Category Archives: 3rd Grade

Math Journals as Formative Assessment

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Whenever it snows, it feels so cozy inside that I just have the urge to read and write. And nothing inspires me more to write than student thinking. And there is no better place to see student thinking than in math journals!

When I was a classroom teacher, my fifth graders wrote in their math journals almost every single day. Sometimes they used them before a lesson to record estimates or predictions. Other times they wrote during class to capture their ideas as they worked through a problem. Often, they ended the lesson with a short prompt. No matter how the journals were used, they were always a safe, ungraded space for students to put their thinking on paper. And no matter the prompt, I learned something new every day about my students’ thinking simply by reading their entries.

Later, as a math specialist, I had the opportunity to see student writing in math classrooms across many grade levels, and it was so fascinating. I could see where it all begins in Kindergarten, when students are representing ideas with drawings and numbers, and how that thinking evolves through fifth grade as students’ written reasoning becomes lengthier and the prompts become more metacognitive. In every lesson I planned with teachers, we would build in a writing prompt. Those student responses, would always give us a new window into each student’s thinking.

For example, when I planned a lesson on arrays with a third-grade team, we intentionally designed an exit prompt that went beyond a simple right-or-wrong answer. The lesson began with a Dot Image, and students spent the rest of the time building arrays and writing equations to represent them. At the end of the lesson, we returned to one of the dot images from the launch. Instead of asking students to write an equation, we asked them to choose two mathematical expressions that had been shared during the Dot Image discussion and explain how those expressions were the equivalent using the image.

When we later looked through the student journal responses, they became the anchor for our reflective conversation. Each journal entry revealed something a little different: how students were making sense of multiplication, the connections they were noticing, and where their thinking was still emerging.

Math journals don’t just show us what students can do; they offer a window into how students are thinking. Let’s take a closer look at some of that student work based on broader mathematical understandings.

The Commutative Property

The majority of students chose two expressions demonstrating the commutative property of multiplication. Often students see that you can change the order of the numbers in a multiplication problem and the product remains the same, however in the journal entries, we were able to see student understanding of this property in a representation. 

16 x 2 = 2 x 16

8 x 4 = 4 x 8

16 x 2 = 2 x 16 and 4 x 8 = 8 x 4 

Changing the Number of Groups and Number in Each Group

A few students noticed that when they changed the number of groups and the number of dots in each group, the product remained the same. While these students are not yet articulating how the groups are changing, this work provides a great opportunity to plan future conversations around this idea. 

Rearranging the Groups

This response is very similar to the previous responses, however this student is beginning to articulate how the groups are changing. Instead of having 10 groups of 3, the student explains he took some dots away and added them to another group to make 16 groups of 2. 

Relating Operations

Some students related expressions based on what they understand about the operations and were able to represent these understandings in the dot image. 

While the team and I heard and observed so much interesting student thinking during the Dot Image discussion itself, the journal prompt allowed us to look more closely at each student’s understanding and see the connections they were making. It served as a important formative assessment, one that extended beyond what we could learn through discussion alone.

Math journals have transformed the way I listen to students’ thinking. I love seeing math journaling used across grade levels, from students who are just beginning to represent their ideas to those who are refining written explanations. Journals give students who may not feel comfortable sharing aloud a space for their voices to be heard, while giving teachers invaluable insight into how students are making sense of the mathematics. I encourage all math teachers to incorporate math journals into their classrooms—not just to see how students arrived at an answer, but to uncover the connections, understandings, and confusions that shape their learning. That insight truly informed every planning decision I made in my classroom and deepened my understanding of the not only the mathematics, but how students build mathematical understanding.

Now, off to make some more coffee, grab a good book, and then follow up with some Fortnite or Zelda gaming time:) Happy snowy Sunday all!

