Maths - Fraction
Introduction
A common, vulgar,
or simple fraction (for example
,
, and 3/17) consists of an integer numerator, displayed above
a line (or before a slash), and a non-zero integer denominator,
displayed below (or after) that line. The numerator represents a number of
equal parts and the denominator indicates how many of those parts make up a
whole. For example, in the fraction 3/4, the numerator, 3, tells us that the
fraction represents 3 equal parts, and the denominator, 4, tells us that 4
parts make up a whole. The picture to the right illustrates
or 3/4 of a cake. Numerators and denominators are also
used in fractions that are not simple, including compound fractions, complex fractions,
and mixed numerals.
Fractional numbers can also be written
without using explicit numerators or denominators, by using decimals, percent
signs, or negative exponents (as in 0.01, 1%, and 10−2 respectively,
all of which are equivalent to 1/100). An integer such as the number 7 can be
thought of as having an implied denominator of one: 7 equals 7/1.
Other uses for
fractions are to represent ratios and to represent division.[1] Thus the fraction 3/4 is also
used to represent the ratio 3:4 (the ratio of the part to the whole) and the
division 3 ÷ 4 (three divided by four).
In mathematics the
set of all numbers which can be expressed in the form a/b, where a and b
are integers and b is not zero, is called the
set of rational numbers and is represented by the symbol Q, which stands
for quotient. The test for a number being a
rational number is that it can be written in that form (i.e., as a common
fraction). However, the word fraction is also used to describe
mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that
contain irrational numbers,
To work with fractions
we need to know the different forms and they take reciprocal fractions as well
as the part they are made of denominators, numerators and fraction bars. Fractions are useful when we deal with numbers
that do not represent whole units.
The
concept of fractions
Fraction
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Part of
a whole.
· the bottom number (the denominator) says how many parts the whole is divided into · the top number (the numerator) says how many you have, |
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Definition of Fraction
There are three types
of fractions;
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Proper
Fractions:
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The numerator is less
than the denominator
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Examples: 1/3, 3/4, 2/7
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Improper
Fractions:
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The numerator is
greater than (or equal to) the denominator
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Examples: 4/3, 11/4, 7/7
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Mixed
Fractions:
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A whole number and
proper fraction together
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Examples: 1 1/3,
2 1/4,
16 2/5
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Improper
Fraction
So, an improper fraction is
just a fraction where the top number (numerator) is greater than or equal to
the bottom number (denominator).
In other words, it is top-heavy.
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4/4 |
Can
be Equal
What about when the numerator
is equal to the denominator? For example 4/4 ?
Well, it is obviously the
same as a whole, but it is written as a fraction, so most people agree it is
a type of improper fraction.
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Improper
Fractions or Mixed Fractions
You can use either an improper
fraction or a mixed fraction to show the same amount. For example 1 3/4 = 7/4, shown here:
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1 3/4
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7/4
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=
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In
arithmetic, a number expressed as a quotient, in which a numerator is divided
by a denominator. In a simple fraction, both are integers. A complex fraction
has a fraction in the numerator or denominator. In a proper fraction, the
numerator is less than the denominator. If the numerator is greater, it is
called an improper fraction and can also be written as a mixed number—a
whole-number quotient with a proper-fraction remainder. Any fraction can be
written in decimal form by carrying out the division of the numerator by the
denominator. The result may end at some point, or one or more digits may repeat
without end.
The concept of
equivalent fractions
Equivalent Fractions have the same
value, even though they may look different.
These fractions are really the same:
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=
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=
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Why are they the same? Because when you multiply or divide both the top
and bottom by the same number, the fraction keeps it's value.
The rule to remember is:
"Change the bottom using multiply or divide,
And the same to the top must be applied"
And the same to the top must be applied"
So, here is why those fractions are
really the same:
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× 2
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× 2
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1
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=
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2
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=
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4
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2
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4
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8
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× 2
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× 2
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And visually it looks like this:
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1/2
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2/4
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4/8
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=
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=
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See
the Animation
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Dividing
Here are some more equivalent
fractions, this time by dividing:
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÷
3
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÷
6
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18
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=
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6
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=
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1
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36
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12
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2
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÷
3
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÷
6
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Choose the number you divide by
carefully, so that the results (both top and bottom) stay whole numbers.
If we keep dividing until we
can't go any further, then we have simplified
the fraction (made it as simple as possible).
