Fractions Fractions Grade 4 Numbers & Operations -- Fractions

Which Mistake Most Surprises You?

Fraction comparison for 4th Graders. They’ve been working a lot with representing fractions as circles and as rectangles. They’ve done some basic addition with fractions. Most aren’t generally able to find equivalent fractions.


What mistakes do you expect to see in the class set?

Make a prediction! Mark it down somewhere. Don’t do that internet thing of just continuously scrolling through a page at half-attention. Take a moment, form a thought. Then scroll on for the full class set of 14.

In the comments, would you please answer this question: Which mistake most surprised you? Why?

Kid 1

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Kid 2

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Kid 3

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Kid 4

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Kid 5

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Kid 6

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Kid 7

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Kid 8

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Kid 9

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Kid 10

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Kid 11

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Kid 12

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Kid 13

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Kid 14 

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16 replies on “Which Mistake Most Surprises You?”

I wonder why they are drawing a separate picture for each of the two fractions being compared. At this what i would call “non-numerical” level fractions are still parts of the SAME whole, so a single picture is appropriate. later, when they finally realize that fractions are NUMBERS, the number line comes in handy.

What surprised me was that so many of them stopped drawing circles and started drawing bars. Why move between the two representations?

Generally I don’t like the use of circles to represent fractions (because it inhibits understanding of questions like what is 3/5 of 70) but the inability to “extend” the circle certainly helps get the right answer here.

(For the record I predicted that the larger numerator would be the “largest”.)

I think it’s because it’s easy to equally divide a circle by hand into 3 or 4 parts, but 5 and above are hard.

I thought that the last one would be the hardest, but I am most surprised that most said that they are the same.

Chester I know for me, as the numerator increases it becomes easier to divide rectangles than circles.

Given that 3/7 is 0.4285 which is only a small distance from 0.4, visually they are the same.

Without using equivalent fractions or a calculator — and armed with a hand as steady as a fourth grader — I doubt I could tell which was larger.

Huh. I guess it had never occurred to me that they would think two fractions with coprime denominators are the same… I guess they haven’t played so much with different lengths of steps along a line, to see that no amount of 5ths lines up with an amount of 7ths until they reach a 1…

I had predicted that the larger numerator would be called the larger fraction (when the kid didn’t just get it), and the one person who seems to be counting the number of pie-pieces remaining (regardless of their size) after removing the fraction (Kid 11), is not a far cry from that same phenomenon, though that’s a neat perspective and one that could work to the Kid 11’s advantage after some tweaking…

I am very curious about Kid #8, who seems to think that “fractions work backwards”. I wonder if that has to do with their answer to “2/5 or 3/10” – they got it correct, but that answer’s not consistent with their working at all. They drew a larger bar for 3/10 and they thought that 2/5 of 10 is 2, while 3/10 of 10 is clearly 3 – do they think they always have to do the opposite of what makes sense to them?

I was pleasantly surprised to see how effectively drawing the models prevented students from making comparisons based solely on the numerators or solely on the denominators or misjudging whether a larger denominator makes a smaller or larger fraction. Yay! The last problem is a wonderful problem to introduce students to the limitations of this kind of modeling. Looks like the students approached that just right given the skills that they had! This would lead nicely into a discussion of other ways to compare fractions.

Two things surprised me. First, the use of wholes that were not the same size. Problem 3 had several wrong answers because kids needed more space to make 10 parts. Because the whole came out larger, the fractional part looks larger. What a revealing little assessment this has turned out to be! It’s revealed that this key idea of comparing fractions as parts of same-sized wholes is not quite clear yet. The bigger surprise was the student who said that 2/5 of 10 was 2. That was a fascinating approach. Don’t know if he’s trying to find some similar units to compare the fractions or if he’s really thinking about sets. It’s some interesting thinking though that could be promising to follow!

The use of wholes that weren’t the same size surprised me, but it makes sense that a student would do that if the notion of “1 whole thing” isn’t clear to them yet.

What really got me was student 4. Yes, fractions and decimals are both used to represent parts of a whole but they’re not the same thing! Somebody needs to work on decimals, methinks.

Using a square or circle image to compare two different fractions was the most surprising issue for me. Since denominators are different in the questions, it seemed that children would have a hard time to visualize some fractions to compare more accurately. Since everyone’s drawing was not accurate enough to compare the fractions, some children made mistakes just based on the sizes of the drawing. I also figured out that most children had hard time on finding/comparing the fractions for question 3. Some children did not (or could not) even draw a circle or a square because it is difficult to compare the size with the drawings. The ways how the child 6 did was another surprising issue for me. The child already knew the concept of factoring or making same denominators through her/his drawing. Even though I do not know if she exactly knows how to solve these kinds of questions by making the same value for denominators, I could see that she already has a basic sense of doing/working on this kind of fraction questions. The last interesting point was that many children said the fractions for the last question have the same values. I guess the limitation of comparing the fractions with circle/square drawings came out at this point. Since there is (1/35) as a difference between 3/7 and 2/5, it would be difficult for the children to figure out the right answer at this moment.

