My third grade class explored integers this week. Over the past few days students have started to become more comfortable in being able to compare and locate integers on vertical/horizontal numbers lines. The next sequence is integer computation. I find this to be more of a challenge for students. Specifically, some students find the concept of subtracting a negative integer to be confusing. Most students have encountered computation at this stage as either addition, subtraction, multiplication, or division. The idea of subtracting a negative isn’t something that they’ve experienced and can cause students to question their own understanding.

This topic was discussed at #msmathchat last Monday night. The consensus was that students need to experience different models to gain a better understanding of how to put together and take apart integers.

Manipulatives, such as counters and the such are always important. I believe most teachers use some type of manipulative to showcase integer computation. Sometimes they’re taken away too early.

The problem that sometimes comes up with this, is that students want to move towards only following a rule/process to find the solution. This “answer getting” mentality can lead to a lack of understanding and isn’t beneficial long-term. Wording also plays a role with integers. Getting caught up with “add” and “subtract” can limit what students perceive. How about find the “difference” between x and y?

Changing the wording and using a number line can make a huge difference and can empower students to rely on their own understanding of computation and integers.

I kept this chat in mind as my third grade crew finished up a lesson on integer computation. Near the end of one lessons I gave each student a blank number line and asked them to find the difference between two integers. The instructions are below.

Students were given dice and headed to work. Students ended up rolling the dice and then created their number lines. They were required to show a number model, the number line and any type of work that was used to find the solution. The number line was initially blank and they had to fill it in with the numbers related to their problem. There were initial questions, but it seemed as though the multiple models/strategies were beneficial.

I believe students are making progress in better understanding how to put together and take apart integers. There’s more work ahead of us, but I’m excited about the growth so far. Next week, the third grade class is scheduled to use a number line to show multiplication and division. I’m thinking of using a similar model for those lessons.

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One of my goals this year was to have students analyze their own work, make observations and improve. These observations have improved this year by a light margin. For example, students get back their graded paper and look over how they did. Most students look at the top for their points or some type of feedback. Some look for where something was marked incorrectly, while others look for a place in their binder to place the paper. The good news is that students are looking at their graded papers with a more critical eye. That’s a win in my book. Students are starting to observe where they needed to elaborate or change a procedure. That’s good, but the time spent looking at what to change is still minimal.

This year I introduced the NY/M model. Students were a bit hesitant at first, but I’m finding some pockets of success. Those pockets are not just related to the new model, but also a whole range of opportunities that have been put in place for students to understand where a mistake might’ve occurred. Ideally, I’d like to have students identify how the mistake or error happened and to curb that action in the future. Don’t get me wrong, I’m all for making mistakes in order to learn, but some errors impact an entire answer and I’d like students to be able to identify where that’s happening. Being able to self-reflect in order to improve is a beneficial skill.

In an attempt to provide multiple opportunities for error analysis, I’ve intentionally planned for students to identify their own math misconceptions. This has taken many different forms. I believe that students that can identify math misconceptions may be better able to proceed without making them in the future. Three tools/strategies that have been helpful in this endeavor are found below.

- Nearpod has been a useful too this year. Specifically, having students show their work using the draw tool has helped other students identify misconceptions within their own understanding. Displaying the work on the whiteboard without a name has been especially helpful, as a student might not be embarrassed, yet the class can still learn from that particular person. I’ve used this as an opportunity to look at positive elements of student work and also look for areas that need some bolstering.

- Lately I’ve been giving feedback on student papers and incorporating that into my agendas. Before passing back the papers I review the misconception list and answer questions then. I then pass out the papers and students complete the NY/M process. Generally, students make very similar errors and I attempt to address this while reviewing the agenda. This has decreased the amount of questions that students ask related to why/how to improve their answer to receive a M.

- On the paper I’m making a renewed effort to write feedback on homework and projects. The feedback takes many different forms and isn’t necessarily in a narrative form. Sometimes I ask question and other times I might circle/underline a specific portion that needs strengthening. This method often elicits student questions as it’s not as clear-cut as other methods. Regardless, it’s another way for students to analyze their work, make changes and turn it back in a second time.

