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SLIDE 6
? Find
? Find the highest common factor of 36 and 84
? Find
? Find the lowest common multiple of 9, 24 and 30
SLIDE 7
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? The highest common factor of 36 and 84 is
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? The lowest common multiple of 9, 24 and 30 is
SLIDE 8
A number is prime if only 1 and itself divide into it. Otherwise, it is composite.
The numbers 9 and 10 are both composite numbers, but they do not have any common factors.
When two numbers have no common factors, they are said to be relatively prime.
When two numbers are relatively prime, their lowest common multiple is found simply by multiplying the two numbers together.
Find the HCF and LCM of
? 4 and 11
? 8 and 20
? 3, 4 and 7
? 9, 27 and 54
SLIDE 9
? 4 and 11 are relatively prime.
? 8 and 20 are not relatively prime.
? 3, 4 and 7 are relatively prime.
? 9, 27 and 54 are not relatively prime.
SLIDE 10
p q
8 14 112 2 56
10 25
24 36
15 24
? Complete the table
? What do you observe about the entries in the third and last columns?
SLIDE 11
p q
8 14 112 2 56
10 25 250 5 50
24 36 864 12 72
15 24 360 3 120
SLIDE 12
? Work out and simplify
? Find
? Find
? Cancel down
? is equivalent to
? is equivalent to
? Convert to a top-heavy fraction
? Convert to a mixed number
SLIDE 13
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? is equivalent to
? is equivalent to
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SLIDE 14
The number on top of a fraction is called the numerator, and the number on the bottom of a fraction is called the denominator.
For example, in the numerator is 11, and the denominator is 14
State the numerator and denominator of each of the following
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SLIDE 15
? numerator is 3, denominator is 8
? numerator is 5, denominator is 9
? numerator is 17, denominator is 32
SLIDE 16
? Using the terms numerator and denominator, what is the rule for the multiplication of fractions? Give an example.
? If you are asked to find a fraction of another number, what do you do? Give an example.
? When you are asked to simplify a fraction, what are you being asked to do? Give an example.
? If you are asked to multiply by a mixed fraction, what is the best way to start? Give an example.
SLIDE 17
? The rule for the multiplication of fractions is that you multiply the numerators and multiply the denominators while cancelling out any common factors. Example
? If you are asked to find a fraction of another number, you multiply the numerators and denominators of the two numbers and cancel down any common factors. Example
? When you are asked to simplify a fraction, you are being asked to cancel it down to its lowest equivalent fraction. Example
? If you are asked to multiply by a mixed fraction, it is advisable to make any mixed fractions into top-heavy fractions first. Example
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SLIDE 24
In these examples the denominators are relatively prime. To find the common denominator, you multiply these numbers.
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SLIDE 26
Is the following statement true or false?
“When you add two fractions, you begin by finding their common denominator. This common denominator is the same as the lowest common multiple of the denominators of the two fractions you are adding.”
SLIDE 27
“When you add two fractions, you begin by finding their common denominator. This common denominator is the same as the lowest common multiple of the denominators of the two fractions you are adding.”
This statement is true.
SLIDE 28
Adding general fractions involves three steps. Explain what is happening in each of these steps.
SLIDE 29
? You find the common denominator of the two fractions. This is the lowest common multiple of the denominators of the two fractions.
? You find the equivalent fractions of the two fractions over their common denominator.
? You add the numerators of the two equivalent fractions.
SLIDE 30
Annotate the above addition of fractions
SLIDE 31
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SLIDE 38
Make these fractions top-heavy first, and then subtract
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SLIDE 39
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SLIDE 40
The reciprocal of is
The reciprocal of is 5
Find the reciprocals of the following
? 8 ? 120 ?
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SLIDE 41
? reciprocal of 8 is ? reciprocal of 120 is
? reciprocal of is 32 ? reciprocal of is 27
? reciprocal of is ? reciprocal of is
SLIDE 42
To invert means to turn something on its head.
For example, if you invert a Martian you obtain an upside-down Martian.
When we invert a fraction, we turn it upside down.
This is the same as taking the reciprocal of a fraction.
Invert the following
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SLIDE 43
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inverted
SLIDE 44
? Draw a diagram to show that divides into 1 cake 2 times.
? How many times does divide into 2, 4 and 8?
? Complete the following table
SLIDE 45
? There are two halves in one whole
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SLIDE 46
The rule for division of fractions is invert and multiply
? Invert the back fraction. This is the same as taking the reciprocal or flipping the two numbers.