Formatively Assessing Student Thinking

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At the beginning and end of a curriculum unit, I find it valuable to learn what students already know and what questions they have, to help guide my planning and instruction. While pre- and post-assessments can provide useful information, they also tend to limit the range of students’ thinking, especially when students show minimal written work. Because of this, I began using a few key routines. One of these routines, which I call “Tell me everything you know and want to know about [topic],” invites students to share their ideas more openly.

For example, after a 3rd grade unit on multiplication and division, Katie (an amazing 3rd grade teacher) and I wanted to gain insight into what students learned, in their own words. We wanted to give them some individual think time first, so we gave them this simple sheet to record their ideas. We decided to leave the page unlined so students could freely draw any representations that made sense to them. Their ideas definitely did not disappoint! (Click on each thumbnail to see the full page)

We only got one question, but it was such an interesting one!

I think since it was the first time doing this routine at the end of a unit, we didn’t get as many questions as we had hoped. I do wonder how changes in wording such as “What new questions do you have about multiplication?” or “What wonderings do you have about multiplication?” would impact the amount of questions we would get next time.

After students had their independent think time, we shared their responses as a whole class and recorded their ideas on chart paper to stay up as an anchor chart we could refer back to throughout the year!

If you would like to try this routine before the holidays to see what students have learned, I blogged the directions here. In my next blog post, I will explore another routine I love to formatively assess student thinking! Until then, I would love to hear some of your favorites in the comments!

Making Sense of Word Problems

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I am sure we have all seen it happen at one time or another in math class. We give a student a story problem to solve and after a quick skim, the student pulls the numbers from the problem, computes them, and writes down an answer. 

If the answer is correct, we assume the student has a grasp of the concept. However, if it’s incorrect, we’re left with a laundry list of questions: Do they realize their answer doesn’t make sense? Did they not understand the context? Did they simply pull the numbers and operate to be finished or did they truly not know what to do with them? Most importantly, we ask ourselves, how can I help students make sense of what they are reading and think about the sensibility of their answer in the context of the problem?

If we’re lucky, we can identify a mathematical misconception and work with that. Oftentimes, though, the answer isn’t even reasonable. Then what do we do?

This scenario has me reflecting on the Common Core Standard of Mathematical Practice 1:

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. 

The best way I’ve found to help students make sense of what a problem is asking is, ironically, to take the question out altogether. Inspired by the wonderful folks at The Math Forum, I do a lot of noticing and wondering with students in this fashion. Most recently, after reading Brian Bushart’s awesome blog post, I have started taking the numbers out as well! Instead of students thinking about how they’re going to solve the problem as they read, they are truly thinking about the situation itself. It’s been an amazing way to give every student entry into a problem and allow me to differentiate for all of the learners in the classroom, while at the same time provide insight into my students’ mathematical understandings. 

Recently, I had the opportunity to work with a 3rd grade class. The class recently finished their multiplication and division unit and will soon be starting their work with fractions. In order for their teacher and I to see and hear how students apply the operations, make sense of contexts, and currently think about fractions,  I thought it would be interesting to take a story problem from their Student Activity Book and take the question and numbers out.

The Planning

I chose the problem below and thought about what I would learn about a student’s mathematical understandings and sense-making after they answered the questions. 

I was curious to observe how students make sense of problems based on the idea of removing the numbers and the question so I changed the problem to this simple statement:

“Webster has boxes of granola bars to share with his class.”

I anticipated the students would wonder about the missing mathematical pieces involved in an open-ended statement like this. I believed their wonderings could lead them to develop questions that could be answered based on the very information they were wondering about. I knew the mathematical ideas of multiplication, division, and/or fractional sharing would arise and that I would learn so much more about their thinking then if I had given them the original problem.

In The Classroom:

I launched the lesson by posting the sentence on the board and recorded things they noticed and wondered. 

They noticed:

“We don’t know how many boxes of granola bars.”

“There is not enough information to know what is going on.”

“We don’t know if it is adding, subtracting, multiplying, or dividing.”

“There are multiple people in the class because it says boxes and share.”

They wondered:

“How many granola bars are in each box?”

“How many boxes he bought?”

“How many kids are in his class?”