Why is this important to me? Well, I believe that students should be provided additional opportunities to showcase their understanding. At times, I feel as though there’s a gap between what math work they show and what they’re capable of showing. Giving feedback, along with another opportunity to improve, tends to help my students show a real-time understanding of a particular concept. Ideally, this would seamlessly work and all students would move from an NYàM. It’s not all roses though. I’d say at least 50% of the students improve on their second attempt, but I’d like to see more. I believe we’re making progress and have more to go, but I believe we’re on the right track. I’m encouraged to see that this model is slowly and slightly changing the review, redo and improve cycle. This has me thinking of how to expand on it for next year. Stay tuned!

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My fifth graders are in the middle of a unit on ratios and proportions. Two weeks ago the fifth grade kids worked through a detective mystery. It was a good start and a decent opportunity for students to explore proportions. This week my students came back from spring break. It took a while to get back to our regular schedule after a week off. We had to complete some review on Monday and started a new project today. I was meaning to start it before break but we ran out of time. I researched a few different sites (1,2,3) and decided to modified my original project.

The class began by talking about proportions and scale models. The discussion lasted around five minutes and then we reviewed the concept and vocabulary with a Kahoot The majority of kids were able to answer the questions using estimation, but many were challenging, which was good because we were able to stop and use different strategies. Most students wanted to cross-multiply for everything, but by the end of the activity students were starting to see the value in diversifying their strategies. I felt like spending time on this was worthwhile. This experience reminded me of a Tweet from #msmathchat from last night.

Afterwards, I introduced the action figure project to the students. Students measured the dimensions of the action figures. They measured the figure in millimeters, converted it to centimeters and eventually to inches.

They then measured their own dimensions with a partner and compared them to the action figure.

Many of the students were able to use proportions as a tool to find a solution. Some students had a bit of trouble tackling the issue of converting the units. Overall though, students are becoming better and using different strategies to solve proportion problems – an #eduwin in my book. You can access the files that I used for this project here.

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Today is the last day before my school’s Spring Break. Generally, my classes end up finishing up a particular unit before a large break. This time is different. Both my fourth and fifth grade classes are in the middle of a unit. I’m also finding that both classes are due for some review. Foundational pieces involving place value and order of operations are tripping up some of the students. While looking around for resources, I came across a BreakoutEdu that corresponds with March Madness. I need to give a huge shoutout to Rita for creating this game. I’ve used Breakout in my class before, but am still in the rookie stage. I printed out the files and started to compile them last week. I figured out which locks where needed and started to compile a few different ideas on how it would work.

What’s great is that my school’s media specialist, the fantastic @mrsread, has a teacher BreakoutEdu box that’s available for checkout. I was able to checkout the box and fiddle with the locks earlier this week. I was able to get most of the locks figured out and reset to the codes needed for the activity. I say most and not all because the multi-lock is still giving me issues. After checking on the forums it seems like this tends to happen more frequently than I originally thought.

After becoming a bit more confident in how to use the set in my own classroom, I decided to use the Breakout with a fourth grade math class this past Thursday. Since I couldn’t use the multi-lock, I decided to use a combination lock that I had at home. I put together a small Google form that coordinated with that particular lock. The next day I spent my planning period organizing the materials. I decided to go with manila envelopes to store the papers and deviously hid them around the classroom. I introduced the game with the slide show in the file at the bottom of this post.

The kids were excited as they already completed a Breakout a few months ago. I told the students that four manila folders were in the room and they had to find them to locate the clues to open the box. I then started the timer and they were off.

**Observations**

The class of 21 split themselves fairly evenly and started working on the tasks. It just so happened the Google form was completed quickly and one of the locks was open in less than five minutes. That wasn’t my intention. I was hoping it would be a bit more challenging. The other tasks, especially the order of operations, took more than 20 minutes to complete. I noticed that around 4-5 students would be working on the sheet while others congregated and tried to find more clues. Some of the kids were making simple errors with the order of operations. The bracket challenge was also tricky, as some students didn’t understand how a bracket worked. Students would complete the bracket and not understand that the larger number would move on to the next section. I could tell that students were getting frustrated as time ticked away. I didn’t interject although I wanted to help. Eventually, students had to use a hint card, but they prevailed. We had a great conversation afterwards using the Breakout reflection cards. This was also great for me to hear, as students gave feedback about which particular tasks were the most difficult and how they contributed to their team.

The Breakout worked so well that I decided to use it with my fifth graders the next day. I changed up the Google form piece and made it more aligned to what we’ve been learning over the past few months. I even added a question where students had to translate a problem from German to English. I may or may not have had a ton of fun helping create questions for the game.