A
fraction is a part of a whole
Slice a
pizza, and you will have fractions:
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1/2
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1/4
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3/8
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(One-Half)
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(One-Quarter)
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(Three-Eighths)
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The top
number tells how many slices you have
The bottom number tells how many slices the pizza was cut into. |
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Numerator
/ Denominator
We call the top number the Numerator, it is the number of
parts you have.
We call the bottom number the Denominator, it is the number
of parts the whole is divided into.
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Numerator
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Denominator
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You just have to remember those
names! (If you forget just think "Down"-ominator)
Equivalent
Fractions
Some fractions may look
different, but are really the same, for example:
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4/8
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=
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2/4
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=
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1/2
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(Four-Eighths)
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Two-Quarters)
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(One-Half)
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=
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=
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It is usually best to show an
answer using the simplest fraction ( 1/2 in this case ). That is called Simplifying, or Reducing the Fraction
Adding
Fractions
You can add fractions easily if
the bottom number (the denominator)
is the same:
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1/4
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+
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1/4
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=
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2/4
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=
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1/2
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(One-Quarter)
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(One-Quarter)
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(Two-Quarters)
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(One-Half)
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+
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=
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=
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Another example:
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5/8
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+
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1/8
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=
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6/8
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=
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3/4
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+
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=
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=
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Adding
Fractions with Different Denominators
But what if the denominators (the bottom numbers) are not the same?
As in this example:
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3/8
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+
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1/4
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=
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?
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+
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=
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You must somehow make the denominators the same.
In this case it is easy,
because we know that 1/4 is the same as 2/8 :
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3/8
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+
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2/8
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=
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5/8
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+
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=
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But
it can be harder to make the denominators the same, so you may need to use one
of these methods (they both work, use whichever you prefer.
The simple reason why learning the various
fraction operations proves difficult for children is the way they are typically
taught in school books. Just look at
the amount of rules there are
to learn about fractions:
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1.
Fraction addition - same denominators
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Add the
numerators, and use the same denominator
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2.
Fraction addition - different denominators
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First
find a common denominator by taking the least common multiple of the
denominators. Then convert all the addends to have this common denominator.
Then add using the rule above.
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3.
Finding equivalent fractions
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Multiply
both the numerator and denominator with a same number
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4. Mixed
number to a fraction
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Multiply
the whole number part by the denominator and add the numerator to get the
numerator. Use the same denominator as in the fractional part of the mixed
number.
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5.
(Improper) fraction to a mixed number
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Divide
the numerator by the denominator to get the whole number part. The remainder
will be the numerator of the fractional part. Denominator is the same.
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6.
Simplifying fractions
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Find the
(greatest) common divisor of the numerator and denominator, and divide both
by it.
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7.
Fraction multiplication
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Multiply
the numerators, and the denominators.
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8.
Fraction division
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Find the
reciprocal of the divisor, and multiply by it.
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If
children simply try to memorize these without knowing where they came from,
they will probably seem like a jungle of seemingly meaningless rules. By
meaningless I mean that the rule does not seem to connect with anything about
the operation - it is just like a play where in each case you multiply or
divide or add or do various things with the numerators and denominators and
that then should give you the answer.
Fraction math can then
become blind following of the rules, tossing the numbers here and there,
calculating this and that - and getting answers of which the kids have no idea
if they are reasonable or not. And of course, it is quite easy to forget these
rules, or remember them wrong - especially after 5-10 years.
The solution: use manipulatives
and visual models (pictures)
Instead
of merely presenting a rule, as many schoolbooks do, a better way is to teach
children tovisualize fractions, and perform some simple operations with
these visual images or pictures, without knowingly applying any given 'rule'.
If a child is able to
visualize fractions in his
mind, they become more concrete - not just a number on top of other number
without meaning. Then the child can estimate the answer before calculating, and
evaluate the reasonableness of the final answer, and perform many of the
simplest operations in his head.
Another example is the
topic of addition of unlike fractions. The teacher can show how the individual fractions need to
be split into further pieces so that they are all same kind of pieces. You
don't need to discuss "least common denominator" at this point. The teacher can simply use
pictures or manipulatives. Then,
the kids will do the same with manipulatives, or by drawing pictures.
After a while, some kids
might discover the 'rule' of what kind of pieces the fractions will need split
into. And in any case, they will certainly remember it better when they have
been able to verify it themselves with numerous examples.
