Category Archives: Fractions

Fraction Talking Points: 3rd Grade

The 3rd grade is starting fractions this week and I could not be more excited. Fraction work 3-5 is some of my favorite stuff. Last year we tried launching with an Always, Sometimes, Never activity and quickly learned, as we listened to the students, it was not such a great idea. We did not give enough thought about what students were building on from K-2 which resulted in the majority of the cards landing in the “Sometimes” pile without much conversation. And now after hearing Kate Nowak talk about why All, Some, None makes more sense in that activity, it is definitely not something we wanted to relive this year!

We thought starting with a set of Talking Points would open the conversation up a bit more than the A/S/N, so we reworked last year’s statements. I would love any feedback on them as we try to anticipate what we will learn about students’ thinking and the ideas we can revisit as we progress through the unit. I thought it may be interesting to revisit these points after specific lessons that address these ideas.

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We were thinking each statement would elicit conversation around each of the following CCSS: 

Talking Point 1CCSS.MATH.CONTENT.3.NF.A.3.C
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

Talking Point 2CCSS.MATH.CONTENT.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Talking Point 3CCSS.MATH.CONTENT.3.NF.A.2.B
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Talking Point 4: CCSS.MATH.CONTENT.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Talking Point 5CCSS.MATH.CONTENT.3.NF.A.3.D
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Talking Point 6CCSS.MATH.CONTENT.3.NF.A.3.C Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

After the activity, we have a couple of ideas for the journal prompt:

  • Which talking point did your whole group agree with and why?
  • Which talking point did your whole group disagree with and why?
  • Which talking point were you most unsure about and why?
  • Which talking point do you know you are right about and why?
  • Could any of the talking points be true and false?

Would love your feedback! Wording was really hard and I am really still struggling with #4.

If you want to read more about Talking Points for different areas, you can check out these posts:

First Fraction Lesson of 4th Grade

The first lesson of a new unit always feels like an entire class period of formative assessment to me, which I love! I think finding out what the students know about a topic, especially if it is the first time it is introduced that year, is so interesting.

Since the first lesson of the 4th grade fraction unit starts with fractions of a 4 x 6 array, we wanted to create an introduction lesson that was more reflective of all of the great work they did with fraction strips in 3rd grade to get a better picture of what they know. In 3rd grade they do all of the cutting of the strips, and since we didn’t feel that was necessary to do again, I created a SMARTBoard file so we could build together. [the file is attached at the end of the post if you want to use it].

I posed this slide to introduce the whole:

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Then I asked this sequence of questions as we built them on the board:

  • If I wanted halves, how many pieces would I have? What is the size of each piece?
  • If I wanted thirds, how many pieces would I have? What is the size of each piece?
  • etc….until they were all built.

I wanted to reintroduce the language of “size of the piece” from their 3rd grade experiences. Every once in a while I would pause and ask how much I would have if I had more than 1 of those pieces to see if they could name fractions over a unit. For example, What if I had 3 of those fourths? How much would would I have? 

Next, we put up the following questions with the picture of the fraction strips we built:

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They recorded them in the journals as a group and then we made a poster to add to as the year progresses. They started with fractions they could show on the fractions strips and an interesting conversation about the fact that we couldn’t list any for 1/8 or 1/12 based on the strips, arose. After talking with their groups, they generated a couple. The conversation about the change in the size of the piece when we make equivalents and how many pieces we would have was really awesome (Yeah, 3rd grade team):)

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This was as much as we could fit into one class period, so we asked them to journal about any patterns they noticed or things they were wondering about fractions.

I apologize for the overload of student journals from this point forward, but there were so many great things to think about in planning the unit from here!

These are things that jumped out at me after reading and leaving notes in their journals, I would love to hear any other things that stood out to you:

  • A lot of talk about “doubling” and “halving” when naming equivalent fractions. Will want to address what is exactly doubling, what that means in terms of the fraction strips, and how it is affecting the numerator and denominator.
  • Interesting noticing and wondering about addition. Some wondering how it works and others thinking they know.
  • Love the even and odd talk throughout!
  • Some wondering about multiplication and division of fraction!
  • The range of fractions – how many we can name, how many unit fractions there are.
  • The size of a fraction in different forms – Is the whole the biggest fraction? Is the numerator smaller than the denominator?

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In case you want to try it out:

SMARTBoard file for 4th grade

SMARTBoard file for 5th grade

PDF file of the file.

Cuisenaire Rods: Fun in 4th

The 4th grade has just started their fraction unit, so I was curious how that may impact their work with the Cuisenaire rods. I started just like I did in Kindergarten and 3rd grade, with a notice and wonder:

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There were two of the ideas that really struck me as things I want to have the students explore further later: First was, “2 of the staircases (the staggered rods in order of size) could make a square.” They had them arranged on their desks like the picture below…but I want them to answer:

  • Is that a square?
  • If not, could we make it a square?

Then I started to wonder, do we call it a square? Should we say square face? Then what about area…would we say, “What is the area of the rectangle?”? That feels wrong because they keep calling the white rod a cube (which it is). But then asking about volume is not 4th grade. BUT, the tiles we use for area in 3rd grade are also 3-dimensional. <–would love thoughts on any of that in the comments!

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One student noticed that the orange rod was the height of the staircase and I thought of area again since it was said right after the comment above. This idea would be really helpful for the students above when they are determining if their figure is a square.

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I loved that one group noticed that any of the rods could be a whole and another group wondered if orange was the whole. Great lead into what I was thinking I wanted them to explore!