Overall, the game went just as well with the fifth grade group, although they had more trouble with the locks. They were a bit confused with the combination lock. Once they figured out at that skill the class opened up the final locks with about 15 minutes or so to spare. The class didn’t have time to review the reflection cards. I’m hoping we can take those out after spring break is over.

Rita’s files for this Breakout can be accessed here. Feel free to use the Google forms (1) (2) that can be copied and used as well.

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My fifth grade class just started a unit on ratios and proportions. This is fairly new vocabulary to them, although students have used proportions before. They’ve used them before to convert measurements and fractions, as well as other items. They weren’t called proportions at that time. Instead, I remember discussing them as conversions or creating equivalent fractions.

So, on Tuesday the class was formally introduced to proportions. I started off by using a Brainpop video that helped introduce the concept. We watched the video twice and answered the quiz as a class. Through this process it seemed as though students were starting to become more familiar with proportions.

The class then picked up their math reflection journals. The class completed a few proportion examples. They were able to use a few different strategies to complete the proportions and seemed most comfortable by using a cross-multiply and solve for x strategy. Maybe so, because that’s what’s fresh in their minds from a past pre-algebra unit. We continued to work on a few different math journal pages. The majority of students were starting to pick a strategy to solve the proportions. Although most were feeling confident about proportions, I had a group of students that were having trouble. I decided to reach out to a few different people about proportions and found the Tweet below.

After reviewing the documents, I decided to try out the Illuminations activity with my kids. I brought all the kids to the front of the classroom and explained that they will be solving a mystery. The kids were stoked. I had to go over the directions multiple times, but after around 1o minutes I believe they were all on board. I put the students in teams. I then told them that not all students would be catching the same culprit. Students were confused about this, but I thought it added a wrinkle to the activity. I passed out the sheets to the students and they were off to working on their own.

Almost all of the teams had questions about how to proceed. I had the teams tackle the problems on their own for the first ten minutes. Fortunately, teams started to show some perseverance and solved the first problem based on the clue that they were provided. The teams used the strategies discussed earlier in their journal pages. After around 20 minutes I had teams starting to come up to me with their final answer. I gave them a thumbs up or thumbs down. If they didn’t have the right culprit I asked them to redo a specific question that would move them in the right direction.

I feel like this was an activity that helped students become more aware of proportions and how to solve them. The overall goal was successful, although I need to reflect on some of the team dynamics that played out. Not all the teams worked well together. Some students were more confident than others, and some students wanted to let others do the majority of the work. I think this tends to happen in varying degrees during group work. Although this happened, I still feel that the student conversations added to activity. The ideas and strategies that were being discussed seemed to benefit all involved.

Next week I’ll be using the questions to have the students reflect on this activity.

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I believe teachers are always tinkering to see what works best. I’m in the midst of doing just that with my NY/M strategy. Near the beginning of the year I noticed that students would take quizzes and focus on the top third of the page. Near the top, often right next to the assignment title would be a score. That score would mean the world to the kid. Students would take a look at the point value and immediately make an evaluation based on that that score alone. The internalization and analysis that I’d hope for wouldn’t happen. So this year I decided to move a bit closer to a standards-based model with assignments. Students would be allowed to redo an assignment for credit.

At first this strategy seemed to make a huge difference. Students took more risks and it seemed to curb some of their anxiety. In the most positive cases, I’d write specific feedback on the students’ papers and place a NY at the top of the paper. Students would redo the sheet and put it back into the turn in bin. The less-than-perfect cases would involve me putting a NY at the top of the page without additional markings. In all honestly, sometimes the errors were careless mistakes and didn’t require much feedback on my part. Regardless, students were turning back in the NYs and around 60-70% would receive Ms in return. This was good news. Although I was glad that this strategy seemed to be working, I started noticing a trend.

Some students would put less effort in completing the assignment the first time knowing that they’d have another opportunity. Also, students would redo an assignment and not truly analyze what they did wrong in the first place, so they ended up netting zero. This started to discourage me and that’s when I started tinkering again. I asked a few different people on Twitter about the logistics behind retakes in their math class. One person stated that her students fill out a form before retaking the assignment, while another mentioned that the student would be required to get extra help before the retake. I don’t believe there’s a right or wrong answer to this. I believe that there needs to be some type of reflection/feedback that occurs before the retake. That could help students become more aware of what happened during the first attempt and prepare them a bit better for the second. Being aware of where a hiccup happened is usually the first step in the reflection process. So starting this week my students will be filling out the sheet below before retaking a quiz.

