Fraction/Percent Equivalents

It goes without saying that I miss talking 5th grade math with my students each day. But I am so lucky this year to have a new, wonderful teacher in 5th grade who lets me plan and teach some lessons with her! This lesson was one of her first lessons of Unit 4, Name That Portion.

Since in 4th grade the students do a lot of work with comparing fractions, we designed a Number Talk string in which students were comparing two fractions. We wanted to hear how they talked about the fractions. In the string we used a set with common denominators, common numerators, and one unit from a whole. On each problem we were excited to hear talking about the “size of the piece” being the unit and the numerator telling us how many of those pieces we have. Our 4th grade teachers really do a beautiful job with this work. They also used equivalents to have common denominators to compare and a few used percents, since they had done a some grid work with that they day before.

We started the lesson by asking them how they could shade 1/4 on a 10×10 grid. The majority of the students split the grid in half vertically and then again horizontally and shaded one quadrant. We heard a lot of the “1/4 is half of a half.” As I was walking around, I heard a pair talking about shading a 5×5 in that grid. I saw this as a beautiful connection to the volume unit they just completed in which they were adjusting dimensions and seeing the effect on the volume. I had her explain her strategy and wrote 5 x 5 under the 10 x 10 that was up on the board already and asked how that could get us 1/4 of the whole thing? One student said it looks like it should be half of it because 5 is half of 10, but then one student said since we were taking 1/2 of both it would be a fourth….this is where I hope Leigh (the 5th grade teacher) and I remember to use this when they hit multiplication of fractions!

They then worked in pairs to shade 1/8 and 3/8 and we came back to discuss. We noticed as we walked around that the shading was wonderful on their papers, but when asked to write the fraction and percent, most were blank. I remember this lesson from last year during decimals where the same thing happened. So, we asked them what they thought the fraction was as we got these three answers…

12 r4/100

12 1/2/100

12.5/100

They were not overly comfortable with any of them so we asked them to journal which one “felt right” to them and why…

We loved to see what they knew about decimal fraction relations, but we especially liked the “it sounds more fifth grady to use 12.5.”

-Kristin

Listening Carefully to Student Thinking

Recently, I have been reviewing a new “CCSS-Aligned” middle school curriculum and find myself completely frustrated with the overabundance of scaffolding and lack of student thinking required on every assignment. Not having the days/weeks it would take for teachers to engage in the mathematics as both learners and teachers, I needed a short, powerful way to show that this is not how students should experience/learn mathematics.

As I looked at the fraction page like this, my thought was “Why just two ways?” quickly followed by “Why those two ways?” quickly followed by “My students are doing this now, flexibly.”

Right then, I realized the perfect proof of why NOT to do this, was the work my students already do when given the freedom to reason about a problem and do more than just procedurally compute an answer. So, I put the proof in their hands.  I simply asked them to solve 2/5 x 7/10 as many ways as they could. Some got creative after a couple of ways, and by no means am I saying some of these are “efficient,” but they show so much flexibility.

This felt perfect. Why would we want to miss out on all of the great conversations that can happen around this work by making them answer in just 2 ways, and more specifically, those 2 ways they show you how to do…step-by-step?

and THEN this happened which validated my thoughts even further and instantly made me reflect on my friend Christopher’s talk at ShadowCon (video coming soon) around listening carefully to student thinking…

The students were working on 2/5 x 7/10 as I was walking around the room observing their work. I glanced over a student’s shoulder and saw “Doubling and Halving” written on her paper with the correct answer. Assuming it was doubling/halving in the sense of doubling one factor and halving another factor, I was excited to see the use of the strategy.

I asked her how she did it, she said, “I double/halved” and I was about to move on to get ready for our sharing. When I glanced down, however, it was not at all like I had imagined. I asked her to explain further…“I halved this numerator and doubled this denominator [points to 2/5] then I doubled this numerator and halved this denominator [points to 7/10].”. Ok, now THIS is much different than I thought!!

I had her share, and others immediately said they had double/halved also but did not get those fractions to multiply and wondered if that worked every time (I love that they ask that now:). I let them play around with it for a bit but since we had some division work to do I told them to keep thinking about that and we will revisit it tomorrow. By the end of the next class period, I had a student come up and say, “She didn’t double/half really, she quadrupled/fourthed.” I asked him to write down his explanation for me because it was lovely.

