Multiplying and Dividing Fractions
Watch the following videos:
This may be a lot to take in for one day... as before we could talk about this for an entire two week workshop.
Here are a few questions to answer in your comment for today: (you don't have to answer them all)
- Discuss one advantage or one disadvantage of using an area model when multiplying fractions.
- Discuss an idea you learned from these videos and could apply in your classroom.
- Discuss some problems students may have and ideas on how to help them based on what you have learned thus far.
- Look back at the Progression document for 4th and 5th grade fractions and explain how creating a story/real-world context might help assist a student in understanding fraction multiplication.
- Look back at the Progression document for 4th and 5th grade fractions and explain how creating a story/real-world context might help assist a student in understanding fraction multiplication.
For video 1...
ReplyDeleteOne advantage of using an area model for multiplying fractions is to show that the fraction x whole number is a part of the whole number. I never thought of it like that. (1/2x5 is 1/2 of 5)
One disadvantage is deciding which number goes where on each side of the area model with correct shading.
Also real world context would definitely help such as slicing apples.
In video 2...
I definitely saw a progression of thinking about equal sized groups that goes back to 2nd grade. And again the importance of the word "of" (2/5 of 2/4 is 2/5x3/4)
In video 3...
Real-world context definitely helps with division. Division can really be though of as "sharing." In the video, they used the example of sharing sandwiches between students. This could even be played out in real life with real sandwiches. Students can associate with real life sharing as opposed to just shaded boxes.
One of the first problems my fifth graders have when we start multiplying a whole number by a fraction is that the quotient is less than the whole number. This throws them for a loop every year. I ask why they think this happens and let them discuss in groups. Most years it is THAT student who is an outside the box thinker, and has a very good number sense, that can explain that a fraction is less than one. When you multiply a number by one the quotient is always that number. When you multiply a number by less than one (the fraction) it stands to reason that the quotient will be less than the whole number. (Then there are some years where it's like pulling teeth:-)
ReplyDeleteOnce students can get comfortable with that idea, they are able to check for reasonableness of an answer.
When multiplying fraction by fraction, I've always used a model and had students use two different colored pencils coloring one fraction going across the model in one color and the other fraction the other way. The part of the model that is colored with both colors is the answer. Students LOVE doing this and seeing it! Errors occur when they don't count the blocks carefully and color too many pieces.
In viewing the video I can see the progression from second to fifth grade. As a fifth grade teacher I see the need for those teaching below me to make sure that students get their fractional number sense to a place where it becomes second nature for them.
Dividing a whole number by a fraction can again be a bit confusing to students if they don't understand what is happening. For example: 5 divided by 1/3. It takes many examples for students to understand that five whole somethings are dividing into thirds and when you count those thirds there are 15 of them. Yep, mind blowing to those little ones. Turn the problem around and divide 1/3 by 5 and .... well let's just say it's a full week but after all our practice they can do it and better yet explain it!
I marvel at the human brain and how it can be stretched.
Becky,
ReplyDeleteI really liked your idea of using the colored pencils. I could see this being used any time there is shading needed too instead of just pencil markings. Great idea.
Becky, I also like the idea of the colored pencils. Anytime we can make math more fun, even if it is just colored pencils, the students love it! Thanks for the good idea!
DeleteAgree!! That was actually going to be one of my suggestions! Thanks Becky!!
DeleteI am a huge proponent of the area model for multiplication. In the first video I thought that how it was described was very confusing. Obviously, this is for educators though, and for students we would go into much more modeling. The second video did a much better job of showing how the area model works and why it works for fractions too. An advantage of the area model is that it is so visual and the students can actually see why the fractions equal what they do.
ReplyDeleteStudents in our district should be able to understand the area model with fractions as we begin using the area model in third grade. Thankfully we had Sue R., our 'math guru', come into the district for curriculum work. She helped us to learn and utilize this strategy.
