The Teacher Studio: Learning, Thinking, Creating
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Writing about math thinking is REALLY challenging.  Students in the intermediate grades aren't used to taking their ideas and transferring them to paper when it comes to complex ideas!  If you have asked students to "explain their thinking" about a solution, you may have noticed them writing things like:

"First I took the 64 and the 49 and I added them.  Then I took the rest away." or

"I could tell it looked like about a half so I wrote that." OR my personal favorite...

"I just knew it in my head." (You've heard this, right?  It's not just me?)
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What do the Standards for Mathematical Practice Say?

The Standards for Mathematical Practice and other rigorous math standards have made it clear that we need our students to get better at explaining their thinking and critiquing the reasoning of others (and themselves!).  The standards mention things like, "make conjectures and build a logical progression of statements" and "justify their conclusions" and "communicate them to others" and more.  So how do we get our students to dig deeper and explain their thinking more clearly? (Want to see how I help my students understand these standards?  Just CLICK HERE.)

Today I presented my students with one of the fraction challenges in my fraction unit to see a few things--one, if they had internalized an important fraction concept we had been working on and, two, how they were doing with their "explaining their thinking".

I learned a few things!  First of all--almost ALL students got the right answer!  #boom

Secondly, our written explanations were in dire need of some work!

I thought I'd share with you my next steps--because I REALLY want to see my students make gains in this area...and stay tuned over the next weeks for an update!

Coaching Students Toward Better Math Explanations

As I was "walking the room" as students were working, I noticed that very few students were writing what I considered to be a quality explanation.  I began to wonder if they REALLY knew what I meant when I said, "Write a clear explanation."  

I have noticed that students tend to write procedures rather than thinking.  Instead of writing, "I read that I needed to find the difference between the two amounts so I needed to subtract.", students write, "I subtracted 53 from 82."  I try to tell them to let their math computation speak for itself and let their explanations explain the WHY...but it's hard!

I started by having students work in trios to share their explanations.  They needed to read it aloud and listen to see if they heard WHY and HOW in the explanations.  After a few minutes, I asked if any groups had heard any explanations that they thought did a good job.  As students nominated other students, I asked their permission to share under the document camera.  I reminded them that we weren't looking for perfection--just for ideas on how to improve our work.  I got six samples that we then looked at together.  We collected words and phrases like "proved" and math words like "equivalent" as we went, talking about how important it is to be specific with our explanations.
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So...we generated a list of fraction words we might expect in this problem...and made it clear that each type of problem will have math vocabulary specific to it that a reader would expect to see.

I also shared with them an anchor chart I whipped up quickly with some sentence stems...phrases to get them "unstuck" when writing about math.  There are so many more--but I wanted them to see that there are different types of writing about math, and you have to choose what makes sense.  We tested these with our fraction problem and realized that the third one might be really useful.  "I knew that the larger shapes would divide the rectangle into fourths so..." or something like that.  
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Revising Their Thinking

So the real learning comes in when students take in these student models, the vocabulary list we generated, and these writing stems to set out to revise and improve their own work.  It's not enough to see other people do it--students need a chance to "give it a go" on their own--so off they went to make improvements.  When they finished, the met back with their original trio to see if the group agreed with the improvements!  Mission accomplished!
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We will continue to refine our anchor chart by adding new phrases we find as we work on different problems and will continue our discussions about how to improve our math writing.

Coming up next?
third grade, fourth grade, 3rd grade, 4th grade, grade three math, grade 4 math, fractions, teaching fractions, standards for mathematical practice, fourth grade fraction lessons, fraction activities, fraction unit, fraction resources, teaching fractions, writing about math, critique the reasoning of others
If you are interested in seeing more of the fraction work we do in our class, just click the image below.
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I know so many of you have been looking for a way to build deep math thinking with your intermediate students--I know this because I get questions about it all the time!   You want your students to be challenged in new and interesting ways—and be easily able to differentiate so that ALL your students can benefit, right?
problem solving, differentiation, addition, subtraction, word problems, math enrichment, math workshop, math stations, guided math, third grade, fourth grade, fifth grade, tiered math, tiered problem solving, teaching resourcesHere's the problem...what's challenging for some is way too tricky for others, right?  Or you want to give a task--and you do--and then 1/3 of your class is finished in minutes and asking for more while others have barely gotten started.  (Tell me that I'm not the only one this happens to!)