So glad I listened carefully and didn’t makes assumptions on her understandings because how amazing is this work? I am also so glad that I can appreciate a curriculum that allows for these reasonings and conversations to happen.

-Kristin

Creating Contexts for Decimal Operations

Sometimes I have students engaging in math within a context, however at other times, we just explore some beautiful patterns we see as we play around with numbers. I see a value and need for students to experience both. This week was one of those “number weeks” and it was so much fun!

Over the past few weeks, we have been working on decimal multiplication. If you want to see the student experiences prior to this lesson, they are all over my recent blog posts….it is has been decimal overload lately:) After sharing strategies and connecting representations in this lesson, I was curious how students thought about this problem in a context because up to this point, I had not given them one for thinking about a decimal less than one times a decimal less than one.After they wrote their problem, I asked them to tell me what they were thinking about as they were deciding on the context.

I anticipated that many would refer back to what they know about taking a fraction less than 1 of a fraction less than 1, like in this example…

I love how this one said she knew she “had to start with .4” That shows the order of the numbers in the problem create a context for her. It mattered to her, taking .6 of the .4.

This student went with two different contexts and again saying that he started with the .4. This must be something we have chatted about quite a bit about because it showed up multiple times. I loved how this student said he thought about an area model in creation of his problems.

This student was great in listing all of things he was thinking about as he thought about a context..

I had students who attempted to create a “groups of” context. I don’t know if I ever realized how difficult this and how much I, as an adult, need to be able to create a visual in my mind of what is happening in a problem to make sense of it. Here is one example (not the sweetest context but she thought the Mary HAD a little lamb was clever…) She worked a bit yesterday to show what the representation would be, but kept running into problems with cutting into “.6 pieces.”

And then I have these two that had my brain reeling for a bit, for many reasons. First, does the context work with this problem? Secondly, I knew it sounded like it should work, but when I tried to make sense of it, I couldn’t create a visual. Also, as I read them, I thought I knew where it was going and the question I would pose, but it wasn’t the way they saw it ending. I asked them to create an Educreations about their problem so I could check out their thinking around the context.

Yes, Rick Astly. But the question at the end, compared to the total time Never Gonna Give You Up, threw me a bit, not where I was going with it….

His Explanation: https://www.educreations.com/lesson/embed/31398809/?ref=embed

The second one tried it out, and wasn’t so sure of his question after messing with it. The wording “.6 as small” was making me think. I was trying to make sense of that wording, do we ever say six tenths times as small? Then does his question referring back to the .4 make sense?

His Explanation:https://www.educreations.com/lesson/embed/31402039/?ref=embed

Definitely a lot for me to think about this week too! I have some amazing work with them connecting representations to write up later…they are just such great thinkers!

-Kristin

Commutativity in Fraction Multiplication

Think about these two expressions…

2/3 x 6              6 x 2/3

Do you think differently about each?

Does your solution approach change?

I had not really given this much thought because we do both in 5th grade, multiply a fraction by a whole number and whole number by a fraction. However, recently, when working with a group of 4th grade teachers and looking more closely at the standards and my curriculum, I am beginning to see a distinct difference. I now look at each expression from a different perspective. Not that both ideas do not arise at multiple grade levels in some form or another, but it is so interesting to me as to which thinking would come before the other.

Let’s first look at the standards…

4th Grade:

5th Grade:

Interesting. For me, taking a fraction of a group feels more “natural” and intuitive than multiplying a whole number by a fraction, however in the learning trajectory of multiplication and building of unit fractions composing a whole, the multiplication of a whole by a fraction feels like the natural next step.

For our upcoming Illustrative Mathematics professional development, I was collecting work samples for the following problem (thanks Jody:)

“Presley is wrapping 6 packages. Each package needs 2/3 of a yard of ribbon. How much ribbon will she use for wrapping the 6 packages?”

As anticipated, I received a wide variety of solutions to arrive at 4 yards of ribbon. Here are just a few examples in what I think is the progression I expect (some of them got finished  quickly and opted to show a few ways to solve).

They all finished fairly quickly and as I was walking around I thought it was really interesting to see such a variety in the equations they used to represent the problem. We came together as a whole group and I asked them for the equations they thought best represented the problem. The most common answers were: 2/3 x 6 = 4, 6 x 2/3= 4 and 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3= 12/3 = 4.

