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how to teach your child numbers arithmetic mathematics

fractions, decimals, percentages and ratios 2

 



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how to teach your child number arithmetic mathematics - fractions, decimals, percentages and ratios 2 is part of the series of documents about fundamental education at abelard.org. These pages are a sub-set of sums will set you free
how to teach a person number, arithmetic, mathematics on teaching reading

 

Fractions, decimals and percentages are particular types of division sum, and are all effectively the ‘same’ thing. It is important to internalise this realisation right from the start. That is why here, as with the first page on fractions, decimals and percentages, all three topics are discussed on the same page. Once you understand one of these topics, you are in a position to understand them all. They are just different ways communicating and writing the same manner of information.

When you multiply by one, the value does not change. That is, 2 x 1 = 2 and 474 x 1 = 474. And when you multiply by more than one, the value grows greater: 2 x 3 = 6 and 474 x 3 = 1,422. However, when you multiple positive integers by less than one, the value becomes smaller, not larger.
Thus, 1 x 1/2 = 1/2,
10 x 1/4 = 2 1/2 and so is less than ten,
and 10 x 3/5 = 30/5
= 6, which again is less than ten.

And when you divide integers by fractions, the inverse occurs - the value becomes greater.
So 1 ÷ 1/2 = 2,
10 ÷ 1/4 = 40
and 10 ÷ 3/5 = 16 2/3.

 

introduction to doing multiplication and divison sums with fractions

counting, addition and subtraction are one-dimensional

Ordinary counting, addition and subtraction are one-dimensional.

Multiplication and division with fractions are two-dimensional.

Multiplication and division with fractions are two-dimensional.

ways of multiplying fractions

nine blocks/hundred blocks fifty-six blocks/hundred blocks 7x8
Each individual block is clearly one hundredth of the total ten by ten square of blocks.

Note that the method for multiplying fractions that follows can be used for any fraction multiplication.

  • As you can see in the left photo, there is a square of a hundred blocks with nine blocks taken out.
    So the nine blocks are nine hundredths of the larger square.
    The block of nine is made up of three blocks by (or times) three blocks, and it is separated from a block that is ten blocks by (or times) ten blocks.

    Three tenths by three tenths are nine hundredths,
    3/10 x 3/10 = 9/100 [above on the left].

    So the top parts of the fractions are multiplied together, and the bottom parts of the fraction are multiplied together, keeping the fraction in balance.

  • In the right photo above, there is a square of a hundred blocks with fifty-six blocks taken out.
    So the fifty-six blocks are fifty-six hundredths of the larger square.
    The block of fifty-six is made up of eight blocks by (or times) seven blocks, and it is separated from a block that is ten blocks by (or times) ten blocks

    Eight tenths times seven tenths are fifty-six hundredths,
    8/10 x 7/10 = 56/100 [above, to the right].

    Again, the top parts of the fractions are multiplied together, and the bottom parts of the fraction are multiplied together, keeping the fraction in balance.
Remember that multiplication is continual addition, but for fractions the multiplication is applied to both the top and the bottom parts of the fraction.
 
So 3/10 x 3/10
= 3/100 + 3/100 + 3/100
= 9/100,
and
8/10 x 7/10
= 8/100 + 8/100 + 8/100 + 8/100 + 8/100 + 8/100 + 8/100
= 56/100

 

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56/100 can be cancelled down, also known as simplifying the fraction, by dividing both the the top and the bottom part of the fraction by the same number, and repeating until both parts can no longer be divided by the same number.

What number to divide by?
Well, if you are not sure, or cannot yet surmise what are probable numbers, then start with the smallest prime number: 2.

56 ÷ 2 = 28; 100 ÷ 2 = 50
or 56/100 = 28/50

Now divide again by 2: 28/50 becomes 14/25. There are no other prime numbers I can see that will divide into both numbers.
So, 56/100 simplified (or cancelled down) is 14/25.

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Seven by seven blocks twelve fortieths
Each block is clearly one forty-ninth of the total seven by seven square of blocks.