? At the same time change the division to a multiplication.
? Multiply the numerators and denominators, just as for regular multiplication of fractions.
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SLIDE 51
is the same as
This is one a fraction divided by another fraction.
The top fraction has both a numerator and denominator, and likewise the bottom fraction has both a numerator and denominator.
Explain what the rule for division, which is invert and multiply, does to numerator and denominator of the top and bottom fractions when one fraction is divided by another.
SLIDE 52
The numerator of the top fraction goes on the bottom. The denominator of the bottom fraction goes on the top.
“The bottom of the top goes on the bottom, and the bottom of the bottom goes on the top!”
SLIDE 53
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SLIDE 55
Example
Place the following fractions in ascending order of size
We compare the equivalent fractions over their common denominator
SLIDE 56
Place each of the following sets of fractions in ascending order of size
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SLIDE 57
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SLIDE 58
Addition, subtraction, multiplication and division are called operations.
? How are fractions added?
? How are fractions subtracted?
? How are fractions multiplied?
? How are fractions divided?
SLIDE 59
? Addition of fractions
1. You find the common denominator of the two fractions. This is the lowest common multiple of the denominators of two fractions.
2. You find the equivalent fractions of the two fractions over their common denominator.
3. You add the numerators of the two equivalent fractions.
? Subtraction of fractions
Subtraction is the addition of a negative number. Subtraction of fractions follows the same procedure as addition of fractions.
? Multiplication of fractions
Multiply the numerators and multiply the denominators, cancelling down if possible.
? Division of fractions
Invert and multiply
1. Invert the back fraction. This is the same as taking the reciprocal or flipping the two numbers. At the same time change the division to a multiplication.
2. Multiply the numerators and denominators, just as for regular multiplication of fractions.
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SLIDE 64
Revision of the sequence of operations
The sequence of operations governs the order in which we evaluate operations. Evaluate in the order: brackets first, multiplication and division second, addition and subtraction last.
Work out
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SLIDE 68
Calculators have fraction buttons.
What are the benefits and disadvantages of using calculators to work out sums in general and sums involving fractions in particular?
SLIDE 69
What are the benefits and disadvantages of using calculators to work out sums in general and sums involving fractions in particular?
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An open question is a question to which no definite answer can be given. This is an open question.
Opinion will vary about the use of the calculator, and when to introduce calculators into the learning process.
We think that calculators are devices for speeding up mathematical processes that one could do oneself. We think that this rule applies especially at the early stages of learning mathematics.
In other words, if you cannot add fractions without a calculator, then you are in danger of destroying your whole later mathematical development if you start using the calculator to “do the thinking for you”. It is like someone who is born with two good legs but starts leaning on a stick at an early age, and by doing so turns him or herself into a cripple.
On the other hand, a calculator is certainly a useful tool. Operations are tricky and one can make slips with them. Besides, once one has the mastery of fractions, why shouldn’t one just get to the answer to a calculation as fast as possible?
You need to learn to use the calculator. It is a useful machine.
SLIDE 70
You should already be familiar with the above buttons and be able to input a sum in true form. This means, using the above buttons, to input exactly as you see it, and obtain the answer 106.
This button is a memory button. It enables you to break down a calculation into steps. You find the answer to one part, and then start over, pressing this button when you need the answer to the previous question.
Example
This gets 106 as the answer to the sum
SLIDE 71
Calculators nowadays are sophisticated machines. Even the simplest calculator contains many buttons that you will not need for some time to come. Input in true form (as you see it), is not always the easiest.
All calculators differ in the way task are executed by pressing buttons. You need the calculator manual and to keep it safe, as the manual may not be available on-line to be downloaded if you lose it.
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Calculators store the answers to previous questions, so when you switch it on, a previous answer may mess up your work.
You need to be able to clear the memory or reset the calculator. On some machines this is achieved by pressing the shift key on the left and the CLR key on the right. The CLR key doubles up as a MODE key, but you do not need MODE just now.
The calculator may then display a series of options. Press the number that will clear ALL.
Task
By checking the manual to your calculator, or by asking a friend, find out how to clear the memory and/or reset your calculator.
SLIDE 72
Your calculator may do things differently. Check your manual to find out how to input fractions. Typically, calculators have a fraction button
The fraction button is used to input a fraction.
This inputs the fraction and displays it as
To input the fraction key in
This displays as
Use the fraction button to compute
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SLIDE 73
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