Category Archives: 4th Grade

Show Your Work vs Show Your Thinking; 4th Grade Division

The Planning

Last week, the 4th grade team and I planned during our Learning Lab for their first division lesson in Unit 3 of Investigations. The book opened with the following problem:

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As we talked during the lab, one of the main concerns expressed by the teachers was student comprehension of the context. This was not necessarily in reference to this specific problem, but story problems in general. This is not surprising and definitely something, I would say, we as teachers face quite often. This is why I love Notice/Wonders and offered that as an idea to take that “number grabbing and compute” feel out of the problem.

As we read the problem aloud, we anticipated how students may respond in two scenarios:

  • Keeping the numbers in the problem and taking the question out.
  • Taking out both the numbers and the question.

We opted for the second one with the thought that we could ask them to anticipate what a question may be for this problem and what information they would need from us. It felt like a bit of a mash up of noticing/ wondering and a 3 act task.

The Launch

I really want to get to the student thinking here so I will not go into the details of the N/W, but here is what we ended our conversation with…

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After this, we gave them the original problem and sent them off too work on the problem. After 5 minutes of individual time, they came together as small groups to share their work and create a poster with all of their strategies.

*We did debate that last sentence for a bit. We didn’t know whether the rows being the same size would arise and/or if it would send the problem in a different direction. Another 4th grade teacher tried it without that sentence in a different class period and it didn’t seem to change the focus of the work so I wonder if student make that assumption? Then I wondered if we give too many problems that are arrays and maybe don’t play around more with the “extra” pieces that could be added on?

Show Your Work vs Show Your Thinking

After the lesson, I wasn’t surprised by the students’ strategies as much as I was left questioning how we posed the prompt for students to show their strategies. The teacher asked them to solve the problem and show their work. This piece is not something we talked about at all in our planning because it is what we all say, but does that make a difference in how students show their solution path? Do we ever make explicit the way we can show our thinking clearly and explicitly either pictorially or numerically?

As I walked around, I would have thought at first glance that over 50% of the students were “passing out the apples” on their paper to arrive at 14, however after asking them to explain their thinking to me, I would have completely assumed wrong.

For example, this student’s work (on the left) was just the rows and dots underneath it when I walked by. I assumed he had “passed out” 56 apples to each of the 4 rows, however when I asked him to explain his thinking to me, it was so much more than that. He said he “gave each row 10 because that would be 40 and then there were 16 left, so half of that is 8 and half of that is 4, so each row got 4 more. I asked him if we could take what he did to write an equation for it? He said 10+10+10+10 and then wrote 10×4=40 and added the 4×4=16 underneath. I asked where the 56 apples appeared in his work and where the apples in each row were. So, the thinking he explained made his understandings so much more clear than the work he had shown. I wondered if he really needed to show all of that work or he just thought he did for the teacher to see “work”? Next, I partnered him with his neighbor (work on the right) to talk about how his thinking could be show within her multiple tower.



When I walked up to this student, she had everything on her paper but the equations on the right. I asked her to explain her thinking and she explained that she divided up the 56 into 4 rows. By the way it looked to me, her x’s were written going down each column which did not indicate to me that she has passed them out so I asked her if she solved using those x’s and she quickly responded, “No, I just got bored…see all of my decorations!” I pressed her a bit more to explain how she got the 14 and she said she knew 8 x 7 = 56 and she used that but was having trouble explaining it. I asked her if she thought it may help her remember if she wrote down the 8×7=56 on her paper and quickly after she wrote it she explained she had halved the 8 and doubled the four. Again, he work would not have matched her thinking. IMG_1292

Going in the other direction, this student had the 56 circles with 28 labeled at the top, 14 labeled on the right and the equations written at the bottom when I walked up. I assumed she had split the 56 in half and then in half again, using what she knew from her equations. However when I asked her to explain her thinking, I found she had drawn the circles one at a time in rows of 4. She said she split it into two groups, counted the 28 on one side and then counted how many were in one column. This was interesting to me and I am kicking myself about not asking more about the 4 rows because she counted by rows of 4 but then found her 4 rows of apples actually in the 4 columns. In this case I would have assumed things that didn’t happen from her work that really didn’t match what she was thinking.


Here are some other great examples of student thinking:IMG_1291IMG_1290IMG_1289 (1)

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Show your work versus show your thinking…do we say both? do we make explicit what we mean by them? I am not sure, but what I do know is that we never learn more about student thinking than we do when we talk to students about their work!


4th Grade Multiple Towers – Pt 2

Due to a schedule change today because of an assembly, Malorie and I did not get a chance to sit and plan together before teaching the lesson. Luckily, she is always so prepared and had read my thoughts on the blog I did yesterday about it so we went into the lesson with a common vision and then talked our way through the shortened class period with the students.

