Category Archives: 3rd Grade

Multiplication: Does Order Really Matter?

Some things I am wondering right now about 3rd grade multiplication…

When students notice 4 x 3 is the same product as 3 x 4 and say, “The order doesn’t matter,” how do you answer that question?
Is there a convention for writing 4 groups of 3 as 4 x 3?
Is there a time, like when moving into division or fraction multiplication and division when the order does matter in solving or in thinking about the context?

Answers I have right now for these questions….

Right now, since they are just learning multiplication, I ask them what they think and why.
I think there is a bit of a convention in my mind because the picture changes. Three baskets with 2 apples in each is different than 2 baskets with 3 apples in each. Also, when reading the CCSS it seems that way.
I am still thinking about division but it makes me think that this would be the difference between partitive and quotative division. I also think when students begin 4th fraction multiplication, they are relating it to what they know about whole number operations, so 4 x 1/2 is 4 groups of 1/2. This seems important.

The 3rd grade teachers and I have been having a lot of conversation about these ideas. The students have been doing a lot of dot images and some feel strongly that the two expressions mean the same thing because they can regroup the dots to match both expressions. Others think they are different because the picture changes. All of this seems great, but then students are taking this reasoning to story problems. For example, given a problem such as, There are 5 shelves with 6 pumpkins on each shelf. How many pumpkins are on the shelves? students will represent that as 5×6 or 6×5. Is that a problem for me, not really if they have a way to get the 30, but should it be? I am not sure.

I went into a 3rd grade classroom to try some stuff out. I told them I was going to tell them two stories and wanted them to draw a picture to represent the story (not an art class picture, a math picture) along with a multiplication equation that matched.

1st story: On a grocery store wall there are 5 shelves. There are 6 pumpkins on each shelf.

2nd story: On another wall there are 6 shelves with 5 pumpkins on each shelf.

I asked them if the stories were the same and we, as I anticipated, got into the conversation about 5×6 vs 6×5 and what it meant in terms of the story. They talked about 5 groups of 6, related the switching of factors to addition and then some talked about 6 rows of 5.

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From this work, many interesting things emerged…

Some students had different answers for the two problems. They obviously did not see the two expressions as the same because they struggled with 5 groups of 6 as they tried to count by 6’s and forgot a row.

One student said they liked the second problem better because she could count by 5’s easier than by 6’s.

Students skip counted by 5’s but added 6’s when finding the 5 groups of 6.
One student noticed the difference between 5 and 6 and could relate that removing one shelf was just adding a pumpkin to each of the other rows.

One student showed how he used what he knew about one to switch the factors to make it easier to solve.

But they keep asking Which one is right? and I tell them I don’t have an answer for them. I just keep asking them:

Is the answer the same?

Is the picture the same when you hear the story?

After chatting with Michael Pershan yesterday, I am still in a weird place with my thinking on this and I think he and I are in semi-agreement on a few things (correct me if I am wrong Michael) …Yes, I think “groups of” is important to the context of a story. I want students to know they can find the answer to these types of problems by multiplying. I want students to be able to abstract the expression and change the order of the factors if they know it will make it easier to solve BUT what I cannot come to a clear decision on is…

If we should encourage (or want) students to represent a problem in a way that matches the context AND if the answer is yes, then is that way: a groups of b is a x b?

Adding & Subtracting: Tools and Representations

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There is always a lot of talk about students using an algorithm, process or set of rules, for addition and subtraction. Whether talking about “any algorithm” or “the algorithm,” I am certain, in most cases, people are talking about a process that is absent of tools such as a 100 charts, number lines or base ten blocks. But, what happens when we see the tool becoming an algorithm in and of itself? Can moving left or right on a number line, making jumps of 10s and 1s, writing problems vertically, or jumping rows and columns become an algorithm where students lose sight of the numbers themselves because the process is one more thing to learn?

This was the exact conversation I had yesterday with two 3rd grade teachers as I was leaving school. The students had been playing a game called Capture 5 and struggled making various jumps on the 100 chart. The teacher, understandably, was concerned students were confused about adding and subtracting tens and ones. After more conversation, we began to wonder if the students saw the 100 chart as a set of rules to follow in order to add and subtract instead of a place to look for patterns and structure as we add and subtract. Were they getting caught up in the left, right, up, down movement and losing sight of what was actually happening to the number?