I set out a quest to try to find a way to get GOOD tasks into all my students' hands...but with room to expand and extend for those who are ready.  I'm hoping you find this idea as appealing as I did.  It took me nearly a year of mulling it over, but I think I have something to help.  I have been excited to see that my students have gotten better at thinking outside the box...better at talking to each other about math...and better at organizing their work.  It looks like this!

My problem solving goal: 

The first thing I wanted to do was to provide teachers and students with sets of quality math problems that not only get students thinking deeply and “doing” math—but talking about it as well. I wanted it geared for grades 3-5--knowing full well that different students in that age range have different skill sets.  Because of this, I wanted to find something that would be perfect for differentiation as well—so for this, not all students need to do all three parts!  In fact--for some of the challenges, we only do the first part.  For other tasks, SOME students move on to other challenges.

Similarly, I wanted problems that could be done as a whole class warm up or with a smaller group, and because our curriculum is plenty full, I wanted something that would maybe last from 10-30 minutes but could be extended even more if the students are encouraged to talk and push their thinking.  It's always interesting to me how the timing of problems works out...sometimes what I think will be a quick task takes so much longer, but if the discussion is good and the students are engaged--it's pretty hard to shut down the math, right?

How can I incorporate this type of problem solving into my math class? 


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I often get people asking me what a "typical" math class looks like in my room--and I have to be honest.  THere is no such thing.  I feel like I operate on a menu system...I have all these "tasty" things and I serve them up when I think it makes the most sense!  That being said, here are a few suggestions for working these quality tasks into your day.

*Use as a “bell ringer” or warm up task. The goal of this should not be getting a correct answer, but the actual WORK of doing the problem solving! Each question has a starting point which can be used whole-class (you choose how much modeling/help) and then parts 2 and 3 can be used for everyone—or just for students who are ready! The colored slides are perfect to project from your computer…you can click to the next slides to show parts 2 and 3…but if students aren’t ready, no big deal! The original problem appears on every slide!

*Print and laminate and use as task cards at a problem solving station. These half-page cards are low ink and are perfect for math rotations, math workshop, guided math, or for fast finishers.  Whether you do rotations or organize your stations differently, having quality problems ready to go really saves your time.

*Use as a reproducible problem-solving journal. I typed these problems up in a journal format with a full page of work space for the first part of the problem--with parts 2 and 3 copied on the next page. Copying the entire journal only takes 12 pieces of paper (without the cover) and is full of the 36 tasks. These can be used in so many ways—and even flexibly within a given classroom. I have some students who only do part of the collections--and others who might have the time and motivation to do much, much more.

*Consider using open-ended tasks as enrichment opportunities for students needing just a bit more. This is perfect--whether you have one student or a handful.  They can work together, practice that accountable math talk, and push each other.

Problem solving is not easy!

Like many of my resources, this set of problems is certainly not meant to be a time filler! It is meant to be a rich and meaningful problem solving experience for you to use with your students. HOW you use it is up to you!  I know we are all busy...but the time we invest in modeling some of the thinking and strategies needed with this type of problem REALLY pays off in the long run as students become more and more independent. 

When I use tasks like this with MY students, there are a few things I like to make very clear and I think really contribute to building a culture for problem solving.  One of the most important things that I think teachers need to keep in mind is that we often "overteach".  We TELL too much.  We push our own strategies and ideas onto them--even if they aren't quite ready for them.  Before I used this problem with my students, I thought about how I would solve this as an adult--and then thought about what might get students off track.  In this case...I did NOT want to show students my "boxes" (although for a few students I did coach them in this direction after they had worked awhile), but I DID want to make sure students understood the task--and the terms (like digits) so that they didn't waste time.  I didn't TELL them what the task was...but students worked in pairs to make sense of it, then we came back as a whole group to discuss it and get on the same page.
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1. Explain to students that there may be multiple answers—and they may not even actually GET an answer—the process is far more important than the solution.  A child can work diligently on these tasks and truly work their brain even if they don’t come up with a viable solution.  This is a big shift for many students (and teachers!).  For this reason, I don't even make solutions available to students.  Organizing work, working with precision, and explaining thinking are other key components that students need to see acknowledged and recognized.  Remember, too, that this type of problem solving can take time—and students may need more than one class time to really dig in.  