We started with choral counting by 3 and then 30. We chose to include 5 multiples in each row in hoping the students would see a pattern in the “friendly numbers” if they struggled to see patterns in other places. After the list was up, they took a few minutes to write  down any patterns they saw in their journals and we shared out. This was the count and then notices Malorie typed as they were sharing:



I could have stayed with these patterns all day long, but we decided to let students revisit these patterns over the course of this unit. There are some really great ones up there like adding the digits and that number is also a multiple of three and if you add any of the numbers up there, the answer will also be a multiple of 3. They made some predictions about numbers that would show up later in the list and talked about connections between the two lists. Multiplying by 10 and the zero “put on the end” was definitely the most popular noticing of the class period, which was a nice lead into our stacking of the boxes of oranges.

We explained the boxes of oranges held 30 oranges and asked them to estimate how many boxes (represented by the post-its) would be needed to stack up to Mya’s shoulder. Malorie and I quickly tried to figure out if we wanted to ask about number of boxes or number of oranges and decided to give them the option of either. It worked out beautifully because the majority of students told us boxes along with how many oranges it would be as well. As Malorie stacked the post-its, the students counted along. Some were counting post its while I heard others counting oranges. We stopped approximately halfway, took adjustments on their estimates and continued. We finished with 480 oranges, 30 boxes. As we were running out of time, we decided to end by asking if there were any equations they could think of to represent the oranges and boxes.

I was surprised to get division first…480 ÷ 30 = 16 and then 16 x 30 = 480.

This was unfortunately where we had to leave off because the classes were shortened and then next class was waiting at the door. So we met during her planning later that day to regroup and find our starting point for the next day.

Tomorrow we have decided to lead with a talk about the orange tower we built today. This is where we had a big discussion around what we want to really have students think about….the book seems to really focus on the multiplying by a multiple of 10, like what would be the 20th multiple, the 30th multiple, etc but we want to play around with the properties a bit more here.

So, we decided to ask students what equations or expressions they could use based on the tower to arrive at the 330 post it. We are thinking we may get some things like this:

30 x 11

(3 x 10) x 11

(3 x 11) x 10

(30 x 10) + 30

(30 x 10) + (1 x 30)

We want to set them equal to one another and ask how they could prove are the same answer. We are hoping to see the associative and distribute properties come out. We definitely could get some division too and that could make it really interesting!

We decided to go with this because Malorie says that she often sees students “putting one zero” on the end of any number when they multiply by any multiple of 10. For example, when multiplying 40 x 30, they will just get in the habit of putting one zero because they don’t see that 10 x10 happening. This is where being able to think about 40 x 3 x 10 is helpful for students.

Next, they will make their multiple towers based on these numbers we will assign to partners: 15, 16, 32, 28, 24, 48, 18, 36, 35, 70, 45, 14. Since Doubling/halving is coming up, we thought this could bring out some of those ideas. Then we will ask them to come up with expressions for any number they choose in their tower. We had planned for that to be in their journal, but I wonder if that is something they could hang up next to their tower for a notice/wonder walk around the room? Hmmm…have to ask Mal in the morning!


Follow Up…today we did a choral count 6 and then 60 and the pattern-finding was even better then yesterday. It was always amazing to me how students notice new things and then based on how we record. This time Malorie did four multiples in each row so in addition to finding new patterns, they also started comparing it back to what they noticed yesterday during the 3 and 30 count. Even though the 3 and 30 count was not up on the board anymore, they still remembered the patterns and numbers like they were right in front of them because they were so excited about them. So cool to see/hear. 

Next, we looked back at the multiple of tower to 30 on the wall. I asked the students if they could find an equation with the answer of 330 (which was on the tower) using numbers on the tower and gave them time to jot some ideas in their journal. 

As the students started to share, I realized it was a really Badly Worded Question. As they shared I was getting “ways to get 330” but not relating anything to the tower, which we were hoping would happen. We wanted to see the associative and distributive properties emerge, but were getting addition and subtraction equations. 

So, as I got “420-90=330 and 270 + 60=330…” etc, I asked “How would would you show that action on the tower….and then they were on a roll. I didn’t want them to think what they had written was not valued so we asked them to share what they had originally written and we built on from there! There is SO much to work with here so I cannot wait to plan for some exciting algebraic work. 


I had to plan with another teacher so I left as the students began working on their multiple towers. After they finished their own towers, we were going to ask them write equations for a number on their tower based on moves on their tower. I have to check back in with Malorie in the morning! 




Multiple Towers in 4th Grade

In the Making Number Talks Matter chapter on Division, one of the strategies described was “Make a Tower.”

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This strategy immediately reminded me of the 4th grade lesson in Investigations called just that, Multiple Towers. To make the timing even better, I am planning and teaching that lesson with a 4th grade teacher this week. The goal of this lesson is not necessarily eliciting a division strategy, but there is so much possibility for so many things in this lesson. Because of that, however, I am struggling with how we can leave it open enough for the many ideas to emerge, but focused enough to have a purpose in our planning.

The structure of the lesson, as it is the book, looks roughly like this:

Students count around the classroom by 3’s to 72 and it is recorded on the board. They then count by 30’s to 720 and it is recorded on the other side of the board, so both lists are visible. Asking students what they notice and what relationships they see. We record these noticings and leave this because the investigation of multiplying by a multiple of 10 continues to emerge throughout the unit.

