The Meaning of Unit Fractions
This 2nd week I would like for us to look at fractions a little deeper in order to gain a better understanding to help students "get" the foundational understanding of fractions.
1. Watch the following two videos on Unit Fractions:
Meaning of Unit Fractions Video 1
Meaning of Unit Fractions Video 2
(You may want to also look back at the beginning of the progression document where the development of the meaning of fractions and the number line is discussed on page 3)
2. Please complete your post by adding any additional comments on what you have learned today and think is worth noting. Also, please answer the following question: What are important aspects of fractions that provide opportunities for mathematical practice of attending to precision?
These videos were very beneficial.
ReplyDeleteI see the importance of the progression. One thing that really jumped out at me was the importance of making sure the students work with the same wholes when first learning. This will clear up confusion. I also saw the importance of showing students various areas. Objects in the world are not always cookie cutter shapes. In 4th or 5th grade, they should have opportunities to work with various shapes and areas when comparing.
For precision, fractions show that things can be broken down and "guessing" can be taken away. Students should be able to check their work.
I think one of the hardest things for third graders would be initially understanding that when comparing fractions, the size of the whole must be the same. This is difficult for me to think about because fifth graders have this down quite well by the time I see them. I also think that unit fractions on a number line requires more "math sense" than seeing a rectangle divided into four pieces with one colored in. To understand that there is something between zero and 1 is huge! In my mind it is as difficult as asking students in fifth grade to name two decimals between 3.3 and 3.4. It takes practice to "see" that there is more between those two digits! When understanding equivalency becomes second nature, this concept becomes a DUH!
ReplyDeleteWhen comparing fractions, students who look at the denominator and assume the bigger denominator must be the biggest fraction, clearly don't have strong math understanding of what a unit fraction is. Again, this takes a great deal of practice. Much like using a new word! Use it 10 times and it is yours!!!
So, as an answer to your question, I think it's through practice of seeing fractions different ways, working with fractions of different kinds both in line segments and regions in the plane, and understanding that the size of the whole is very important that students will develop precision.
Becky,
DeleteWe were both thinking the same thing about making sure the size of the wholes were the same when comparing.
I think actual objects would be beneficial to use at first when first introducing this to the students. Then they can move the creating their own drawings.
I also think we should show students how they could make a mistake with not using the same size whole. Use the objects to show this and demonstrate. People often learn best when they see a mistake in action.
Becky, I have noticed the same thing about numbers between numbers. The idea that there is something (actually an infinity of somethings) between numbers doesn't come automatically. Many students need to see this in all kinds of ways and places to begin to believe it... Maybe they need more cutting and tearing and folding of things to see that the pieces can keep getting smaller, and the number of pieces keeps getting larger?
DeleteI agree Dana!! The more they see it the more it becomes real and understandable.
DeleteJen,
DeleteYes! The non-examples are so important. I love showing students work that is NOT correct and having them tell me what "the student" did incorrectly. When they can do that - that's when you know they've got "it". Whatever the "it" happens to be:-)
RANDOM INFO: I know this is one line course is all about fractions, but I am in the middle of digging into the Numbers and Base Ten Domain as that is what I always start the year doing! I came across two web sites that I found quite interesting. Don't ask me how I found them. It is always a surprise when I click on something :-) The first site is beyondtraditionalmath.wordpress.com, the second one is thegriddle.net.
ReplyDeleteI was looking for blank number lines and unit squares divided into tenths and hundredths. BINGO. But to my surprise, so much more.
If you have time I would encourage you to take a peek. Or at lease write the sites down so that you can look at them later.
Have a wonderful day everyone!
Thank you for the websites.
DeleteYes, thank you for the websites! I am always looking for new ones to share with my students.
DeleteAwesome Becky!! Im doing another workshop in October on that topic and these will be great to share!!
DeleteThank you for sharing these websites, Becky.
DeleteLike Becky and Jen, I noticed the emphasis on making sure the whole being considered is the same. I think that the way the whole was differently divided was a great concept, too. Different shapes can have the same area might take a bit of convincing for some students, but it would help them to see that. Especially since many seem to lack some spatial math sense, too. Activities that force them to see things differently and manipulate (especially mentally) would probably benefit them later on as well, when they work with geometry.
ReplyDeleteAs for attending to precision opportunities with fractions, communication of the concepts to a classmate or teacher would be a key one: How do they explain the importance of using the same whole, or how unit fractions are made, or how different shapes might have the same area? And in calculations, are they partitioning accurately, are they comparing appropriately, etc.?
