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green (REFERENCE) sheets

A topic I do a lot of PD (professional development) on is how to support students who arrive to our classrooms with academic holes to engage in grade level mathematics. At these trainings, I share LOTS & LOTS of things I do to support these students. One idea I share gets by far the most requests for more information & it is something I call ‘GREEN SHEETS‘. A couple of times a week for the last several years I get emails that say, “Sara, can I get a copy of all your green sheets?“. I usually respond, “here are a few I have – I am not organized enough to have the rest available yet“. Well, in this blog I am going to share four of these sought after Green sheets with you that you can adapt as well as let you know how you can get more. Read on!

Why did I start making ‘GREEN SHEETs’?

Catalyzing Change in High School Mathematics: National Council of ...

I am passionate in my pursuit to assure all my students, regardless of the skills and background they brought with them, make growth on grade level mathematics. If I am teaching Algebra 2, then all my students – even if some test says they are not ready for Algebra 2 – will be supported in making growth on Algebra 2 standards. I am a big believer in heterogeneous grade level classes for all. I support NCTM’s ‘Catalyzing Change’ books with recommendations like ‘Dismantling structural obstacles that stand in the way of mathematics working for each and every student.’ Specifically, I am against taking students out of grade level courses and putting them in courses that slow down the curriculum. I am against teachers teaching ‘7th grade math’ and only covering 60% of the curriculum because they feel their students are not ready to go faster. I believe that all students in my mathematics courses can grow and learn grade level material.

Challenging my beliefs my entire career has been a large portion (more than 50%, at times much higher) of students enrolled in my courses each year that are not academically ready to be there. Each year I have large percentages of students who arrive – due to EL status or homelessness or _____________ – with gaps in their education – due to no fault of the own. I have large numbers of students that are not proficient on our state exams. Regardless of this, I know that all of these students can do grade level mathematics. BUT – they can not do grade level mathematics unless I have a plan to support them in filling the academic holes/gaps they arrive with AND to do this with minimal time lost to grade level work.

It is so tempting to slow down my pacing and teach the things I know many of my students missed out on prior to the courses I teach. I don’t do this for 2 reasons. One, slowing down means that they will not have time for grade level work and this will put them further behind. Two, it is not fair to the students who are academically ready for the course they are enrolled in. (& again, as I said above, I refuse to create separate courses for students who are ready and those who are not for grade level work.

SOOOOOOOOOOO……….I created a plan to support all students being able to do grade level work in my courses. This plan has lots of parts. Green Sheets are just one of them. Green Sheets alone will not solve your challenge of supporting students who are not academically ready – but they will give you a tool that can be part of a larger overall plan (and they may be a great way to start). Sooooooo…….

What is a ‘GREEN SHEET’?

Simply, a ‘green sheet’ is one sheet of paper reviewing mathematical concepts, connected to current learning, for students to use in the math courses I teach. I call this one-pager review sheet a ‘green sheet’ because it is printed on green paper, specifically green card stock. Why green card stock? Have you ever tried saying, “Students, please get out your review sheet on ‘solving systems of equations‘. ” and then watched students spend 5 minutes digging through their folders looking for the right sheet only for 1/2 of them to get out the wrong one? I reserve the color green in my classroom to only be used for the 1-2 review sheets I make for students each unit. When I say, “Everyone, please get out your green sheet on solving systems of equations“, it takes moments for the entire class to have out the correct sheet. Saying ‘green sheet’ led to nearly 100% of students being successful with my request. It also allowed me to look across the classroom and see if all students had a green sheet out.

‘Green sheets’ are one of several ways I support the academic needs of my students who arrive in my courses with academic holes that cause them to struggle with grade level mathematics. For example, an Algebra 2 Unit on Solving Systems of Equations assumes my students are proficient already at solving systems of linear equations with 2 variables using a variety of methods (graphing, substitution, elimination and others). I know many of my students are not proficient with this yet. To support students I created a ‘green sheet’ reviewing multiple ways to solve systems of linear equations. A common green sheet I use at the start of each year, because I know students struggle with this is on using the ‘Order of Operations’. Each unit I create 1-2 one page review sheets on topics connected to the learning in the current unit.

How I create ‘Green Sheets’.

