Nice to Meet You Area Model

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This week my second grade students have been exploring multiplication strategies.  We started off early in the year looking at arrays and using doubling strategies.  Then we moved to helper facts.  These are still used to this day, but we introduced a new tool this week.  Enter the area model.  Hello!

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Students transitioned from arrays to squares, but didn’t sit at that spot long.  Through the area model, students take apart numbers and partition (yes, we say partition at second grade) the rectangle into parts.  Each part is a partial product.

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I’m fortunate in my position to see this strategy used at multiple grade levels.  The rectangle evolves over time.  As students progress, I find that place value and advanced decomposing strategies become more prevalent.  You can learn quite a bit about a student’s understanding by checking out their math work with an area model.  How they split up the numbers can also tell a story.  Why did they split up the rectangle that way?

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I find quite a bit of value in using this strategy.  For one, it doesn’t immediately move students towards the standard algorithm and it helps build/show conceptual understandings.  My 2-6th grade math students use it in a variety of capacities. My 5th grade crew has recently been using them to multiply fractions. Short story: It makes an appearance at every grade level.  It’s also a a fairly smooth transition to using the partial-products strategy.

Even though it’s a useful resource, I find there are a a couple things that irk me about using this tool.  Sometimes organization skills can hamper the effectiveness of drawing and organizing.  I’ve had more than a handful of students draw boxes that overlap or numbers that might not be decomposed correctly.  Also, it’s not to scale, but that’s not a game changer for me.

As students progress through elementary school they encounter a variety of math tools and strategies.  Manipulatives are generally used to help students build a better understanding of math concepts. The CRA model is often emphasized at this level. Many tools are brought out to help fill gaps and others are continually used.  At some point, I’m assuming the my students will rely on the standard algorithm to quickly multiply numbers (if they don’t have a calculator handy).  They probably won’t understand why the algorithm works, but it just does.  The area model shows multiplication in a concrete way.  Don’t get me started on lattice.

Surface Area and Improvements

Last year I taught a lesson on surface area that bombed.  I thought it’d be great to have students measure the surface area of a state using a scale model.  This task was found in my course adopted resource pack. Looking back, it wasn’t a bad idea or problem but the execution was far less than stellar.  The problem asked students to find the surface area of the state of Nevada.  They were given a model and a scale at the bottom.

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The class completed this mostly in whole group (which in hindsight was not the greatest idea).  I asked students to use the scale to find the surface area.  Students used rulers and decided to find the area by dividing the shape into one rectangle and one triangle.  After giving students about 10 minutes I surveyed the class and the answers were all over the board. Some debated on the word “approximate” as the class was asked to find the approximate surface area.  Other students thought the 0-100 km was a guideline and could be rounded. While others decided to neglect the missing piece near the southern border of the state.  Needless to say it didn’t go as well as planned.  Looking back, one of the problems was that this activity was completed whole group.  Students didn’t get time to discuss with each other what or how to measure.  There wasn’t a determination of what to do with the missing piece in the south and how to divide up the state.  The class eventually came to a consensus that there was one right answer and we moved on.  I put a note in my planner to do things differently next year.


So it is now next year (2020) and I have a different class.  This year I gave the same problem, but did things a bit differently.  I first front-loaded information about the state itself as a whole class discussion.  The class discussed the shape of Nevada and how it’s not exactly one rectangle and one triangle.  I reinforced that we can’t just neglect the small corner of Nevada.  It may be helpful to find that area as well.  Students were then randomly selected and placed in small groups of 2-3 students per group.  I asked the students what was meant by the scale in the bottom left and how they could use it to help them find the area.  Student groups had time to discuss and report out how they would use it.  Some students even found that the 0-100 km was actually 1 centimeter. I then gave each group a ruler/straightedge to help construct shapes within the state itself. Students had approximately 15-20 minutes to discuss and find the surface area using the tools that they were provided.  Students were busy slicing up the state and using a straightedge to find the approximate surface area.