The illustration on the left shows forty-nine blocks in a seven by seven square.
On the right, a block of three by four, which is twelve blocks, has been ‘bitten out’ of the block of forty-nine.

Thus, three sevenths by (or times) four sevenths is the same as (equals) twelve forty-ninths:
3/7 x 4/7 = 12/49.
Recall that both the top and the bottom parts of the fractions are multiplied together:

3/7 x 4/7 = 12/49

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twenty-five hundredths

Five tenths by five tenths are twenty-five hundredths,
5/10 x 5/10 = 25/100.

Of course, twenty-five hundredths is an equivalent fraction to one quarter, illustrated in the photo above. Twenty-five hundredths can be reduced to one quarter by cancelling, by dividing both the top and bottom parts of the fraction 25/100 by 5:

25/100 = 1/4

And now an example of multiplying fractions when the bottom numbers are different:

 

ways of multiplying fractions by whole numbers

The examples of multiplication with fractions so far have illustrated multiplying fractions by fractions. When multiplying a fraction by a whole number (an integer), such as the sum 1/3 x 3, convert the whole number into a fraction - a whole number is a fraction where the bottom number is one, a number divided by one remains the same number.

So 1/3 x 3 can be multiplied like this:

 

ways of dividing fractions

And division?
As with integer division, fractional division is constant subtraction.

Here’s an example, nine tenths divided by three tenths, 9/10 ÷ 3/10.
This equivalent to subtracting 3/10 three times, giving a sum of 9/10 - 3/10 - 3/10 - 3/10 = 0
That is, taking away 3/10 from 9/10 three times leaves nothing.
And taking away three times is like dividing by three, so 9/10 ÷ 3/10 = 3.

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Now, there are many permutations of dividing whole numbers by a fraction, a fraction by a whole number, a fraction by another fraction, and then there’s numbers that are combined whole numbers and fractions as well. So how to not be completely confused as what to do?

Well, if all the numbers, fractions, or whichever combination, are all converted into fractions and you then work out their biggest common bits (lowest common multiple) the task becomes simpler.

With this following example, 2 ÷ 2/5, firstly the whole number 2 is converted to 10 fifths (2 x 5). Next the division is done to the top part and to the bottom part. The photo of blocks illustrated how 2/5 divides five times into 2, or 10/5.

2 divided by 2/5 = 5   2/5, 2 halves, 10 tenths

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Often, a easier way to do divisions involving fractions is to invert the fraction doing the dividing, and then multiply the first fraction by the inverted fraction. Now, explaining this in words is not easy to understand, so after giving a short explication why this process works, we will give a couple of worked examples.

And why does multiplying with the inverted dividing fraction work? Well, multiplying is the inverse of dividing, just as subtraction is the inverse (the opposite) of addition. So when a fraction is inverted (or turned upsidedown) the action being done (dividing) with that fraction is also inverted (to become multiplying).

If you divide seven eighths by two, first consider dividing one eighth by two, the result of which is one sixteenth.

So when you divide seven eighths by two, you have seven sixteenths. Thus doubling the bottom part of the fraction halves the value of the fraction, that is it divides it by two.

7/8 divided by 2 = 7/16   3/5 divided by 6 = 1/10

Examples of a fraction divided by a fraction and a mixed fraction divided by a mixed fraction:

1/4 divided by 1/2 = 1/2   5 2/3 divided by 2 1/2 = 2 4/15

 

decimals

Until we reach putting in details in this section, 1/10 is written as .1, that is a point (or dot) before the one, and one quarter is written as .25, that is 25/100, and so on.

converting decimals to fractions, and fractions to decimals

Originally, the decimal was called the decimal fraction (the Latin word for ten being decem). Decimals are a convenient way of writing and using tenths, or other fractions divisible by ten, such as hundredths, thousandths and so on.

.1 (or more commonly, of less accurately, 0.1) is the same as one tenth, 1/10.
.01 is the same as one hundredth, 1/100.
.001 is the same as one thousandth, 1/1000.