In preparation for each unit I teach in a course (6-8 units per year), I read through the goals of the unit. I read through the mathematical standards students are expected to master. I make a list of mathematical vocabulary for the unit. I ask myself, ‘Why might my students struggle with this unit?’. Ultimately, I ask myself…

What does this unit assume my students already know how to do?

What mathematical terms and concepts should my students already be proficient with before this unit?

I make a list of the prerequisite skills/concepts/vocabulary and then prioritize the ones I think are most important for success in our new unit. I assume many of my students will need support with this list, so I create a ‘green sheet’ reviewing these items.

For example, in my Algebra 2 unit ‘Solving Systems of Equations & Inequalities’, I knew that to be ready for the mathematics my students needed to have flexible skills for solving systems of equations with 2 variables, so I created a ‘green sheet on this’ skill.

Note: If you want an editable copy of this sheet, scroll to the bottom to download several of my green sheets.

After using these sheets in my classroom for over a decade, I have found that ALL students find these green sheets useful. If I don’t have one of these sheets in a unit – students often beg for me to create one. Students don’t find these useful though unless I help them to find these useful. Keep reading to find out how I do this.

When do I use ‘Green Sheets’ in class?

I pass out a unit’s new green sheet on the day of the previous units end of unit assessment. The new green sheet is usually the homework the night of the assessment. Students pick up the green sheet AND a one page set of practice problems. For example. If the topic of the new green sheet is ‘order of operations’ then students will pick up the green sheet pictured to the left. I will ask them to review/read it for 5-10 minutes first. (note: we’ve modeled & discussed ‘how to read for understanding’ prior to me asking them to do this). In addition to the green sheet, I give students 6 problems to do. The problem set includes ‘hints’ and ‘solutions’ to each problem. I ask students to spend a maximum of 20 minutes working on these problems. I do NOT teach them how to do each problem. I have selected problems that will draw out common mistakes that students will make with the topic.

Students arrive to class on the first day of the unit having looked at the new green sheet and attempted a few problems. (Note: 95% of my students arrive to class each day with complete homework – but that it another post on how I get such high percentages of homework completion). When the arrive in class I hand them an annotated answer key I’ve made of the 6 practice problems. I have modeled for students how to use ‘answer keys’ to maximize learning. Most of my students find the 5 minutes I give them to look over the answer keys extremely valuable. Many students take photos of the answer keys (I totally allow this) and I often find them pulling up these photos later on to study from.

How much class time to I give to reviewing previous learning?

Somewhere between 50-80% of the students in each class I teach arrive to my course with small to huge academic holes that get in the way of their current learning. If I were to slow down to fill all these holes, I would never have enough time to teach the grade level standards outlined in the courses I teach. To assure I can teach all grade level standards well, I have given myself a budget of TIME for each unit to spend reviewing. This budget is ONE 55-minute class period per unit will be used to review previous learning. ONE and only one class per unit. My students often need way more than this, but I limit myself to 55 minutes. I may split this 55 minutes into 30 minutes one day and 25 another, but I force myself to prioritize the most important ‘stuff’ to review for each unit.

Most of my one day is spent reviewing prior mathematics learning needed for success in the upcoming unit. This review is always connected to the new ‘green sheet’. I believe if you don’t consistently use the green sheet with your students, they will never use it on their own.

How I teach students to use their ‘green sheets’.

At the start of each unit, I give students one (at most two) green sheet for the unit. Simply giving students a ‘green sheet’ does not mean they will use it. In fact, most papers I pass out or share with students are never read or used. If I want students to use the ‘green sheet’ then I need to TEACH them how to use the ‘green sheet’.

One part of doing this is to give students a small number practice problems they can do with the green sheet.

When students do problems side by side with their green sheet they start to see a usefulness to the green sheet. In the case of the green sheet above on ‘order of operations’, students work on the 6 practice problems independently (given hints and solutions). The next day in class I start class by saying, “Everyone get out your ‘green sheet’ on order of operations (I wait until I see every student with a green sheet out) and your set of 6 review problems – even if you did not complete them.”. I then give every student an annotated answer key with all my work shown to the practice set. Students independently review this answer key for 5 minutes. Then I take questions. Usually most students have very few questions at this point. I at most cover one of the 6 problems to review together.

