Shouldn’t the correct answers be simple fractions in lowest terms?

I like it! It shows some good reasoning about fractions, with the understanding that an eighth is half of a fourth. Very nice! It’d be good to multiply top and bottom by 2 at the end, and to show some explanation of where those answers come from, but still, this is pretty neat.

The third is wrong, and I think it might indicate a big issue with letting them work with this more complicated notation instead of making things streamlined. The best thing about that work was the observation that 1/2 + 2/8 = 3/4, presumably made by working with a unit of fourths. Then when trying to add that last eighth back in, the big blunder happened. I think the student is clearly confused about the meaning of this notation and applied a tempting “symbol-pushing” rule to perform the final addition rather than reasoning out what the fractions being added represent.

Perhaps the first two are indicative of good reasoning. Its hard to say without seeing any work. I certainly approve of letting students devise and reason with this more cumbersome notation, but much of mathematical notion that has survived through the ages has done so because of the particular efficiency it provides once its proper use is learned. Perhaps after working with the other notation for a while, students might begin to appreciate the efficiency of standard fraction notation.

The procedure for adding fractions is very simple: 1) convert to a common denominator, 2) add the numerators, 3) reduce to lowest terms. For problem #3 this gives you: 1/2+3/8 = 4/8+3/8 = 7/8. What could be simpler? How could the students miss it?

The first student didn’t show any work, and it’s possible they used this basic procedure but converted to a common denominator of 4 (why?) which produced the complex fraction. But the second student did show some work which suggests they used a more complicated procedure. It looks as if they split the 3/8 into 2/8+1/8, reduced the 2/8 to 1/4, then added it to 1/2 giving 3/4. In other words, they did this: 1/2+3/8 = 1/2+(2/8+1/8) = 1/2+(1/4+1/8) = (1/2+1/4)+1/8 = (2/4+1/4)+1/8 = 3/4+1/8. Then they added the 1/8, mistakenly adding it to the numerator to get (3+1/8)/4. The first student didn’t show their work, but it’s possible they followed the same procedure, except that they converted the 1/8 to the complex fraction (1/2)/4 before adding it at the end.

Why would the students use this convoluted procedure? Is it possible that they used a common denominator of 4 because they have seen many examples of addition using fourths, which are easily visualized with squares or circles divided into quadrants?

I wonder how these students would handle 1/3+2/5?

The procedure for adding fractions is very simple…Why would the students use this convoluted procedure?

I’d say that this post is evidence that the procedure for adding fractions isn’t so simple to a 4th Grader.

But that would be a bit unfair since, yeah, it is a simple procedure, once you have a sophisticated notion of equivalence. But that’s an important rider — you need to have a sophisticated notion of equivalence, and these kids obviously don’t.

What you see in this batch of student work, I’d say, is that sense of equivalence in transit. You see kids trying to figure out how many fourths 3/8 would be equal to. Sure, we could tell these kids how to find equivalent fractions and they’d be doing these problems cleanly in an hour. But, frankly, who cares if these kids know how to add fractions of any denominator efficiently in 4th Grade? A lot of these students are still working out issues with the language and basic concepts of fractions, and these problems are giving them a great chance to work out those issues, and giving them informal experience with something that we can formalize with them basically whenever we want to.

I have to admit it’s been a very long time since I was a fourth-grader. I don’t remember much about it, but I’m pretty sure back then the teachers cared whether students could do math accurately and efficiently. Times have changed, I guess. I definitely agree that students need to understand equivalent fractions before they try to understand adding fractions. Not sure how much sophistication is required. Fractions are equivalent if they represent the same portion of the whole. The key observation, found by subdivision, is that if you multiply the numerator and denominator of a fraction by the same number, the resulting fraction will be equivalent to the first. That tells you how to convert fractions to a common denominator, which allows you to progress from addition of fractions with like denominators to addition of fractions with unlike denominators. That knowledge may be what the students are missing here.

Making all denominators equal to 4, gives 1/4 + 1,5/4 + 1/4. So total is 3,5/4. (1,5 = 1 1/2, 3,5 = 3 1/2). Not the simpler result though as right as 7/8.