A story/real word context may help students because they can see an example of something that they most likely have experienced. For example, using candy and grouping it to show multiplication, or 3 times as many. When we taught the bar model for multiplication of whole numbers, we had the student write their own word problems. They loved this. Having them do this for multiplication of fractions would also be valuable because they could make their own scenarios and be more able to relate to it and understand it.
One area I would like to have seen with fractions is on the ruler. I know this is in the measurement standard, but our students have such trouble reading a ruler and connecting the measuring of fractions to the fractions in multiplication. Although in a way I already do this, I think I will make a larger connection between a bar model and dividing it up into parts and connecting it to the ruler that is divided into parts.
Thank you, Jessi, and all of my classmates, for all of the great ideas and web information that you have shared!
Thank you!! And you bring up a very interesting component, i would love to see the visual of a ruler being used more, that is definitely an area they really struggle with, plus what a great way to get science , technology and art teachers involved too!!
DeleteUsing the area model to illustrate multiplication of fractions seems like a natural progression of what is taught in third grade (relating area to multiplication). In third grade we stress the words "groups of" and "rows of" when multiplying, so I think this model would make sense for students. The disadvantage would be when you get into fractions with larger numbers in the denominator. I think students would work with smaller denominators first to understand the pattern and then hopefully come to the generalization of how multiplication of fractions works.
ReplyDeleteSomebody mentioned using 2 different colored pencils to shade the different fractions in the area models. I think this is a great visual idea to help students understand this concept. I also think that using some concrete examples would help. Going back to third grade with the folding and shading of the strips of paper seems like it would be a good concrete example. And then of course, some kids of a food treat is always a fun concrete example.
The modules are very good at giving story problems with every concept. In fact, that is one of the differences between how parents learned and how our students are learning. Parents might have a worksheet with 20 calculation problems and one dreaded "word problem" . Now our students often have only 6 "word problems" for homework, which really make them think and understand. I also give at least one "problem of the day" based on the previous day's lesson using a student's name from the class and something they like (pokemon cards, baking cookies, reading books, etc). The kids love this and look forward to seeing what story I made up about them!
Lori, I also have found that specific problems with students and their interests will catch some attention. Your post was a good reminder to do this more, as I have't been consistent in doing it despite it's appeal. A problem of the day is a great venue for doing this.
DeleteLori, I agree with you about the area model concept being a natural progression from 3rd up to 5th grade. In fact, It was 2 years ago that I worked as a CT for math in both 3rd grade and 5th. During that year there were so many times that I left a third grade lesson only to teach the exact same concepts to my 5th graders. The word problems just had bigger numbers and decimals! So I do feel that the Common Core has a very well planned and organized progression built into it.
DeleteLori, I liked what you shared regarding the modules and story problems. Story problems really do force students to "think and understand." I think they address the "deeper, rather than wider" understanding that the modules promised to develop. I think the abstract/thinking about the work approach that we see more frequently now also helps students attack story problems. Students are required to have a deeper working knowledge of concepts to attack these problems.
DeleteIt seems like a broken record, but models like the area model, make math more accessible for most of us. Even for those who pick up an algorithm quickly and don't like to "waste time" with pictures and manipulatives (students who kind of just want to get it over with and move on), I would think models would provide opportunity for a depth of understanding that may prevent them from hitting the wall later on when conceptual understanding and ability to creatively use tools becomes more integral to problem solving than mere facility with algorithms.
ReplyDeletePiggy backing on some of the word problem comments, first of all, this is a big part of the exams they will take, both in class and for the state assessments, so practically speaking, they have to have as much exposure and practice as possible. More importantly, we are solving math-related problems daily in our real lives, and they take the form of word problems even if we don't recognize them as formal math problems. For example, a student may calculate how much time they have before they have to leave to catch the bus and if they can play one more survival game in Minecraft or not. Or we may calculate the speed and distance of an oncoming car before we pull into traffic. Students just need to get used to contextualizing and decontextualizing the math that they must work with. So when they are given a fraction problem in the abstract, and if they are having trouble with it, we have to give them the confidence and experience working with giving those numbers a context, be it an actual picture or model they make on paper, or one they "draw" in their head. This also can help them when they are presented with a pre-packaged word problem that they may have trouble with understanding - maybe because they lack the cultural background for it (like the hummus problem in the progression document 5th grade section). They can try to re-contextualize the information to something that makes sense to them (using the lemons for lemonade rather than hummus, for example) and thereby make the math more concrete and less frustrating to solve. Like "katydid" referred to, things like the ruler, or as Jen wrote about sandwiches, when they can make some word problems with these parts of their real-life, we are building confidence and problem solving ability, I think.
