICALS AND LOGALITHMS
Key Unit competence: Solve problems related to powers, indices, radical and common logarithms
7.0. Introductory Activity
7.1 Powers and radicals
7.1.1 Definition of powers/ indices and radicals
7.1.2 Properties of indices and radicals
7.2 Operations on indices and radicals
7.3. Decimal logarithm
7.3.1 Definition
Activity 7.3.1
What is the real number for which 10 must be raised to obtain:
1) 1 2) 10
3) 100 4) 1000
5) 10000 6) 100000
d. Logarithms are used to determine growth decay
Example:
7.4. END UNIT ASSESSMENT 7
UNIT: 8 :PARAMETERS OF CENTRAL TENDENCES AND DISPERSION
Key Unit competence: Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to the standard deviation
8.0. Introductory Activity 8
1. During 6 consecutive days, a fruit-seller has recorded the number of fruits sold per type
Which type of fruits had the highest number of fruits sold?
b) Which type of fruits had the least number of fruits sold?
c) What was the total number of fruits sold that week?
d) Find out the average number of fruits sold per day.
2. During the welcome test of Mathematics out of 10 , 10 student-teachers of year one ECLPE scored the following marks: 3, 5,6,3,8,7,8,4,8 and 6.
a) Determine the mean mark of the class.
b) What is the mark that was obtained by many students?
c) Compare and discuss about the mean mark of the class and the mark for every student-teacher. What advice could you give to the Mathematics tutor?
d) organize all marks in a table and try to present them in an X-Y Cartesian plane by making x-axis the number of marks and y-axis the number of student-teachers.
8.1 Collection and presentation of grouped and ungrouped data
8.1.1. Collection and presentation of ungrouped data
Activity 8.1.1
1. Observe the information provided by the graph and complete the table by corresponding the number of learners and their age
CONTENT SUMMARY
Every day, people come across a wide variety of information in form of facts or categorical data, numerical data in form of tables.
For example, information related to profit/ loss of the school, attendance of students and tutors, used materials, school expenditure in term or year, student-teachers’ results. These categorical or numerical data which is numerical or otherwise, collected with a definite purpose is called data.
Statistics is the branch of mathematics that deals with the collection, presentation, interpretation and analysis of data.
A sequence of observations, made on a set of objects included in the sample drawn from population, is known as statistical data.
Statistical data can be organized and presented in different forms such as raw tables, frequency distribution tables, graphs, etc.
Qualitative data
Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description. In statistics, it is often used interchangeably with “categorical” data. Categorical data represent characteristics such as a person’s gender, marital status, hometown, or the types of movies they like. Categorical data can take on numerical values (such as “1” indicating male and “2” indicating female), but those numbers don’t have mathematical meaning. It couldn’t add them together, for example.
Quantitative data
Quantitative data is a numerical measurement expressed not by means of a natural language description, but rather in terms of numbers. These data have meaning as a measurement, such as a person’s height, weight, IQ, or blood pressure; or they’re a count, such as the number of stock shares a person owns, or how many pages you can read of your favorite book before you fall asleep.
Numerical data can be further broken into two types: discrete and continuous.
• Discrete data represent items that can be counted; they take on possible values that can be listed out. The list of possible values may be fixed (also called finite); or it may go from 0, 1, 2, on to infinity (making it countably infinite). For example, the number of heads in 100 coin flips takes on values from 0 through 100 (finite case), but the number of flips needed to get 100 heads takes on values from 100 (the fastest scenario) up to infinity (if you never get to that 100th heads). Its possible values are listed as 100, 101, 102, 103, . . . (representing the countably infinite case).
• Continuous data represent measurements; their possible values cannot be counted and can only be described using intervals on the real number line. For example, the exact amount of gas purchased at the pump for cars with 20-gallon tanks would be continuous data from 0 gallons to 20 gallons, represented by the interval [0, 20], inclusive. You might pump 8.40 gallons, or 8.41, or 8.414863 gallons, or any possible number from 0 to 20. In this way, continuous data can be thought of as being uncountably infinite.
After the collection of data, tally is used to organize and present in a frequency distribution table. Tally means to count by grouping the number of times an item has occurred. When the data are arranged in this way we say that we have obtained
the frequency distribution.