I’ll probably continue to tinker with the wording, but it’s a start. I want students to be able to analyze their first attempt and find the reason behind the retake. Sometimes, the reason is because of a simple mistake, but I’d like kids to move beyond that as most reasons for the second attempt are mathematical. I’m looking forward to seeing how this works during the last third of the school year.

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This year I’ve had the opportunity to work with a reading enrichment group. It’s different for me, as I’m used to working with math students for most of my day. The class has been exploring a few different novel studies this year and just finished a unit on Hamlet. We spent about two months discussing the characters in Hamlet and the play in general. They read this book to initially get an overview. We also watched portions of RSC’s rendition of the play. Students enjoyed this so much. They kept on asking to see the next part of David Tennant’s dialogue. I’ll admit it, its been challenging for the students, but there’s been so much growth. Looking back, I’m impressed with how well the students have persevered while learning about the play.

Introducing new content and then reviewing that content with some sort of project has been a good recipe for this Hamlet unit. It’s a complicated subject and reviewing content periodically has helped students remember Hamlet a bit better. We had two projects and one writing prompt that seemed to help improve the students’ understanding of the play.

**Storyboard**

Students used Storyboardthat for one of the Hamlet projects. I actually stumbled across the site when someone mentioned it at a local edcamp. Student teams were assigned particular pages to complete with the rubric below.

Students worked in groups of two to complete each storyboards. The majority of student teams used Chromebooks to complete their boards. The boards were put together to make one complete storyboard.

**Character Map**

The second activity related to creating a character map. Students were asked to create a character map on large paper or in a digital form. Students needed to include the character name, relationships, cause of death, significance to plot and a hashtag that summarized the role. The kids enjoyed putting together the map and loved the hashtag piece.

I’m in the process of laminating the posters and will be hanging them up in the room in the next week or so. You can find the character map activity here.

**Writing Prompt**

The third activity emphasized the writing component of the class. I believe in giving students a choice in their assignments. Sometimes that’s possible, and other times it’s not. Students are asked to reflect on the play and answer one of the three questions.

Students were given a rubric and asked to give evidence from the text to support their reason. Every student finished the writing prompt in two sessions. The next day I was able to review the assignment with the kids. Some had to revisit and redo, but the majority met the expectations. The prompt and rubric are available here.

I’m hoping the activities above helped students become more aware of the fascinating play and world of Hamlet. Hopefully they’ll remember these experiences when they encounter Hamlet again at some point in their life.

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My fifth grade students are in the midst of a unit on probability. This is one of my favorite units to teach for a couple different reasons. One is that it follows a massive pre-algebra unit and it’s so different than what students have been working on for the past few months. I feel like it’s time students see a different strand of mathematics. Another reason, is that students have to think logically about probability and it’s something that impacts their daily life. Also, students haven’t had a lot of time to discuss probability in math class.

Near the beginning of the week students started to explore the different terms related to probability. They completed a random selection activity the week prior and students are starting to have a better understanding of the terms. Around mid-week students investigated tree diagrams and their usefulness in determining actual probability. One of the highlights on Tuesday was a maze activity. Students were given a scenario where they needed to find the probability that students would win or exit the maze without running into a dead end. They used number cards 1-4 to accomplish this. It looked similar to the image below.

Students first estimated the probability that they’d win and then created a tree diagram to find the actual results. They tested out the game by playing six times with a partner. The class was asked what they found and if their estimations were in the ballpark. For the most part they weren’t, which was good news because the class used a tree diagram to find the actual probability.

Students were then asked to use the maze as a fundraising activity. The next question is below.

If 100 students entered the maze, how many would end up being the winner? Let’s say that the winner receives $25. How much profit would be made If students were charged $5 to enter the maze?

This was a turning point in the lesson because students started to become even more vested in what was happening. I gave them about 3-5 minutes to work independently and then they shared their findings with their table group. Most groups were right on target and were able to explain their math reasoning.

On Thursday, students were asked to use their probability skills with spinners and tree diagrams. I found an amazing resources in this book that spurred me to recreate a diagram that my students could use. I gave a copy of the diagram to each student.

