# Fraction Number Talks

Two days a week we have a Math RTI period built into our school schedule. It is 50 minutes in which students receive additional math support through Marilyn Burns’ Do The Math Program, as well as the use of Number Talks. The groups are smaller than the regular core classes, allowing for more individual time with each student. In 5th grade, we focus heavily on the fraction module and building reasoning within the structure of our number system. When we implemented this structure about four years ago, the majority of the students in the more intensive groups had an extreme aversion to fractions and really just a lack of confidence in their ability to do math. They were just looking for a “way to solve” the problem to get it over with, rather than reasoning and working through a problem.

The fraction module, through the use of fraction strips, encourages the students to think about the size of fractional pieces, creates a visual for fraction equivalence and looks at the relationships between fractions. Students use these understandings to compare, add and subtract fractions and most importantly build their confidence in their ability to do math. The Number Talks I do with fractions really focus on getting the students to THINK about the fractions before just operating left to right and looking for a common denominator each time. This week I was doing a number talk on adding fractions with my group and put up this problem: 3/4 + 5/10 + 1/8 + 2/16. My thought in choosing the problem was there was some great decomposition and equivalence that could happen.

We usually do these problems mentally, so I don’t typically give them white boards but since I really wanted to see their thinking, I did this time (and I am so glad). Seven students came up with six different answers. It was awesome. I had them lay their boards down and look at them all before they started to explain their strategy. It was all of the great decomposition, equivalence, and addition I was hoping it would be. I especially love 3/4 + 5/10 = 1 1/4 and the bottom left where the student rewrote 5/10 as 4/8 + 1/8 to add to the 6/8.

I started to hear a lot of “Oh”‘s and “They are the same”‘s but the student who got 24/16 thought she was wrong because hers “looked different.”  They all agreed the others were equivalent but I asked them to explain to their strategy and discuss the 24/16.

It was such a great discussion and as I was listening to them, I wondered how in the world any teacher could ever want to teach a group of students how to solve problems in only one way when there is such rich conversation in their individual thinking. They loved matching their answer to the others and proving how it was the same. Not to mention the confidence, independence and reassurance in their own math ability when they arrived at the correct answer.

-Kristin

# Number Talks – Fractions

Through doing Number Talks with students K-5, I started to realize that one thing I look for students to use in our whole number computation discussions is using known or derived facts to come to a solution. I feel like the problems I have been using are crafted to use the answers from previous problems to reason about the ending problem.

In the younger grades, I would like to see students using the double known fact of 7+7=14 to know 7+8=15. I want them using 23 + 20=43 to get 23 +19 = 42. I don’t want them treating every problem as if they have to “start from scratch” adding all or adding on.

An example in the upper elementary:

18 x 2

18 x 20

18 x 19

This progression leads them to use a known or derived fact (18 x 20) in order to solve 18 x 19. To build efficiency, I don’t want them to the treat the final problem in the progression as a “brand new” problem in order to reason about an answer.

Along these lines of thinking, as I observed students working the other day, I realized that students weren’t using this same use of known/derived facts when working with fractions. For example, a student was adding  3/4 +  7/8. He used 6/8 as an equivalent of 3/4, added that to 7/8 and ended with an answer of 13/8. Don’t get me wrong, I loves his use of equivalency and I am a fan of improper fractions, however I started wondering to myself if it would have been more efficient (or show that he actually thought about the fractions themselves) if he used a fact he may have known such as 3/4 + 3/4=1 1/2 to then add an 1/8 on to get 1 5/8? Or used 3/4 + 1 = 1  3/4 and then took away an 1/8? Is that the flexibility I want them using with fractions like I do with whole numbers?

I thought I would try a Number Talk the following day to see….

1/2 + 1/2

Thumbs went up and they laughed with a lot of “this is too easy”s going around.

1/2 + 1/4

Majority reasoned that 1/2 was the same as 2/4 and added that to 1/4 to get 3/4. Some said they “just knew it because they could picture it in their head” I asked if anyone used what they knew about the first problem to help them with the second problem? Hands went right up and I got an answer that I wish I was recording. It was to the effect of,”I know a 1/4 is half of 1/2 so the answer would be a 1/4 less than 1.”

1/2 + 3/4

Thumbs went up and I got a variety here. Some used 2/4 + 3/4 to get 5/4 while others decomposed the 3/4 to 1/2 + 1/4, added 1/2 + 1/2=1 and added the 1/4 to get 1  1/4.