As I mentioned previously and the teachers above have mentioned, one of my biggest take-aways is remembering to focus on the size of the wholes. I like Dana's ideas on attending to precision. Having students explain to another takes a higher level of thinking that would be so valuable to them.
ReplyDeleteI think that because I am not in the classroom during direct instruction I miss out on exactly what is being taught and how the teachers are modeling it.
Some use Smartboards with websites and You-Tube. Others are using the text books. This could result in the students not getting all the same information. Because of this I would like to have some of these key concepts added to our curriculum map so that the teachers remember to focus on them.
I cannot wait to share what I have learned through this class so that all of our students can have a better understanding of fractions.
You are all exactly right, precision is definitely the size in which we are comparing is SO VERY important for students to see and understand this, and i think we can all agree that practice is key! I think as a younger student the teacher should always help with the beginning pieces to show the same size and as they progress the students can then begin to do this on their own. All about PROGRESSION!
ReplyDeleteTwo important concepts from the videos that I noticed include: 1. Highlighting the importance of the whole for students. It is such a basic concept but also such a common misconception for students. It would be beneficial to invest time in some concrete activities that would support instilling this concept in students. 2. The second concept that was mentioned in both videos is the ability for equal fractions that are represented by area models to be NON-congruent. This could really be a fun lesson to teach to third grade students. They would enjoy creating and coloring different designs that represent this concept.
ReplyDeleteGood points Heidi. I love it when we can do anything concrete with the students and let them think and explore. As you know, the modules suggest a couple of really good exploration activities with several different mediums (clay, colored water, paper, string...) in order to have the students display unit fractions. The kids love this activity. Honestly, I bypassed this lesson the first year I used the modules because I was trying to "fit everything in", but after you pointed out the usefulness of these lessons, I would never skip these again. And the kids really enjoy being math "scientists"!
DeleteHeidi, that NON-congruent model is so difficult to see for some students (and me sometimes). It's fun to watch them cut apart models to see, "oh yeah!"
DeleteI agree the non-congruent models are definitely much more challenging. I, thankfully, do not believe I have seen these in 3rd grade but correct me if I am wrong, please.
DeleteAs for precision, third graders need to be math artists! Having them focus on creating neat and organized models for fractions is so important. Also, third grade is the perfect place to teach them how to PLAN and use their work properly so their final answer can be easily communicated to a peer.
ReplyDeleteI agree Heidi. Mathematics needs to be neat and precise!
DeleteGreat point Heidi!!
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DeleteI like how the video pointed out that students have been exposed to a whole number as the sum of units (5=1+1+1+1+1). I am going to build on this concept as I teach unit fractions to my third graders, as this can be tricky for some!
ReplyDeleteThe concept of starting with the same whole is an important one for third graders. I try to use concrete examples of why, showing how unfair it would be to share half of a delicious large candy bar or half of a snack size candy bar.
I also let the students draw their fraction pictures to "prove" their comparisons using any of the strategies they have been taught, but I stress the importance of being precise in their drawings paying attention to the size of the wholes and drawing like a math artist so that any onlooker can understand their hard work!
Having the students draw fraction pictures to make comparisons is an excellent way to help students visually comprehend the meaning of fractional parts and determine which one is larger.
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ReplyDeleteI thought it was important to emphasize in third grade that the whole unit must be the same size as another when comparing fractions. I think it is a common mistake on the part of the students to look at the larger denominator and think that it is larger. For example 1/2 is smaller than 1/3. The progression of these skills requires precision in each grade level. I also think it is important to show students various ways that fractions can be represented such as in pictures, number lines, or sets.
ReplyDeleteI agree that one common misconception students develop early on is that, when comparing equally sized wholes, the greater the denominator the greater the amount represented. Using visual models (and manipulatives) students begin to discover that unit fractions with greater denominators represent "lesser" pieces of that whole.
DeleteI agree with so much of what has been shared here; especially the need (or "issue" as it was referred to in the second video) to compare the size of the wholes. It seems that comparing the size of the whole should be step one on any list created to help students tackle fractions! I am a big believer in connecting math with the outside world whenever possible. However, I wonder if we are forgetting that students may not always remember that the world is full of inconsistencies. (Not all pizzas, pies, oranges are created equal.) With that in mind, I believe studying fractions provides a beautiful opportunity to attend to precision. For example, students grasp early on that if I have 2 cookies and you have 2 cookies, we each have the same number of cookies. However, when we begin discussing fractions, students learn that if I have 1/2 a cookie and you have 1/2 a cookie, we do not necessarily have the same amount.
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