I'm not saying that the
rules are not needed - because they are. You can't get through algebra without
knowing the rules for fraction operations. But if 10 years from now the child
has maybe forgotten the fraction rules, hopefully he will have retained the
simple fraction pictures and is able to "do math" with the pictures
in his mind, and not consider fractions as something he just "cannot
do".
An
improved approach in teaching equivalent fractions
An improved technique to generate high yields of
relatively pure seminiferous tubule-enriched fractions from mouse testis by
manual isolation is described. Our laboratory had previously developed an
isolation method based on mild enzymatic digestion to separate individual
constituents of each compartment of the testis, namely, the interstitial tissue
and the seminiferous tubules. Although the method had the advantage of allowing
the production of seminiferous tubule-enriched fractions in large amounts, we
show here that this approach does not allow optimal preservation of the
integrity of the proteins in the samples, in particular of the phosphorylated
and/or glycosylated forms of the proteins. In an attempt to solve this problem,
we developed a novel mechanical approach to generate interstitial tissue- and
seminiferous tubule-enriched fractions that does not require the use of
enzymatic digestion. This approach has the advantages of providing relatively
pure seminiferous tubule-enriched fractions in large quantities and in a short
amount of time. In addition, and more significantly, the approach allows a more
faithful detection of the phosphorylated and glycosylated forms of the
proteins.
There are many ways
how to teach fractions, and the success or failure of the student to understand
fractions will depend on the teaching method used. The methods outlined
below to teach fractions have been proven to be easy for the student to grasp,
and easy for every math teacher to implement.
It's been said that
if a student understands fractions, then they can understand any mathematics
concept. It is then very important for every math teacher to know how to
teach fractions in the most approachable way possible.
*
When deciding on a
method of how to teaching fractions, we need to use fractional analogies that
the student will immediately recognize. Thus enters the pizza as the
perfect instrument needed to teach the concept of the fraction.
Students will never understand how to
add or subtract a fraction until they truly understand the concept of a
fraction. Use fractions that every student understands. For
example, take the fraction:
Take the time to explain that this
fraction represents a pizza that has been cut into two pieces, but you only
have one of the pieces. The numerator of the fraction is the top number
and the denominator of the fraction is the bottom number. Always keep in
mind that using the best methods of how to teach fractions involves moving very
slowly, giving multiple examples involving fractions until every student is
comfortable. Use familiar examples involving fractions such as 1/4, 1/3,
and 3/4 to reinforce the concept of what a fraction is.
Children are visual learners and the best technique of how to teach fractions involves drawing pictures of fractions. For example, take the fraction: This fraction can be best described by drawing a pizza cut into four pieces, but you only have three of the pieces such as in this picture of the fraction:
Children are visual learners and the best technique of how to teach fractions involves drawing pictures of fractions. For example, take the fraction: This fraction can be best described by drawing a pizza cut into four pieces, but you only have three of the pieces such as in this picture of the fraction:
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Other fractions can also be visually
taught in this way. For example, if we cut a pizza into 10 pieces but we
only have 7 pieces, we can teach the fraction as follows:
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Most students have a
very hard time understanding the concept of equivalent fractions, otherwise
known as fraction simplifcation. The best way how to teach fraction
simplication is to show pictures of many such fractions that we know to be
equivalent. For example, we know that:
For a student who is
new to learning fractions, this concept can seen very hard to understand.
As always, when it comes to how to teach fractions, pictures are almost always
the superior teaching tool. To show how these fractions are equivalent,
we draw each case separately. The teacher can then show the student that
the fraction 2/4 is "two out of four pieces" and that 1/2 is
"one out of two pieces". It is a very powerful method of
teaching fractions to show visually how these two fractions represent the exact
same amount of pizza:
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The solution: use manipulatives
and visual models (pictures)
Instead of merely
presenting a rule, as many schoolbooks do, a better way is to teach children to visualize fractions, and
perform some simple operations with these visual images or pictures, without
knowingly applying any given 'rule'.
If a child is able to
visualize fractions in his
mind, they become more concrete - not just a number on top of other number
without meaning. Then the child can estimate the answer before calculating, and
evaluate the reasonableness of the final answer, and perform many of the
simplest operations in his head.
Teaching with
understanding
Equal shares
To be based upon halves because children are confident
with this concept. Using this method the
child starts by splitting each composite unit 2 parts. Examples;
Sharing 3 chocolate bars
between 4 children would be achieved by 1st halving all the
bars. It also can be shared with one
whole bar remaining.