I asked them to find values for the rods based on their relationships. Of course the very first group I call on had 2 as the whole, which blew a lot of students minds, so I want to revisit that a bit later and ask them to explain how that works.

All of the other groups had orange as either 1 or 10, so I asked them to find the other values if the orange was 5 and 100. They played with that for a bit and then I began to hear a lot of aha’s, so I set them off to find more and they could have gone on forever.

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I left them with the prompt, “Tell me about the patterns and relationships you notice.” and for those who looked like they were struggling to answer that question, I added, “If you are struggling with that, tell me how you could find the rest of the values if I gave you one of them and which one would you want?”

I loved how this student chose the orange, white and yellow as the easiest end, beginning, and half. I also like the red x 2 is purple, but we need to talk through that notation a bit.

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This was the most common response, seeing the numbers get smaller as the rod got shorter.

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This student’s noticing could be an interesting number choice question to pose: Why do you think groups chose numbers for orange that were doubles or halves of the other numbers we already had?

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This student disagreed with the student who gave the responses in the first column because he is determined the white is 1/10 because the orange is 1. Would be great to pair them up and have them come to an agreement.

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This student is seeing the white value adding to each value above it to get the next. I also love how she writes notes about how neat her handwriting is:)

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I would love to have them play around with this first pattern in this entry! What other relationships could they find after they explored this one?

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So much fun! Cannot wait to get into other grade levels to see if I can begin to find a progression of ideas with these rods!

 

My Beginnings With Cuisenaire Rods

I have never been more intrigued with using Cuisenaire rods in the classroom until I started reading Simon’s blog! I admit, I have read and watched his work from afar…not knowing really where to start with them and was afraid to just jump into another teacher’s classroom and say, “Hey let me try out something!” when I really didn’t know what that something may be. However, after Kassia reached out to Simon on Twitter asking how to get started with Cuisenaire rods and Simon wrote a great blog response, I was inspired to just jump right in!

I am a bit of an over-planner, so not having a really focused goal for a math lesson makes me a bit anxious. I am fairly certain I could anticipate what 4th and 5th graders would notice and wonder about the Cuisenaire rods because of my experience in that grade band, however I wanted to see what the younger students would do, so I ventured into a Kindergarten and 3rd grade classroom with a really loose plan.

Kindergarten (45 minutes)

Warm-up: Let’s notice and wonder!

  1. Dump out the bags of Cuisenaire rods in the middle of each table of 4 students.
  2. Tell them not to touch them for the first round.
  3. Ask what they notice and wonder and collect responses.

Things they noticed:

  • White ones looked like ice cubes.
  • Orange ones are rectangles.
  • End of blue one is a diamond (another student said rhombus)
  • Different colors (green, white, orange..)
  • They can build things (which is why we did no touching the first round:)
  • Orange is the longest.
  • They are different sizes.
  • We can sort them by colors.
  • We can sort them by size.

Things they wondered:

  • What do they feel like?
  • What can we make with them?

Activity 1: Let’s Sort!

  1. Tell the students to sort them by size or color. (they quickly realized it was the same thing)
  2. Discuss their sort/organization and check out how other tables sorted.

I was surprised to see not many sorted them into piles because that is normally how they sort things. I am wondering if the incremental size difference between each rods made them do more of a progression of size than sort into piles? Some groups worked together while others like making their own set with one of each color (and size) and keep making more of those!

Activity 2: Let’s Make an Orange!

Since a lot of students kept mentioning that the orange was the longest, I decided to see if they could build some trains (as Simon calls them) that made an orange.

My time was running out, but it left my mind reeling of where I wanted to go next! My inclination is to ask them if they could assign numbers to some of the rods or if they could build some trains the same length as the different colors? I would love to hear which piece is their favorite piece because a lot of them found the smallest cube really helpful when building the orange.

3rd Grade (60 minutes)

Warm-up: Let’s notice and wonder!

Things they noticed:

  • Groups were the same color and length.
  • Blue and white is the same length as the orange rod.
  • Kind of like adding.
  • White is 1 cm.
  • Go up by one white cube every time.
  • Odd + even numbers
  • 2 yellows + anything will be bigger than 0.
  • 1 white + 1 green = 1 magenta

Things they wondered:

  • Is red 1 inch?
  • How long are the rods altogether? (Prediction of 26 or 27 in wide)
  • Is orange 4 1/2 or 5 inches?
  • Why doesn’t it keep going to bigger than orange?

Activity 1: Let’s build some equivalents!

I found 3rd graders love to stand them up more than Kindergarteners:)

Activity 2: Let’s assign some values!

After they built a bunch, I asked them to assign a value to each color that made sense to them…this was by far my favorite part – probably because it was getting more into my comfort zone!

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Again, time was running out, but next steps I am thinking…

  • What patterns and relationships do you see in the table?
  • What columns have something in common? Which ones don’t have anything in common? Why?
  • What if I told you orange was 1? What are the others?
  • What if orange was 2? What happens then?

Thank you so much Simon for all of inspiration and Kassia for the push into the classroom with these! Reflecting, I was much more structured than Simon and Kassia, but I look forward to a bit more play with these as the year goes on! I look forward to so much more play with the Cuisenaire rods and continuing Cuisenaire Around Ahe World!

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:

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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.

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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:

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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.

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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…”

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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!

5th Grade Fraction Clothesline

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:

  1. 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!
  2. 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.
  3. 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!!

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

-Kristin

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…

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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