## 7 replies on “3 and a half fourths”

Shouldn’t the correct answers be simple fractions in lowest terms?

I like it! It shows some good reasoning about fractions, with the understanding that an eighth is half of a fourth. Very nice! It’d be good to multiply top and bottom by 2 at the end, and to show some explanation of where those answers come from, but still, this is pretty neat.

The third is wrong, and I think it might indicate a big issue with letting them work with this more complicated notation instead of making things streamlined. The best thing about that work was the observation that 1/2 + 2/8 = 3/4, presumably made by working with a unit of fourths. Then when trying to add that last eighth back in, the big blunder happened. I think the student is clearly confused about the meaning of this notation and applied a tempting “symbol-pushing” rule to perform the final addition rather than reasoning out what the fractions being added represent.

Perhaps the first two are indicative of good reasoning. Its hard to say without seeing any work. I certainly approve of letting students devise and reason with this more cumbersome notation, but much of mathematical notion that has survived through the ages has done so because of the particular efficiency it provides once its proper use is learned. Perhaps after working with the other notation for a while, students might begin to appreciate the efficiency of standard fraction notation.

The procedure for adding fractions is very simple: 1) convert to a common denominator, 2) add the numerators, 3) reduce to lowest terms. For problem #3 this gives you: 1/2+3/8 = 4/8+3/8 = 7/8. What could be simpler? How could the students miss it?

The first student didn’t show any work, and it’s possible they used this basic procedure but converted to a common denominator of 4 (why?) which produced the complex fraction. But the second student did show some work which suggests they used a more complicated procedure. It looks as if they split the 3/8 into 2/8+1/8, reduced the 2/8 to 1/4, then added it to 1/2 giving 3/4. In other words, they did this: 1/2+3/8 = 1/2+(2/8+1/8) = 1/2+(1/4+1/8) = (1/2+1/4)+1/8 = (2/4+1/4)+1/8 = 3/4+1/8. Then they added the 1/8, mistakenly adding it to the numerator to get (3+1/8)/4. The first student didn’t show their work, but it’s possible they followed the same procedure, except that they converted the 1/8 to the complex fraction (1/2)/4 before adding it at the end.

Why would the students use this convoluted procedure? Is it possible that they used a common denominator of 4 because they have seen many examples of addition using fourths, which are easily visualized with squares or circles divided into quadrants?

I wonder how these students would handle 1/3+2/5?

I’d say that this post is evidence that the procedure for adding fractions

isn’tso simple to a 4th Grader.But that would be a bit unfair since, yeah, it is a simple procedure, once you have a sophisticated notion of equivalence. But that’s an important rider — you need to have a sophisticated notion of equivalence, and these kids obviously don’t.

What you see in this batch of student work, I’d say, is that sense of equivalence in transit. You see kids trying to figure out how many fourths 3/8 would be equal to. Sure, we could tell these kids how to find equivalent fractions and they’d be doing these problems cleanly in an hour. But, frankly, who cares if these kids know how to add fractions of any denominator efficiently in 4th Grade? A lot of these students are still working out issues with the language and basic concepts of fractions, and these problems are giving them a great chance to work out those issues, and giving them informal experience with something that we can formalize with them basically whenever we want to.

I have to admit it’s been a very long time since I was a fourth-grader. I don’t remember much about it, but I’m pretty sure back then the teachers cared whether students could do math accurately and efficiently. Times have changed, I guess. I definitely agree that students need to understand equivalent fractions before they try to understand adding fractions. Not sure how much sophistication is required. Fractions are equivalent if they represent the same portion of the whole. The key observation, found by subdivision, is that if you multiply the numerator and denominator of a fraction by the same number, the resulting fraction will be equivalent to the first. That tells you how to convert fractions to a common denominator, which allows you to progress from addition of fractions with like denominators to addition of fractions with unlike denominators. That knowledge may be what the students are missing here.

Making all denominators equal to 4, gives 1/4 + 1,5/4 + 1/4. So total is 3,5/4. (1,5 = 1 1/2, 3,5 = 3 1/2). Not the simpler result though as right as 7/8.