I placed five minutes on a timer and gave students that time to work independently to read the prompt and start to find a solution. Students wrote on the sheet and attempted to put together a cohesive tree diagram that made sense to them. I had a few students that thought it was impossible After the five minutes were up, students were asked to share their strategy with partners. The answers were interesting and all over the place. Some students were confused with the spinners as they had to convert them to fractions. Other students had issues with the actual directions. I helped answer questions and students presented their ideas on the solution. This entire activity took 30+ minutes to discuss. Students finished up their ideas on the paper and turned it in. I’m reviewing the results right now and can tell that I need to follow-up with the class. The majority of students did very well, although simple mistakes seem to be evident in quite a few. The class will be discussing this on Tuesday.

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My fifth grade classes started their data analysis and probability unit this week. On Monday the class had a conversation about the terms we use when discussing data. The words, likelihood, probability, experiment and chances were all discussed. After reviewing the terms we dove into the first lesson of the unit.

One of the first activities I generally use asks students to draw a card (between 1-5) 20 times. The data is supposed to be collected and then shared. The class then looks at the predicted probability compared to the actual results.

I decided to change the lesson a bit by incorporating a technology component and possibly save some time in the process. The class also just finished a pre-algebra unit and I thought the formulas used in a spreadsheet could reinforce some of the learning. I’ve had success with using Excel with my fifth grade class so I decided to use that medium for this lesson. Also, my students now all have Google Drive passwords so they’re all able to login with a Chromebook.

Earlier in the day I put together a Google Sheet with a tab for every student in the class. I shared it with all my students during our math block. Students retrieved a Chromebook, logged in and found the shared document. I modeled the formula within Sheets and the students followed along.

Students were able to randomly select the digits between 1-5. Students observed their data and how it changed. We had a classroom discussion on how the sample that they created was based only on 20 trials. They were then able to observe their personal total.

After reviewing their total, they could view the tab called data set. This showcased the data of the entire class. The total, over 300, was much closer to the predicted results.

After students compared the two they filled out a writing prompt asking them. to compare their individual results to the class. What were the similarities or differences? How does a larger data set impact reliability? Students wrote down their responses. I’m in the midst of grading those right now.

The activity was great, but also had some issues. Getting everybody to stick to their individual tab took some work. Some students were caught viewing other students’ tabs. Also, the data sets kept changing when someone clicked certain cells. This was tedious near the beginning. Regardless, once those two kinks were taken care of it was smooth sailing. I ended up freezing some of the cells so students couldn’t change them.

At some point the class will revisit the spreadsheet to discuss tree diagrams. Click the image to copy and use the spreadsheet in your own class.

I changed the names to S1, S2 … so you can change them as needed.

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I’m straying a bit away from my usual math posts to highlight a literacy connection. This week my school kicked off their One School One Book campaign. It’s been an annual tradition to have all students in the school read one book for a period of about a month or so. Every students gets a copy of the book and classes engage in questions and conversations about the book. We’ve been participating in OSOB for the past few years. In the past, we’ve used Charlotte’s Web and The World According to Humphrey for the school-wide event. This year the school is using the Lemonade War for OSOB. One aspect that’s helped make this successful is an organized process that’s been used before and during the reading. One of our school’s fifth grade teachers, Vicki, has helped organize a team and the process since its beginning. Here’s the process that’s been used over the past few years:

1.) To generate student curiosity, cutouts of certain items are placed around the school. Sometimes the school uses an Ellison letter machine for this. For example, when our school used Charlotte’s Web, cutouts of a pink pig were placed all over the school. They were placed on doors, windows, in the hallways and even on the ceiling. Teachers didn’t say a thing and let students ask questions and wonder why they were placed all over the school. This year coins were placed all over the school since we’re using the Lemonade War. This year students had an idea that it was related to OSOB, but they weren’t exactly sure what the book was.

2.) In the meantime, teachers read certain chapters of the book in front of a green screen. For the past two years my school has been using Touchcast to record the readings. The background is changed based on the chapter’s contents. Since I teach mostly math, I took the chapters involving math and graphs.

3.) A group of teachers participate in a skit from the book. All students attend an assembly to see the skit. The students don’t yet know what the book will be until the skit concludes. At the end of the skit the book is revealed. The kids tend to get a kick out of seeing their teachers as characters in the book and it also generates additional interest in the book.

4.) Students read the first chapter of the book the evening after the skit. A reading schedule is sent out to all the students and the principal includes information about OSOB in their newsletter. After reading the book they can also watch the Touchcast videos. This year questions are embedded within the videos.

Every year it seems that OSOB engages students in reading a book as a community. The curiosity and engagement that seems to follow OSOB continues and that benefits stakeholders. The success of this has me wondering how school’s can use a similar model to promote math.

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