As I thought more about it last night, I wondered about other tools and representations students learn that could easily turn themselves into an unhelpful set of procedures. I also wondered how often we make connections between these tools and representations explicit. Like, when is one helpful over another? How are they the same? How are they different?

I emailed the teacher my notes (below) and we decided we would try this out this morning.

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If you can’t interpret from my notes, the plan was to have each student in a group using a different tool or representation as I called out a series of operations to carry out. After the series of addition and subtraction, they compared their answers and discussed any differences. They rotated seats after each series so they had a chance to try out each of the roles.

We came back together to discuss their favorite one. The recording is below…what do I have against writing horizontally, really?? I found this entire conversation SO incredibly interesting!

They found the base ten blocks to be “low stress” because they were easy to count, move and trade, but did agree that bigger numbers would be really hard with them because there would be too many.
They really did not like adding and subtracting on the number line with multiple jumps. It got messy.
They liked mental math because there was nothing (tool) to distract them and they could focus but they didn’t like that you couldn’t check your answer.
The 100 chart’s only perk was they didn’t have to write the numbers in, they were already there for them.
I really loved that they mentioned the equations were they only way they could track their work. So if someone in their group messed up, the equation person was the only one that could help them retrace their steps easily.

I am not sure what I learned today, I am still thinking a lot about this. I know that I loved having them compare the tools and representations and that the teacher felt much better about their ability to add 10s and 1s. I feel like there are so many other cools things to do here, but my brain is fried today so that will have to wait!:)

3rd Grade Dot Image

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Since the 3rd grade classes are about to begin their multiplication unit, the teachers and I wanted to hear how they talk about equal groups to get a sense of where they are in their thinking. What better way to do that than a dot image? I chose the first image because of the 3’s and “look” of 5’s and the second image because of the 2’s,3’s, and 6’s, all of which students can count by easily.

Image 1 went relatively the same in both classrooms and much like I anticipated. There were two things that stood out to me as a bit different between the class responses:

There were more incorrect answers shared in the 1st class than the 2nd class.
In the second class, multiplication came out during the discussion. The “4 groups of 7 and 4 x7 = 28” in the 1st class came out after both images were finished and one student said she knew some multiplication already. She asked to go back to the first image and gave me that.

1st Class

2nd Class

After the first image, I anticipated Image 2 would go much the same, however it was quite different.

1st Class

2nd Class

After the 1st image, I was really surprised at the difference in responses and I have to say it even felt really different. My assumption at this point is that in the 2nd Class one of the early responses was multiplication.

I am left wondering:

Does that early multiplication response shut down other students who don’t know anything about multiplication yet? While I asked her to explain what she meant when she said 4 x 12, I wonder if that intimidated others?
How could I have handled that response differently so others felt OK using skip counting or addition to count the dots?
Can we anticipate that type of reaction from other students when someone starts the discussion with something that may be beyond where the majority of the class is in their thinking?
Was this even the issue at all? Did the 2nd class just see so many more dots and wanted to avoid adding and skip counting?

The 2nd Class ended with a journal entry after a student remarked, “If we know it is 8 groups of 6, then it is also 6 groups of 8.”

I asked if 8 groups of 6 is the same as 6 groups of 8 and the class was split on their response, so they set off to their journals.

The yes’s went with multiplication expressions representing the same product and commutative property:

I loved this no because the picture changes:

I am not sure about this argument but I would love to talk to the student a bit more about the bottom part!

After that talk, I am excited to see what these guys do when they actually start their multiplication unit!

Which One Doesn’t Belong? Place Value

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Since the 3rd grade team begins the year with an addition and subtraction unit in Investigations the teachers and I were having a conversation about how students understand place value. While I don’t see teachers using the HTO (hundreds/tens/ones) chart in their classrooms, students still seem to talk about numbers in that sense. For example, when given a 3-digit number such as 148, students are quick to say the number has 4 tens instead of thinking about the tens that are in the 100. I think a lot of this is because of how we as teachers say these things in our classrooms. I know I am guilty of quickly saying something like, “Oh, you looked at the 4 tens and subtracted…” when doing computation number talks, which could lead students to solely see the value of a number by what digit is sitting in a particular place.