Next, as a class, we introduce multiple towers through a context of stacking boxes of 30 oranges. Using post-its on the wall, students place the post-its one above the other, labeling the amount of oranges as the tower grows until it reaches the shoulder height of one student.  We then ask if there is anything in the tower that would help us figure out how many oranges are there without counting each one? They estimate how many to the top of the student’s head, ask how many multiples are there, and record equations for it.

Choosing from the list below, students work with a partner to create their own multiple towers as tall as one of them on a strip of paper:

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After creating their tower, they answer questions in their activity book about the number of multiples, how they could find the 20th multiple without counting, and writing equations for some pieces of their work. They will continue to reference these towers in the following lessons in which they multiply 2-digit numbers and discuss various strategies and representations for multiplication.

I love this lesson, but as always, I like to play around with different ideas when I am lesson planning and these are some things I cannot wait to chat with Malorie, the 4th grade teacher, about on Monday:

  • Instead of a number talk do we open with the count around the classroom to save time for the multiple towers?
  • How do we record the multiples? Will different recordings draw out different noticings?
  • Do we start with estimation of how many cases of oranges will it be before we just start the tower? Could that draw out their use of multiplication combinations they know?
  • Do narrow the list of starting numbers to choose from so there are some really great relationships that emerge, like one factor that is 1/2 of another so the towers have a relationship? or a factor that is 1/3? or do we let them choose their own off of that list within a range?
  • Do we put the towers up, walk around and do a notice/wonder before they jump into the activity book pages?
  • Do we want to end with a journal prompt? The answer is obviously yes, but what do we want it to be?
  • How will this leave her for the next day’s lesson?

Feel free to chime in with thoughts/suggestions and I will post the plan we decide upon after our planning on Monday!


Follow Up Post

Commutativity in Fraction Multiplication

Think about these two expressions…

2/3 x 6              6 x 2/3

Do you think differently about each?

Does your solution approach change?

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

Let’s first look at the standards…

4th Grade:

cc25th Grade:


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

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

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

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

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

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

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

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

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

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This 4th grader was the most interesting..

IMG_9736She solved it as 2/3 of 6 and arrived at 4. I asked her if she could write an equation for the problem she solved and she wrote 2/3 of 6 = 4. Completely because I am so nosy, I asked her to write 6 x 2/3 under that. I asked how she thought about that problem? Would she solve it the same? She said, “No, that is 6 of the 2/3’s so I have to multiply the 2 and 3 by 6.” She proceeded and ended with 12/18. She saw the numerator and denominator as numbers in and of themselves and used the distributive property to arrive at her answer instead of thinking about the 2/3 as a number. This was something I had never thought of before! I wish I had more time with her because I SO wanted to ask if that makes sense, but since my planning runs into dismissal, she had to get back to class! Argh!

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

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


The “One Frame”

I love this introduction of decimal addition so much from last year, that I had to relive it again:  It was just as amazing this time!

I opened with the same discussion about the ten frame, why we call it a ten frame, and then changed it to a one frame. We discussed the value of each box and were on our way. This year, I really pushed the students more into the equations that matched the frame on the board. We did .9 as a group in a number talk setting with a lot of revoicing and restating to be sure the students could explain how their equations matched the one frame image. I then put up a frame showing 0.7 ( four tenths on the top row and 3 tenths on the bottom row) and sent them to their journals to write some equations by themselves before sharing out. Here are some examples… (Some went crazy:) I think it is so interesting that without any formal work with decimal multiplication, students intuitively can see that any number of groups of some tenths can be written as multiplication.

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The one below was so interesting when he said, “.35 x 2” I asked him how that matched the picture and he said, “..since I like symmetry, I took the fourth dot on the top row, split it in half, and put the other half on the bottom row.” I asked the class how that gave him .35 and another student explained that because half of a tenth was 5 hundredth, it became .35 on each row. YES!

IMG_9275 - Version 2I think put up two frames, one with .9 and the second with .3 and asked students to write down how much was represented in the picture. Like last year, it was a mix of 1.2, 12/20 and .12. I asked students to prove the one they got as their answer and then explain where they think someone got confused with one of the answers they do not agree with. They did a beautiful job with this.

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It was so nice to kick off our adding decimals with students identifying what the whole it, looking at decomposing numbers, being aware of place value and reasoning about what makes sense. I am SO looking forward to the rest of this work!


Student Work with Fractions

I do not have much time to write this morning, however I know how much I love looking at student work, so I thought I would give some of my friends who love doing the same some stuff to look at this morning!

For a future PD I am doing on fraction progression, I wanted some thinking around this Illustrative Math problem: This is a 3rd grade CCSS, so I had some beginning of the year 4th graders this task to try out. Here are some samples:

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And this Illustrative task: It is a 2nd grade CCSS so I gave it to beginning of the year third graders this year.

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This task ( a 6th grade CCSS, these are my students from last year who are now in 6th grade:

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This task is 4th grade CCSS:, this is my current 5th graders:

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Same group of students on this task:

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And then finally, in my class we were comparing fractions. I asked them which was greater 7/8 or 5/6 and how they knew…

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Have fun math peeps! I would love to chat in the comments or on Twitter about any/all of them!