PS I also like the coloring idea for the area model. It helps keep things straight and it adds a little spice to math, too.
First off ~ the concept of calculating fractions has gotten such a bad wrap! I can't even count the number of parents, over the years, who have announced to me that they don't have the ability to help their child with homework when fractions are involved! With that said, starting with models gives fraction instruction a whole new twist! Shading and labeling is the basis for much of the fraction instruction at the 5th grade level. The kids are so involved in the models that the calculation work becomes the easy and obvious part! One thing I have noticed is that the problems can be very involved and even at fifth grade we still need to encourage students to plan out their work space!!!
ReplyDeleteA huge change that the Common Core presents is the absence of the "put it in lowest terms requirement" I remember how we used to hammer kids by taking off points if they failed to do this. We would also focus on using the least common denominator. These are still important, but I think that dwelling on these two concepts blocked that opportunity for there to be more than one way to solve a problem.
I agree Heidi! You bring up very important points. The "old" way isnt the only way anymore. A lot of people dont like change (parents) but the kids learning it know no different. Its so powerful to show them both way!!
DeleteI agree with you, Jessi. Sometimes it is easy to get caught up with the same old ideas. I think it is important to present both ways to reinforce learning.
DeleteI participated in a workshop where the presenter advocated for introducing math concepts with pictures, followed by symbols and then reinforced by verbal explanations. I was reminded of the above progression when considering how area models can be used advantageously in the classroom. The models provide an excellent visual, and allow students to use their hands and some of their creativity. I think this is another example of when tissue paper (and colored pencils) might really come in handy.
ReplyDeleteI also like the idea of having the model represent something in the real world. A Kit Kat for example. It seems that when students have a real world object to relate to, they are able to better engage and retain the big idea of the lesson. I think this is partially true because, as noted in the progression document, "when solving problems students learn to attend carefully to the underlying unit quantities." This, in turn, allows students to continue developing mathematical precision through their engagement with the material.
Carrie, I like the idea of using a Kit Kat bar to represent fractional parts. I know the students always relate well to food and it makes the lesson more meaningful. Thanks for the idea!
Delete- Discuss one advantage or one disadvantage of using an area model when multiplying fractions.
ReplyDeleteOne advantage of using an area model when multiplying fractions is that the students can easily visualize parts to whole relationships. The students are able to connect the two skills of multiplication of fractions and area by utilizing the commutative property as well. The use of shading to represent the two fractions is an excellent way for students to increase their understanding of this skill.
- Discuss an idea you learned from these videos and could apply in your classroom.
In my classroom this year I will use the shaded techniques that were mentioned in the videos, while relating it to area. The commutative property will be reinforced at the same time. Students will make connections when multiplying whole numbers with fractions. I will teach the students that if the fraction is greater than one, the product will be greater than the whole number. Whereas, if the fraction is smaller than one, the product will be less than the whole number. Students automatically assume when you multiply a whole number with a fraction, that the product will always be larger.
- Discuss some problems students may have and ideas on how to help them based on what you have learned thus far.
As previously mentioned, the students may have difficulty processing too much information, but I feel the visual models provide great examples. Class discussions of these representations can lead to a better understanding of the concepts of multiplying and dividing fractions.
- Look back at the Progression document for 4th and 5th grade fractions and explain how creating a story/real-world context might help assist a student in understanding fraction multiplication.
I would relate real-world examples to answer questions relating to each topic using food/pizza to reinforce part to whole relationships. To make the lessons meaningful, I would connect the lesson to something they can relate to in their environment.