“What kind of granola bars are they?”

Based on their noticings and wonderings, I felt everyone had a strong grasp of the context and sense of where this was going. Based on their noticing that there is not enough information to know what is going on, I asked what more they would want to know. They responded that they wanted the answers to the first three of their wonders: bars per box, number of boxes, and number of kids in the class. 

I asked them what questions they could answer if I gave them those pieces of information and they responded:  

How many bars does he have? 

How many bars does each kid get? 

How many does he have left?

At this point, I could have given them the information they wanted. However, I thought it would be so much cooler to allow them to choose that information for themselves. I was curious: how they would go about choosing their numbers! Would they strategize about the numbers to make it easier for themselves? Would they even think that far ahead? What would they do with the leftovers?

When I told them I was not giving them the information and that instead they were choosing their own numbers along with the question they wanted to answer, they were so excited! 

Some partners chose their numbers very strategically to make it easier for themselves. To me, this demonstrated a lot of sense-making and forethought of what was going to happen in their solution path. And as an added bonus, while only asked to answer one question, the group answered all three questions! (Teacher note: if students chose numbers strategically and therefore finished quickly, I gave them extra bars to factor into their problem to see how they dealt with the leftovers.)

Other students chose the opposite route and strategically picked numbers to make it “harder for themselves.” Check out the way these two students showed strong reasoning and perseverance through division of numbers larger than any they’ve ever worked with. 

Others chose numbers without much forethought and dealt with some amazing leftovers. This was a great way to formatively assess students’ thinking related to fractions before they began that unit.

And then there are always the surprises. Who would have thought third graders would reason about the leftovers in terms of percentages?

Reflecting on what the students would have been asked to make sense of and the work they would have had to do based on the original problem versus the reasoning and work they did related to this one simple sentence, I’m amazed by the difference. I learned so much more about what each of the students know beyond simply multiplying 5 and 6. Taking out the numbers and question allowed every student to think about the meaning of the sentence, the implied mathematical connections, and plan a solution pathway before jumping into a solution attempt. 

I highly recommend everyone try this strategy with a word problem from your current text. It’s a wonderful way to give every student access to the math and freedom to think beyond just getting an answer. 

If you know me or have ever read my blog, you know I could talk for days about student math work! You can visit my blog for a more detailed description of the work shown in this post as well as additional work captured from the lesson.

Leveraging Digital Tools for Problem Posing

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I have blogged a few times about problem posing using print materials and lately I’ve become really interested and excited about the potential for digital tools in this work!

If you are new to problem posing, below are a few slides from Jinfa and my NCSM presentation for background – each image is linked to an associated research paper.

What is problem posing?

Many activities can easily be adapted to provide opportunities for problem posing by removing task questions (left) and replacing it with different prompt options (right).

original curriculum activity
problem-posing prompt options

How can digital tools enhance problem-posing experiences?

Being relatively new to both problem posing and digital lessons, I have learned so much trying things out in math classes this year. As always, the more I learn, the more questions/ideas I have. Below are two digital lessons that involve different flavors of problem posing.

Lesson 1: Our Curious Classroom

You can click through the lesson screens to see the full flow, but in a nutshell, students answer questions about themselves and explore different data displays.

After answering the first survey question, we asked students for problems they could answer about their class data and recorded their responses (sorry for the blurry image, I had to screenshot from a video clip:).

The students then worked at their table to answer the questions based on their choice of display.

The lesson continues with more survey questions, data display exploration, and ends with students personalizing their own curioso character (see bottom of post for unrelated, cute idea).

Things I learned:

  1. Student responses can be collected and displayed so quickly with digital which saved us more instructional time for posing and solving problems.
  2. The capability to see data displays dynamically change from one to another enhanced the discussion about which display was most helpful to answer the problems and why.
  3. Students were so motivated to answer questions about themselves, learn about their classmates (audio clip below), and ask and answer questions about their own class, not a fictitious one.
“What did you like about the lesson?”

