Category Archives: Comparing Fractions

3rd Grade: Comparing Fractions

I was so excited just walking into Jenn Guido’s room today and seeing this awesomeness on the board from the day before:

We chatted with the class a bit about their responses on the board before jumping into our Number Talk. One thing Jenn and I both noticed during this chat was the use of the word “double” when talking about equivalents such as 2/4 and 4/8. We had the chance to ask them what exactly was doubling and kept that in the back of our mind as something to keep revisiting. Even in 5th grade, I would hear the same thing being said each year. I would always have to ask, “What is doubling?” “What is 1/2 doubled?” “What exactly is doubling in the fraction?” “What happens when we double the numerator? denominator?”

After this chat, it was time to move into our planned activity. The class has been doing a lot of work with partitioning (and they used that word:) circles, rectangles and number lines so we planned a Number Talk consisting of a string of fractions for the students to compare. We were curious to hear how they talked about the fractions themselves and how they used benchmarks and equivalents. The string we developed was this:

1/6 or 1/8 – Unit Fractions

5/8 or 3/8 – Same Denominator (same-sized pieces in student terms)

3/8 or 3/4 – Common Numerator, Benchmark to 1/2, or Equivalents

3/3 or 4/3 – Benchmark to 1

The students shared their responses and did an amazing job of explaining their reasoning very clearly. In all of these problems and actually in all of their work thus far, they have always assumed the fractions referred to the same whole. We decided to change that up on them a bit and see what they would do with the statement, “1/2 is always greater than 1/3.” We thought the word “always” would make them second guess the statement, but we could not have been more wrong…they all agreed. A few students shared their responses, and it was great to see such a variety of representations.

This student was interesting because he used 12ths, and although he could not articulate why, it was labeled correctly. I am assuming it was because 1/2 and 1/3 could be placed on 12hs, but I am not sure because his reasoning sounds like he is comparing the 1/2 and 1/3 as pieces not in 12ths.

Jenn, Meghan (another 3rd grade teacher with us in the room) and I chatted while they were working about how to get them to reason about different-sized wholes. A picture would have been a dead giveaway so I just went up and circled the word always and asked, “Does this word bother anyone?” and one lone student said it made him feel like there was a twist. I love those skeptics. I asked them to talk as a table about what the twist could be in this statement, and then we had some great stuff! They talked as tables, and while only two of the tables talked about different wholes (in terms of number lines which was not what I expected either), there was so many great conversations trying to “break the statement.”

This is an example of the number line argument:

This group kept saying it would be a different answer if they were talking about “1/2 of” or “1/3 of”…then said, “Like 1/3 of 1/2” and THEN KNEW IT WAS 1/6 when I asked what that would be! They said 1/2 is 3/6 so 1/3 of that is 1/6. Wow. Then, of course I could not resist asking what 1/2 of 1/3 would be and they kept saying one half thirds, but could figure out how to write it and then questioned if that could even be right.

After having the tables share with the whole group, they all agreed the statement should be sometimes instead of always. Jenn asked them to complete two statements…

“1/2 is greater than 1/3 when….”

“1/2 is not greater than 1/3 when…”

A great day! We are doing the same thing in Meghan’s classroom tomorrow and are changing the first problem in the string to 1/2 and 1/3 so we can revisit that at the end. Can’t wait!

Fraction & Decimal Number Lines

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Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and it felt like perfect timing as 3rd and 4th grade would soon be starting their fraction unit and 5th would be entering their decimal unit. Prior to the meeting, we read the NCTM article, “Identify Fractions and Decimals on a Number Line” by Meghan Shaughnessy, so we started the meeting discussing ideas in the article. We then jumped into playing around with clothesline number lines and double number lines, discussing what they could look like at each grade level based on where students are in the fractional thinking.

I have co-taught number line lessons in both 5th grade and Kindergarten this year, but both were a bit different in not only number, but organization. In 5th grade we used one clothesline with the whole class, while in Kindergarten we used tape on the floor and students worked in small groups. Leigh, 5th grade teacher, was interested in trying the small group number lines on the floor. As we planned the lesson, the one thing we thought would be difficult about having small groups is getting around to each group to hear their conversations, especially when we were planning cards purposefully to address misconceptions and misunderstandings. However, knowing we would have the two of us circulating, as well as two 3rd grade teachers who wanted to see the lesson (yeah!), we knew we had plenty of eyes and ears around the room to hear the math conversations.