Raw data
Data which have been arranged in a systematic order are called raw data or ungrouped data.
For Example, the following are marks out of 20 for 12 student-teachers.
13 10 15 17 17 18
17 17 11 10 17 10
Frequency distribution
A frequency distribution is a table showing how often each value (or set of values) of the collected data occurs in a data set. A frequency table is used to summarize categorical or numerical data. Data presented in the form of a frequency distribution are called grouped data.
Example
The following data of marks, out of 20, obtained by 12 student-teachers can be presented in a frequency distribution table.
13 10 15 17 17 18
17 17 11 10 17 10
The set of outcomes is displayed in a frequency table, as illustrated below:
Cumulative frequency
The cumulative frequency corresponding to a particular value is the sum of all frequencies up to the last value including the first value. Cumulative frequency can also defined as the sum of all previous frequencies up to the current point.
Example:
The set of data below shows marks obtained by student-teachers in Mathematics. Draw a cumulative table for the data.
Solution:
The cumulative frequency at a certain point is found by adding the frequency at the present point to the cumulative frequency of the previous point.
STEM AND LEAF DISPLAYS
Is a plot where each data value is split into a leaf usually the last digit and a stem the other digit. The stem values are listed down, and the leaf values are listed next to them. This way the stem groups the scores and each leaf indicates a score within that group
Example: The mathematical competence scores of 10 student-teachers participating in mathematics competition are as follows:15,16,21,23,23,26,26,30,32,41. Construct a stem and leaf display for these data by using1, 2, 3, and 4 as your stems.
This means that data are concentrated in twenties.
Example: The following are results obtained by student-teachers in French out of 50.
37, 33, 33, 32, 29, 28, 28, 23, 22, 22, 22, 21, 21, 21, 20, 20, 19, 19, 18, 18, 18, 18, 16, 15, 14, 14, 14, 12, 12, 9, 6
Use stem and leaf to display data
Solution:
Numbers 3, 2, 1, and 0, arranged as a stems to the left of the bars. The other numbers come in the leaf part.
3|2337
2|001112223889
1|2244456888899
0|69
From this, data are concentrated in ones
Application activity 8.1.1
1. At the beginning of the school year, a Mathematics test was administered in year 1 to 50 student-teachers to test their level of understanding. Their results out of 20 were recorded as follows:
Construct a frequency distribution table to help the tutor and the school administration to easily recognize the level of understanding for the Year 1 student-teachers.
2. Differentiate qualitative and quantitative data from the list below: Product rating, basketball team classification, number of student-teachers in the classroom, weight, age, Number of rooms in a house, number of tutors in school.
8.1.2. Collection and presentation of grouped data
Activity 8.1.2
The mass of 50 tomatoes (measured to the nearest g) were measured and recorded in the table below
Construct a frequency distribution table, using equal class intervals of width 5g and taking the lower class boundary of the first interval as 84.5g.
CONTENT SUMMARY
When the range of data is large, the data must be grouped into classes that are more than one unit in width. In this case a grouped frequency distribution is used. Data in this case are grouped in a frequency distribution using groups or classes.
• Class limits: The class limits are the lower and upper values of the class
• Lower class limit: Lower class limit represents the smallest data value that can be included in the class.
• Upper class limit: Upper class limit represents the largest data value that can be included in the class.
• Class boundaries: Class boundaries are the midpoints between the upper class limit of a class and the lower class limit of the next class. Therefore, each class has a lower and an upper class boundary.
Example: The following data represent the marks obtained by 40 students in Mathematics test. Organize the data in the frequency table; grouping the values into classes, starting from 41-50
54 83 67 71 80 65 70 73 45 60 72 82 79 78 65 54 67 64 54 76 45 63 49 52
60 70 81 67 45 58 69 53 65 43 55 68 49 61 75 52
Solution:
Example
Using the frequency table above, determine the class boundaries of the three classes.