The class then came back as a whole and each group submitted a response.  I received all the responses and students were given time to think about their submission and possibly make a change.  It’s interesting how peer pressure and consensus will sometimes make you second guess a decision.  In this case students mostly received affirmation and there was justification that came along with that decision.  All but one group was in the ballpark and that group didn’t initially convert the scale.  There answer ended up being extremely small compared to others.  Some of the groups decomposed like this:

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The majority of the class was within the approximate range and it was a productive discussion.  If you’re wondering, the surface area is approximately 278,000 square kilometers.  So now you can win a trivia contest.


I put a note in my planner to use this method next year.  Last year it bombed and this year was much better. Part of teaching is improving your craft and I had more than a couple pieces of humble pie last year. I tend to hear the phrase best practice thrown around in the field of education. I’m more of the mindset of emphasizing better practices and looking forward to tweaking this even more to make it a better experience next year.

Exploring Perimeter

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My third graders have been investigating perimeter and area for the past week.   I find when the terms are isolated, students are able to define them fairly accurately.  When put together it’s a different story.  Students tend to switch them around or heavily rely on one term based on what the class has been working on for that day.  So this year my students worked on a project that focused specifically on perimeter.  Area is part of it, but only if a teacher wants to pursue that avenue.

Students were put in groups and given two sheets.

Students outlined the map and personalized the city.  The construction zone is intended to be used for the actual city piece.  After the maps were distributed, each group received a centimeter grid.  The grid was used for students to cut out and create a city based on a certain criteria.  Each group received one sheet that indicated certain dimensions.

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Students then filled in the rest of the grid to match the dimensions.  Some of the dimensions were non-negotiable, like the height for the school or perimeter for the police department.  Others had some leeway.  There was a lot of erasing and rewriting for this sheet.  Once they completed the sheet students started tracing and cutting out the centimeter grid paper.  Trial-and-error was part of the process.  Students then cut out the buildings, put together some supports and glued them to the construction zone.

Students put together the cities and attached the dimension sheet to the bottom.  I’d say that around half of the class is finished and the rest are making some great progress.  I’m looking forward to seeing how the rest of them turn out and the gallery walk that will happen afterwards.  Here are the files that I used and feel free to use it in your own classroom.

Class-generated Quizzes

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One of my classes explored multiplication, factors, multiples and arrays in December. Students solved problems involving using different multiplication strategies and we thoroughly discussed how arrays can be arranged using the rows and columns as factors.  A unit assessment is scheduled for January so students were given a task before break.  Students were asked to create a problem involving multiplication. They had to write out the problem, provide three close but incorrect answers, and one correct answer.  Students could use dice or a random number generator to create the problem.  Most opted to create an original problem.

Students picked problems involving arrays, while others decided to add to the challenge and have students identify factors and distinguish between the product and factors.  Other students created square array problems.  I found that in the creation process many students had to erase their model and start over.  They had to be clear and I reiterated that everyone in the class would need to be able to clearly distinguish the rows, columns and total of an array.  Students realized that the array had to be in the form of a rectangle or square and some used a ruler for precision purposes.  Yay!  Others didn’t. I collected all of the potential questions and answers and brought them home over winter break.  I didn’t look at them again for another two weeks.  I

Yesterday evening I took pictures of the drawings with my iPad and inserted each question into a quiz.  It didn’t take as long as I originally thought to put all the questions and pictures into the quiz.  Feel free to access the actual quiz here.

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Today I paired students up and they took the class quiz.  Students were stoked to see their question on the quiz and the excitement was contagious.   It took the students around 15-20 minutes to complete the quiz and the class reviewed each question together.  The author of each question revealed themselves as we went through the questions and drawings.  Students gave feedback on the questions and I was impressed with how close the incorrect answers were to the actual solutions.   They wanted to make sure the students actually read the questions carefully.  Good call!

