1. Module :MA0001NI Foundation Mathematics Lecture Week 1

2. Agenda Module Introduction Your Lecturer and tutors Module Objective
Your Module Leader
Your Lecturer and tutors
Module Objective
Tutorial/ workshop
Module outcomes after successful completion
Module Assessments and Syllabus Summary
Recommended booklist

3. Module Leaders’ Roles Every module has two module leaders:
Creates the main lecture/tutorial notes
Writes coursework and examinations
Moderates the coursework and examinations results
Serves as a lecturer for module (usually in London)
Writes localised lecture/tutorial notes
Marks the coursework and examinations (lecturers/tutors might also be involved in marking)
Serves as a lecturer for that module

4. Your Module Leaders Are…
Mr.David Brown
(LondoN Metropolitan University)
Mr.Ashok Dhungana
(Islington College
Lecturer)

5. Your Lecturer/Tutor Mr. Ashok Dhungana (MSc IT, TU, Nepal)
Phone # 977 (1) ext. 26
977(1) ext. 26
Mr. Ashok Dhungana
(MSc IT, TU, Nepal)

6. Module Assessments Assessments: 40% CW1(Course Work ) 60% CW2 (EXAM )
Note:- Students should obtain 40% pass on aggregate from examination and coursework.

7. Syllabus
Number Fractions, decimals, percentages, ratio, proportion, scientific notation, estimation, calculator use
Basic Algebra Algebraic notation, manipulation of algebraic expressions. Transposition and evaluation of algebraic formulae. Formulation of problems in algebraic form. Solution of linear equations, simultaneous equations and quadratic equations.
Graphs Plotting linear and non-linear graphs. Gradient and intercept.
Indices and Logarithms Simple indices, Exponentials and Logarithms

8. Integers Classifying numbers Calculating with integers
Multiples, factors and primes
Prime factor decomposition
LCM and HCF

9. Classifying numbers Natural numbers
Positive whole numbers 0, 1, 2, 3, 4 …
Integers
Positive and negative whole numbers … –3, –2, 1, 0, 1, 2, 3, …
Rational numbers
Numbers that can be expressed in the form n/m, where n and m are integers. All fractions and all terminating and recurring decimals are rational numbers, for example, ¾, –0.63, 0.2.
4/21/2017

10. Classifying numbers Irrational numbers
Numbers that cannot be expressed in the form n/m, where n and m are integers. Examples of irrational numbers are  and 2.
Even numbers are numbers that are exactly divisible by 2.
The nth even number can be written as E(n) =2n.
4/21/2017

11. Classifying numbers
Odd numbers leave a remainder of 1 when divided by 2.
The nth odd number can be written as U(n) = 2n –1
Triangular numbers are numbers that can be written as the sum of consecutive whole numbers starting with 1.
For example, 15 is a triangular number. It can be written as 15 =
4/21/2017

12. Classifying numbers So, for any triangular number T(n)
T(n) =n(n + 1)/ 2
We can now use this rule to find the value of the 50th triangular number.
T(50) =50(50 + 1)/2
T(50) = 1275
T(100)=???
4/21/2017

13. N1.2 Calculating with integers
Contents
Integers A
N1.1 Classifying numbers A
N1.2 Calculating with integers A
N1.3 Multiples, factors and primes A
N1.4 Prime factor decomposition A
N1.5 LCM and HCF

14. Adding integers
We can use a number line to help us add positive and negative integers.
–2 + 5 =
= 3 -2 3
How can we use the number line to work out –2 + 5?
Start at –2 (click to highlight the –2 on the number line) and count forwards 5.
On a number line we move to the right for forwards (in a positive direction) and to the left for backwards (in a negative direction).
Explain that when we add or subtract integers the signs tell us whether we move up or down the number line. Unlike multiplication and division, they do not generally tell us what sign the answer will have. This depends on the starting point and the size of the numbers. It is important to stress this fact because when pupils learn the rules for multiplying and dividing negative numbers they often confuse these rules with the rules for addition and subtraction.
To add a positive integer we move forwards up the number line.