We thought it would be interesting to get a vibe of how this new group of 2nd graders talked about numbers since their first unit deals with place in terms of stickers. A sheet of stickers is 100, a strip of stickers is 10 and then there are the single stickers equal to 1.

I designed a Which One Doesn’t Belong? activity with four numbers: 45, 148, 76, 40

I posted the numbers, asked students to share which number they thought didn’t belong, and asked them to work in groups to come up with a reason that each could not belong. Below is the final recording of their ideas:

I loved the random equation for 148 that emerged and the unsureness of what numbers they would hit if they counted by 3’s or 4’s. One student was sure she would say 45 when she counted by 3’s and was sure she would not say 76 or 40, but unsure about the 148. I wrote those at the bottom for them to check out later.

Since the teacher said she was good on time, I kept going. I pulled the 148 and asked how many tens were in that number. I was not surprised to see the majority say 4, but I did have 3 or 4 students say 14. As you can see below a student did mention the HTO chart, with tallies, interesting.

As students shared, I thought about something Marilyn Burns tweeted a week or so ago…

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So, I asked the students to do their first math journal of the school year (YEAH!):

“For the students who answered 14, what question did you answer?

“For the students who answered 4, what question did you answer?

After the students shared, I revisited the Hundreds, Tens, Ones chart. I put a 14 in the tens column, 8 in the ones column, and asked if that was right. The light bulbs and confusion was great! It was as if I had broken all rules of the HTO chart! Then I put a 1 in the hundreds, 3 in the tens, and they worked out the 18. I look forward to seeing them play around with this some more and wonder if when they go to subtract something 148-92, they can think 14 tens -9 tens is 5 tens.

I had to run out because I was running out of time, but snagged three open journals as I left! (I especially love the “I Heart Math” on the second one!
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My One Hundred Hungry Ants Obsession

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Lately, I have been obsessed with children’s literature across K-5. My most recent obsession is the book One Hundred Hungry Ants. I did this in Kindergarten and this in 4th grade and today I invaded a 3rd grade classroom with it!

I followed the same pattern I usually do, I read the story aloud and did a notice/wonder. These are all of the things they noticed:

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The last one led perfectly into asking about the ways the ants rearranged themselves. I wrote the combinations they recalled from the book and asked them to chat with a neighbor about patterns they see.

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The discussion started with the 50+50=100, 25+25=25 and 10+10=20. Another student said they had the same things but it sounded different because she saw 50 was half of 100. They moved away from that and went to divisibility by the numbers that did not show up like 3,6,7,8, and 9 and pointed out that all of the second factors were multiples of 5. At this point they were focusing primarily on the second factor until someone pointed out the increasing and decreasing pattern happening. Then we got into the doubling and halving, quadrupling and dividing by 4 and multiplying and dividing by 10 of the factors.

I asked them if that would work with any number I gave them. They were quiet so I threw a number out there for them to think about, 24. They had to move into another activity so I left them with that thought. Before I left, however, one student said yes for 24 because 2×12, 4×6,8×3. Another student said it could be sixteen 1 1/2s and then thirty-two 3/4s! Wow!

Tomorrow they are going to investigate this further to see if they can come up with a conjecture about this work! So excited!

Making Sense of Problems: Part 2

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This post is an extension of a previous post. For the background story to this post, it will be helpful to read THIS POST first.

The original Noticing and Wondering from the launch of the lesson:

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Here are some expanded descriptions of the student work:

Chose numbers strategically to make it easier for themselves:

These two girls were great because they wrote out the paragraph first with the blanks left to fill in after they made a decision on their numbers. You can see the erased 5 in the second blank. When I asked them about it, they said 25 in a class seemed like too many but they couldn’t make the class too small to each get more than 1 bar.

These two were concerned with the number in each box. They said they knew 6 usually came in a box so they just did 4 boxes and then wrote the students in last to make it easy division.

These two were done SUPER fast so I gave them 5 more bars to try and decide what they wanted to do with them. They didn’t do any written work, but asked me how they divide something up into 5 pieces because then each student could get a piece. “We know halves and fourths, but that is not 5 pieces.” After playing around with “fiveths” I gave them the word fifths and they wrote down 1/5.

