Equivalent Fractions and Comparing Fractions
This post will require two comments, label within your comment 1 and 2 so that it is easier to follow
1. Watch the following video on equivalent fractions:
Equivalent Fractions Video
Answer the following question in your comment: How can the use of area models and number line diagrams solidify a students understanding of fractions?
2. Watch the following video on comparing fractions:Comparing Fractions Video
After completing the above exercise, answer the following question in your comment: What rules about the relative sizes of fractions can you state from these examples? Be precise in expressing rules trying not to use words such as; "numerator," "denominator," "top number," or "bottom number."
Comment 1 (Equivalent Fractions)
ReplyDeleteArea models show that the quantity of shaded area does not change even though the fraction is written is different numbers.
The number line shows that the same point on the line can have different names.
I also liked the use of folded paper with shading. The students could see again that the amount of shading does not change, just the numbers.
I like the use of area models both for the fractions, as well as, multiplication. Our students find the area model easy to understand since they can see what is actually going on!
DeleteI agree with you that folded paper presents a wonderful opportunity for students to investigate fractions. Folded paper also serves as an excellent tool because it only requires 1 piece of paper and a pre-determined amount of colored pencils for shading. I say this because some students can quickly become overwhelmed with (or distracted by) fraction tiles.
DeleteComment 2 (Comparing Fractions)
ReplyDeleteHope I am correct in my answer here and what the question is asking. Please offer help if needed. Tried not to use numerator and denominator.
In the video, there was a great explanation of what each number represents...
A/B
A = how many pieces we have
B = number of pieces that make a whole
I think if students have this engrained in their head, the will stop trying to immediately compare the numbers.
There was also a general rule stated as
a/5 > a/12
so for the first example of 4/5, 4/7 this would work because 4/5 > 4/7 when plotted on the number line.
Perfect!
DeleteA. This video discussed the different ways you can have the students show equivalent fractions by using the number line or tape diagrams and having them actually show the work themselves. They need to understand and show how to break up a whole into the unit fractions either by folding a paper (for the tape diagram) or showing it on the number line and writing the equivalences on them. This helps to solidify this for the students because they get to see the actual fractions that are equal because the are at the same spot on the number line or same area in the tape diagram.
ReplyDeleteSorry, this is Comment 1, not A.
ReplyDeleteComment B. First I would like to say that some of these ways to explain fractions are totally new to me and I can see where they will help some of my struggling students.
ReplyDeleteFor the comparison of 4/5 and 4/7: because you have the same number of unit fractions, the students need to look at the size of the unit fraction. In this case, the unit fraction 1/5 is larger than the unit fraction 1/7, therefore making 4/5 a larger fraction than 4/7. It was also described as 5 long steps across the number line as compared to 7 short steps. Students will like that one I think!
For the comparison of 5/6 and 7/8: This is described as fractions that are one short of a whole. Students need to visualize or draw tape diagrams divided into the unit fractions for each one. Since 7/8 needs a shorter piece to complete the whole, then 7/8 is greater than 5/6. So 'kid friendly' in its explanation!
For the comparison of 3/8 and 2/9: Going to say this is about the Benchmark fractions. Since 3/8 is one unit fraction short of the benchmark fraction 1/2 and 2/9 is one unit fraction short of the benchmark fraction of 1/3, then 3/8 must be larger than 2/9 because 1/2 is larger than 1/3.
And if all else fails you can find common numerators, hadn't thought of that in the past, or common denominators!
I will definitely be referring to this video in the future!
Great thing to point out, always try to relate to the benchmark fractions when applicable!
Deletekatydid
DeleteI never heard of - or thought of common numerators either. But it makes sense.
I'm thinking about all these new things I'm learning and how at times I need to rewind the video and slow down and really think about it as well as try different examples on my scrap paper. As I have really good number sense! Imagine those kiddos.
The use of area models and number line diagrams solidify a students understanding of fractions by giving them a concrete experience to build on. At the third grade level, area models are a nice beginning but as you move towards number lines some accuracy in drawing also needs to be the focus. Third graders need to organize their space and write small!! to fit all their fractions amounts in the number line space.
ReplyDeleteFor the second question I would like to comment on something different. I really like the "missing pieces strategy" I have discussed this thinking with 5th graders in the past, but I have never seen it officially mentioned before. The video is correct in saying that kids really need to have a good understanding of "number sense" to quickly grab onto this strategy.
I agree Heidi! Thank you for pointing it out, it is definitely worth mentioning and needs to be used more in classroom instruction.
DeleteHeidi,
DeleteYou brought up a great point that third graders need to organize their space. Many of us, including myself, have said that sometimes students get confused because they're drawings are not accurate. We need to make sure they can do this instead of just saying well they're just messy drawings and maybe we should just use a template. Kids can do it if they are expected to do it.