3/4 + 3/4

Got some grumbles on this one, because it was “too easy” – 6/4…Duh! The class shook their hands in agreement and they were ready to move on to something harder.  I noticed that when the denominators are same, they don’t really “think” about the fractions too much. I waited….finally a student said, “It is just a 1/4 more than the previous problem so it is 1  1/2″ and another said each 3/4 is 1/4 more than a 1/2 so if you know 1/2 + 1/2 = 1 then you add 1/2 because 1/4 + 1/4 = 1/2.” I had to record that reasoning for the class bc it was hard for many to visualize.

3/4 + 5/8

Huge variety on this one and I thoroughly enjoyed it! From 6/8 + 5/8 = 11/8 to decomposing to combine 3/4 and 2/8 to get the whole and then 3 more 1/8s = 1 3/8.  There were many more students who used problems we had previously done.

What I learned (and questions I still have) from this little experiment:

– Students LOVE having the same denominator when combining fractions.

– Do they really “think” about the fractions when the denominators are the same? Can they reason if that answer makes sense if they are just finding equivalents and adding.

– Students can be flexible with fractions if you push them to be.

– Subtraction will be an interesting one to try out next.

– I would much prefer if I remembered to use the word “sum” instead of “answer”…. I tell myself all of the time, but in the moment I always forget.

– Using known or derived fact and compensation are invaluable for students when working with both whole numbers,  fractions and decimals.

– Are there mathematical concepts that present themselves later in Middle School or High School in which known and derived facts would be useful?

Happy Thanksgiving,

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…

# 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

# Fraction Talk

It has been forever since I have blogged, and although I have been so inspired from many things I read this summer, nothing inspires me like talking to my 5th graders!

As we begin our venture into fractions, I have to first give some props to my 4th grade teachers. I have never heard so many “Yeah, fractions” and “I love fractions!” ever.  I attribute this to a lot of hard work and dedication by Nancy (math specialist), the fourth grade teachers, and the Marilyn Burns’ Do the Math fraction units.

Yesterday in class, to get a feel for what my students know about fractions, we did a “Show What You Know” with problems involving writing, comparing, and adding fractions. They seemed very comfortable with writing fractions, comparing fractions using benchmarks, and finding fraction of a group.

Then we get to the problem asking students if the expression 2/3 > 2/6 is True or False. As they shared their reasonings, I heard many anticipated strategies such as “2/6 is equivalent to 1/3 so 2/3 is bigger than 1/3” and “The pieces are bigger in 2/3 and you have the same amount of each so it has to be more.”

As the conversation was coming to an end, one student raises her hand and sets my wheels spinning.  She said “I know that if I just subtract the numerator from the denominator, whichever fraction has the the smallest difference is the larger fraction. But it only works when the numerators are the same.” Huh. I asked her why she thought that worked and she said she didn’t know but proclaimed it would work every time.  I told her we would think through that one and revisit it soon because I needed time to think it through. Being the thoughtful student she is, I had this work from her by the end of the day:

I was proud she gave examples and tested even and odd numbers to be sure that didn’t effect the outcome.

So my next question for myself (and anyone else who is reading and feels like offering some advice) was what to do with this…

Nancy and I sat and talked about why this works…here are some points to our discussion:

– When you subtract the numerator from denominator you could finding the fractional piece the fraction is from a whole, assuming you put it back over the denominator.

– But since the denominators are different this would not give you a piece of information that would make this “trick” valuable.

– As the denominator gets larger and the numerator stays the same the fraction gets smaller.

– So the bigger the difference between the numerator and denominator, the smaller the fraction.

– Does it work with improper fractions? Yes.

– Is it worth revisiting in class yet because some students may pick up the “trick” and not be ready for the reasoning behind why it works.

– But isn’t it really simple? 3/4, 3/5, 3/6, 3/7…and so on…the difference of the numerator and denominator is getting greater, so the fraction is getting smaller.

So in closing I have no answer of what to do with this information. I am thinking I will revisit it with the student alone because she is anxious for why this works. I may save it for the rest until I have a better grasp on where they are with their understanding of numerator/denominator relationships, but am I being too cautious? I just don’t want “tricks” to be used because they are easier for some students than the reasoning piece.

Would love any thoughts!

-Kristin