2. The process of problem solving is challenging and is not black and white.  For this reason, I like students to familiarize themselves with self-reflection and assessment.  (The resource I have showcased has a student self-assessment checklist that can be used as a part of the reproducible journal (if used) or to use separately to help students understand how important some of these math “behaviors” are to the learning process.)  You will also see “prove it” as a part of many of these tasks.  We want students to be developing their critical reasoning and learn to defend their thinking.
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3. You will notice a variety of different problem “types”—from find the “misfit” to “What are ALL the ways…” to “What is the smallest…”.  These problem types are meant to get students thinking outside the box and to really have to make SENSE of the problem and not fill in an answer blank.  If this type of problem solving is new for your students, you may need to do more scaffolded instruction and modeling.  As students try more and more, you can take a less structured role as the teacher.

Push Yourself!


For real.  If you want to feel what students feel, try some challenging problems yourself!  This can help you find areas where students might struggle--and can help you prepare to coach them better.  Before I give students tasks, I spend a little time experimenting myself.  I really feel this helps me get in that best teaching zone--and I challenge you to do the same!

Interested in checking out this set of problems?  Just click HERE or the image below.
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Research shows that students learn best when they are collaborating and talking about math.  My 25+ years in the classroom tells me that this is, indeed, true--but I'll go a step farther and make the claim that math discourse can raise the level of engagement, motivation, and excitement as well.  I thought I'd share some details about some recent area and perimeter work in my classroom to see if I can show you what I mean.  The resources used in these photos are my Area and Perimeter Activities and my Area and Perimeter Task Cards.
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Working in Pairs with Split Responsibilities

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 One of the first area and perimeter tasks I do is in pairs as I want to set the stage for a couple of things.  One, during this unit we work almost ENTIRELY collaboratively.  Two, I want students to realize the power of having a second set of eyes!  We worked hard on this challenge and I gave each partner a different task--one was in charge of  "building" the rectangles and one was in charge of recording.  I pulled my "recording" group to help them become experts on the difference between inches and square inches and told them they were responsible for making sure that correct language was used throughout the lesson.  I pulled the "builders" to tell them that their job was to ASK their partner for suggestions and to then follow through.  It was a ton of fun to walk around and eavesdrop as they worked to fulfill their jobs!

(Side note: I am often asked where I get these tiles.  I use them for SO MANY things...they are foam and they are inexpensive and they are quiet!  My affiliate link is below if you want to see them...)

The Challenge of Working in Larger Groups

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Let's be honest.  Working in pairs is hard enough for elementary students.  That being said, I DO like to occasionally push them to work with larger groups so we can have discussions about the challenges with it.  For this area and perimeter challenge, students worked in teams of SIX (I know!).  The rules?  They had to follow the challenge (which involved designing a shape with an area of 24 square inches that would fit on a certain grid) AND no two members of the team could have the same area.  It meant they needed to talk.  It meant they needed to ask questions.  It meant they may need to give and take.  They were only given their grid paper to record their solution AFTER they had all worked together to make sure every member of the team met the challenge.  It took some coaching to be sure...but we had a great discussion when we finished about the challenges of working in larger teams, the "give and take" needed to make sure everyone's voice was heard, and so on.  I don't do it often--but this was a great forum to explore large group work!
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Restating Directions to Improve Focus

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Another way to get students talking about math is to ask them to restate directions before a task.  Because so many of our area and perimeter projects had multiple steps, I would have partners take a minute to talk before we got started to make sure they understood the task, the expectations, and how to get started.  It was great to be able to clarify these things BEFORE they started working!  These students were talking about how to tackle the challenge shown in the next section.