Things I wonder:

  1. While having the teacher record the questions on the board worked perfectly, I wonder if or how younger students might digitally input their own questions w/o wearing headphones for voice to text or having spelling errors that are challenging for others to interpret? Maybe something like a bank of refrigerator magnets to choose from?
  2. During the lesson, could the teacher input student questions onto cards in the Card Sort in Desmos so they could then sort the problems based on structure before solving?

Lesson 2: Puppy Pile

In this lesson, students generate a class collection of animals, are introduced to scaled bar graphs, and create scaled bar graphs. This one has a different problem-posing structure than the the first lesson which was interesting!

In this lesson, students use the Challenge Creator feature. In order to pose their problem to the class, students create their own set of animals (left) and then select a scale and create a bar graph (right).

After submitting their challenge, students then pick up one another’s problems and solve them.

Things I learned:

  1. Students were extremely motivated to create their own problems and solve the problems of others.
  2. This version of problem posing allowed students to have more control over the situation around which they were formulating problems, which they really enjoyed.
  3. Challenge Creator is an amazing tool for repeated practice that is MUCH more engaging than a worksheet of problems.

Things I wonder:

  1. How could this activity structure support or extend the problem posing experience in Lesson 1?
  2. What other K-5 math concepts would be great candidates for a Challenge Creator problem-posing activity?

Final thoughts

I think problem posing is such an important instructional structure whether done in print, digital, or a hybrid of the two. It is important, however, to also consider the math, student motivation, and amount of time students spend engaging in the problem-posing process when choosing the format we use.

I would love to hear about what you try, learn, and wonder whether you try these lessons or adapt other lessons for problem posing!

Unrelated by Adorable Idea…

After Lesson 1, Katie printed out their personalized Curiosos for the wall;)

Gallery Walks: Engaging Students in Other’s Ideas

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One instructional strategy that I love for collaboration and public sharing of student ideas is a gallery walk. In a gallery walk, students create displays of their thinking on chart paper or white boards and then the small groups walk around the room and visit each other’s posters. And even though students create such beautiful displays of their ideas, it is always challenging for me to structure the walk in a way that actively engages them in one another’s ideas. Like any problem of practice, it takes trying out new ideas to see what works, when, and for whom.

The Lesson

Last week, it was the first 3rd grade lesson about division. We decided to launch by mathematizing Dozens of Doughnuts to set the stage for the subsequent activities. If you haven’t read the book before, it is about a bear named LouAnn who keeps baking 12 doughnuts to share with a different number of guests who arrive at her door. We read the book and did a notice and wonder, anticipating we would hear something about LouAnn sharing doughnuts and the number of doughnuts, friends, or plates, which we did.

Student Displays

We then asked small groups to record all they ways that LouAnn shared her doughnuts. We purposefully didn’t specify the representation so they could look for different ways during the gallery walk.

As we walked around it was great to see the various ways students were representing the situations, but some small groups seemed to have settled on only one way. We had planned for them to look for similar and different ways during the gallery walk, but that can be so passive, with no opportunity for them to connect those new ideas to their work. So, instead of waiting for the gallery walk at the end, we decided to engage them mid-activity with each other’s ideas and allow time for them to use those ideas.

Taking a page from Tracy’s book, Becoming the Math Teacher You Wish You Had, we opted for a Walk-Around to cross pollinate ideas. We asked students to walk around and look for ideas they wanted to add to their poster. These could be new ideas or just a different way of representing an idea they already had.

You would have thought we gave them a chance to ‘cheat’ as they walked around with such intention to other’s posters. I wish I had captured the before and afters of all of their posters, but here are just a few where you can see the new addition of ideas.

After they finished adding to their posters, we paused to discuss the ideas they found from others – both new ideas they hadn’t thought about and ideas they had, but were represented in different ways.

Next Activity

Students then independently solved a few problems. It was great to see the variation we saw on the posters in their work. So many great representations to share and connect in future lessons!