These two partners were so interesting because when I walked by the first time they had chosen their numbers together, but when I went back the second time, their computation was completely different. I absolutely loved that and asked them to explain their strategy to one another and asked how they were the same and how they were different. The difference was more about the look of their work, but they agreed they were the same because it was still how many 30s were in 63.

Chose numbers randomly:

…and then they worked together on breaking the extra 3 into 10 pieces. Because they didn’t know how to name tenths, they went to something they obviously knew something about…percents! We ran out of time to ask how they knew that was 10%, but I have to make a point to go back and ask!

Dealt with the leftovers using fractions:

This one took a while for me to figure out. These two girls finished rather quickly, so I asked them if they could share the leftovers equally among the kids in the class. It looks like they multiplied the 22 by 2 to get how many pieces they would have if they split them in halves. They each person got an extra 1/2 and they were left with 18 halves. They multiplied by 2 to make them wholes again and ended with 9 bars left over. The sense-making in this one was so incredible to me.

Chose numbers strategically to make it harder for themselves:

When I asked these two girls why they chose the numbers they did and they said they wanted to make it hard! There are so many things I think continued to be fabulous after their initial number choice. The partial products for multiplication and then the repeated subtraction were amazing. I asked them why they were subtracting 26 every time. They said each time they subtracted 26, each student got 1, hence the growing list of 1,2,3,4,5… Absolutely awesome and something I would have never seen if I had given the original problem and the sense-making of what is happening whey you repeated subtract to divide just blows my mind.

To me, every one of these examples, along with all of the papers in the class that day, demonstrated to me how we need to look critically at our math textbooks, think deeply about what we learn about students as they do that work, and adapt materials to allow students to make sense of problems and allow us to learn more about their understandings.

Formative Assessment

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Assessment always seems to be such a broad, hot topic There are rubrics to help create assessments, rubrics for reviewing assessments, and tons of reading about the benefit of assessments. While I agree assessment is an important topic of conversation and all of these things can be helpful, I just lose a bit of interest when it becomes so cumbersome. I feel the longer the rubric and steps to create an assessment, the more detached the assessment becomes from student thinking. This could be completely be my short attention span speaking, however the way assessment is discussed feels either like data (a grade or number-type of data) or a huge process with tons of text in rubrics that I really, quite honestly, don’t feel like reading. Not to mention, I just love looking at student writing and listening to student thinking when planning my immediate next steps (formative) or checking in to see what students have learned over a longer period (summative). This is why I find the work we are doing each month in our Learning Labs such a wonderful way to think about formative assessment in an actual classroom context, in real time.

This passage from NCTM’s Principles to Action really captures how I feel about the work we are doing in our Learning Labs:

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In this most recent Learning Lab in 3rd grade, we planned the activity together using the 5 Practices model and reflected after the lesson. Since this blog is always my thoughts about student work, I thought it would be great to hear what the teachers took away from the activities we are doing in terms of the students’ understandings and impact on their future planning, formative assessment.

The teacher mentioned in the blog said, “I was surprised by how quick many of the students defended their responses that 1/2 will always be greater than 1/3, and then proving this response using visual representation of the same whole ( which is an idea that we have made explicit). I was impressed with “skeptics” in the crowd that were looking to deepen their understanding around the concept by asking those “What if” questions. Going forward, I want to create opportunities that push and challenge my student’s thinking. I want them to continue to question and explore math – especially when it uses the word “always.”

Another teacher who taught the same activity after watching it in action in another classroom said, “I learned that almost half of my students assumed they were comparing the same size wholes. They agreed with the statement, and each student gave at least two different ways to prove their thinking (area and number line model were most common). The students that disagreed almost all provided their own context to the problem, such as an example with small vs large pizzas, or a 2 different-length races being run. I found it so interesting that almost all students confidently chose one side or the other, and were able to defend their thinking with examples (and more than one-yeah!) I was excited to see that they could be so flexible in their arguments as to why they felt as they did. Three students responded that they were unsure, and gave reasons to support both sides of the argument. This impacted my instruction by giving me such valuable formative assessment information with a simple, non-threatening prompt. It took about 5 minutes, and gave me tons of information. It was accessible and appropriate for all. Students were comfortable agreeing or disagreeing, and in some cases, saying “unsure-and here is why.” I was most excited about that!”