Solution
For the first class, 41-50
The lower class boundary is the midpoint between 40 and 41, that is 40.5
The upper class boundary is the midpoint between 50 and 51, that is 50.5
For the second class, 51-60
The lower class boundary is the midpoint between 50 and 51, that is 50.5
The upper class boundary is the midpoint between 60 and 61, that is 60.5
For the third class, 61-70
The lower class boundary is the midpoint between 60 and 61, that is 60.5
The upper class boundary is the midpoint between 70 and 71, that is 70.5
Class width
Class width is the difference between the upper class boundary and lower class boundary
Application activity 8.1.2
Suppose a researcher wished to do a study on the distance (in kilometres) that the employees of a certain department store travelled to work each day. The researcher collected the data by asking each employee the approximate distance is the store from his or her home. Data are collected in original form called raw data as follow:
Since little information can be obtained from looking at raw data, help the researcher to make a frequency distribution table (use 1-3 km and 16-18 km as class limits) so that some general observations can easily be done by the researcher.
8.2 Measure of central tendencies: Mean, median and mode
Activity 8.2
1. Observe the following marks of 10 students in mathematics test: 5, 4, 10, 3, 3, 4, 7, 4, 6, 5 and complete the table below
Calculate the mean, median, mode and range of this set
2. In three classes of Year three in TTC, during the quiz of mathematics out of 5 marks, 100 student-teachers obtain marks as shown in the table below:
• What is the marks obtained by most of the students?
• You are asked to calculate the mean mark for the class. Explain how you should find it.
• What is the marks obtained by most of the students?
• You are asked to calculate the mean mark for the class. Explain how you should find it.
CONTENT SUMMARY
For the comparison of one frequency distribution with another, it requires to summarize it in such a manner that its essence is expressed in few figures only. For this purpose, we find most representative value of data. This representative value of the data is known as the measure of central tendency or averages.
a) Mean, mode , median and range of ungrouped data
Thus for any particular set of ungrouped data, it is possible to select some typical values or average to represent or describe the entire set such a descriptive value is referred to as a measure of central tendency or location such as mean, mode, median
The median:
The data is arranged in order from the smallest to the largest, the middle number is then selected. This really the central number of the range and is called the median.
Mean, mode, median and range of grouped data
For any particular set of grouped data, it is possible to select some typical values or average to represent or describe the entire set such a descriptive value is referred to as a measure of central tendency or location such as mean, mode, median and range.
Range
In the case of grouped data the range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class.
Application activity 8.2
1. A group of student-teachers from SME were asked how many books they had read in previous year; the results are shown in the frequency table below. Calculate the mean, median and mode number of books read.
2. During oral presentation of internship report for year three studentteachers the first 10 student-teachers scored the following marks out of 10:
8, 7, 9, 10, 8, 9, 8, 6, 7 and 10
a) Calculate the mean of the group
b) Calculate the median and Mode
8.3 Graphical representation of grouped and ungrouped data
Activity 8.3
1. In a class of year 1, student-teachers are requested to provide their sizes in order to be given their sweaters. The graph below shows the sizes of sweaters for 30 students. Observe and interpret it by finding out the number of students with small, medium, large, extra-large. Guess the name of the graph.
2. The mass of 50 tomatoes (measured to the nearest g) were measured and recorded in the table below.
a) Determine cumulative frequency
b) Try to draw a histogram, the frequency polygon and the cumulative frequency polygon to illustrate the data.
3. A pie chart shows the mass (in tonnes) of each type of food eaten in a certain month. Observe the pie chart and make a corresponding frequency distribution table by considering the number of tones for each type of food as frequency.
CONTENT SUMMARY
After the data have been organized into a frequency distribution, they can be presented in a graphical form. The purpose of graphs in statistics is to convey the data to the viewers in a pictorial form. It is easier for most people to comprehend the
meaning of data presented graphically than data presented numerically in tables or frequency distributions.
The most commonly used graphs are: Bar graph, Pie chart, Histogram, Frequency polygon, Cumulative frequency graph or Ogive.
A bar graph
A bar chart or bar graph is a chart or graph that presents numerical data with rectangular bars with heights or lengths proportional to the values that they represent.
The bars can be plotted vertically or horizontally. A vertical bar chart is sometimes called a line graph
Pie chart
A pie chart is used to display a set of categorical data. It is a circle, which is divided into segments. Each segment represents a particular category. The area of each segment is proportional to the number of cases in that category.
Example:
A TTC tutor conducted a survey to check the level of how student-teachers like some subjects (English, Mathematics, French, Entrepreneurship, Kinyarwanda) taught in SME option. The survey was done on 60 student-teachers and the table below summarizes the results.