More Ideas and Resources

Want to learn more about mathematizing? Check out Allison and Tony’s book, Mathematizing Children’s Literature.

Want to read more mathematizing blog posts? I have written about some of the books I used when coaching K–5.

Want to share your children’s book ideas for math class? Join me on IG!

Students’ Brilli-ANT Connections in Math Class

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This past week we planned for a 3rd grade lesson on arrays. The curriculum lesson goal was for students to build and describe arrays, in particular connecting the array structure to their understanding of multiplication as equal groups. The first activity in the lesson was written to encourage this connection, however having taught this lesson in previous years, we knew that the workbook examples could have come from students if we gave them the chance. Since we are always looking for ways to better amplify and leverage student thinking, we made some significant adaptations.

The original lesson

Learning goal: Build arrays with physical objects and describe them in terms of multiplication.

We decided that that the diagrams and questions in activity 1 (left image) would easily come from students’ prior understandings and experiences if we launched with a context that encouraged array thinking. Once we did that, it was then about selecting which problems in activity 2 (right image) we wanted to use. We figured we could do that on the fly depending on student work and our timing.

Adaptations

New Learning Goal: Make connections between multiplication as equal groups and arrays.

We read 100 Hungry Ants to open the lesson and asked students to mathematize the situation in a notice and wonder.

The notice and wonder elicited all the ways the ants rearranged themselves which was the perfect launchpad into the activity.

Each student had a cup of 30 ‘ants’ (beans) and a sheet of graph paper if they wanted to use it. We asked them to organize the ants into 4 groups of 6 and then captured pictures of student work to share and connect. They did not disappoint! They built the same images and made the exact connections as Activity 1, however in this version, students got to decide on the arrangement based on their understandings and experiences.

We first shared a picture of discrete groups next to an array and asked how they were the same/different and where the 4 and 6 were in each. Then, we shared arrays with 4 rows of 6 and 6 rows of 4 and discussed the same questions.

At this point we could have used Activity 2 problems, but decided that since they already have 24 counted out we could save time counting out a new set by just asking them to arrange the 24 ants in a different way. We wrapped up the lesson by asking students to write multiplication equations they used today when arranging 24 ants. It was a beautiful lesson.

Takeaways

While this is one really specific example of adapting, there are some general instructional ideas that work like this makes me think more about:

  1. When using a new curriculum, teaching the lessons as is the first year is extremely helpful in making productive adaptations. Having experienced the math goal in action and understanding what students did with the lesson activities was invaluable in adapting to better center students and their ideas.
  2. When we plan for lessons, we not only need to understand the content, goal, and lesson flow, we also need to look for places in the lesson where students are bringing their ideas and understandings to the table, especially when we are asking them to make new connections between concepts and representations. Side note: This is one of my favorite papers on students practicing connections.
  3. Unsurprisingly, students are so much more motivated to look for similarities and differences between their own work than a workbook example. The more we can do this, the better!
  4. Mathematizing children’s literature is such an incredibly engaging and powerful way to elicit and discuss math ideas. While this book is overtly mathematical, students still noticed things about the storyline and illustrations that showed wonderful sense making around the context. If you want to learn more about mathematizing, Allison and Tony wrote a beautiful book about mathematical read alouds with underpinnings, examples, and structures.

-Kristin

*If you are on Twitter (I can’t call it X yet), join me and others in sharing lesson ideas and learnings like this: https://x.com/LeahBaron03/status/1710305997472797074?s=20

Keeping Math Conversations Alive

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Math routines are such a powerful tool for eliciting student ideas and making connections between them. The challenging part for me has always been ending them. Once I ask students for strategies or things they notice and wonder, the ideas are so uniquely interesting that I want to explore them all! However, when each idea can lead down a different path that may or may not be related to that day’s lesson, it is hard to know what to do in the moment. And the last thing I want to do is abandon the wonderful math ideas on the board.

