Division of Fractions

Mathematics is the study of patterns and general rule is derived on the observation of those patterns. So, while teaching any concept we should find the patterns through different activities and then derive the general rule on the base of the patterns.

Division of Fractions

What does we do, to find 6 ¸ 2, here we find how many 2 are there in 6, i.e. 3

So, 6 ¸ 2 = 3

Similarly consider the example of a fraction 3/4 ¸ 2/5

Here we have to find how many 2/5 are there in 3/4.

We have represent 3/4

So, we will take a shape and divide it into 4 equal parts and color 3 parts.

Again represent 2/5

So, we will take a shape (same) and divide it into 5 equal parts and color 2 parts.

Now we have to find, how many times 2/5 are there in 3/4 ?

Here we compare the how many times the yellow area (2/5) are there in light orange area (3/4), 

We found that that there is 1 complete 2/5 and 7/8 of 2/5 are there in 3/4.

So, there are (1+ 7/8) time 2/5 in 3/4.

Þ  There are 15/8 time 2/5 in 3/4.

Þ  3/4  ¸  2/5 = 15/8

If we will solve, some other problem relating to division of fraction, we will find a pattern in the result.

The patterns in the above example is

So, the general rule of division of fraction is

Least Common Multiple

Least Common Multiple (Activity)

Objectives: Students will be able to understand the meaning and how to find LCM.

Questions:

• How do you define multiples?
• Find the multiples of 3, 5, 7?

Materials Required:

• Meter tape/rod.
• Small Circular Boards.
• Chalk

Activity:

Teacher tells a story to students.

(Once there was running a completion between the animals in a forest. In that completion the lion win the race. Deer, the friend of lion was very happy for the win of his friend lion. He congratulate his friend for his success, the lion also impress with the running of deer. A monkey came to them and told them that you both should not be both friend as you have different speed. The monkey blames the deer for his running speed. The lion told the monkey it’s true that we have different speed but we can meet at particular points in out running. The monkey said no never, you both cannot meet at any point while running. (Hints: the lion and deer jumps at a distance of 3 foot and 2 foot respectively).

 

What do you think, whether the lion and deer meet at any point or not?

Collect the response of students.

Lets us play an activity to clarify our doubt. Teacher takes all students to veranda.

Teacher select two students one named as Lion and the Deer. Asked the students named as Lion and Deer to jump at 3 foot and 2 foot respectively. With the support of meter tape he draws a line and marked 1 feet distance in that line. Instruct the two students to walk side the line with the given distance. He instruct the lion to wait after each step whether his friend can reach at that point or not. They have to put the circular card board on that point, where both meet each other.

  

After walking on the side of the line teacher asked the students..

·         Which are the points where the lion meets?

·         Which are the points where the deer meets?

·         Which are the points at which both lion and deer touches?

So the multiples of 3 are:   3, 6, 9, 12, 15, 18 .....  (Steps covered by lion)

The multiples of 2 are      : 2, 4, 6, 8, 10, 12, 14, 16, 18.....  (Steps covered by lion)

The common multiples are: 6, 13, 18....

The least of the common multiples is: 6

The LCM is named as Least Common Multiples.

All squares are rectangles but not all the rectangles are squares.

It is an interesting statement "all squares are rectangles but not all the rectangles are squares.

I have discussed on the truthness of  this statement with many children of different schools. I got the result NO, the statement is wrong. Because square and rectangle have different definitions and are also different types of quadrilateral (they know the definition of square and rectangle).

As a teacher if we will think on the response of children,

• where is the problem?
• whether the children did not understood the concept or we are unable to teach the concept in a proper manner?

If we will analysis the problem, we will find that children are unable to visualize the relation between both the concepts (WHY?).

Discussion:

Let us take an example of a statement: All the girls are people but not all the people are girls. Is the statement is true? Look at the venn diagram.  Is the representation in true? If yes, then we can say that the statement is true.

Similarly, consider the statement: All the squares are rectangles but not all the rectangles are squares.

If we draw the venn diagram of the statement, we will observe as in fig.

For a square:     all sides are equal in length

                         Angles are right angles.

For a rectangle: opposite sides are of equal length

                        All angles are right angles.

If we compare both the conditions we will observe that square has one more condition (all sides are of equal length). In square also opposite sides are of equal length.

So, the statement is true.

Nature of Mathematics: All the concepts in mathematics should teach in a sequence (hierarchical nature), as there is a linkage of concepts between other concepts.