In order to clearly present his/ her findings, the tutor presented the data on a bar graph and then on a pie chart as follows:
1. Bar graph
Histogram
Histogram is a statistical graph showing frequency of data. The horizontal axis details the class boundaries, and the vertical axis represents the frequency. Blocks are drawn such that their areas (rather than their height, as in a bar chart) are proportional to the frequencies within a class or across several class boundaries. There are no spaces between blocks.
Example: Draw a histogram for the frequency distribution given in the table below.
Step 2: Draw Histogram (frequency against class boundaries)
Frequency polygon
In a frequency polygon, a line is drawn by joining all the midpoints of the top of the bars of a histogram.
A frequency polygon gives the idea about the shape of the data distribution. The two end points of a frequency polygon always lie on the x-axis.
Example
Use the histogram above to draw a frequency polygon.
Solution
Example:
Construct a histogram, a frequency polygon, a cumulative frequency graph or ogive to represent the data shown below for the record high temperatures for each of the 50 states
Application activity 8.3
1. At the beginning of the school year, a Mathematics test was administered in year 1 for 50 student-teachers to test their levels in this subject.. Their results out of 20 were recorded as follow:
Present the data using a bar graph to help the tutor and the school administration to easily recognize the level of understanding for the Year 1 student-teachers.
2. The cumulative frequency distribution table below shows distances (in km) covered by 20 runners during the week of competition.
a) Construct a histogram
b) Construct a frequency polygon
c) Construct an ogive.
8.4 Measure of dispersion: Range, variance, Standard Deviation and coefficient of variation
Activity 8.4
CONTENT SUMMARY
The word dispersion has a technical meaning in statistics. The average measures the center of the data. It is one aspect of observations. Another feature of the observations is how the observations are spread about the center. The observation may be close to the center or they may be spread away from the center. If the observations are close to the center (usually the arithmetic mean or median), we say that dispersion, scatter or variation is small. If the observations are spread away from the center, we say that dispersion is large.
The study of dispersion is very important in statistical data. If in a certain factory there is consistence in the wages of workers, the workers will be satisfied. But if some workers have high wages and some have low wages, there will be unrest among the low paid workers and they might go on strikes and arrange demonstrations. If in a certain country some people are very poor and some are very high rich, we say there is economic disparity. It means that dispersion is large.
The extent or degree in which data tend to spread around an average is also called the dispersion or variation. Measures of dispersion help us in studying the extent to which observations are scattered around the average or central value. Such measures are helpful in comparing two or more sets of data with regard to their variability.
Properties of a good measure of dispersion
i. It should be simple to calculate and easy to understand
ii. It should be rigidly defined
iii. Its computation be based on all the observations
iv. It should be amenable to further algebraic treatment
Some measures of dispersion are Quartiles, variance, Range, standard deviation, coefficient of variation
Variance
Variance measures how far a set of numbers is spread out. A variance of zero indicates that all the values are identical. Variance is always non-negative: a small variance indicates that the data points tend to be very close to the mean and hence to each other, while a high variance indicates that the data points are very spread out around the mean and from each other.
The variance is denoted and defined by
Example 1
Calculate the variance of the following distribution: 9, 3, 8, 8, 9, 8, 9, 18
STANDARD DEVIATION
The standard deviation has the same dimension as the data, and hence is comparable to deviations from the mean. We define the standard deviation to be the square root of the variance. thus, the standard deviation is denoted and defined by
Example 1
The six runners in a 200 meter race clocked times (in seconds) of 24.2, 23.7, 25.0, 23.7, 24.0,4.6
a) Find the mean and standard deviation of these times.
b) These readings were found to be 10% too low due to faulty timekeeping.
Write down the new mean and standard deviation.
Example 2
The heights (in meters) of six children are 1.42, 1.35, 1.37, 1.50, 1.38 and 1.30.
Calculate the mean height and the standard deviation of the heights
Coefficient of variation
The coefficient of variation measures variability in relation to the mean (or average) and is used to compare the relative dispersion in one type of data with the relative dispersion in another type of data. It allows us to compare the dispersions of two different distributions if their means are positive. The greater dispersion corresponds to the value of the coefficient of greater variation.