Based on my observations of the case and experiences, if I was a teacher in this classroom, I would definitely bring an idea of using number lines to compare two different fractions with one model. It would be better for the children to visualize and to figure out the size of each fraction more easily and clearly.

I was wondering if there is another way to support the children to do fraction questions more accurately and easily!

It was actually good to see how students usually compare two fractions by drawing a square or circle. Drawing a circle or square and compare two different fractions somehow indicates that students are aware of the methods to compare these fractions. Even though what students drew were not accurate, I was astonished by their capability of using strategies. What I could observe from students’ mistakes are mostly about making the same value for denominators for two different fractions, because they just draw literally what fractions value is, they just compare what they drew as. For example, if it is 2/5 and 3/4, they just drew 5 boxes filled with 2 boxes and 4 boxes that filled with 3 boxes. After that they just drew horizontally which is actually hard to compare two drawings. Because of these mistakes, most children get the answer for last question as “two fractions are same” which is quite interesting result. For the answer for kid 9, I found out that the kid tends to just circle the answer if the boxes were more filled up with color. From this situation, I realize they are still not familiar factoring the fractions and make the same denominator for comparison.

I was so astonished by almost every student here demonstrate the ability of drawing circles or squares to compare 2 fractions. It seems to me that they all have realized drawing pictures and looking up for amounts or sizes of pieces are shaded are good strategies for solving these kind questions. Even though what they draw are not accurate to compare fractions, the efforts they take and the connections between drawing and fraction comparison they make really surprise me.
It makes sense for me that most children do not realize the way to make the denominator same since they just draw what fraction value is and compare with it directly, for example, for 2/5 and 3/4, kid would just draw 2 shaded out of 5 box and 3 shaded out of 4 box and then compare what he/she drew as. And I can observe some of them even compare the sizes of shaded parts. But, for question 3, I know it is hard and some of them even leave it or cross out the picture they draw. However, I was surprised that even if some kids have no clue/method to compare question 3, they still come up the answer by circling the fraction they thought was bigger, which makes me wonder their logic reasoning behind this guessing.

Therefore, to this point, I think as a teacher, i hope to teach my kids that while comparing fractions, we need to take care of the same-sized wholes.

The image that resonated with me the most was from Kid #11. For me, growing up, I had a really hard time visualizing math. From fractions to decimals, I had to have the teacher write everything down in sentence and word explanations. I was not one of those kids who could just create a rough sketch of squares and circles to demonstrate which one was bigger or smaller.

I believe that what was a mistake that the teacher made was that he or she should have instructed the students to change all of the denominator to make them the same. This lack of instruction prevented the students from learning how to correctly judge fractions by the numerator and denominator. At first, I thought that all of the students were correct on their endeavors. Yet, when I looked closer at one of the student’s work, specially Kid #11, I noticed that he or she misjudged that 3/7 is the same at 2/5. He stated, and attempted to prove it with simple arrays, that the two fractions were the same. When I did the math myself, I found that 3/7 is lager than 2/5 and 3/7 can be transformed into 15/35 and 2/5 is 14/35. Attending to precision, which is the 6th Standards for Mathematical Practice, was not used in this thought process. I wonder why it was not very stressed, but I understand that this was just a short worksheet to probably warm up the students.

When I was in elementary school, my math teacher didn’t really talk about how to draw picture to represent a numerator is bigger than the denominator. Thus, I will say some of the students may make mistake when they need to draw picture to represent this type of fractions. But it seem like the 4th students in this example did well, and understand the concepts. I think drawing circle/ square is a good/ common way for elementary students to compare two fractions. Usually students can find out the answer easily when they compare two pictures. I was surprised when I see most of the students wrote 3/7 and 2/5 are the same (3/7 is actually larger). This question was tricky because the difference between 3/7 and 2/5 are very mild, it is hard to tell the different even though the students draw out pictures to compare. Instead of drawing out circle/ square pictures, I think for this question it will be better to use a number line. By using number line, it will be much easier to compare this type of fractions which their difference are mild.

I would have to say what surprised me the most is the way the students choose to draw the pictures when comparing the two fractions. I think a good way to start comparing this making sure that your whole is the same and that way you have a base and anchor to how ones bigger than the other or not. Also they should stay consistence with the method of drawing they choose, if it circle then stay with that if its squares then that fine. I think it will just confuse them if they keep going back in forth with different shapes until they really understand the whole and how they want to draw it out. I also think it is important for students to finish their work even if they think they know the answer like kid 14 for the ones they thought they knew the answer they only shaded in that one and not the other so technically that not correct interpretation of the fractions.

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