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In the future I’d like to add more topics to the quiz.  Adding variety will also give students more options to review the topics discussed in class.  I feel like this idea has legs and I might use it again later in the year.

Pixel Reflections

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It’s nearing the end of 2019.  It’s hard to believe, but in just a few days it’ll be 2020.  Near the end of the year I like to take stock and think about last year and what ended up being successful and what didn’t.  Last year I came across an image on Twitter that showed a different way to reflect on the year.  Not sure where I exactly found it, but I believe the image was pulled from this site.  In 2019 I wanted to purposefully reflect more throughout the year.  I decided to jump on this trend of using highlighters, a notebook and pixels to analyze how I felt throughout the year.  I ended up creating my own sheet with rectangles modeled after some of the pictures that I came across.  I wanted to originally use squares, but that didn’t happen when I printed it to fit to the page so I just went with it.

Highlight per day

I tried to stay consistent with filling out the sheet daily and at the same time.  That didn’t happen every time, but I became better as 2019 rolled forward. As the year progressed I started to notice a few trends in my own analysis of how a day went.  I became more clear on what events/activities/notices indicated an amazing day compared to a frustrated day.  The list below is certainly not all-encompassing and isn’t perfect, but used as general guidelines as I filled out each rectangle.


Amazing (orange) – Feel well-rested, vacation time, visits with family, able to get outside in the sun, time to read, drinking my coffee slowly,

Really good (purple) – Feel productive, time to plan, able to get outside, exercised, get to bed on time

Normal (blue) – Feel good, able to accomplish what’s needed for that day.  Feeling a bit tired but productive, sleep patterns are a bit irregular,

Exhausted (yellow) Lack of sleep, too much or too little coffee, traveling day, didn’t exercise, wasn’t able to get outside, too much work, evaluations, not feeling as productive, starting to feel sick

Frustrated (green) – Sick, bad news about family, medical issues, rejection letters

Sad (red) – I ended up not using this one and probably won’t in 2020


Moving forward, I’m thinking of continuing this process in 2020.  I have my highlighters ready I think there’s power in being able to reflect and categorize how the days make you feel.  Taking the time to write it down has been a valuable experience.  I also have to be a more critical in how I categorize a day.  Everyday isn’t going to be perfect.  I need to be reasonable with expectations.  For example, does getting bad medical news for 5 minutes negate having a terrific day? Not sure and I don’t have an answer here.  Looking at the data is interesting as I can make generalizations, but my takeaway is the time spent being mindful of how daily events impact my perspective.

Here’s to 2020!

Distributive Property Discussions

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My fifth grade class is in the middle of a unit on pre-algebra concepts.  We explored the different associative, commutative and distributive properties earlier in the week.  Students were able to use the first two with modest success, but the distributive property was causing some issues.  I believe some of the reason is because students were confused with what the parentheses meant, while others needed a visual model to make a better connection.

The class reviewed a few different examples and we went back to a concrete representation.  I find this is the place where solid understanding is developed before we  move to more abstract models.

Students have been use to using base-ten blocks, counters and unifix cubes to put together and take apart numbers.  Students were asked to use cubes to show an understanding of the distributive property.  They used a dice to create a multiplication problem and then split it into two parts.  They then wrote on the desks (who doesn’t love that?) to show multiple number models to indicate the total.

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This student didn’t show the partition, but displayed the different number models

One issue was trying to figure out how to divide or create the partition.  Some students used dice to indicate where to split the model.

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I appreciate how this model shows the different representations

From here, students transitioned to problems involving larger numbers and in an abstract form.  They were more successful this time around.  Students then worked in groups to complete this OpenMiddle problem.  They worked in this task for about 15 minutes using whiteboards in the process.  This was a quality activity that has students trying out multiple numbers to make the equation work.

 

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I’m looking forward to revisiting the distributive property when schools starts back up in January 2020.


