Category Archives: 3rd Grade

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.

Screen Shot 2017-06-01 at 8.22.07 AM.png

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:

Rhombus? Diamond? Square? Rectangle?

It happens every year, in what seems like every grade level…students continually call a rhombus a diamond. Last year, when we heard 3rd graders saying just this, Christopher helped the 3rd grade teachers and me put the students’ thinking to the test with a Which One Doesn’t Belong he created.

CayX_LdUUAEIQb5.png

 

This year, at the beginning of the geometry unit, we heard the diamond-naming again along with some conversation about a rectangle having to have 2 long sides and 2 short sides. What better way to draw out these ideas for students to talk more about them than another Which One Doesn’t Belong? We changed the kite to a rectangle this time, hoping we could hear how they talked about it’s properties a bit more.

Screen Shot 2017-01-05 at 6.30.31 PM.png

Overwhelmingly, the class agreed D did not belong because it had “5 sides and 5 corners” and eventually got around to calling it a diamond, which in their words was “not a real shape.”

While we knew a lot of things could arise, our purpose was diamond versus rhombus conversation, so of course the students had other plans and went straight to the square versus rhombus.We wouldn’t expect anything different!:) For every statement someone had about why the square or rhombus did not belong, there was a counter-statement (hence the question marks in the thought bubbles).

IMG_3730.jpg

Jenn, the teacher, and I were really surprised at how much orientation of A and B mattered to the name they gave the square and rhombus but did not matter for the rectangle. That was just a rectangle, although one student did wonder if a square was also a rectangle (he heard that from his older sister). The students had so many interesting thoughts that we actually had to start a page with things they were wondering to revisit later! That distributive property one blew me away a bit!:)

IMG_3731.jpg

We then sent them back to journal because we wanted to hear how they were categorizing a square and rhombus. It ended up being really interesting just seeing them try to explain why they were different and change their mind because they just started turning their journals around!

Some stuck with them being different..

img_3738

 

Some thought they were different, but one could become the other…

 

img_3744

Some were wavering but the square was obviously the “right way.”

img_3737-1

Some argued they were the same…

So much great stuff for them to talk about from here! I left wondering where to go from here? In thinking about the math, is it an orientation of shapes conversation? or Is it a properties conversation? In thinking about the activity structure, would you pair them up and have them continue the conversation? Would you throw the rectangle into this conversation? Would you have some playing with some pattern blocks to manipulate? Would you pull out the geoboards? I am still thinking on this and cannot wait to meet and plan with the 3rd grade team!

However, before I left school today, I went back to the 3rd grade standards to read them more closely:Screen Shot 2017-01-05 at 7.07.05 PM.png

and read the Geometry Learning Progressions, only to find this in 1st grade:

Screen Shot 2017-01-05 at 5.58.18 PM.png

Would love to hear any thoughts and ideas in the comments!

 

 

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!

IMG_3396.jpg

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!

Number Talk Karaoke

It is always so fun when I have the chance to hang out with my #MTBoS friends in person! This summer Max was in town, so I not only got to have lunch with him but also meet his amazing wife and puppy!  Of course, during lunch, we chatted a lot about the math work we are doing with teachers and some of the routines we are finding really valuable in their classrooms. From these two topics of conversation, Number Talk Karaoke emerged.

We both agreed that while Number Talks are invaluable in a classroom, it can be challenging to teach teachers how to use them in the classrooms. As much as we could model Number Talks during PD and show videos of them in action, it is still not the same as a teacher experiencing it for themselves in their classroom with their students. There is so much to be said for practicing all of the components that are so important during the facilitation with your own students.

That conversation then turned into two questions:  What are these important components? and How do we support teachers in these areas?  We discussed the fact that there are many books on mathematical talk in the classroom to support the work of Number Talk implementation, however the recording of student explanations during a Number Talk is often left to chance. What an important thing to leave to chance when students often write mathematics based on what they see modeled. We brainstormed ways teachers could practice this recording piece together, in a professional development setting, where students were not available.

Enter Number Talk Karaoke.

During Number Talk Karaoke, the facilitator:

  • Plays an audio recording of students during a Number Talk.
  • Asks teachers to record students’ reasoning based solely on what they hear students saying.
  • Pair up teachers to compare their recordings.
  • Ask teacher to discuss important choices they made in their recording during the Number Talk.