I asked them if there was a difference between the equations and there was a unanimous “No” because they mean the same thing. “They all get 4.” In my head I was very excited that commutativity was something they see when finding a solution, but I was also curious if it worked the same in the opposite direction. I asked if we could narrow it down to two equations and they all agreed that the repeated addition was the same as 6 x 2/3 because it was “six groups of 2/3.” Interesting, so they see that in the numeric representation but not contextually?

I then asked them to write 6 x 2/3 and 2/3 x 6 on the top of their journal page and think about them without the previous context.  I posed, “If I gave you these two problems to solve, would you think about them the same way? Do you think about them differently?” I was curious to hear their thoughts on the commutativity.

The conversation after was so great and interesting! There is a difference when going from number to context, however when put in context, I think students use whatever strategy is easiest for them to arrive at the answer. Is this what is truly meant by contextualizing and decontexualizing in the SMPs?

To further intrigue me, I went and pulled a few fourth graders to interview during my planning period. It was so interesting that they saw this as a whole number times a fraction because it was “six 2/3’s.” Their connection to multiplication and “groups of” was evident. I did love how they did 3 of the 2/3s first to get 2 and then doubled that to get 4.

This 4th grader was the most interesting..

She solved it as 2/3 of 6 and arrived at 4. I asked her if she could write an equation for the problem she solved and she wrote 2/3 of 6 = 4. Completely because I am so nosy, I asked her to write 6 x 2/3 under that. I asked how she thought about that problem? Would she solve it the same? She said, “No, that is 6 of the 2/3’s so I have to multiply the 2 and 3 by 6.” She proceeded and ended with 12/18. She saw the numerator and denominator as numbers in and of themselves and used the distributive property to arrive at her answer instead of thinking about the 2/3 as a number. This was something I had never thought of before! I wish I had more time with her because I SO wanted to ask if that makes sense, but since my planning runs into dismissal, she had to get back to class! Argh!

This progression (to me) now seems to be more about building on student’s understanding of multiplication then about what is more intuitive for students to do. That is such a revelation to me. In second and third grade students do so much in “sharing” situations, that I had assumed it was en route to this skill of taking a fraction of a number when in fact it is more about the operations. It builds multiplication and division. Those operations then progress from operations with whole numbers to operations with fractions and from there students start to build deeper understandings of the properties of operations.

This is of course, all my interpretation based on my experiences and perspective of the student work, but how awesome! I cannot wait to share this with the 4th grade teachers along with the video of the kids chatting with me about this, awesome stuff!!

-Kristin

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

Fractions as Division…Say What?

Last year I learned to appreciate the Investigations lesson in which students explore fractions as division in a Division Table: https://mathmindsblog.wordpress.com/2014/03/18/fraction-to-decimal-division-table-noticings/. However, as I was planning this year, I thought I really missed the mark in making it an explicit understanding that fractions represent division before exploring this table. I think I completely just assumed that students understood this from previous years and investigations with sharing situations involving fractional answers. I decided to check it out this year to see what they students knew/understood before beginning the division table work. I thought it could make some really nice connections evident.

I started by putting a few sharing problem on the board: 6 subs shared by 4 people, 9 subs shared by 4 people, 3 subs shared by 5 people, and 6 subs shared by 9 people. I asked how much each person would get if they shared the subs equally.  I gave the students some individual time to work through the problems and, after that, an opportunity to share their answers and strategies with their group.  In the majority of the class, I saw the work I had anticipated based on their third grade brownie sharing work in Investigations. A lot of drawing of subs, people, and “passing out” of the pieces.

One student thought about the whole being the number of subs, divided it into the number of people sharing and arrived at decimals, however struggled when he got to the 6 shared by 9. (The side written piece is after I asked them to write what they noticed and then he proved it worked with fractional subs to start).

I had a few students that provided the perfect transition between the visual drawings and the fraction being division. They intuitively wrote the problem as a division problem and solved it using what they know about multiplication. After sharing some of the visual representations, I had these students share their equations. They explained to the class that is felt like division because they were dividing it up among people.

After our sharing, I revisited the original problems, with the students proven answers, and ask them to write what they noticed about the problems. After a few moments, I heard so many “Oh My Gosh”s and “It was really that easy”s echoing about the room. One student exclaimed, “Why did I do all of that work?” pointing to his beautiful sub and people drawings.

