Reflecting on Formative Assessments

Every Story has a Graph / Target Quiz

Earlier this week, I gave a short Target Quiz – just one big idea.Students were given three scenarios and asked to create a graph to model the situation.  Out of the class, there were 4 students I felt I needed to pull over to the side for some one on one time.  I found they were often drawing the “shape” of what was happening rather than comparing the distance from home to time.

The one most missed had Tom walking up a hill, quickly across the top, then ran down the other side.  Yes, most kids draw the shape of the hill.  As opposed to the distance continuing to increase as he ran down the other side.

Whiteboarding Examples / Non-examples

The second Target Quiz was on whiteboards – students had to create an example of a graph, set of ordered pairs and a table of values with a function and not a function in each example.

I laughed as one table was begging me to give “real quiz” and take a grade because they knew that they knew!!  As I walked around the room, observing, asking questions – there were 3 students with some minor mistakes and 3 who were really struggling.  Upon questioning, they were able to identify when the example was given, but unable to create examples on their own.  With some “funneling”  – they were able to get examples of each, but I have them * to keep an eye on and requiz next week.

Deltamath Practice – immediate feedback from tech;

Teacher observation & questioning

We had a very brief introduction to writing domain and range of graphs in interval notation.  We spent some time in the computer lab today practicing this on deltamath.com.   I appreciate the immediate feedback they are able to see if they miss the question.  Also, how he has programmed the many different options for defining domain and range.

Many misconceptions were cleared as we learned whether to use the endpoints or extreme values (if they were not the same).  There was discussion about the open circles and closed circles and which inequality symbols were correct to use and when.  And yes, a few realized they were mixing up the x and y for domain or range.  I look forward to practicing this skill Monday after their experiences today.

Desmos Activity – Inequalities on a Number Line – Matching Tasks

For my other class, we will be solving and graphing inequalities next week.  So while in the lab today, we worked on Desmos – Inequalities on a Number Line and Compound Inequalities.  The first task was a good review and learning opportunity for the direction of the symbols.  I still had some students exchanging those up.  Most were correct in open versus closed circles and what that meant in symbol terms.  Though I did not make it to all of the students in the second task – I was trying to catch students on the two sorting pages of the first activity as they were going through.  For some it was as simple as a brief discussion about why one was the correct choice and comparing it to their wrong match.  There are about 4 students still having troubles on the first task.  And several have not completed the last task.

I feel like looking at their responses, I can use their examples as discussion pieces while we are looking at our notes next week.

I almost feel like there were not as many issues in the second task.  However, I still have several that have not completed them yet.  But I feel like using live examples from their work and discussing maybe two stars and a wish they would have for each student – may help them steer away from making their own mistakes.

I love the real time feedback I get as a teacher and how I am able to grab kids before they move on too far and help erase some of their thinking and replaced it with correct ideas immediately.

Someday – I’ll get to have a classroom lab… I hope.  Until then, we will keep on doing what we can.

 

January #MTBoSblog18 – Formative Assessment Strategies

From Jennifer Fairbanks…

 Jennifer Fairbanks‏ @HHSmath 8h8 hours ago

Happy #mtbosblog18 Day – Our 1st one of 2018! Join us and blog today! Share anything you want! When you blog, tweet out the link.

Join us as we all blog on the same day – the 18th of each month!  Blog about anything.  Write at any time.  #Pushsend on the 18th.  Then, we will all have a plethora of good things to read.  We can use Twitter and the hashtag #MTBoSBlog18 to encourage and remind people to blog.  If you are interested, record your name and Twitter Handle, areas of interest/teaching/coaching, and a link to your blog’s website.  Don’t have a blog? 2018 is as good a time as any to start.  Take a risk and dive in! Or, read and add comments!

Hmmmm.  WElllll.  Okkkkkkk.  What do I blog about?  Here goes…

Our schoolwide focus this spring is on revisiting what we know about good formative assessment and putting it into practice.  Eventually, we will be encouraged to ensure we are utilizing the practice of PA on a daily basis – for those not already doing it.  After speaking with our SLC, we thought it would be a good use of time for our department virtual PLC – on our NTI (aka Snow Day) – to work on ensuring that each learning target in an upcoming unit has a quality FA in place.  And if not or if it really doesn’t measure what the target is intending, then plan a better one!

As we began building the document for Algebra I unit on Functions, I was reminded of so many great strategies  learned through the years and new strategies shared by others.  Most of these have been learned through trial and error, they didn’t “just happen.”  When trying new things, sometimes you need take NIKE’s advice and Just Do It!  See what happens, reflect and try it again!  So here is a list of a few things we ran across while working this morning:

• Every Graph Has a Story –

  When given a graph with no labels, numbers, etc. – can students devise a story that will related key features of the graph to the context of the story?