Max and I decided to get a recording and try it out for ourselves. So, the next week, I found two of the 3rd grade teachers in my building who were willing to give it a go!

They wanted to try out the recording piece themselves, so they asked me to facilitate the Number Talk. They sat in the back of the room, with their backs to the students and SMARTBoard so they could not see what was happening. All they had in front of them was a paper with the string of problems on it.

Before seeing our recording sheets below, try it out for yourself. In this audio clip of the Number Talk, you will hear two students explain how they solved the first problem, 35+35. The first student explains how he got 70 and the second student explains how he got 80.

Think about:

  • What do you think was really important in your recording?
  • What choices did you have to make?
  • What question(s) would you ask the second student based on what you heard?

The talk went on with three more problems that led to many more recording decisions than the ones made in just those two students, but I imagine you get the point. I have to say, when I was facilitating, I tried to be really clear in my questioning knowing that two others were trying to capture what was being said. That makes me wonder how this activity could be branched out into questioning as well!

Here was my recording on the SMARTBoard with the students:

Screen Shot 2016-11-13 at 9.12.41 AM.png

Here are the recordings from the two teachers in the back of the room:

screen-shot-2016-11-13-at-9-13-22-am

screen-shot-2016-11-13-at-9-12-58-am

We sat and chatted about the choices we made, what to record and how to record certain things. We also began to wonder how much our school/district-based Number Talk PD impacted the way we record in similar ways.

Doesn’t this seem like a lot of fun?!? It can be done in person like mine was, or take the audio and try it with a room of teachers, like Max did! <– I am waiting on his blog for this:) Keep us posted, we would love to hear what people do with this!

 

Writing in Math: After a Number String

Many people ask me when and how I use journals in math class. At those moments,  I always seem to have so many reasons that it is hard to pinpoint just one to focus on during the conversation. And even when I seem to find a coherent way of explaining when and how I use journals, I often forget the reasons that seem to happen naturally in the classroom. The other day I had one of those moments that I think Joan Countryman, author of Writing to Learn Mathematics, would classify as continuing the conversation.

During Number Talks or Number Strings it always seems to happen…one student has a way of solving the problem that, as he or she gets midway through the explanation, the rest of the class begins to disengage either because it is a long explanation or they are lost in what is being said mathematically. Journals help me continue that conversation with the student who is sharing. I attempt to clearly capture what is being said, but ask the student to tell me more in their journal because I am so interested to hear all of their thinking.

This particular string was in a 3rd grade class who has been working with multiplication. I wanted to see how they thought about changing one of the factors in a particular way. This was the string:

3 x 4

4 x 4

6 x 4

12 x 4

The majority of students shared strategies that involved either skip counting or using repeated addition of one of the factors. Some used previous problems (which was my goal) to help them with the new one, however there was one student who started using 5’s for the last two problem instead of either of the factors. He had a very clear way of explaining it, but I could tell many students were beginning to get lost in the explanation. I encouraged the students to ask him some clarifying questions, but that conversation began to stretch this number talk a bit too long time-wise. Not to mention, many had stopped listening at this point.

I was so curious to hear more about his strategy because to be honest, I was getting a bit lost in his explanation of 12 x 4 using 5’s. I told him I wanted to hear all of his thinking but we needed to finish up the number talk to get started with class. I asked him to explain to me what he as doing with 12 x 4 in his journal and I would be sure to check it out later! He went right to work and knocked out this beautifully clear explanation, not just for 12 x 4 but EACH of the problems!

fullsizerender-58fullsizerender-57

The thing that I appreciated most was the opportunity if gave me to continue this conversation with him. I could feel he wasn’t done explaining his strategy during the talk and this also gave him the chance to think about how he could clearly communicate it to me in his writing. What a powerful thing for a student to be able to do! It was amazing to me he had done all of that decomposition, adjusting, and adding in his head!

So, if you asked me this week for a reason I have students write in math class, it is to continue the conversations that are not quite ready to end during our class time together.

 

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.

IMG_3124 (1).jpg

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.

IMG_3135.jpg

  • Students skip counted by 5’s but added 6’s when finding the 5 groups of 6. IMG_3127.jpg
  • 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.

IMG_3125.jpg

  • One student showed how he used what he knew about one to switch the factors to make it easier to solve. IMG_3150.jpg

 

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

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.

Scannable Document on Sep 27, 2016, 7_06_18 PM.png

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.
    img_3048

 

 

 

 

 

 

 

 

 

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!:)