# You Never Know What They Know Until You Push Their Thinking….

Last Friday, at a state math meeting, we had so much fun diving deep into a fraction lesson of a 6th grade teacher. The lesson was on multiplying fractions by fractions and while the conversation started with thoughts about the lesson itself and areas for discussion for the math coach, the lesson really brought to light the fraction progression. I cannot even begin to recap all of the insightful discussion points such as using models and the importance of the representation in mathematics, teacher pedagogy and mathematical understanding, vertical articulation across grade levels….I could go on and on, but I had one brief conversation that leaked its way into my classroom the following Monday.

While we were “doing the math” the students would be doing in the lesson video, a colleague and I were talking about where our 5th graders leave off with fractions and how that is built upon in 6th grade. She made the comment that if the students truly understood taking a fraction of another fraction and fraction of a whole number (both 5th grade standards), then they could reason their way through mixed number times mixed number, which is introduced in 6th grade.  She quickly drew out 3 1/3 x 3 1/3 and we talked through the context in which our book uses and how students could reason about that problem.

So, of course, I have to throw it out to my students on Monday because I am curious at this point if they could work their way through the problem and the various ways they would think about it. This is where that “engaging” vs “not engaging” or “real world” vs “not real world” conversation seems void. I used no context, no real world example, I simply said, “I was talking to some middle and high school teachers at my meeting on Friday about your fraction work and they were wondering how you guys would solve this problem. 3 1/3 x 3 1/3.” They went to work and I started walking around to chat with them.

Here are some strategies I saw…

She started with 3 1/3 x 3 and then added another 3 1/3 and found 1/3 of that to be 1 1/9.

He used partial products. When I asked him how he figured that out, he wrote the 25 x 25 and explained how he gets his partial products there so he did the same thing with wholes an fractions. Wow. Did not expect this one!

Same partial products, just a bit neater!

She used separate bars for each 3 1/3 and then divided the bottom up to find the 1/3 of 3 1/3.

I was so impressed by the work of these kiddos and they were so proud of themselves! They connected understandings of whole number operations to fractions, applied properties of mathematics, used what they knew conceptually about fractions to model the situation, and most importantly persevered through the problem and constructed arguments about their answer.

Don’t get me wrong, it wasn’t all picture perfect….I did have some who initially gave me 9 1/9 (as I anticipated they multiplied the whole numbers then the fractions and put them together) but that led to a great “reasonableness” conversation. A context in this case helped some students see that if you did 3 laps that were 3 1/3 miles long it was 10 miles, so if you did a 1/3 longer, can your answer be less than 10?

Needless to say, I don’t know how anyone doesn’t just love hearing students talk about math and reason about problems. I find it energizes me, my students, and the climate in my classroom. So, thank you to MSERC (University of Delaware Math & Science Education Center) and the Delaware Math Coalition for all of the hard work that is put into making these professional development opportunities so rewarding for both myself and my students! I think you all are AMAZING!

-Kristin

# Fraction of Fraction Day 2

As I mentioned in my previous post: https://mathmindsblog.wordpress.com/2013/11/15/fractions-of-fractions/
I had wondered about fraction multiplication being introduced without a context when the students were coming from lessons in which a fraction of a whole/mixed number had a context. Feeling like the students had a solid grasp on how to find a fraction of a fraction on a fraction bar, I thought I would try having them develop a story context for fraction multiplication problem. They had free reign of the fractions they used and context they chose. Needless to say, it was a learning experience for me. Some showed understanding of what they were doing when finding a fraction of a fraction of something while others unveiled some things I need to go back and revisit.

I have included clips from some of their videos and what I learned from each…. (turn your volume up bc they whispered on these)

This one was SO interesting (and a little humorous) because she cut the fraction bar to find 2/3s of 1/2, however when she is explaining her reasoning she used the commutative property. Saying that the answer is 2/6 because that is half of 2/3 was something I had never thought of exploring with students when reasoning about whether the answer made sense. I loved it and definitely added to my lessons for next week!

When she introduces the scenario, she says “1/2 of 1/4” so I don’t know if she misspoke or not really understanding the context. I can see she has the process but I don’t know if the understanding is there. I do love how she says “He wanted to find how much of the whole bread stick that was” because she is relating her answer back to the whole. This was difficult for many students. Maybe picky on my end, but I would have liked for her to label the pieces 1/8, 2/8, etc instead of by whole number, even though I know she is counting the pieces.

I was impressed how she used a class of students as the whole and did not get confused with the fraction of the class as opposed to the number of students. Many others got caught up in “How many students…” instead of “What fraction of the class.” One thing that just bothered me in watching it was the empty seat in the class! I just wanted to draw a person in for her!

This one has such a great context and division of the Hershey Bar that I was so excited, until the end. She seemed good with the context, decontextualized to solve, but then struggled to recontextualize to explain the answer.

I could post and comment all day, but needless to say there is other work to be done and papers to be commented on! It was a great first day with our 1:1 iPads using Educreations! I learned so much that now I must work on readjusting my math plans for next week!

-Kristin

# Fractions of Fractions

This is day 1 of multiplication of fraction by a fraction and I can already see this will dramatically increase my blogging! So much to write about (for reflection, excitement and possibly confusion). With the implementation of CCSS this year, this is new in the Investigations curriculum and I am finding some things I love about it already and some things I am struggling with just a bit.

Before this lesson, students have worked in the context of a bike race of “x” number of miles and found a fraction of the race various bikers have completed.  Looked like this:

This lesson went very smoothly and I found it was more of a struggle to have them model what was happening on the fraction bar since finding the fraction of the whole number was an action they could do mentally.  To some, it seemed like an unnecessary step and to be honest, I wavered between unnecessary and yet completely necessary to make their thinking visual. I knew how important it would be in fraction x fraction, so I made them construct the model of what was happening in the story.

Today we started fraction of a fraction. It incorporates the same visual image of the fraction bar, so I love that continuation from previous lessons. It did lack a context, which at first bothered me but as we continued working, and heard the discussions, I moved past that.  Tomorrow, I am actually going to have them come up with a story to go along with a few problems to see if they can contextualize the math they are doing.  We started with a fraction of a half and then a fraction of a third, writing the expressions (some equations) as we went:

Of course, you always have the students who fly through the work and finish early as I am walking around and having discussions with the students who need some extra help, so I asked those who finished early to think about the denominator each time. Why is the product’s denominator changing from the denominators of the factors? Did you have an idea what the denominator would be before you used the fraction bar?  There thoughts were so interesting:

Absolutely LOVE all of this scratching out, changing her reasoning!

This one brings up the issue of vocabulary….fours instead of fourths, eights instead of eighths. Something I have to bring out in our discussions.

This one I struggle with because of the words double and triple. I know the number itself is doubling and tripling, but I would like to have them expand that is it happening because there is another half to split or two other thirds to split.

I love that this makes the fractions factors and products are just like whole number factors and product.

Again with the “double” word. Is it just me that struggles with this one??

I am thinking this will be one of MANY multiplication and division of fraction posts! I am just amazed at the ease the students work with the fraction bars and I like what Investigations has done thus far with these lessons. One tweak I would like made would be the directions…students are asked to “stripe 1/2 of the shaded portion” and it is becoming a tongue-twister for me 🙂 I keep saying shaded when I mean striped, minor detail but they keep correcting me!

These conversations are so rich and valuable for this understanding that it blows my mind that a teacher could just say “multiply the numerators. multiply the denominators. That is multiplication of fractions.” If I had learned fractions this way, it would have all made SO much more sense!

To be continued…