During the lesson, in groups of three, students placed 24 cards on their number line. There were two sets of cards, so after placing all of their cards, each group visited a number line with a different set of cards to discuss. Instead of boring you with all of the number choices we made, here are a few of the choices in cards and the reason(s) we chose them:

1/3 and .3: Students often think these two are equivalent so before the decimal unit we were curious to see how they were thinking around that idea and how they used what they knew about fractions or percents to reason about it.

0.3, 0.33, 0.333, 1/3: The 1/3 and .333 were there to think about equivalency, while the others were there to think about what is the same in each and how much more each decimal has to make it larger. Which you can see caused some confusion here:

2.01 and 2.08: We were curious about the distance they put between these two cards.

1 6/9 and 1.6: We wanted to see how students compared fractions and decimals when they can’t easily convert 6/9 into a decimal. Then, if they began thinking 6/9 and 6/10, how would they decide on the larger fraction and then how much distance do they put between them?

The group below practically had them on top of one another:

While this group had a bit of a space between them:

2.8 and 2.80: Tenths and hundredths equivalency. They all seemed to handle this with ease.

.005 and 1/100: Curious to see the placement in relation to the other numbers. This 1/100 is close to 0 but I wonder about it in relation to the .2. Definitely a conversation worth having!

2.8 and 2 7/8: To see how they compared the 7/8 to the 8/10.

After they visited other number lines, they had a chance to meet with that group and discuss card placements they agreed with and placements they did not. Groups then made adjustments accordingly…

Here was a group’s completed number line and my first stab at panoramic on my phone!

The journal entry we left them with was, “Which cards were the most difficult to place on the number line? Why?” Many were just as we suspected.

The conversation as I walked back over to the other building with the 3rd grade teachers was, what does this look like in 3rd grade? Could we use array images to place on the line instead of the fractions? Could the pictures include over 1 whole? What whole numbers would we use? Do we play with equivalent pictures with different partitioning? Being mindful of the students’ second grade fraction exposure, below, we are planning on trying out something very soon! I am thinking the cards like these on Illustrative, with the pictures but no fraction names at this point.

CCSS.MATH.CONTENT.2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

CCSS.MATH.CONTENT.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

…and then could the journal could be, “Can you name any of the fractions on your number line? How do you know?”…or something like that!

How Planning Mistakes Can Lead To Great Student Thinking….

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The other day I did this fraction clothesline activity with a 5th grade class and today I had the chance to do it again with another 5th grade teacher, Leigh. It is always so nice to get to have a do-over after having time to reflect and think more about what the students thought about both during and after the activity.

I really thought the conversation was great during the clothesline activity, but it took too long the first time. We noticed that some students began to disengage. To try and improve upon that, Leigh and I decided to give only one card to every pair of students instead of each student having one. However, due to us wanting to keep a few important cards we wanted to hear them talk about, some pairs had two.

I also did not like my placement of 0 being at the very end of the left (when looking at it) end of the string. I moved it out some and talked about the set of numbers that falls on either side of the 0. I felt much better about that this time around!

In the planning of the first clothesline activity, we took fractions from the work the students had been doing with percents and decided on putting 100% in there, completely thinking it would be at 4/4. As the student placed it, however, I started realizing that I never thought about the difference of 100% in terms of the area representations the students had been using versus 100% when talking about distance on a number line. But now, having time to reflect on the card, I thought it would make a great journal entry!

As we neared the end of the card placements, I handed the 100% card to a student and told her it was going to probably cause a lot of discussion but just put it where she thought it went. She said she got it, walked up there and placed it on top of the 2 (the highest number on the line). There were some agree signals going on and some other hands that shot right up to disagree. We talked about it a bit and then we asked them to journal their ending thoughts so we could move on with the rest of the lesson about different sized wholes.

Some thought that 100% was at 4/4 on the number line because it equals 1….

Some thought it was at 4/4, but because of the conversation became a bit unclear…

Some thought it goes on the 2 because it is the biggest number on the number line…

Some related it to different contexts with different wholes…

And one student said it can be anywhere with beautiful adjustments as it moves….

What a great day revisiting my planning mistake!

-Kristin

5th Grade Fraction Clothesline

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Today, I had the chance to plan and teach with a 5th grade teacher and it was awesome! Last week, this class had just finished a bunch of 100s grid shading in thinking about fraction/percent equivalencies, so we picked up planning the lesson in Investigations with the fraction/percent equivalent strips. Instead of the 10-minute math activity, we thought it would be really interesting to do the clothesline number line to kick off the class period.