Meeting Different Learning Styles

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 This project is one of my favorites.  Much of the work I do with area and perimeter is very "hands on", but this one, in particular, requires students to "build" rectangles--and then to create an artistic representation.  Students who "see" math differently often are very successful at this--and the fact that they "build" rectangles by measuring side length and then assembling the rectangle really helps with cementing that concept of perimeter.
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I also love how much dialogue happens among students...from giving feedback on solving the math challenge and clarifying the directions, to helping with the physical formation of the rectangles.  It is definitely a collaborative effort--and I love watching how engaged they are.  It really frees me up to TEACH and coach when they are so actively engaged in the math.
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Task Cards, Proving Thinking, and More

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As our studies of area and perimeter moved forward, it was time to use my area and perimeter task cards to see if students were really internalizing the math.  Again, we started with partners working and talking about the math--but I let them choose their level of challenge.  These task cards have some rectangular shape...some with the squares visible...some "open" shapes...and some irregular shapes for students to try their hand at dividing into workable rectangles.  The discussion was AMAZING and we then came together to talk about some of the strategies students used.  When it was time to assess, I simply started pulling some of my area and perimeter exit slips to see what students could do independently--and I gave a different slip at the start of each class time for about 5 days.  This really helped me know who to focus my teaching time on.

Displaying Math Work to Build Pride and Discussion

I think it goes without saying that when students see their work displayed, it shows that we value what they have done.  There is an added benefit as well--that students use it like a gallery walk...every time we walk by, they are pointing out things they notice or explaining why they did things a certain way.  And another thing?  Students from OTHER classes are looking as well! It's one reason I like to hang a sign up by the work explaining what the challenge was so that other students can see what we did--and can work to make sense of the task as well.
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I hope you see how providing students with opportunities to TALK about math can lead to some amazing things!  If you are interested in any of the resources I used, you can find them by clicking on the images below.
area and perimeter, area, perimeter, problem solving, hands on learning, math discourse, fourth grade, third grade, 4th grade, 3rd grade, geometry, measurement, cooperative learning, constructivist learning
area and perimeter, area, perimeter, problem solving, hands on learning, math discourse, fourth grade, third grade, 4th grade, 3rd grade, geometry, measurement, cooperative learning, constructivist learning
area and perimeter, area, perimeter, problem solving, hands on learning, math discourse, fourth grade, third grade, 4th grade, 3rd grade, geometry, measurement, cooperative learning, constructivist learning, assessment

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So many times I am asked questions about why many students seem so disengaged in math.  I ask these teachers to flip through their resources and see how many of them truly help students see the real life application of math--and solve rigorous, engaging problems that students can relate to.  PBL (project based learning) can help make the connections between the math skills and the real world, and my students are always BEGGING for more!  It's always fun to tie this work to the seasons, so I have tried to make a number of these projects that are the perfect break from routine to do at those special times of the year.  This one?  A "feast" project that is perfect for Thanksgiving!
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Benefits of Project Based Learning

It would be hard to list all the benefits of having students dig into real-world math challenges and other project-based learning opportunities.  Let me just leave you with a few.
  • They are engaging!
  • They nurture collaboration and math communication.
  • There are often many solutions.
  • These experiences provide students with a chance to struggle (and this is so valuable!).
  • Connections to the real world help students realize WHY we do math!
  • Students who learn in different ways can shine.
  • They allow time for teachers to coach.
  • They are perfect for differentiating!

How to Differentiate

It may seem contrary to what one might thing--how can a challenging, rigorous task be easy to differentiate?  Aren't they best for our top students or fast finishers?

Not at all.  I love using these tasks in a number of ways--and I DEFINITELY want ALL my students to have exposure to and experience with quality, real world tasks.  How can we help them succeed?


  • Strategically partner/group them with a supportive team.
  • Allow use of "tools" such as calculators and manipulatives (remember--we want them to be problem solvers so don't let the computation get in their way!)
  • Spend more time scaffolding directions or simplifying directions if needed.
  • Reduce the amount of writing or allow them to use technology.
  • Use the tiered problems included so they can do the same work at a slightly more simplified level.
  • Provide more frequent coaching opportunities.
  • And so many more ways!  
  • Provide the emotional encouragement and support to keep them hungry to solve the task.
And don't forget our super capable students!  A list of additional challenges and extensions are included as well so they can take this project as far as they want!
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Remember, self-esteem and mathematical mindsets are elevated by tackling challenging work--not by be given simple tasks!  

How Can I work These Tasks Into My Math Schedule?

I just wanted to share a few ideas on how to work this type of problem solving into your schedule--which I'm sure is already packed!

Here are just a few ideas to get you thinking...

1.  Consider introducing this as a whole-class activity where you give students/groups some time to get started and get clarity on the project.  After that, it becomes something they work on when they have extra time.