15. Subtracting integers 5 – 8 = = –3
We can use a number line to help us subtract positive and negative integers.
5 – 8 =
= –3 -3 5
How can we use the number line to work out 5 – 8?
Explain that this could also be written as 5 – +8 (five minus positive eight), but that we don’t usually need to write + in front of a number to show that it is positive.
In this example, we start at 5 (click to make the 5 orange) and count backwards 8.
To subtract a positive integer we move backwards down the number line.
5 – 8
is the same as
5 – +8

16. Adding and subtracting integers
To add a positive integer we move forwards up the number line.
To add a negative integer we move backwards down the number line.
a + –b is the same as a – b.
To subtract a positive integer we move backwards down the number line.
To subtract a negative integer we move forwards up the number line.
Stress that adding a negative integer is the same as subtracting and that subtracting a negative integer is the same as adding.
Unlike multiplying and dividing integers, when adding and subtracting integers the answer is positive or negative depending on the start number and how much we move up or down the number line.
a – –b is the same as a + b.

17. Rules for multiplying and dividing
When multiplying negative numbers remember: + × = – + × = – + × = – + × =
Dividing is the inverse operation to multiplying.
When we are dividing negative numbers similar rules apply:
These rules have been drawn graphically to make it easier for pupils to spot the pattern.
As each rule appear read it as, for example,
A positive number multiplied by a positive number always equals a positive number.
Remind pupils of the meaning of ‘inverse operation’ – one ‘undoes’ the other.
For example, if 4 × –3 = –12, then –12 ÷ –3 must equal 4.
Tell pupils that easiest way to remember these rules is that when we multiply together (or divide) two numbers with different signs (a positive number times a negative number or a negative number times a positive number) the answer will always be negative. If we multiply together (or divide) two numbers with a different sign (a positive number times a positive number or a negative number times a negative number) the answer will always be positive.
Encourage pupils to first work out whether their answers will be positive or negative and then multiply or divide.
Ask pupils to write down rules for multiplying (or dividing) three numbers.
For example, negative × positive × negative = positive and negative × negative × negative = negative. + ÷ = – + ÷ = – + ÷ = – + ÷ =

18. N1.3 Multiples, factors and primes
Contents
Integers A
N1.1 Classifying numbers A
N1.2 Calculating with integers A
N1.3 Multiples, factors and primes A
N1.4 Prime factor decomposition A
N1.5 LCM and HCF

19. Multiples
A multiple of a number is found by multiplying the number by any whole number.
What are the first six multiples of 7?
To find the first six multiples of 7 multiply 7 by 1, 2, 3, 4, 5 and 6 in turn to get: 7,
14,
21,
28, 35
and 42.
Discuss the fact that any given number has infinitely many multiples.
We can check whether a number is a multiple of another number by using divisibility tests.
Any given number has infinitely many multiples.

20. Factors
A factor (or divisor) of a number is a whole number that divides into it exactly.
Factors come in pairs. For example,
What are the factors of 30?
1 and
30,
2 and
15,
3 and
10,
5 and 6.
So, in order, the factors of 30 are:
Ask pupils to tell you what a factor is and reveal the definition on the board.
Remind pupils that factors always go in pairs (in the example of rectangular arrangements these are given by the length and the width of the rectangle). The pairs multiply together to give the number.
Ask pupils if numbers always have an even number of factors. They may argue that they will because factors can always be written in pairs. Establish, however, that when a number is multiplied by itself the numbers in that factor pair are repeated. That number will therefore have an odd number of factors.
Pupils could investigate this individually.
Establish that it follows that if a number has an odd number of factors it must be a square number. 1, 2, 3, 5, 6,
10, 15
and 30.

21. Prime numbers
If a whole number has two, and only two, factors it is called a prime number.
For example, the number 17 has only two factors, 1 and 17.
Therefore, 17 is a prime number.
The number 1 has only one factor, 1.
Therefore, 1 is not a prime number.
Establish the 2 is the only even prime number.
Ask pupils to name all the prime numbers less than 20.
There is only one even prime number. What is it?
2 is the only even prime number.