We chose fractions (and one percent I will talk about later) based on the fractions the students had been working with on the grids. We chose fractions based on different comparison strategies that could arise such as:

Partitioning sections of the line
Distance to benchmarks
Equivalent Fractions
Common Denominator
Greater than, Less than or equal to a whole or 1/2

We settled upon the following cards:

1/4, 3/4, 4/4, 1/3, 4/3, 5/10, 2/5, 100%, 3/8, 1 5/8, 1 7/8, 4/5, 11/6, 1 6/10, 1/10, 9/8, 12/8, 2

To start, I put the 0 toward the left of the line (when you are looking at it) and we practiced with a few whole numbers. One student volunteered to be first and I handed her a card with the number 7. As she walked up, looked around, walked up and down the line, looked at me like I was playing some type of trick on her, we immediately had the conversation about how knowing the highest numbered card would be super helpful. She settled on putting it toward the far right side and had a seat. I gave another student the 10 card. He put that at the far right and adjusted the 7 to be “about 2 cards away” from the 10, leaving a really long distance from 0-7 for them to think about. We had some students disagree so we talked about distance and adjusted the cards to be more reflective of distance. Since the conversation of half of the distance to 10 came up, I handed another student the 5 card and he placed it right in the middle. The discussion went back to the 7 and they decided that since 7.5 would be halfway between 5 and 10 that 7 had to be a little bit less than the halfway of 5 and 10.

Then, we moved into the fraction cards. We gave each pair of student two cards. In hindsight, for times sake, I would probably only do one card per pair. I gave them one minute to talk about everything they knew about the fractions they had and then we started. I asked for volunteers who thought their card would help us get started and called on a boy with the 1 7/8 card. He went up and stood all of the way to the right and said he couldn’t put his on. I asked why and he said that since the cards were all fractions the line could only go to 1 so his is more than one and can’t go on here. I asked if anyone in the class had a card that may help us out and a student with the 2 card raised her hand. She placed her card all of the way to the right, said “maybe it goes to two” and the other student placed it just to the left of it because, “it is only 1/8 from 2.” Awesome!

We went along with the rest of the cards and so many amazing conversations, agreements and disagreements happened along the way. There are a few things that stand out in my mind as some great reflections on the activity:

A student had placed 5/10 halfway between 0 and 1. The next student placed 2/5 just to the left of the 5/10 because, “I know 2 and a half fifths is a half so that means that 2/5 has to be less than 5/10. It is a half of a fifth away.” The NEXT student volunteered and placed 3/8 overlapping just the edge of the 2/5 card on the left. I was expecting percentages to come out, since that was their most recent work with those fractions, however the student said they knew 3/8 was an 1/8 from a half and 2/5 was a 1/10 from a half and an 1/8 and 1/10 are close but an 1/8 is just a little bit further away. Awesome and definitely not what I expected!
I wish I had not put the zero so far to the left. Looking back I am wondering if that instills misunderstandings when they begin their work with negative numbers on a number line similar to the original misconception that launched the activity with the 1 7/8.
Oh, the 100% card….complete mistake on my part, although it may have been a great mistake to have! In the first class, the student with the 100% card came up and said, “I have 100% and that is 100/100 which is 1” and put it in the appropriate place on the line. Just as she did that, I started thinking how I never really thought about the distinct difference between percent in relation to area (like the grids they had been shading) and 100% when dealing with distance on a number line. No one seemed to notice and since I didn’t know exactly what to ask at that point because I was processing my own thoughts, I waited until another student placed 4/4 on top of it and erased it from my immediate view!
- I stayed for the next class and this time I was prepared for that card and now really looking around to see what students’ reactions were when it was placed. As soon as the student placed it at the 1 location, I heard some side whispers at the tables. I paused and asked what the problem was and they said, “100% is the whole thing.” The next student who volunteered had the 2 card, picked up the 100% card on the way to the right side and put the 2 down and the 100% on top. Lovely and just what I was thinking.

I have never had students reflect on the difference of talking about percentages with distance versus area because I had never thought about it! It definitely feels like an interesting convo to have and a great mistake that I am glad I made!!

I will be back in another 5th grade class tomorrow and will see what happens…it could make for a great journal writing!