Practice sets for new green sheets don’t all look the same. For example, in Algebra 2 my students needed to review ‘how to solve a systems of linear equations’ – past learning from 8th/9th grade Algebra. The green sheet had 8 different ways to solve a system of linear equations. I knew that my students may be overwhelmed with this working independently, so the first ‘practice set’ was just built to give students choice. I asked them to review the new green sheet and select 2 problems only to solve using any method they preferred.

In class the next day I said, “Everyone, get out your new green sheet.” and together we used this green sheet to solve 1 system of linear equations 6 different ways (graphing, substitution, elimination – 2 ways -, desmos…..). Student’s homework that evening was to do the same thing with a new system of equations using the green sheet. The next day they arrived in class and had a few minutes to look over an answer key. This was our one, and only one, day of review.

Everyone, get out your green sheet! – Modeling the use of Green Sheets in the classroom.

Using the ‘green sheets’ with students in my one day of review is NOT enough for students to continue using them independently. Students who most need to use the green sheets, will not use them. This is for a couple of reasons.

  1. They don’t know how to use the green sheets.
  2. They don’t want to use the green sheet and reveal they are not prepared (their is a sense of embarressment in some)

For these reasons, I feel it is important to teach all students how to use these sheets AND normalize the use of the sheets. One of the ways I do this is to make it mandatory for myself to use the green sheets in class, with all students, at least once every day the first 4-5 days of the unit. It sounds like this….

“Everyone, get out your green sheet.”

I then wait & watch until I see everyone taking out their green sheet.

“Everyone put your finger on the box labeled ‘grouping symbols’.”

I wait until I see everyone pointing at right box. Then I ask them to interact with this information in some way, like… “Turn to your neighbor and tell them what grouping symbols are – hint there are more than 1 or 2 things to say, go”

I have found that all my students, both those academically ready for the course they are enrolled in and those who are not, love using the green sheets. My students, of all types, love these green sheets. I have also found that using a new green sheet 3-5 days in a row is enough for most students to use them on their own the rest of the unit. Later in the unit, if I know that the green sheet would benefit some students, I will again say, “EVERYONE, get out your green sheet and….” – again, normalizing their use.

How students feel about ‘GREEN SHEETS’.

My students loved, loved, loved the green sheets I made. Not some students, but all. I have lots of evidence of this.

Linus' security blanket | Peanuts Wiki | Fandom

Students asked to use green sheets (review of previous learning) on assessments in my class. At first I leaned towards not letting them use them – but I decided to allow it. It turns out that the green sheets were like Linus’ blanket – it was a safety thing for them. They had it near by if they needed it, but 95% or more of the students never used it during the assessments. I found it relaxed their testing anxiety to have it nearby.

I would see my students using green sheets outside of class. My classroom was on 2nd floor, near the commons space that overlooked the lunchroom. During lunch, I would often see students doing homework and would have their green sheets out as part of this work. Our school had a ‘Math Center’, staffed by a math teacher, open all day that students could stop in and get help. The teachers in this center reported that my students always had their green sheets out, without prompting, while working on math. They even would loan them out to students from other courses, without green sheets, and use them to tutor other students.

At the end of each unit, my students would clean out their folders. I recommended saving their green sheets (along with a couple of other items). At the end of the year, my students had 8-12 green sheets and 6-8 yellow sheets (review of current learning – but that is a different post) that they could use to study for and use on the year final. Again, I found my students loved having these, but almost never took them out during the final. The pre-calculus teachers, the students next course after their course with me, reported seeing green sheets used the next year.

What I found is that if I made green sheets useful to them in class and normalized their use by everyone, students would use them, over and over again on their own.

Making your own ‘GREEN SHEETS’

I get asked all the time to share ALL my green sheets. If you are a reader of my blog, you know that I am willing to give away for free what I create. At the end of this blog I will share several green sheets I’ve created with you. My plan is to clean up green sheets I’ve created in the past and also create NEW green sheets on math topics I wish I had when I was teaching. Anything new I create I will share with you. My goal is to create about one new sheet a month for the next year to share with you. I will send new green sheets out to all those who have entered their email at my site (either on my ‘About Me’ page or by downloading any resources at my blog).

When I create green sheets, as mentioned above, I have two guiding questions.