Scales showing decimal tenths and hundredths

Notice that with the fractional and decimal parts of numbers, all the exciting action takes place in
a very narrow part of the total number scale/line. When you reach one hundredths (the .0x part of the decimal),
the number x is in the first one-tenth of the decimal/fraction area - and so on.

Fractions are, in part, called rational numbers, not because they are particularly sane (although they are) but because they consist of ratios - ratio-nal. The integers also come within the class of rational numbers, for any integer can be expressed as a fraction or ratio. For example, 2 can be expressed as 2/1, or as 6/3, or even as 50/25, while 1734 can be written as 1734/1.

Any fraction can be converted into a decimal form (13/19 = 13 ÷ 19 = .6841...) and any decimal can be converted into a fraction ( .731 = 731/1000), but there comes a time when this starts to become a bit more difficult, and even mathematicians have problems keeping their heads straight. If you do want to go deep-diving, see comparing predicates, relational strengths and irrational numbers.

The diagram above gives help for converting a decimal into a fraction. As you can see, the decimal number goes on the top of the fraction, and the bottom part is dictated by how many places there are after the decimal point (how many numbers to the right of the dot). So .3 = 3/10 and .03 = 3/100, while .38 = 38/100.

To convert a fraction into a decimal, you divide the bottom part into the top part, with the result being written (expressed) as a decimal.
    ¾ = 3 ÷ 4
= .75
 
4 divided by 3 = .75

Four will not divide into three (four into three won’t go), so the first division in this sum is four divided into three point zero (3.0).
3.0 ÷ 4 = .7 r.2.

The remainder, 2, is used as part of the next sub-sum. The zero place marker for the decimal hundredths (.00 position) is brought down to complete the next number into which four is divided.
.20 ÷ 4 = .05

Multiplications and divisions involving decimals are like doing those sums with integers, but you must make sure that the decimal point is in the right position. Such sums are easy to check using a calculator, or the educational counter below.

    multiplying decimals

  • 120 x .75 = 90

    1.2 x .75 = .9
    [Note how the decimal place has moved two places left from 120 to 1.2, so the decimal place in the answer has also moved left two places.]

  • 414 x 3.75 =
    This multiplication process is almost exactly the same as multiplying whole numbers. While teaching this process, it is important to lay out the numbers with the decimal points lined up. In order to help the learner to keep clear in their own mind what is happening, it can be helpful to add extra zeroes as follows:
    414 x 3.75 = 1552.5
    You probably notice that this is a long multiplication sum, as described in writing down sums.


    dividing decimals
  • 120 ÷ .75 = 160
  • 1.2 ÷ .75 = 1.6

 

abelard.org maths educational counter

[This counter functions with javascript, you need to ensure that javascript is enabled for the counter to work.]

The full version with more detailed instructions, go to the introduction page.

Here is how to practise sums with decimals, for example .25 x 4,

  • Reset Counter Value to 0;
  • Change Step to .25;
  • Switch Direction (if necessary) to Increasing;
  • Now click on the Manual Step button four times. The red number counts to 4.

The counter counts up: 0, .25, .5, .75, 1. Thus .25 x 4 = 1.

Now help the learner to try other multiplication sums. Each time, click the red Reset button to return Manual Steps (the red number) to zero.

Below is a concise version of the abelard.org eduacational maths counter. For an expanded version with more detailed instructions, go to how to teach your child number arithmetic mathematics - introduction.

 
[This counter functions with javascript, you need to ensure that javascript is enabled for the counter to work.]
Is the counter Manual or Automatic? :

You have done manual steps since the last reset

Decimal Places
[between 0 and 5]:
the counter is displayed up to decimal places
Reset Counter Value:

[enter number in base 10]

Change Step:
Enter step size:
[enter: step size in base 10]

change step size:

is added or subtracted on each update

 

Direction:
Counting up/counting down

Base [between 2 and 32]:
the counter is displayed in base
Change Speed:
the counter changes every seconds.