22. N1.4 Prime factor decomposition
Contents
Integers A
N1.1 Classifying numbers A
N1.2 Calculating with integers A
N1.3 Multiples, factors and primes A
N1.4 Prime factor decomposition A
N1.5 LCM and HCF

23. Prime factors A prime factor is a factor that is a prime number.
For example,
What are the prime factors of 70?
The factors of 70 are: 1 2 5 7 10 14 35 70
Ask for the factors of 70 before revealing them.
Then ask which of these factors are prime numbers.
The prime factors of 70 are 2, 5, and 7.

24. The prime factor decomposition
When we write a number as a product of prime factors it is called the prime factor decomposition or prime factor form.
For example,
The prime factor decomposition of 100 is:
100 = 2 × 2 × 5 × 5
Verify that 2 × 2 × 5 × 5 = 100
= 22 × 52
There are two methods of finding the prime factor decomposition of a number.

25. Factor trees 36 4 9 2 2 3 3
Explain that to write 36 as a product or prime factors we start by writing 36 at the top (of the tree).
Next, we need to think of two numbers which multiply together to give 36.
Ask pupils to give examples.
Explain that it doesn’t matter whether we use, 2 × 18, 3 × 12, 4 × 9, or 6 × 6, the end result will be the same.
Let’s use 4 × 9 this time.
Next, we must find two numbers that multiply together to make 4.
Click to reveal two 2s.
2 is a prime number so we draw a circle around it.
Now find 2 numbers which multiply together to make 9.
Click to reveal two 3s.
3 is a prime number so draw a circle around it.
State that when every number at the bottom of each branch is circled we can write down the prime factor decomposition of the number writing the prime numbers in order from smallest to biggest.
Ask pupil how we can we write this using index notation (powers) before revealing this.
36 = 2 × 2 × 3 × 3
= 22 × 32

26. Dividing by prime numbers2 96 2 3 2 48
96 = 2 × 2 × 2 × 2 × 2 × 3 2 24 2 12
= 25 × 3 2 6 3 3
Explain that another method to find the prime factor decomposition is to divide repeatedly by prime factors putting the answers in a table as follows:
To find the prime factor decomposition of 96 start by writing 96.
Click to reveal 96.
Now, what is the lowest prime number that divides into 96?
Establish that this is 2. Remind pupils of tests for divisibility if necessary. Any number ending in 0, 2, 4, 6, or 8 is divisible by 2.
Write the 2 to the left of the 96 and then divide 96 by 2.
Click to reveal the 2.
This may be divided mentally. Discuss strategies such as halving 90 to get 45 and halving 6 to get 3 and adding 45 and 3 together to get 48.
We write this under the 96.
Now, what is the lowest prime number that divides into 48?
Establish that this is 2 again. Continue dividing by the lowest prime number possible until you get to 1.
When you get to 1 at the bottom, stop.
The prime factor decomposition is found by multiplying together all the numbers in the left hand column. 1

27. Prime factor decomposition
Use the prime factor form of 324 to show that it is a square number. 2
324 2 3
324 = 2 × 2 × 3 × 3 × 3 × 3 2
162
= 22 × 34 3 81
This can be written as:
(2 × 32) × (2 × 32) 3 27
or (2 × 32)2 3 9
If necessary break this down further to show that 324 = (2 × 3 × 3) × (2 × 3 × 3).
Ask pupils to use the information on the board to tell you the square root of 324.
324 = 2 × 32 = 2 × 9 = 18.
If all the indices in the prime factor decomposition of a number are even, then the number is a square number. 3 3 1

28. Using the prime factor decomposition
Use the prime factor form of 3375 to show that it is a cube number. 3
3375
3375 = 3 × 3 × 3 × 5 × 5 × 5 3 5
= 33 × 53 3
1125
This can be written as: 3
375
(3 × 5) × (3 × 5) × (3 × 5) 5
125
or (3 × 5)3 5 25
Ask pupils to use the information on the board to tell you the cube root of 3375.
33375 = 3 × 5 = 15
If all the indices in the prime factor decomposition of a number are multiples of 3, then the number is a cube number. 5 5 1

