More and more evidence suggests that visualization helps children develop a more conceptual understanding of mathematics. Division involving fractions seems to remain a stumbling block for many. Liping Ma noted in her 1999 work Knowing and Teaching Elementary Mathematics that teachers in the United States had difficulty generating situations that would involve division with fractions - such as sharing the remaining portion of an apple pie or cutting a piece of rope. It was in teaching fraction division that I realized how fragile my own understanding was - and how much I needed to listen to my students and work to connect their thinking. "When we don't ask students to think visually, we miss an incredible opportunity to increase students' understanding and to enable important brain crossing." ~Jo Boaler, Visual Math Improves Math Performance I began with thinking about division with whole numbers. What did we know about division with whole numbers and how might that apply to division involving fractions? Partitive division seemed easiest to model for my students. We began with friendly numbers that we could represent with fraction circles, such as 3/4 divided by 1/3: Students noticed quickly they could overlap the blue pieces on top of the 3/4 to see that 2 blue pieces (each representing 1/3 of a whole circle) would fit. They discovered that the green piece (representing 1/12 of the whole circle) would fill the remaining space. We then needed to figure out how to describe that amount since it no longer represented 1/12 with this model. Students worked to determine that four of the green pieces would equal a blue piece - deciding that the green piece represents 1/4 of the blue piece. We then connected the fraction pieces to symbols, talking about the use of common denominators: As our exploration continued, we wondered whether this method would work with fractions greater than one. In addition to fraction pieces and Cuisenaire rods, we considered tools such as number lines: Using a variety of tools and representations allowed time and space for students to link the meaning of division with the symbolic notation. They had a mental image to manipulate and connect with expressions. And, ultimately, WHY does "keep, change, flip" or "invert and multiply" work? Can we visualize this? This is one way I've thought about it using 3/4 divided by 2/3. I was intentional in using numbers that would be friendly in using an area model but this method would work with any numbers - it just might become too cumbersome to draw! If we multiply both the dividend and divisor by 3/2 (the reciprocal of the divisor), we maintain the relationship between the quantities. We can confirm the quotient of 9/8 using the idea of common denominators described above: I'd love to hear your thoughts about this progression. What are ways it might benefit student thinking? What might potentially interfere with student understanding?
21 Comments
9/11/2022 04:44:48 pm
Really informative article, I had the opportunity to learn a lot, thank you. https://kurma.website/
Reply
9/12/2022 03:51:21 am
Really informative article, I had the opportunity to learn a lot, thank you. https://odemebozdurma.com/
Reply
9/14/2022 07:38:09 pm
Really informative article, I had the opportunity to learn a lot, thank you. https://bit.ly/site-kurma
Reply
9/30/2022 06:14:27 am
It's great to have this type of content. Good luck with your spirit. Thank you. https://bit.ly/site-kurma
Reply
10/5/2022 12:06:55 am
I think this post is useful for people. It has been very useful for me. Looking forward to the next one, thank you. https://escortnova.com/escort-ilanlari/duzce-escort/kaynasli-escort/
Reply
10/5/2022 02:31:53 pm
It was a post that I found very successful. Good luck to you. https://escortnova.com/escort-ilanlari/amasya-escort/suluova-escort/
Reply
10/6/2022 02:26:09 am
I follow your posts closely. I can find it thanks to your reliable share. Thank you. https://escortnova.com/escort-ilanlari/van-escort/gevas-escort/
Reply
10/6/2022 01:31:34 pm
I support your continuation of your posts. I will be happy as new posts come. Thank you. https://escortnova.com/escort-ilanlari/kirsehir-escort/akpinar-escort/
Reply
10/8/2022 03:03:04 am
Thank you for your sharing. I must say that I am successful in your content. https://escortnova.com/escort-ilanlari/hakkari-escort/cukurca-escort/
Reply
10/8/2022 02:34:53 pm
Thoughtful and real content is shared. Thank you for these shares. https://escortnova.com/escort-ilanlari/sirnak-escort/
Reply
12/10/2022 08:34:00 am
Düşmeyen bot takipçi satın al: https://takipcialdim.com/
Reply
12/10/2022 11:04:35 am
Tiktok takipçi satışı burada: https://takipcialdim.com/tiktok-takipci-satin-al/
Reply
1/8/2023 10:07:07 pm
100 tl deneme bonusu veren siteleri öğrenmek istiyorsan tıkla.
Reply
Leave a Reply. |
AuthorSharing what I'm learning from others. Collaborative experiences and communication are essential in building a shared understanding of mathematics! Archives
September 2018
Categories |