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.