29. Contents Integers N1.5 LCM and HCF A N1.1 Classifying numbers A
N1.2 Calculating with integers A
N1.3 Multiples, factors and primes A
N1.4 Prime factor decomposition A
N1.5 LCM and HCF

30. The lowest common multiple
The lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers.
For small numbers we can find this by writing down the first few multiples for both numbers until we find a number that is in both lists.
For example,
Multiples of 20 are :
20,
40,
60,
80,
100,
120, . . .
Multiples of 25 are :
You may like to add that if the two numbers have no common factors (except 1) then the lowest common multiple of the two numbers will be the product of the two numbers.
For example, 4 and 5 have no common factors and so the lowest common multiple of 4 and 5 is 4 × 5, 20.
Pupils could also investigate this themselves later in the lesson.
25,
50,
75,
100,
125, . . .
The LCM of 20 and 25 is
100.

31. The highest common factor
The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists.
For example,
Factors of 36 are : 1, 2, 3, 4, 6, 9,
, 12
18,
36.
Point out that 3 is a common factor of 36 and 45. So is 1. But 9 is the highest common factor.
Factors of 45 are : 1, 3, 5, 9,
15,
45.
The HCF of 36 and 45 is 9.

32. Using prime factors to find the HCF and LCM
We can use the prime factor decomposition to find the HCF and LCM of larger numbers.
For example,
Find the HCF and the LCM of 60 and 294. 2 60 2
294 2 30 3
147 3 15 7 49
Recap on the method of dividing by prime numbers introduced in the previous section. 5 5 7 7 1 1
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7

33. Using prime factors to find the HCF and LCM
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7 60
294 2 7 2 3 5 7
We can find the HCF and LCM by using a Venn diagram.
We put the prime factors of 60 in the first circle. Any factors that are common to both 60 and 294 go into the overlapping section.
Click to demonstrate this. Point out that we can cross out the prime factors that we have included from 294 in the overlapping section to avoid adding then twice.
We put the prime factors of 294 in the second circle.
The prime factors which are common to both 60 and 294 will be in the section where the two circles overlap.
To find the highest common factor of 60 and 294 we need to multiply together the numbers in the overlapping section.
The lowest common multiple is found by multiplying together all the prime numbers in the diagram.
HCF of 60 and 294 =
2 × 3
= 6
LCM of 60 and 294 =
2 × 5 × 2 × 3 × 7 × 7 =
2940

34. The LCM of co-prime numbers
If two numbers have a highest common factor (or HCF) of 1 then they are called co-prime or relatively prime numbers.
For two whole numbers a and b we can write:
a and b are co-prime if HCF(a, b) = 1
If two whole numbers a and b are co-prime then:
LCM(a, b) = ab
For example, the numbers 8 and 9 do not share any common multiples other than 1. They are co-prime.
Explain that if two numbers are co-prime, that is they do not share any common factors other than 1, their LCM is equal to the product of the two numbers.
This rule is also true if more than two numbers are co-prime. For example, the LCM of 3, 5 and 8 is 3 × 5 × 8 = 120.
Therefore,
LCM(8, 9) =
8 × 9 = 72

35. The LCM of numbers that are not co-prime
If two numbers are not co-prime then their highest common factor is greater than 1.
If two numbers a and b are not co-prime then their lowest common multiple is equal to the product of the two numbers divided by their highest common factor.
We can write this as:
LCM(a, b) = ab
HCF(a, b)
Repeat the activity on slide 62 verifying using this rule to check each solution.
Rearranging this formula, we can also say that the LCM(a,b) × HCM(a,b) = ab.
In other words, for any two whole numbers a and b, the LCM of a and b multiplied by the HCF of a and b equals the product of a and b.
This rule is true for any two whole numbers. If the two numbers are co-prime we have LCM (a,b) = ab/1, as on the previous slide.
For example,
8 × 12
HCF(8, 12) = 96 4
LCM(8, 12) =
= 24