Practical Learning Classes
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Practical
7 Classes: 1.1, 1.2, 1.3, 1.4, 2.1, 2.2, and 3.1. Take notes and answer questions as you read and learn.
Series of Classes on Fractions
Further Fractions 2 (Classes 2.1 and 2.2)
Adding and Subtracting Fractions (Class 3.1 Final Fractions Class)
(Fractions Test for when Classes are completed)
Understanding Fractions
Fractions Class 1.1
What is a Fraction?
Whole numbers are numbers like 1, 2, 3…
A fraction is a number that is less than a whole number or in between two whole numbers.
Fractions have been in use since ancient times. They were necessary for making calculations in trade and design.
The concept of a fraction is shown with this illustration. One whole full cup of water, could be summed up with the integer, 1.
If half of the water is poured into another cup there are now two half full cups of water.
By dividing the whole cup of water into two, two fractions were made. Since the fractions are equal in amount they are both the same. In this case the fractions are both one half.
Fractions are the separate equal portions of one whole thing that has been divided up.
In the case of the cup of water, there can be two equal half portions to every whole cup.
But we could also divide one whole cup of water into three separate equal portions, which would be called thirds, or into four separate equal portions, which would be called quarters.
How Do You Write One Half?
Two half full cups of water can be added together to get a whole cup full of water as follows;
Since there are two equal portions to every whole, one half is written as a fraction like this:
$$\frac{1}{2}$$
The 2 on the bottom of the fraction is found by looking at how many equal parts or portions the one whole has been divided up into. In this case, since there are two halves in a whole, 2 is the number on the bottom.
The number on the top is 1 because it is only 1 half we are talking about.
And in maths, the sum above would look like this:
$$\frac{1}{2}+\frac{1}{2}=\frac{2}{2}=1$$
This sum literally means one half plus one half equals two halves, which equal one.
Just as with the cups of water, once the two halves are added together, we no longer need to speak in terms of halves, and say there are two halves, but instead simplify the two halves back to the whole number 1.
Using filled in circles to show the same equation, it would look like this:
Try the following exercises to complete this class.
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$$a)\;True$$
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$$a)\;True$$
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$$a)\;0\;\;and\;\; 1$$
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$$d)\;4 $$because the whole circle has been divided into four parts.
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$$b)\;2 $$because the whole circle has been divided into two parts.
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$$c)\;3 $$because the whole circle has been divided into three parts.
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$$d)\;1$$
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$$b)\;2$$ because there are two parts.
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$$a)\;One$$
Fractions Class 1.2
Recognising Fractions
The number on the top of the fraction is called the numerator and the number on the bottom of the fraction is called the denominator. Remember that the denominator is the number of equal segments that the whole has been divided up into to make the fraction.
You will recall the illustration in the first class in which two half full cups of water were added to make one whole cup of water. But what would happen if more than just two halves were added?
For instance, four half cups of water would add up to two whole cups of water, as shown,
Initially the result would have been four halves or 4 over 2, $$\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=\frac{4}{2}=2$$
4 represents the overall number of portions and the 2 represent how many portions are in one whole, being the two halves. When the numerator and the denominator are the same , you have a complete whole or one. For instance five fifths is equal to one, and four fourths is equal to one, and so on. Here we have,
$$2\;times\;\frac{2}{2}\; =\frac{4}{2}\; =\;2$$
Since four half cups of water added together make a whole number, 2, the answer is simplified again from 4 halves to 2 whole.
Improper Fractions
Whenever a fraction’s numerator is greater than its denominator, it is called an improper fraction. Improper fractions always exceed one whole in value. In this case, since it is possible to simplify the answer by dividing the numerator and the denominator by a common factor, we do so. To simplify four halves, we divide both the numerator and the denominator by the common factor of 2, and we get 2 over 1 as the answer, which simplifies to the final answer of 2 as follows, $$\frac{4}{2} = \frac{2}{1}=2$$
It is worth remembering that all whole numbers have a denominator of 1.
Another Improper Fraction
Another example of an improper fraction, (which has a numerator with a greater value than the denominator) is eight halves or,$$\frac{8}{2}$$ Eight halves can be simplified because the numerator and the denominator have a common factor of 2, $$\frac{8}{2} $$To simplify eight halves, divide the numerator, 8, by 2 and the denominator, 2, by 2, and the answer is 4 over 1, the equivalent of 4, as shown here, $$\frac{8}{2} =\frac{4}{1}=4$$
The following picture, which shows that eight halves are equal to four whole or complete circles, confirms this answer.
Try the following exercises to complete this class.
Exercises For Fractions Class 1.2
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$$b)\;6$$
$$a)\; \frac{8}{2} $$ $$b)\;\frac{5}{2} $$ $$c)\;\frac{2}{6} $$ $$d)\;\frac{6}{2} $$
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$$d)\;\frac{6}{2} $$
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$$c)\;3$$
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$$a)\;Six \;\;halves\;\; is\;$$$$\; an\;\; improper\;\; fraction,$$ $$because \;the \;numerator $$$$is \;larger\; than $$$$the \;denominator. $$
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$$c)\;2,$$ because there are two segments altogether in the resultant fraction.
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$$b)\;4 $$because there are four segments (quarters) altogether in one whole.
$$a)\;\frac{2}{3} $$ $$b)\;\frac{2}{2} $$ $$c)\;\frac{3}{3} $$ $$d)\;\frac{4}{3} $$
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$$c)\;\frac{3}{3} $$
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$$c)\;Seven$$
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$$c)\;$$
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$$b)\;2$$