We all know that 12/25 + 18/30 is not 30/55 yet students are happily told that they got 12 out of 25 in their first maths test and 18 out of 30 in their second maths test giving a total of 30 out of 55 So in this case 12/25 and 18/30 is 30/55 So should we be surprised when students say 2/5 + 3/4 is 5/9 ?
Err, 12/25 can't be added to 18/30 just like that :D Fractions are part of a whole. The two tests are 2 different wholes. I was taught this as a kid - they said "what would I get if I cut a cake and a pie in half and added the halves together?" I said, you would get a half of the pie and a half of the cake. In your example, the two tests have different maximum points, so 1 point is worth more in 1 test than in the other - that's why you use fractions -> 12/25 < half, 18/30 > half. On average the tests have 54/100 correct results or 54%. I don't know man, the most obvious stuff in maths, is the most difficult to explain :/
Well in my opinion, this indicates either a tremendous misunderstanding, or miscommunication between the teacher and the student. If the students were properly taught, they would understand that the two fractions (or ratios) could not be added in that manner. Even the most contemporary teachers introduce the 'method' of utilizing the LCD in order to add the fractions. However, as a teacher, you'd have to explain that adding these two fractions is equivalent to adding for instance: 1/4 to 1/3. Both fractions are basically division equations. So you would start by explaining that (after going through the process of division), .25 =/= .33, in the same way .48=/=.60. Because of this, you cannot simply add the numerator across, as well as the denominator! So similar to your students' claim, 1 slice out of 2 slices of an orange plus 2 slices out of 8 slices of an orange equals 3 slices out of 10 slices of an orange is completely incorrect! You can even picture this pictorially. Because a fraction is another way of explaining an equation of division, you cannot possibly add the two fractions provided the top in that way. The point that I'm making, is that it's the teachers' responsibility to point this crucial aspect out.