Fractions same or equal?

Is the fraction 2/4 equal to or the same as 1/2?

2/4 and 1/2 are

1. Equal but not the same.
2. Equal and the same.

2/4 and 1/2 are

Equal and different

1. Equal and different.
2. Equal and not different.

2/4 and 1/2 are

1. The same and different.
2. Not the same and different.

2/4 and 1/2

1. Have the same action but are different.
2. Have different actions that produce the same number.

So 2/4 acting on 8 is 2/4 acting on 2*4 is 4

While 1/2 acting on 8 is is 1/2 acting on 2*4 is 4.

2/4 acting on 8 cancels out the 4 in 8 and multiplies the 2 left in 8

by 2 to get 4?

1/2 acting on 8 cancels out the 2 leaving the 4?

1. Are the same as operators.
2. Are equal as operators.

We can restrict two fractions to acting on multiples of their denominator.

We could also think of numbers as fully factored.  That could apply to the acted on numbers, the test numbers, as well as the fractions as an operator.

Let us agree to call restricted equality of operators the case when the operators produce the same effect acting on a restricted set, called the test set.

(Weak equality can mean equal if both exist, so this can be considered a version of weak equality.  If we apply fractions to numbers that are multiples of their denominator, then the result exists as a natural number. We can then apply equality of natural numbers to such results.)

So we can take two fractions m/n and p/q and they are equal if acting on numbers that are a multiple of both n and q they produce the same result. Acting on means multiplying in effect.  Note test numbers need not be a multiple of n times q if n and q share any common factors.

If we fully factor the test numbers, why not just factor the numbers in the fractions and cancel out?

2/4 times 8, we can first write as 2/(2*2) * 8, cancel out in the fraction andget 1/2 * 8 which then is 4.

Is 2/4 the same operator as 1/2?  If we apply them to factored numbers as test data?  Do we factor the 4 in 2/4 first?  Then cancel with the 2?  In that case, the procedure would be the same applied to a factored test number.

Does this matter?  All that matters is the same number results.  For sameness or for equality?  If we do two different procedures that produce the same number, are the two procedures the same?  Or do the two procedures have the same result?  Or they have equal results, if they produce different expressions which are equal?

2+2 = 4

Is 2+2 the same as 4?  Or are they just equal?  Or both?

2/4 and 1/2 are the same fraction?  Or they are equivalent in their effect?  So they are the same operator?  Does that depend on how we actually apply them as operators?  They are equal as operators?

(2,4) and (1,2) are different ordered pairs?  But the same rational number?

2/4 and 1/2 are different fractions?  But the same rational number?

(2,4) and 2/4 are the same fraction?

Is the following correct?

1. (2,4) and 2/4 are the same fraction, and
2. (1,2) and 1/2 are the same fraction, and
3. (1,2) is different than (2,4), and
4. 1/2 and 2/4 are the same fraction

Or is the following correct?

1. (2,4) and 2/4 are the same fraction, and
2. (1,2) and 1/2 are the same fraction, and
3. (1,2) is different than (2,4), and
4. 1/2 and 2/4 are different fractions

In classrooms, 2/4 and 1/2 are called the same by teachers?

Is there anyway to say that 1/2 and 2/4 are different?

1/2 and 2/4 are

1. The same fraction
2. But somehow different?
3. What are they different as?

If 1/2 and 2/4 are the same fraction, then what is it that 1/2 is called by itself as distinct from 2/4?

Why this is so important?   The logic of ordered pairs and of fractions is simple if we have simple names and rules.   If we confound these by typical classroom language, then we confound the concepts as well.

If 1/2 and 2/4 are the same fraction, then we need a name for what 1/2 and 2/4 are that are not the same.  Pre-fractions.  Fraction representatives.

If 1/2 and 2/4 are different fractions, but the same rational number, then we have the terminology to explain the logical structure. But if teachers and parents are going to call them the same, we can’t overcome it.  Thus students lack the language to hear the precision of the 19th century concept of an ordered pair as a representative of a rational number.

Pre-rationals and pre-fractions have the advantage that no one has used them. So we can say that 1/2 and 2/4 are pre-fractions that are different.   They are the same fraction but are different pre-fractions.  Fraction representative is an alternative term.  Fraction Rep can be the short version.

If this language is confused, the concept of fraction is confused and thus the student can’t learn the precise definition or meaning.  That means they miss the concept from sloppy language and the lack of precise terms.

A standards group is useful after this problem is exposed and discussed.  If it blunders in too early, it prevents developing the correct solution.