Friday was our last day of school for about two weeks.  We’re officially on winter break, but it doesn’t feel like it yet.  I’m sure it’ll sink in on Monday when my alarm clock doesn’t go off at the normal time.

Student Reflections

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My fifth graders finished up a math unit around a week ago.  The unit took around 1-2 months and students explored topics pertaining to decimals, percentages and box plots. Near the end of November I had a discussion with Jack about student reflections and previewing units.  Jack shared a Tweet by Chrissy about cool-down bins.

I thought the idea had potential and decided to use it as part of my end-of-unit reflections. My students completed the test reflections and I added the bin language near the bottom.

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The class discussed what each category meant and I answered questions.  I think the most challenging part was communicating the difference between practitioner and expert.

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Some students mentioned that you could complete the task without any help, but still not be an expert.  Other students said that they still wouldn’t consider themselves an expert even if they could teach other students a skill.  Still tweaking this idea.  The class is still working on this type of reflection, but I believe we’re making progress.  I’m hoping to use this throughout the rest of the school year.  Students can then reflect back and look at the progress that was made.  One of the goals this year is for student to become better at accurately assessing their math understanding compared to the standard.


Side note:  One small win during the past week. I was able to combine two second grade classes to complete an array polygraph last week.  Another teacher and I had around 35 students complete the polygraph together for around 20 minutes.  It was great to see partners use math vocabulary to try to guess the arrays.  Words like factor, row, column and product were all be used during the process.

We have two more weeks of school and then two weeks of break. Let’s finish 2019 on a strong note!  

Monitoring Progress Towards Goals

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Around two weeks ago my students finished up a math unit and started a new one.  Students previewed the next unit by reviewing a study guide and looking ahead at the skills in their consumable math journals.  They then made appropriate goals based on the preview.  I spoke with the students afterwards and helped them reshape the goals to be more aligned to what’ll be explored during the next unit.  I wrote about this process here and then started to think about next steps.

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Some students created goals related to topics that we haven’t explored yet, while others felt more prepared to answer.  My students around about a 1/3 through the current unit. This week students checked on their progress towards the goal.  My intention was for students to become more aware of their goal and the progress made towards its completion.

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I think in the future I’d like to create some type of scale where students identify the progress made towards the goal instead of a met/not prompt.  I’m looking forward to revisiting and refining this idea as the year progresses.

 

Math Reflections and Sentence Stems

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My students just finished up the first trimester yesterday.  So we’re about a third of the way through the school year. In my mind this is a perfect time to reflect on the progress that has been made over the last couple months.  All of my classes started near the end of August and many of my classes have recently completed the second or third unit. It’s been a great journey so far and we’ve made progress.

Last week I had a class conversation about progress and what it looks like in math class. We discussed growth and how it doesn’t look the same to everyone.  To help facilitate the conversation I had students reflect on their unit assessments.  Usually, I’d have students fill out a form indicating questions that were incorrect and then they’d code the errors.  Students would then set a goal for the next unit.  That process is detailed here.

This time around I wanted my students to recognize their growth and how their perceptions change over time.  I also wanted students to preview the next unit and set a goal based on the preview. I modified a journal prompt from a colleague and decided to add sentence stems with space to write.  I didn’t give students much advice or guidance on how to complete this, but I told them that I wanted them to be honest with their responses.  The prompts are meant to have them reflect on their progress.

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Students were able to follow the sentence stems a bit easier than past reflection prompts.   The wonder question was left vague for a reason as it presents a way to indicate student interest and curiosity.

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Most students were able to analyze their unit assessment and look for trends that were positive.  I wanted to communicate that they should be proud of what they accomplish. Some students even looked beyond the test and wrote down that they were proud of how they improved their understanding of x skill.  Other students stuck with the grade on the test and being proud of that aspect.  I really like the “something I want to remember …” piece as it reinforces that what students are working on and developing will be used in the future.