She also said, “From this activity, I learned that I really needed to revisit the third grade standard to see what is actually expected. It says they should recognize that comparisons are valid only when the two fractions refer to the same whole. My statement didn’t have a context, so how cool that some were at least questioning this! This impacted my planning and instruction by reminding me how thinking/wondering about adding a context to the statement would influence their responses. I am also reminded that I need to stress that students must consider the whole in order to make comparisons accurately.”

Earlier in their fraction unit, the third grade teachers used the talking point below to hear how her students were talking about fractions. (This work is actually from another teacher’s class, but you get the idea;)

A teacher who did this activity reflected, “From this activity, I learned my students had only ever been exposed to a fraction as a part of a whole (and wanted to strictly refer to fractions in terms of pizza). This impacted my instruction by being sure to have the discussion that fractions can represent parts of a whole, but we can also represent whole numbers with fractions.”

To me, these reflections are what assessment should be….the teachers learn about student thinking, the students think about their own thinking, and what we learn helps us plan future lessons with our students’ understandings in mind!

More examples from different grade levels where the teachers and I learned so much about student thinking that impacted future instruction:

Kindergarten: Adding

Kindergarten: Counting

1st Grade: Fractions and Adding

2nd Grade: Counting and Leftovers

4th Grade: Division

5th Grade: Fraction Number Line

3rd Grade: Comparing Fractions

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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!

Rhombus vs Diamond

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Every year in 5th grade, when we begin classifying quadrilaterals, students will continually call a rhombus a diamond. It never fails. While doing a Which One Doesn’t Belong in 3rd grade yesterday, the same thing happened, so Christopher’s tweet came at the most perfect time! (On Desmos here: https://t.co/rZQhu2SGnR)

Which one doesn't belong? (Diamond edition)

cc @MathMinds pic.twitter.com/VmMh8rakXg

— Christopher Danielson (@Trianglemancsd) February 9, 2016

Of course I had to pop into the same classroom today and try it out! The lower right was so obviously a diamond to me that I was curious to see if students saw the same thing and if it changed their reasoning about the rhombus as a diamond.

Here are pictures of the SMARTboard after our talk:

After great discussions around number of sides, rotations, decomposition and orientation, they finally got to the naming piece. Honestly, I was surprised names didn’t come up as one of the first things. It started with a student saying the square didn’t belong because it is the only one that doesn’t look like a diamond. The next student said the lower left was the only one “that didn’t have a name.” When I asked him to explain further, he named the square, rhombus, and diamond. Because I knew at the end of our talk I wanted to ask about the diamond vs rhombus, I wrote the names on the shapes. Another classmate added on and said the lower left “may not have a name but it is kite-shaped and looks like it got stuck in a tree sideways.” I asked the class what they thought about the names we had on the board and it was a unanimous agreement on all of them. Funny how quickly they abandoned their idea from yesterday, so I reminded them….they were not getting off the hook that easy;)

“Yesterday you were calling this rhombus a diamond, what changed your mind?”

Students explained that the lower right actually looks like a real diamond and the rhombus doesn’t now that they see them together.

“Can we call both of them a diamond?” I asked. I saw a few thinking that may be a great idea. I had them turn and talk to a neighbor while I listened to them.

We came back and they seemed to agree we couldn’t call them both a diamond because of the number of sides. They were really confident in making the rule that the quadrilateral one had to be a rhombus and the pentagon was the diamond. I pointed to the kite and asked about that one, since it has four sides. “Could we call this a rhombus?” They said no because the sides weren’t equal, so not a rhombus. And because it didn’t have five sides, not a diamond either.

Thank you Christopher! All of these years of trying to settle that rhombus vs diamond debate settled right here with great conversation all around!

Next up, this one from Christopher…

@jacehan @MathMinds Here's v 3.0. pic.twitter.com/3klfsv2fGJ

— Christopher Danielson (@Trianglemancsd) February 10, 2016

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!