Last week in 3rd grade we did a parallel choral count. Students counted by 2’s and then by 5’s as I recorded. I asked them to look for patterns they notice in either the individual counts or between the two. The lesson that followed was on multiplication, so the skip counting was helpful to lead into that lesson, but as more ideas started to emerge I found myself wondering where to go and what to do with all of these amazing ideas.

If you cannot follow my recording (how have I not gotten better at this after all these years:), here are some of the great math the students brought forward:

  • There are some of the same numbers in both counts, but in different locations.
  • All numbers in the 2 count are even and every other number in the 5 count is even.
  • The 5 count gets to a larger number faster than the 2 count.
  • Every number in the 2 count is the same number being added together – doubles.
  • In the 5 count, there are always 2 numbers with the same digit in the tens place.
  • At the top there is 2 + 5 = 7 and that is similar to the bottom row of 20 + 50 = 70
  • Even + even = even, odd + even = odd, and odd + odd = even
  • Someone added on that the bottom row is the same as 2×10 = 20 and 5×10 = 50

Every time I am in this situation I think about Joan Countryman’s book Writing to Learn Math. In there she describes math journals as a way to keep math conversations alive. That is exactly what I want to do with these ideas, keep them alive for more discussion. I am also a HUGE fan of math journaling, so I don’t need much of a nudge to use them!

Since we need the dry erase board for other things, the ideas cannot live forever on that board. I wondered about giving each student a copy of this picture to tape in their math journal. Then, when students finish up something early, they could find one of these ideas to explore further. I am thinking prompts like “The pattern I am exploring is…..” and “This pattern happens because….” might help students structure their explanations a bit.

Another idea that is more collaborative could be to replace an upcoming lesson warm-up with an idea from this count. We could display the picture on the board, highlight one of the patterns and ask students to work together to figure out why that pattern is happening and decide if they think it will always be true.

I would love to hear others’ ideas for not losing all the great math there is to explore in routines like this!

Embedding Problem Posing in Curriculum Materials

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In mathematics education, problem posing refers to several related types of activities that entail or support teachers and students formulating and expressing a problem based on a particular context, such as a mathematical expression, diagram, table, or real-world situation (Cai & Hwang, 2020).

Because problem posing is so dynamic, multi-faceted, and varied between classrooms, I understand why it is hard to write into published curriculum materials. However, understanding and trying out the structure of problem posing makes it a really impactful teacher tool for adapting curriculum materials.

Why adapt ?

Curriculum activities typically require students to jump right into solution mode which explains why many students pluck numbers from word problems and operate without first making sense of the context. However, when students have the opportunity to pose their own mathematical problems based on a situation, they must make sense of the constraints and parameters that can be mathematized. They then extend from that sense-making activity to build connections between their existing understanding and the new context and its related mathematical ideas.1 This provides opportunity for increased student agency and sense making in any lesson.

How to adapt?

Last week, the third grade teachers and I planned for a lesson that involved students answering questions about data in a scaled bar graph from the prior lesson. Here is the data and graph they were working from.

Instead of asking students to jump right into answering the questions in their workbook, we removed that day’s warm-up to make time for problem posing and adapted the activities that followed.

First, we displayed the graph and asked students “What math questions can we ask about this group of students?” Below are 2 different class examples.

Having such a rich bank of questions, we could have asked students to jump into solving them, however we decided to spend some time focusing on the structure of their questions. We asked them to discuss, “Which questions are similar and why?.” The discussion ranged from similarities based on the operation they would use to solve, whether they could just look at the graph and answer the question without any operation, and the wording problems had in common or not. Such great schema for solving future word problems!

Now that students had made sense of the context and problems, we asked them to solve as many problems as they could. As they solved, we asked them to think about which problems they solved the same way and which ones they solved differently. As we wrapped up the lesson, we shared student solutions and focused on their solution strategies leading to an amazing connection about using addition or subtraction to solve the ‘how many more or less’ problems.

What was the original activity?

If we had followed the curriculum, these are the questions students would have solved. As you can see the students came up with similar, if not the same, questions and SO much more!