Here are some of their noticings (I love that they automatically start proving it to see if will always work without me even asking anymore).

This one just absolutely cracked my up and proved once again that I cannot make assumptions about student understandings….

From this point, we tested out a bunch, talked about why it will always work and then starting looking at representing our “benchmark” answers as decimals. Tomorrow, I feel great knowing we will start looking into the division table with a deeper understanding of fractions as division. The word “explicit” sometimes makes me cringe in the way of “telling” students things, however I feel in this case the understanding of fractions as division was made explicit to the students through their own work group sharing and noticing today. I think that may be the piece I have missed before… I assumed they knew and could arrive at an answer, however never made the idea explicit as a whole group.

Today was a great day in math…Say What?

-Kristin

Fractions As the Denominator

As I was organizing my student work pictures this morning, I realized I had tweeted out this awesome work, but never blogged about it.

My students are very comfortable with putting fractions in the numerator. They use them all of time when decomposing, adding and comparing fractions like in these two examples…

The other day, two of my students finished early and as an aside asked me if there could be a fraction as the denominator. I asked them to try it out and see what they thought.

They wrote their question and then started playing around with some fractions in the denominator. At first they were writing a bunch of fractions with a fractions as the denominator in attempt to find one that jumped out and made sense to them. They tried drawing some pictures of them along the way to see if they could illustrate what it would look like.

The first one they drew was 1/1.5 in which the rectangle was cut into thirds and had 1.5 shaded. When I asked what they would name what they just drew, they said 1.5/3. Hmmmm, back to the drawing board. They moved to 1 / 2/8, drew a rectangle cut into 8ths and shaded 2 of them. After shading, one student wrote 1, 2, 3, 4 over each 2/8 and said that there were four of the 2/8’s in his picture, so 1/ 2/8 must be 4. I asked what the whole was in the picture and left them to play around with that idea for a bit.

I came back to these additions to the work:

When I came back they said they realized that 1/ 2/8 was really a fraction more than 1 since 2/8 / 2/8=1. When I asked them to show me where that thinking was in their representation, they said since 2/8 was really 1 in their picture, it took four of them to make four wholes. I especially liked how they multiplied the numerator and denominator by 4 (the reciprocal of the denominator) to get to 1 in the denominator. Interesting to think about the algorithm for dividing fractions at play here.

As others in the class finished their work, they started to mess around with this question, trying to make sense of it. This student attempted to put it into a context using the meaning of a fraction we use a lot, “a pieces the size of 1/b,” however with b as a fraction, it is not helpful here.

One student wrote this as his thought about the fraction as the denominator.

I am left thinking a lot about the progression in which students learn complex fractions.

Fraction Flexibility in Number Talks

In my RTI (Response to Intervention) class, I use Marilyn Burns’ Do The Math program, which is wonderful for building conceptual understanding of fractions through the use of fraction strips. Students use the fraction strips to build equivalencies, make comparisons and add/subtract fractions. It does not take long for students to be able to “see” the equivalencies without having the strips in front of them and develop fluency and flexibility with fractions. In addition to this module, I do Number Talks with the group. I do a combination of whole number operation talks and fraction number talks.

This Thursday, I did a Fraction Number Talk in which I wanted students to think about the fractions and make friendly combinations when adding. I never like to pose a problem with one solution path, so each can be solved using another strategy, however my goal was making friendly combinations. Next to each problem I put my thought in brackets so you have an idea of what i was thinking:) This is the string I planned:

2/4 + 2/3 + 6/12      [(2/4 + 6/12) + 2/3]

2/3 + 1/4 + 1/4 + 2/6  [(2/3 + 2/6) + 1/4 +1/4]

1 3/8 + 5/10 + 3/4  [(5/10 + 1/2)+1 3/8 + 1/4]

They did so wonderful with these and some began whining that these were too easy and to give them something really hard. So I gave them my final problem:

2/3 + 1/2 + 3 + 1/4

There were a few groans and “this isn’t hard“s because they went to 12ths and had the answer quickly. I told them if they had the answer, to try to use the strategy they had used in the previous problems to see if they got the same answer. I was thinking they would use a piece of the 3 to make the 1/2 and 1/4 a whole, but of course there is always one who surprises me! He had a beautiful explanation so I asked him to write it down for me so I could remember. He got a little mixed up in his wording, so I will do a translation after you check out his reasoning.