I assume many of my students will struggle with the items I made answering the questions above. I create green sheet(s) to support students learning. My goal is to teach them how to learn. I create my green sheets based on what I know about my students. They are designed to support the students I teach. Because I’ve personally made all green sheets I use with students, I know them inside and out and it helps me use them with my students. If I used green sheets others created, I am not sure they would be as effective.

For this reason, if you want to use green sheets I’ve created, great. BUT if you plop them in front of your students – they will not be as effective as creating your own to match the needs of the students you teach. For that reason I HIGHLY encourage you to go through the same process I do. Before every unit you teach, ask yourself, “What does this unit assume my students already know how to do?” Make a list of every concept, skill and/or vocabulary term students should know. Then use the green sheets I’ve made as exemplars of possible ways for you to create your own green sheets. Make your green sheets to match the needs of your students. I know this takes time – but do it anyway. Make it a goal next school year to create your own green sheets for at least 1 or 2 of your units. The next year make them for other units. After a few years, you will have them for all units.

I am sharing my green sheets below as editable word documents. Feel free to take what you like of what I created and tweek it to make even better green sheets for your students.

I promise to continue adding new green sheets to what I am sharing today. If you create one you would be willing to share with other teachers, send it to me and I will add it to the folder of resources. (giving you credit).

Although I plan to create additional green sheets – the very best green sheets for your students will be created by YOU, not me. I encourage you to make your own green sheets.

I’d love to hear from you. Do you have a topic you’d like to see me create a green sheet on? If so, let me know by filling out this quick google form.

A resource for creating ‘GREEN SHEETs’

Ultimately, a green sheets is a review of previous material that is connected to the new learning. Some curriculum resources come with pre-assessments that test this previous material. One FREE resource that I work with is Illustrative Mathematics. You can find the FREE versions at Kendall Hunt’s site (6-12) and at Open Up Resources (6-8) site. At both sites if you register with a teacher email, you can obtain access to all assessments. One way to create a green sheet if you are new to them is to use the unit pre-assessments at these sites to help you craft a new green-sheet. There are 2 ways to do this…

Number 1 – Use ‘Check Your Readiness’ pre-assessments as is…

Illustrative Mathematics Assessments

Even if you do not use the IM curriculum (it’s great, by the way), you could use their pre-assessments (they call them ‘check for readiness’). Find a unit with topics similar to what you are teaching. Print out the pre-assessment. Give the pre-assessment to your students as a homework assignment with the directions, “Spend no more than 20 minutes on this assessment. Try every problem. Show your thinking on each problem. I am more interested in your thinking than the answers. Circle the 1-2 problems that are the most challenging for you.” The next day give students an annotated copy of the pre-assessment with solutions printed on green paper. Have them spend 5-10 minutes reviewing this. Take 1-2 questions from the class. Have students keep the green answer key and find ways to use it every day for the next 4-5 class periods so that students find it useful.

Alg 1 item

For example – Here is an assessment item from an Algebra 1, Unit 2 IM pre-assessment. This unit assumes students can solve 1 & 2 step equations. I would have my student’s try this item on their own and then the next day give them an annotated copy of those same assessment items printed on green paper (AKA – their new ‘green sheet’). Look at my annotations below. What do you notice? How might this be helpful to students? How could I improve this?

Annotated Algebra 1 ‘Green Sheet’ (IM Check your Readiness Assessment item)
Alg 2 item

Here is an Algebra 2 assessment item. Think about how you might make an annotated answer key for this item. What could assist your students in understanding this previous material without you having to reteach this item. My attempt is below. I don’t think it is perfect and honestly I prefer creating my own green sheets as they are more organized , but in a pinch, this is better than nothing.

Annotated Algebra 2 ‘Green Sheet’ (IM Check your Readiness Assessment item)

The Geometry Pre-assessment had items asking students to do constructions. In order to make an annotated answer key, I did a photo shoot of myself doing a construction to explain my steps. It looked something like this:

The geometry ‘green sheet’ also included a glossary to support students.

partial glossary on ‘green sheet’ for geometry.

This glossary will only be used by students if I model it’s use in class. For example, every day you will hear me say, “Everyone, get out your Unit 2 green sheet. Put your finger on the ‘types of transformations’ box. Turn to your neighbor and tell them 4 types of transformations.”

