# 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…

# A Fraction of our Time in Math Class…

I absolutely love fraction work with my students because there is always something interesting that leaves me pondering the whys and hows of my practice….

Being a K-5 Math Specialist for a couple years offered me the opportunity to really see the trajectory of our fraction work. Now being back in the classroom, I feel I have a much better grasp as to the work the students have previously done within our math program.  In third grade, they work tremendously with halves, thirds, and sixths using polygons to represent fractions of a hexagon whole for comparison and addition/subtraction. In fourth grade, students use arrays and known equivalencies to compare and add/subtract fractions with unlike denominators by choosing an appropriate array that works for both fractions (common denominator). In addition, at each grade level, students in need of RTI enrichment, work in Marilyn Burns’ Do The Math Program which utilizes fraction strips to compare and add/subtract fractions. All of this work focuses heavily on the students’ understandings of equivalencies.

Knowing all of this still never prepares you for the power of a new model….time! I have to admit, I am a huge fan of fraction strips and array work, however today I felt the power of clocks in developing equivalencies.  I have taught this lesson in previous years and to be completely honest, never really liked it. It felt contrived, like a pizza divided into slices in another form. This year I have realized it was not the context that was lending itself to the “pizza feel,” it was me.

The class began with a discussion of a blank clock face. I asked the class if the minute hand stayed at 12 and the hour hand moved to the 1, what fraction of the clock did it turn? They said 1/12 and we chatted about how we can prove that, divided it up and went from there. Next I asked if the hands were reversed, would that give us a different fraction? Some said no, some said yes and we talked about the equivalency of 5/60.

The student questions that followed took my appreciation of the clock to another level:

“Is this the same as degrees since it is a circle?”

“Could we do the fraction for a whole day (24 hours)?”

“Can we split the minutes in half to do eighths?”

“What fraction does the clock go at the time we go to lunch?”

Holy cow, how many directions could I take this lesson??  I moved forward with having the students work with partners to find all of the fractions they could represent on the clock.  Then I asked them to use that model to add 1/3 and 1/4 on the clock. It was interesting to see the students who know how to “find common denominators” by multiplying the numerator and denominator by the same number were challenged to make a proof of their equivalencies on the clock face, while the students who needed the clock as a tool had it as their disposal to see that 1/4 is 3/12 and 1/3 is 4/12.  That clock face immediately went from something I saw as just one more pizza, to both a tool and model at the same time in my classroom.

The follow up activity is called Roll Around The Clock (http://tinyurl.com/p8sm7wa). It has fantastic variations to the game and I have student work on the positive/negative scoring system that I will post soon, it was the perfect extension for the students who needed it!

So today, in just a fraction of time, I found a new appreciation for the analog clock and hopefully improved my practice by a fraction!

-Kristin