2.  Dedicate a few days to the task and let students get as far as they can.  There are logical stopping points in each task--so some students might only solve the main challenge, others may tackle some of the computation work, and still others may try some of the extensions.  You can let students choose their path or you can steer them the way you wish!

3.  Use this as an enrichment activity for students who are "fast finishers" or who may need compacting out of their current unit.  This is a perfect, meaningful independent math task for students to do if they already have mastery of the current math topic.

4..Give students a small chunk of time each day to work on this--either in a center, as a warm up, or at the end of class.  Over time, students can get as far as their initiative and math will take them!

So what does this really look like?



I hope these ideas get you intrigued enough to dabble in some high level math--for ALL your students!  This task focuses on a feast...but there are SO many other options out there!

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Project based learning, PBL, Thanksgiving, math projects, problem solving, elapsed time, Thanksgiving math, fourth grade, fifth grade, math workshop




One thing I really like to do with my literacy instruction is to make sure that I weave reading and writing together whenever possible.  This is especially true when working with narrative writing--in our independent reading workshop and our narrative writing unit. One of our phrases this year is "Read like a writer and write like a reader!" and we are working to dig into what that really means.
lesson ideas narrative

Studying Characters in Narratives

As we started our narrative unit several weeks ago, we spent a great deal of time studying our main read aloud, Fish in a Tree, along with other picture books I selected.  We studied the characters, their traits, their actions, and then--finally--their words.  We actually then worked to create our own characters that we would later insert into some "mini" narratives that we wrote.  The students were SO engaged and I truly saw them thinking more deeply about the characters in their own books.  (Note:  This student had a bad experience with a pigeon...I guess my "hint" that authors often work their real life into their books paid off!)
invent a character

Creating a Story Arc or "Story Map"

From story arc to detailed plan...this TRULY helps the students plan ahead, make sure they have a clear ending, and gives a place for them to add quality sensory details.


prewriting strategies narrative writing

Learning how to write dialogue

As we started to realize all the ways writers help us get to know characters, I told my students we were going to work hard to do this with our own narratives--we were going to write so our reader can really get to know our characters.  We went back to our own books (great because no matter WHAT level a child reads at, this activity can be done) and searched for dialogue "tags".  We jotted them on sticky notes and then came back to the large group to do some sharing.  I recorded their findings on a chart and then we talked about the messages writers can send by carefully choosing tags.  What does "mumbled" show about a character instead of "shouted"?  If a characters "demands" something, what does it say about them?
writing lessons
If you have taught writing dialogue before, you know how challenging it can be to get students punctuating it correctly.  There is a LOT to remember!  I decided to go back into our read aloud, "Fish in a Tree", and do some practice with this sentence frames...I gave them permission to change the "tag" to match the characters, but I really wanted them to think about what these characters might say--and to use the guides to punctuate these made-up sentences.  They worked in pairs for a while and then shared with others--they had a GREAT time imagining they were these characters and came up with some great dialogue examples!
teaching writing lessons
I then wanted to give them some more practice, so I used this "sort" activity where they had to manipulate the parts of the sentences and mindfully add the punctuation.  It was challenging for some, but I saw many of them referring back to the pink strips from the lesson before!
writing lessons dialogue

punctuating dialogue
Finally, I talked to the students about how a true dialogue involves characters speaking back and forth--and how their discussion helps move the story forward.  It often reveals character feelings or plans and should have a purpose.  We worked in partners to try writing some dialogue scenarios based on my dialogue task cards and we had a blast.  It was a great chance for me to walk around and do some coaching on the punctuation, remind about indenting for new speakers, and so on.
writing dialogue

punctuating dialogue
 It was so much fun to see how creative they got--and how their dialogues REALLY started to show their understanding of how characters can be revealed.
dialogue lessons

Using "Fish in a Tree" and other books to put all the pieces together!

 With all the pieces in place, we were ready to write our narratives!  It's funny...we spent 2 weeks prepping for them and three days writing them--and it was amazing to see how all the "quick writes" we did paid off--and the students REALLY saw how the planning made it so their story essentially wrote itself.  When I asked how many of them felt it was the best thing they had ever written, it was almost unanimous!  Also, if you are working on narratives, I can't recommend "Fish in a Tree" highly enough to really look at character development, character change, and more.

Want to see the resources I used?
My novel study for Fish in a Tree...
Teaching Fish in a Tree
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