Stuck On counting
In fragmenting the child treats the parts as individual
unit items. So, when asked to determine
what share each child will receive if 3 chocolate bars are shared equally 4
children. No conceptual distinction is
being made between the’2’ as in number of pieces and the 2 pieces as fractions
of each unit, or the whole 3 bars that is being shared. These children are aware of the fragments as
only individual unit items. The complexity
here is also whether the ”whole” refers to the individual bars, in which case
each child gets 3 quarters of the bars or the “whole” refers to the 3 bars.
Partitioning
Partition regions or composite units approximately
equally among 2 or 3 recipients. Partion
the region into x number of parts and then designate y of the parts to indicate
y/x where x is greater than 2.
The lesson plan
DAILY LESSON
PLAN YEAR 3
SUBJECT:
Mathematics
CLASS: Year 3 Amanah
Number of
pupils: 28 of pupils (13 boys and 15
girls)
Date : 14th of July,2012.
Time: 3.00 – 3.30, 3.30-4.00pm
Topic: 3 Fraction
Learning area:
3.1
Learning
Objectives: Pupils will be taught to
1) Recognize
the parts of fraction, one whole, one half, one quarter and 3 quarters.
Learning
Outcome: By the end of the lesson,
pupils will be able to;
1) Read
and write fractions
Previous knowledge: Pupils have studied the parts of fraction.
Moral values: Collaborating and helping each other.
Thinking skills: Compare and Contrast
Teaching aids: cake, manila card and worksheet
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Steps
(Duration)
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Contents
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Activities
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Remarks
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Introduction
(3
min)
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Introduce
the concepts of fraction
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1) Teaching
using a cake to introduce the concept of fractions.
2) Teacher
introduce the numerator &
denominator, one whole, one half, one quarter and 3 quarters.
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Cake
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Lesson
Development
(24
minutes)
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Teacher
paste some diagrams of fractions
Teacher
asks pupils to identify the fractions and say aloud
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Manila
card
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Closure
(3
minutes)
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Conclusion
of the lesson
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Teacher
emphasized the parts of fractions
2
Pupils will do some exercise on worksheet
Teacher
will respond to their answers
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Worksheet
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Reflections:
Pupils can make the difference by saying what type of fractions is that with
the questions 3 answered correctly out of 5.
DAILY LESSON PLAN
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WEEK
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Day Wednesday
Date 07/04/2013
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15
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Subject: Mathematics
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Class: 3 Cekap
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No. of Pupils :
30 / 30
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Time : 03:00 – 04:00pm (1 hour)
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Focus : Number and
Operation
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Title: Fractions
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Content Standard:
Pupils will be guided by the teacher to
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6.1 Naming
equivalent fractions
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Learning standard: Pupils would be able to;
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6.1 (i)
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Learning outcome : By the end of this lesson, pupils would be able to;
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i.
Know and emphasize or find
out , the denominator within 10
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Activities :
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2.
Pupils can find out more
about proper fractions, with denominator within 10.
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3.
The pupil do some exercise in
their writing book.
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EMK : Innovation and creativiti
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Teaching aids : flash cards,, number cards, worksheet, activity book
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Values: Dare to make decision, courage
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Reflections:
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Most of the pupils can defined and make a list about improper
fractions within 10 by answering 5 out of 10 question correctly.
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Conclusion
Generally it is very important for
us to understanding fractions. It is a
very useful skills that can be used in daily life. It is true that some professions use it more
than other but understanding fractions is a necessary skills to have not only
during our time in school but most important thing that it can be used in our
daily life.
As a conclusion,
learning how to teach fractions in the best way possible is very much worth the
teachers time because when we form a good foundation in the essentials of
fractions, it will make the future topics much easier to understand. The
student will then have a good foundation in fractions to understand how to add,
subtract, multiply, and divide fractions without problems. And by using
the methods outlined above, the teacher will know how to teach fractions to the
student in the most approachable way possible.
Actually children
should be exposed to a wide variety of situations so the children will have the
opportunity to reflect and abstract critical relations in different contextual
situations. They have to see a whole in
all its representational forms. This will
aid them in developing a more robust grasp of the concept of a fractions. It is
in terms of properties of being discrete or continuous, definite or indefinite.
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