  1. How many students are represented in the graph?
  2. How many students chose spring or fall as their favorite season?
  3. How many more students chose summer than winter?
  4. How many fewer students chose spring than fall?

While having solid curriculum materials is extremely important, they can be made so much better by adapting lessons in ways that provide the space for students to make sense of problems and have ownership in the problems they are being asked to solve. I am so grateful for the teachers, admin, and Jinfa’s partnership in this work and look forward to sharing our work and learnings at NCSM DC!

  1. (PDF) Making Mathematics Challenging Through Problem Posing in the Classroom(opens in a new tab) ↩︎

Purposeful Warm-up Routines

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As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading →

CCSSMashup – Fractions

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I never tire of conversations about the 3rd – 5th grade fraction progression because after each one, I leave with the desire to reread the Standards and Progressions with a new lens.

A few weeks ago, a conversation about 3rd grade fractions sent me back to the Standards with a #pairedtexts type of lens. Unlike the hashtag’s typical MO of pairing contrasting texts, I was looking for standards that connected in a meaningful, but maybe unexpected way. By unexpected, I don’t mean unintentional, I mean the two standards are not necessarily near-grade or in the same strand, so the connection (to me) is not as obvious as one standard building directly toward another.

The conversation focused on this standard:

Screen Shot 2017-10-12 at 3.01.16 PM.png

With that standard in mind, imagine a 3rd grade student is asked to locate 3/4 on a number line on which only 0 is marked.

I expect a student would mark off the 1/4’s starting at 0 and write 3/4 above the point after the third 1/4 segment. What exactly is the student doing in that process?

Is the student adding?

Is the student counting?

Is the student doing both?

How does adding and counting look or sound the same in this scenario? different?

This is where I find pairing two standards fun and interesting to think about because it demonstrates how important seemingly unrelated ideas work together to build mathematical understandings. It is also really fun to think about how a standard in Kindergarten is so important for work in grades 3-5 and beyond.

In this scenario, I think we instinctively believe students are adding unit fractions when asked to place 3/4 on the number line because the standard is in the fraction strand and therefore we consider all of the work to be solely about fractions. We also sometimes impose our thinking on what students are actually doing in this task. For example, you could imagine the student marking off the fourths, stopping after the third one, writing 3/4 and say the student was adding 1/4+1/4+1/4 to get to the 3/4 because they moved along the number line. If this is the case, then the standard would pair with this 4th grade standard:

Screen Shot 2017-10-12 at 3.36.21 PM.png

Don’t get me wrong, those standards definitely pair as students move from 3rd to 4th grade, however, since the scenario is about a 3rd grade student, pairing it with a higher grade level standard doesn’t seem to make sense in terms of what students are building on. Right here, it is really interesting to pause and think about how building fractions from unit fractions, locating a fraction on a number line, and adding unit fractions are slightly different things a progression.

When I think about the student locating 3/4 in 3rd grade, I hear counting (with a change in units) and would pair that 3rd grade standard with this Kindergarten counting and cardinality standard:

Screen Shot 2017-10-14 at 8.58.36 AM.png

However, because the 3rd grade work is on a number line and the arrangement and order does matter, I would have to add this 2nd grade measurement standard into the mix, but take off the sum and differences part:

Screen Shot 2017-10-12 at 3.35.21 PM.png

So, instead of a #pairedtext, I now think of it more as a #CCSSMashup to create this standard:

Screen Shot 2017-10-14 at 8.59.26 AM.png

With that mashup in mind, I went back to the progressions documents to look for evidence and examples of this.

In the 3rd Grade NF Progression these parts jumped out at me as being representative of this standard mashup:

The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.

The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1 /3 .

There is also a great “Meaning of Fractions” video on the Illustrative Mathematics site that explains this idea with visuals.

There are so many of these great mashups in the standards, especially in the fraction strand, that I find incredibly helpful in thinking about how students coherently learn mathematics.

I look forward to hearing your favorite #CCSSMashup!