Tag Archives: 5th Grade

How Planning Mistakes Can Lead To Great Student Thinking….

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….

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

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

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Some related it to different contexts with different wholes…

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

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What a great day revisiting my planning mistake!

-Kristin

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

#IntentTalk Chapter 1

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Since it was a bit too much for me to continually tweet, I decided to do a quick blog at lunch!

Principle 1: Discussions Should Achieve a Mathematical Goal

The first week of school my mathematical goals revolve around discussions about students’ mindset in terms of math, as well as the mathematical practices. I found this year that Talking Points and one of the tasks I found on Fawn’s blog fostered those goals. I blogged about it here. My tables are all arranged in groups and the students know from the very first day that discussions will be a huge part of our work together.

Principle 2: Students Need to Know What and How to Share

To support this in my classroom at the start of the school year, I have the students agree upon our class norms. They originate after doing a Number Talk together and reflecting on what we expect as a group during our discussions. I reference these norms throughout the course of the school year.

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Principle 3: Teachers Need to Orient Students to One Another and the Mathematical Ideas

I find a lot this happens during our Number Talks and in then daily in our journal reflections. This is such a focus on my planning of questioning. Asking things such as, “Can you re-explain their thinking in your own word?” or “Did something ____ say change your mind about that way you were thinking?”

Principle 4: Teachers Must Communicate That All Students Are Sense Maker and That Their Ideas are Valued

I think this principle emerges during our very first round of Talking Points of the year when the students my go around their circle with no commenting from others. It allows students the opportunity to speak their ideas without judgement or comment. Being able to change their response on the second round, lets the group know that as we make sense of problems and listen to others, we change our mind, just as we do when we make errors. The freedom I see in their journal entries also reinforces the idea that I value their thinking and know that there is reasoning behind everything they write and do in my class.

“Talk is an important way to build that sense of community and to help children grapple with important mathematical ideas.”

-Kristin

Last Day of Math Class :(

Today was the last official day I had my students for math. It was a bit sad for me and it was nice to hear some of them say it was “kinda sad” for them too. In moving into a K-5 Math Specialist position next year, I know it will not be the same experience watching a group of students grow over the course of the school year.  It will be great in different ways, but I am really appreciating all of the amazing work my students have done this year.

So…what to do on the last math day after they just had field day yesterday followed by our PBS bowling field trip tomorrow? It is a tough planning!

I first had them look through their two math journals, one from the first half of the year and one from the second half. As their last writing piece, I asked them to write things they noticed in their work over the course of the year after looking through their journal. I only had time to grab one journal today because the end of the year craziness is kicking in, but I plan on following up with a more detailed post later. This one was so powerful and truly gave me goosebumps….

IMG_0955_2After they finished that, I asked them to revisit some of the claims they had written over the course of the year and see if they still thought they were true and could be proven or were not true and needed to be revised. This student had written a claim that when you are multiplying fractions, you could multiply the numerator and denominator to get your answer. As he was proving it just worked for multiplication, he stumbled upon the realization that it worked for division as well. He then worked through a few more division problems and it was such an amazing explanation!

He revised his claim…

IMG_0956_2IMG_0957_2I promise to follow up with some really amazing work they did on the last day when summer is here and there is a chance to breathe 🙂

-Kristin

Patterns and Perseverance

Today in math was a test in perseverance. The students were working on the growth pattern of an animal called the Fastwalker. It was fairly easy for them to complete up to the 10th year, graph it and answer the questions regarding the line they graphed. The book did not require them to do any generalizing of a rule, however they had other plans! Here is a completed table of one of my students:

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We worked on this up until lunch, discussed the line and how it was different than the constant rate of change graphs we had seen earlier. They kept asking if there was a rule for this one, so I asked them to play around with it and see what they thought. One of the students noticed that if you added all of the terms before with the term number you were trying to find, it gave her the height, so she started adding to see if it worked for the 100th term (nothing like starting small:).

IMG_0826While she was working on adding, another student, who had done a consecutive sums task earlier in the year in RTI w/me, realized there was an easier way to add those numbers, and you can see on the top of the page where he started playing around with pairing up the numbers:

IMG_0823It was so interesting to see the groups working with them and asking questions as they tried different ideas. The two of them finally saw that pairing up the numbers was working and began to generalize based on what they had done with the numbers. It was awesome because they began generalizing based on an even or odd number term because of the pairings and needing to divide the term number by 2. At the bottom of the first paper earlier in this post, you can see she wrote an even and odd rule for the pattern, while this student realized that if should work with even and odd because the decimal didn’t make a difference.