He took 1.5/3 from the 2/3 to add to the 1/2 and make a whole. He then added the 1/2/3 to the 1/4. I, of course, asked him how he added that and his response was so beautiful as he explained it to me. I mean how amazing is it that he knew 1/2/3 is equivalent to 2/12…and this was all mentally!

Let me assure you that this student CAN add these fractions in a much more efficient way, and this was him challenging himself to play around with the fractions. THIS is what I would consider flexibility in operations and also where I want students to see math as fun…playing around with numbers!

– Kristin

What Happens When You Divide by Zero?

This question got thrown out on Twitter the other day (full conversation here). It was something I had never thought about and struggled to think about where in our curriculum or standards it showed up. As always, I thought I would ask my students the following day what they thought happened when we divided by zero. Here are some responses:

It was interesting because most went to breaking into groups, but depending on how they reasoned about it,resulted in different answers. Sharing something between zero people or putting things into zero groups was either zero because there was nothing to put the number of things in OR it was that beginning number because they weren’t put anywhere and were leftover.  Some also thought about inverse relationships which was nice and that is when our conversation got really confusing and people started questioning what in the world it was! One student punched it into his calculator and got Error, even more confusing while another asked Siri and got this, which they wrote in their journal…

Fawn tweeted a piece of student work that was really interesting in which the student had said 5 divided by zero was the same as 5/0, so (trying to quote this as accurately as I can) you cannot take five pieces of something with an area of zero. I am thinking that was like thinking something like 5/6 is 5 pieces when the unit is broken in to 6 pieces. In his case, it is five pieces with the unit cut into zero? Fawn, please correct me if I completely mess that one up!!

Very interesting and something I want to be sure I keep in the back of my mind. I love when a tweet can spark something I had never thought of before in elementary math work! Thanks Tina!

-Kristin

Volume with Fractional Dimensions

Before I began our volume work this year, I blogged about my planning process here: https://mathmindsblog.wordpress.com/2014/10/20/unit-planning/. As anticipated, I had many students who quickly developed (or already had) strategies for finding volume and could articulate a conceptual understanding of what was happening in the prism. In my previous post, I was throwing around the idea of giving those students dimensions with fractional length sides, so the other day I thought I would try it out. I did this Illustrative Task as a formative assessment of student understanding. Many students were done in a couple of minutes, with responses for part b that looked like this:

As I walked around the room and saw they were finished quickly, I asked them to revisit part b and think about a tank with fractional dimensions. Because of the great work they had done here I thought they would have some interesting thoughts. These are a few of the responses I got:

So, what did I learn from this work?  I saw they had some great understandings about taking a fraction of one factor to make a number that they knew they needed to multiply by a third whole number factor to get 240.  In the first two pictures, there is a great pattern happening that I want to explore further with the whole class. I also loved seeing that a student took the question “fractional length sides” to include decimals in his work. In my question, however, I had wanted them to consider more than one side in fractional lengths, however not being more explicit, they took it and ran with one side being fractional.  In the next lesson, I thought I would push them a bit with this.

In the following lesson, students were finding the volume of an unmarked prism in cubic centimeters. They had rulers, cm cubes, and cm grid paper available to them, and went to work. Every year this happens, the Investigations grid paper works with the box to be whole number dimensions, however the cm are a bit “off” when using a ruler or cm cubes. I knew this, however, I do love the discussions that evolve from students who used different tools. I also thought this is the perfect opportunity for my students who were beginning to think about fractional sides. What transpired in the whole class lesson is a blog post in and of itself, however this is what came about from the fractional sides work…

Sooo much great stuff here! I had a group who was using the cubes, coming out with halves, but not wanting to round because it was “right in the middle” of the cube. I let them go and came back to see they were multiplying whole numbers, multiplying the fractions, and then adding them together to get their product. I asked them to think about another multiplication strategy to see if they got the same product, then came the array. Another student in the same group solved mentally to get the products. Unfortunately, the class had to leave me to go to their next class, also leaving me with so many things to think about. From here, I want to be sure students start to think about reasonableness of their solutions, compare their fraction multiplication strategy to whole number multiplication strategies, and think about how we multiply three numbers (Associative property). So much to do, I need full day math classes!

-Kristin