It is difficult to make a number line accurately - especially if you try to partition one number line two different ways. (I tried this for one of the exercises). But we can teach them a few strategies that help make it at least a little easier, like using a different color or symbol for different partitions, and for even denominators starting the partitioning in the center (1/2) and partitioning from there... Like Jen said, they can do it if we give them time and tools...We tell them "difficult" is not equal to "can't", but sometimes our time crunch forces us to skip the difficult and leave them believing they actually can't. :(
DeleteComment 1 (Equivalent Fractions)
ReplyDeleteBy using the number line and bar models to help students understand fractions the students are able to develop a better understanding using two methods. One thing that stood out during the video is when they were comparing equivalent fractions, they used two examples for each one. In other words, they did not just ask what one fraction is equivalent to 1/2. I felt by providing at least three examples for each, led to a better understanding of the skill. I know that Pearson does exactly this on their homework page. I have been guilty of allowing the students to provide one equivalent fraction at times. This makes me think it might be better to assign less problems, while requiring the students to show more equivalent fractions. The reason being it is a long homework page normally. I have also used paper folding as an example of finding equivalent fractions. The students are surprised at what they have discovered.
I agree Joanne, some times less is more! The idea of homework has definitely shifted with the new common core standards
DeleteThis comment has been removed by the author.
ReplyDeleteComment 2 (Comparing Fractions)
ReplyDeleteWhen looking at the first two set of fractions, the students could look at the values/numbers that are the same and then just focus on the two numbers that are not to determine that 4/5 is larger than 4/7, because 1/5 is larger than 1/7.
For the last two sets of numbers I would use the number line or bar models to provide a more visual comparison as shown in the videos to determine the larger fractions. Using benchmark fractions and equivalent fractions should also be emphasized when making theses comparisons. I did like the use of the missing piece strategy to have the students visually make comparisons. I always go back to pizza when teaching fractions and ask the students which piece would they rather have.
Comment 1 I love the use of the shaded piece then folding the paper to see that the amount has not changed but the number of pieces in that shaded piece has changed. This is huge! Personally, I'm still needing to see the number line video multiple times to understand it myself :-) I am going to have to practice on that before I lead anyone astray.
ReplyDeleteIn third grade, the modules use the shaded strips of paper and folding in several lessons and this seems to be a real concrete activity as the students learn to compare fractions!
DeleteComment 1
ReplyDeletea. 4/5, 4/7 Since there are four pieces in each example, I know that 4 out of 5 pieces is more than 4 out of 7 pieces
b. 5/6, 7/8 Both of these fractions are one piece away from being one whole. But 1 piece out of six is larger than one piece out of 8. (Assuming the wholes we are comparing are the same :-)
c. 3/8, 2/9 By using the benchmark 1/2 I know that 3/8 is almost 4/8 or 1/2, 2/9 is much farther away from 1/2 on the number line so I know that 3/8 is larger than 2/9.
To answer the first question: I think area models and number lines force students to see fractions for what they are in real life - from the abstract/complex to the concrete/simple: a fraction is a number representing the relationship between two numbers. That sounds/is complex, so it needs to be exemplified in concrete and pictorial ways. Though fractions aren't always a part/whole relationship, this is a major part of what they are, so modeling helps make this apparent.
ReplyDeleteTo answer the second question:
ReplyDeletea. 4/7 is less than 4/5 because when I broke/partitioned a number line into 7 pieces and took four jumps, I didn't go as far as when I broke/partitioned the same number line into 5 pieces and took four jumps.
b.5/6 is less than 7/8 because both are one piece less than a whole. 1/6 is larger than 1/8, so there is a bigger piece missing from the whole. That means that 5/6 is smaller since a bigger part of the whole is missing.
c. 3/8 is greater than 2/9. 2/8 (1/4) is a benchmark fraction. 3/8 is more than 2/8 because they both have the same number of pieces to make a whole (8), but 3/8 has more pieces out of the whole than 2/8 does. But 2/9 is less than 2/8 because it has the same number of pieces out of the whole (2), but the number of pieces that makes the whole is greater for 2/9 than it is for 2/8.
I think giving the students many opportunities to represent fractions is so important. I like the use of the area model as well as the number line. Again, I understand the importance of the number line, but it is very difficult for third graders to divide one number line into different unit fractions. Even using colors, etc, some kids, like my special ed or OT friends really struggle. However, I have found the paper folding technique a very eye opening experience for third graders. This activity is used often in the modules.
ReplyDeleteAs far as the fraction examples...
ReplyDeleteExample a : students can relate to the size of the pieces to compare
Example b: Students can relate to the size of the "missing piece" - I really like this strategy and I have not used it before, but I will try it now, especially since the kids are getting so proficient with unit fraction comparisons!
Example C: eighths are bigger pieces than 9ths and you have more pieces!
1. Area models and line diagrams can help to solidify a students' understanding of fractions by allowing students to manipulate and visually investigate the meaning of fractions. Especially in the pre/early multiplication stages, students may be unaware or confused by how a "different" fractions can represent the same amount. Creating an area model or a line diagram allows the student to discover the connection between, for example, 1/2 and 2/4 with more understanding and confidence.
ReplyDelete2. When considering the "rules" about relative sizes of fractions, I would say that Example A invites students to notice that 4/5 is closer to the whole than 4/7. As such, students would conclude that 4/5 is greater. Example B invites students to scale each fraction back to its unit fraction; 1/6, 1/8. As 1/6 is greater than 1/8, students can determine that 7/8 is greater because it is closer to the whole than 5/6. Finally, Example C invites students to once again compare unit fractions -1/8 vs 1/9. As 1/8 is greater than 1/9, and there are more pieces in 3/8 than 2/9, the student can determine that 3/8 is greater.
ReplyDelete