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IMG_0828Shew…..they were sooo proud of themselves (and I was so proud of them) at the end of all of this hard work! The student who did the paper above said, “Wow, that one problem took us almost two hours!” And it was SO worth it to see the accomplishment on their faces. THIS is the reason we must make time for students to investigate their own mathematical curiosities and give them the time they need to persevere through these problems!

-Kristin

Growth Patter Number Talk….3rd Times a Charm

Over the past couple of days, with my homeroom, I have tried a few strings of numbers to bring out the different ideas that are important when thinking about growth patterns and finding any term in a sequence, Here and here. Both days brought out many great ideas, conversations, and disagreements, however I couldn’t help but feeling the ideas we talked about in two days, could have been achieved in one and felt a little more connected. I knew it was completely the way I posed the problems, so when my second class came in yesterday, after missing a couple days of math due to testing, I was excited to try and adjust my previous work.

Apologize for the messy board, but I still cannot seem to get a handle on that recording thing…

I started with having a student count by 6’s and wrote that in blue. I stopped them at 4 because I was asking about the 10th term and wanted to see if some would figure our 5th and double. I think that is an interesting thing to think about when the start is different so I wanted it up there. I asked 10th term? 60. Ways to get there? 6 x 10 and, unlike my prediction of doubling 30, one student said 24 x 2 because two group of 4 of them is 8 and then two more 6’s (12) is added to that to make 60. I asked 100th term? 600. 2,000th term? 12,000. I asked them how they were getting those without counting and I got “I did 6 times 100” “I did 6 x 2000” and then one student said you could do any number by multiplying it by 6. I asked how he wanted to write that and I wrote that in green. Another student, who has done Visual Patterns with me in our RTI group, said, “We can also write that as 6 times n = Answer.” I asked them to turn and talk to a neighbor if they thought that meant the same thing. We had all yeses and I had some student prove it. I did the same thing with 8’s and wrote that in orange. They started using “A” for “Answer.”

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After that, I asked them to to count by 3’s starting with 6 and stopped them at 15… Asked for the 10th term and got, as expected, 30 and 33. Then the conversation took off with proofs and some really important ideas that was hoping would emerge. I love it when the class is practically divided in half on an answer, we had the 30’s and the 33’s. I asked a 30 to explain how he got the answer and he quickly said 3 x 10=30. I saw a lot of agreement, so I asked for a 33 to share their reasoning. A student said that we “need the beginning number, three, to find out where the tenth one is. 3 x 10 is 30 but then you started three ahead of that so you add 3 to 30.” I wrote that down on the board.

A student then said something that made me have a realization, “It shouldn’t change because you are still doing 10 jumps of 3, so it HAS to be 30. 33 is 3 x 11.” In my last class I had a student who kept insisting that the 10th term remain the same no matter where we started and I could not figure out what they were trying to articulate. NOW, I understand. 30 will always be the distance between wherever we start in the sequence and the 10th term, but not the tenth from the true beginning. AH HA!

IMG_0774_2So, the beginning number was suddenly becoming very important and articulating “10th term from where” was having students agreeing that the 10th term starting from the 6 was going to be 33 but when thinking about a rule for the pattern we needed the true beginning. We were just about to head back to our desks to continue our work when a student (different than the one who had originally said it) said that we could write this one “3 x n + 3 = A” because you have to “add the three you are missing from the beginning to get the answer.” I had them turn, talk and try a few terms out and see what they thought. It was all wrapping up nicely (I was excited about it) when another student said, “You could also write 6 + (3 x n) since you are starting at 6” ….oh goodness, they just don’t ever let it end and I love it:) A disagreement arose that it would have to be “6 + (3 x n -3) because of that extra jump of 3 to start at 6.”

I always hate to say that time got the best of me, but I had missed this group for 2 days of math and I saw this conversation going lonnnnng so I had them write those ideas down in their journal to kick off our class on Monday!

I love when I have the chance to refine ideas that don’t go exactly as I had hoped they would, especially when I know it was completely how I posed the problem or asked the question. After a couple days of talks not connecting as I hoped they would, third time was a charm!

-Kristin