Number 2 – Use ‘Check Your Readiness’ pre-assessments to inform how you write assessments.

I much prefer making my own ‘green sheets’. I want them to be at most one piece of paper front & back. In my ideal world they would be one sheet with only one side. Remember from above the 2 guiding questions I ask myself when I prep a new green sheet?

The great thing about using IM’s ‘check your readiness’ pre-assessments is they have done the thinking on these questions for you. The items on these assessments can give you the ideas for the content of any green sheet you may want to create. Don’t try and create ‘green sheet’ content for every single item on the assessments, Be selective. Which item(s) are most crucial for your student’s success in the upcoming unit?

How would I use a ‘GREEN SHEET’ in the era of COVID-19?

As I write this, everyone’s classrooms look different due to the pandemic. We are meeting virtually with our students. I can’t simply look around the classroom and make sure students have their green sheets out. What I do know for sure is that now, more than ever, our students need support for their learning. If I were teaching during the pandemic I would still be using ‘green sheets’. Here are a couple of thoughts on how I would be doing this…

  • At the start of the unit I would still give out a green sheet. In the virtual sense I would share a document in a shared google folder or via my district’s LMS. I would encourage students to print this out. Knowing that many of my students who need this sheet the most would not have the ability to print it out, I would work with my school to print out the ‘green sheets’ and find ways to get them to students. (Even if I had to drive to students houses to deliver them myself – that is how important this is to me).
  • When I meet with students virtually in a class meeting, I would normalize the use and say, “Everyone hold up your ‘green sheet’ – even if it isn’t green and show it on camera.” I would say, “point at _____”. I would normalize it’s use virtually, just like I would in class.
  • For the many times I am not meeting with my students via ZOOM or other methods, I would write into the daily directions things like “Step 1: Prepare yourself for today’s work. Take out your green sheet and….”. If I were using Desmos, I would put in statements like, “Take out your green sheet.” into the directions. I would include pictures of the green sheet to normalize it.
  • I would create a short 1-3 minute video for families introducing them to the green sheet and talking about how it can support their child.
  • I also would be flexible with students not having it printed out and using electronic copies. If this were true, I would have them bookmark where their green sheet was, or download it as a document onto their ipad or….anything to make it easier for them to find and use.

You ask, “Sara, How can I get copies of all the green sheets you have created?”. Let me tell you how!

As I said above, I get asked all the time for copies of the green sheets I’ve created for my own students. Making your own green sheets for your own students will always be best, but I want to give you several copies of green sheets and practice problems I’ve created so that you can use them as potential models (although I know you can create even better content than I) for the ones you create for your students.

By clicking the orange button below, you will be connected to a google folder of green sheets and practice problems (all the ones above and more). I will be adding to this folder over time. If you are subscribed to my blog (clicking the button below does this) I will send you emails when I create new green sheets. My plan is to create one new green sheet (hopefully with practice problems) a month for the next year.

One more note on ‘green sheets’

I mentioned this in what I wrote above, but it bears repeating. I use green sheets as one of many supports for my students who arrive to the courses I teach not academically ready for the course. I have found that ALL students find green sheets helpful – but only after I have taught them how to use them and also normalized their use in class.

But……

If you think green sheets will magically make your students who struggle, to not struggle, you would be wrong. Green Sheets alone are not a magic bullet. Green Sheet’s are though part of the solution.

I am on a mission to build a classrooms where all students will grow mathematically, feel safe and empowered. A classroom where positive mathematical identities are built. I have 11 specific things I do to support students that arrive academically behind thrive in a grade level math class. Green Sheets are just one of these things. In my blog, I’ve written about other things I do. I’d love for you to use Green Sheets – but alone they will not magically make all students successful.

I’d love to hear how you support students in your courses. Comment below or tweet at me @saravdwerf. Until next time….

Sara VanDerWerf
 

I am Sara Van Der Werf, a 24-year mathematics teacher in Minneapolis Public Schools. I have taught math in grades 7-12 as well as spent several years leading mathematics at the district office. I currently teach Advanced Algebra at South High School and I'm also the current President of the Minnesota Council of Teachers of Mathematics (MCTM). I am passionate about encouraging and connecting with mathematics teachers. I'd love to connect via twitter.  Join the community.  Tweet me @saravdwerf.

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