The coefficient of variation is a calculation built on other calculations: the standard deviation and the mean as follows:
Example:
One data series has a mean of 140 and standard deviation 28.28. The second data series has a mean of 150 and standard deviation 24. Which of the two has a greater dispersion?
Solution:
Range
In the case of ungrouped data, the range of a set of observations is the difference in values between the largest and the smallest observations in the set. In the case of grouped data the range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class.
Example
Calculate the range of the following set of the data set: 1, 3, 4, 5, 5, 6, 9, 14 and 21
Solution: From the given series the lowest data is 1 and the highest data is 21
The Range = highest value −lowest value
Range = −= 21 1 20
Application activity 8.4
1. Out of 4 observations done by tutor of English, arranged in descending order, the 5th, 7th, 8th and 10th observations are respectively 89, 64, 60 and 49.
Calculate the median of all the 4 observations.
2. In the following statistical series, calculate the standard deviation of the following set of data 56,54,55,59,58,57,55
3. In the classroom of SME the first five student-teachers scored the following marks out of 10 in a quiz of Mathematics
5, 6, 5, 2, 4, 7, 8, 9, 7, 5
a) Calculate the mean, median and the modal mark
b) Determine the quartiles and interquartile range
c) Calculate the variance and the standard deviation
d) Determine the coefficient of variation
8.5 Application of statistics in daily life
Activity 8.5
Using internet or reference books from the school library, make a research to provide in written form at least one example of where the following statistical terminologies are needed and used in real life situations:
• Frequency distribution
• Statistical graphs like Bar graph, Histogram, Frequency polygon Mean,
• Variance
Make a presentation of your findings to the whole class
CONTENT SUMMARY
Statistics is concerned with scientific method for collecting and presenting, organizing and summarizing and analyzing data as well as deriving valid conclusions and making reasonable decisions on the basis of this analysis
Today, statistics is widely employed in government, business and natural and social sciences. Statistical methods are applied in all field that involve decision making.
For example, in agriculture, statistics can help to ensure the amount of crops are grown this year in comparison to previous year or in comparison to required amount of crop for the country. It may also be helpful to analyze the quantity and size of grains grown due to use of different fertilizer.
In education, statistics can be used to analyze and decide on the money spend on girls education in comparison to boys education.
Nowadays, graphs are used almost everywhere. In stock market, graphs are used to determine the profit margins of stock. There is always a graph showing how prices have changed over time, food price, budget forecasts, exchange rates…
Mean is used as one of comparing properties of statistics. It is defined as the average of all the clarifications. Mean helps Teachers to see the average marks of the students.
A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean. Standard deviation is often used to compare real-world data against a model to test the model
In finance, standard deviation is often used as a measure of the risk associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets, Standard deviation provides a quantified estimate of the uncertainty of future returns.
Application activity 8.5
1. Using internet or reference books from the school library, make a research to provide in written form at least one example of where the following statistical terminologies are needed and used in real life situations:
• Frequency distribution
• Statistical graphs like Pie chart, cumulative frequency polygon or ogive
• Mode and median
• Standard deviation
Make a presentation of your findings to the whole class
2. Collect data on 20 students’ heights in your class and organize them in a frequency distribution. Calculate the mean of 20 students’ heights, variance as well as standard deviation. What conclusion can you make based on the calculated mean, variance and standard deviation?
8.6. END UNIT ASSESSMENT 8
1. Use the graph below to answer the questions that follows
a) Use the graph to estimate the mode.
b) State the range of the distribution.
c) Draw a frequency distribution table from the graph
2. In test of mathematics, 10 student-teachers got the following marks:
6, 7, 8, 5, 7, 6, 6, 9, 4, 8
a) Calculate the mean, mode, quartiles and interquartile range
b) Calculate the variance and standard deviation
c) Calculate the coefficient of variation
3. A survey taken in a restaurant shows the following number of cups of coffee consumed with each meal. Construct an ungrouped frequency distribution. 0 2 2 1 1 2 3 5 3 2 2 2 1 0 1 2 4 4 0 1 0 1 4 4 2 2 0 1 1 5
4. The amount of protein (in grams) for a variety of fast-food sandwiches is reported here. Construct a frequency distribution using six classes. Draw a histogram, frequency polygon, and ogive for the data, using relative frequencies. Describe the shape of the histogram.
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