You are welcome to reproduce this configurable counter. However, all pages that include this counter must display a prominent and visible link to:
http://abelard.org/sums/teaching_number_arithmetic_mathematics_introduction.php, with the following text:
“The Configurable Practice Counter was developed by the auroran sunset on behalf of abelard.org and is copyright to © 2009 abelard.org”.
This text and the code, including all comments, must not be altered.

© 2009 abelard.org

 

percentages

Until we reach putting in details in this section, 100% (one hundred percent) is the whole cake. 1% is 1/100th (one hundredth) of the cake.

1/10th (one tenth) is written as 10%, that is a percent sign (%) after the ten, and one quarter is written as 25% and so on.

Multiplications and divisions involving percentages are like doing those sums with integers and fractions.

    multiplying percentages

  • What is 125% of 120?
    120 x 125% = 150
    (125% = 1 1/4, or 1.25)

    dividing percentages
    What is 25% of 120?
    120 x 25% = 30
    (25% = 25/100 = .25 = 1/4)

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end notes

  1. Integers are whole numbers, but definitions can vary according to purpose. Sometimes reference is made to the ‘natural numbers’, these being only the positive integers, or the non-negative integers. When called non-negative integers, zero is included.

  2. The least common multiple [LCM] of some numbers is the smallest number (but not zero) that is a multiple of all of them. For instance, the LCM of 4, 6 and 12 is 12.

    Meanwhile, the lowest common denominator [LCD] is the least common multiple of the bottom numbers of a group of fractions, such as 3/4, 5/6 and 17/12. In this case, the LCD is 12 because 4,6 and 12 will all divide into 12 without a remainder.

  3. The technical names for the parts of a fraction are:
       numerator
    —————
     denominator.

    The numerator is sometimes called the dividend, while the denominatior is also called the divisor.
    When the top part of the fraction (the numerator or dividend) is smaller than the bottom part (the denominator or divisor), the fraction is called a ‘proper fraction’. A proper fraction is always of a value that is less than one.
    When the numerator or dividend is larger than the the denominator or divisor, the fraction is weirdly called an ‘improper fraction’.

  4. It is common to put a zero before the decimal point (dot), as in 0.1. As people become used to mathematics, they tend not to put the zero in front as it has no meaning, any more than they tend to put a zero after a number like 1 or 15, as in 1.0 or 15.0. But it is common, at times, to use leading or trailing zeroes in lists of figures.

    Note that, although English-speaking and Asian countries usually use a dot as a decimal position marker, either above the writing baseline: 5, or more commonly nowadays on the baseline: .5, several continental countries such as France, use a comma as the decimal separator: ,5.

  5. There are all sorts of categories of numbers dreamed up by mathematicians. It is useful to be aware that the categories we are using here form a type of hierarchy, starting with the natural numbers, then the integers (which include the natural numbers), then the rational numbers (which include the previous categories), and then the real numbers (which include the all previous categories).
    • natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and so on, the numbers we use for counting. Sometimes, zero is included in natural numbers.
    • integers: all the natural numbers and their opposites: -1, -2, -3, -4 etc, and zero.
    • rational numbers: numbers that can be written as a fraction (or ratio). A decimal that can be written with a finite quantity of numbers after the decimal point is a rational number: .10 or 1/10.
      Some rational numbers, like 1/3, can only written approximately as a decimal with an imaginary, infinite number of digits .333333 recurring. Thus .33333 recurring is not a rational number, it is called an irrational number (ir-rational, that is not a ratio-nal number), or should I say it is mad?! But you may prefer not to confuse a learner with this at the moment.
    • real numbers: All integers, natural and rational numbers, and irrational numbers, but not imaginary numbers or several other exotic creatures. [Imaginary numbers are real numbers multiplied by i: the square root of minus one (-1). No number when squared makes -1, so mathematicians invented the imaginary unit, i.]
sums will set you free includes the series of documents about economics and money at abelard.org.
moneybookers information e-gold information fiat money and inflation
calculating moving averages the arithmetic of fractional banking

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