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Some students asked for more lines to write additional pieces that they learned.  Again, I found there tended to be two camps of students.  One group focused on the math concepts/skills, while others focused on the points/questions.

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The “PA” is a pre-algebra activity that we complete to start the math class.  It was interesting to read what students felt was the most difficult as some were more vulnerable than others. This year I’m emphasizing the idea that this class is part of their math journey and that we’re all mathematicians.

The next step was to preview the next unit and start to set a meaningful goal.

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Student went through their math journals and looked for words or skills that didn’t ring a bell.  At first students thought that everything looked fine and confidence was brimming a high level, but then they started to look at the wording.  The next unit explores box plots and percentages.  Based on the words/topics, students made a goal that they’d like to accomplish.

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I appreciate how the above student extended the skill to learn about percentages and sports.

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This particular student wants to become better at the “LCM box method” as it was explored last unit.


Students completed the page and then we discussed it together 1:1.  I asked each student why they felt that the goal was relevant and meaningful.  I’m looking at adding a progress monitoring piece to this goal as the class progresses through unit three.  Ideally, I’d like to revisit the goal every 2-3 weeks to see what progress has been made towards the goal and make adjustments as needed.  By doing this, I believe students are taking more of an ownership role as they can see progress made towards the goal.

You can find the entire template for the sheet here.  Feel free to leave in the comments how you’d use this or if you have questions.

Feedback and Reflection Opportunities

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Over the past few years I’ve emphasized the use of student feedback, reflection and goal setting in my classroom.  Trying to find an adequate balance for this has been a challenge.  This school year I’m looking at different mediums in which students can reflect on their math progress.  Reflecting on individual progress and determining where students are at in the progression can lead to powerful outcomes.  This year I’m looking at different ways to encourage students to make more meaningful reflections that lead to actionable goals.


Math journals

Each student has a math journal.  These journals take the form of a 70 count spiral notebooks. The journals are primarily used to reflect on assessment performance.  My class has eight unit assessments per grade level and there’s a reflection sheet designated for each one.  I’ve written about the different sheets and journals: (1)(2)(3).  Students then complete the sheet and bring their test along with their reflection to the teacher to review.  At that point the teacher and students work together to develop a math goal for the next unit.

Peers

Students work together often in my class.  I use a randomizer and group students so they tend to have different partners daily.  After they complete a task I give the students a couple minutes to discuss with their partner the effort level and challenge of a particular assignment.  I tend to use a timer and encourage students to be honest in the process.  Hearing how other students feel about the effort and challenge level can help students be more aware of the struggle that is part of the process.  Certainly not all, but I’ve seen some students thrive with this type of reflection.

Independent

Before students complete a project teachers give a sheet indicating the criteria for success. This looks like a checklist with statement indicating what’s needed to meet the expectation of a particular assignments.  As students complete each component they reflect on whether they’ve met the criteria and place a check.  Students will attach the criteria for success sheet to their assignment

Digital Platform

Along with the criteria for success, students may submit a project to a digital platform like SeeSaw or Canvas. Other students in the class are randomly selected to give feedback on other projects.  Students follow a prompt that asks whether a student has met the criteria for success or not.  They then re-check the work and ask a question or provide a positive comment about the work.  After the comments are submitted, the original poster reviews the comment, reflects and responds to the question or comment.


 

Each one of these mediums has pros and cons.  I’m finding that the classroom environment plays a major role in how comfortable students are in sharing their thoughts with the class.  Some students are much more willing to independently reflect and speak with the teachers than others.  I find that students take more of an ownership role when they analyze their own performance in context and make steps to improve.  Bypassing the “I’m good at math” or “I’m bad at math” type of thinking can be a challenge, but providing ample opportunities to reflect can move students away from generalizing and be more specific about their analysis.

The next step in my process is to take these reflection opportunities and transition it to meaningful individual goal setting.