Here is @heather_kohn’s Ambiguous Sports Graph 

• Thumbs Up, Thumbs Down –

Was reminded of this one by my colleague.  Basically, you can pose a question to the entire class, then ask for a Thumbs Up or Thumbs Down as to if it is true/false, example/noonexample, linear/nonlinear, function/not a function

• Green Pens –

  I am super excited that my green pens arrived today!  I plan to use Amy’s idea for Bell Work, but integrate into independent practice time.  Students will have a brief practice page – when one finishes, I will check – if all good, they will receive a green pen and help me mark other papers.  After I have 3 or 4 Green Pen Helpers, I will have time to visit each table group for one-on-one help.

 

• Give One, Get One –

  I believe the first time I ever used this was out of a Kagan book in Geometry.  In this unit, I plan to give students graphs of functions.  Before we begin, I will ask them to list 3 things they notice about the graph.  They will then have 4 or 5 True/False statements to respond to.  Here’s the GO-GO:  They will write one more True statement about the graph, then go visit someone else across the room, sharing / discussing their true statement, and receiving/discussing/recording their friend’s new statement.

 

• White Boards & Summary Notes

  Individual to practice writing inverse function equations.  Nothing new here, I give them the function, they practice rewriting the inverse on the whiteboard, I walk around the room observing and noting…  Then I will address any common errors I see.  After reading this tweet:

and a discussion a few weeks ago with @druinok about student notes from the teacher – I was reminded…  we will discuss big ideas we noticed in our white boarding, then turn to our INBs and generate our own Summary Notes.  Since these are 9th graders, I will likely give them a few unworked Functions / Inverse examples to help them get started.  Once they have completed their Summary Notes, there will be some time later for independent practice.  Maybe even pull out those green pens again!

•  Open Sort & Card Matching –

Years ago, I was taught about open sorts from a colleague who had attended John Antonetti training.  I plan to use this structure by giving students cards with several types of graphs, in the discussion with their noticing and sorting and support of reasoning – I am anticipating something coming up about dotted / point graphs and connected graphs.  In the debriefing of the sorting task, this will allow me to introduce / review the idea of discrete vs. continuous graphs.

The second part of this sort will be to place those cards inside the ziploc bag and get the other color cards out.  These cards will have various domains and ranges listed.  Again, in the discussion of their reasoning for their sorts and debriefing of the task,  I am anticipating someone sorting based on listed numbers vs. intervals, which will allow me to make the connection between the different notations for domain and range.

Finally, the matching task will be for students to match the correct domain and range to the correct function graph.  The best way for FA assessment to happen here – is to walk around, listen/observe and ask questions, never telling them, but helping them think on their own.

After some practice and discussion, I feel like this might be another great spot to have students create their own Summary Notes of the ideas shared / discussed.

• 2-Minute Assessment Grid

Goodness, this may be one of my favorite student reflections.  You can read about it here.  You can copy the grid and have students fill it in.  However, I like creating a large grid on my board and giving students 4 sticky notes on which to respond.  Basically Students are asked to tell ! Something they want to remember.  ? A question they still have.  @ An A-ah – lightbulb moment and + One improvement they can still make / need ot study.

• Class Closer Reflection

An easy, quick one sentence reflection – have students choose one of these sentence starters and complete it…  Something I’ve learned,…, Something I realized….  OR Something was reminded was of…

Follow-Up Action is what matters most.

As with any FA – its not about the strategies – they only provide a vehicle for the information you get from student learning.  What happens next is very dependent on what information you receive.  In class strategies, you must be present, listening, allow yourself a few seconds to think through their responses / questions before responding to them with a question.  With reflections, exit tickets, target quizzes, we have the opportunity to filter through all of their responses, looking for commonalities and misconceptions – that will help us plan our next actions.  Do we need to address with the entire class?  Are there a handful we need to pull to the side while others are completing bellwork the next day?  Is everyone on the right track and ready to move forward?

Feedback with Flair #greenpens

In recent weeks, I have been re-reading Wiliam & LEahy’s Embedding Formative Assessment.  @druinok has shared that ASCD Express had many articles on Feedback in the most recent issue.  Somewhere in the reading / discussion, I recalled hearing about a green pen.  Who was it?  Almost immediately, @druinok shared a link, which led to other’s posts as well.

From Amy Gruen over at Square Root of Negative One…  a progression of her posts as her use of green pens evolved…

Green Stars, 2010

Bell Work Bliss Gone Bad 2011

Green Pen is the New High Five 2012

Giving Immediate Feedback without Breaking a Sweat  2015

And from Simplifying Radicals

Amy mentions Frank’s Orange Pens… in one of the posts.  My colleague and I have a similar system set up with our semester long spiraling Equations & INequalities Units.  You can read about it here.  I have everything updated and ready to begin next Monday for the Spring Semester!

Yes – this was the idea I was looking for – I believe it was @marybourassa who shared they use a pen on quizzes / assessments when giving students support, this way she knows who / how much she has given input.  I have tried this on several occasions and love that it feels like I have annotated our conversation right on the student’s paper.  AND  I can quickly see what the student did with our conversation as well.

Sometimes bellwork seems so, uh, I don’t know – useless.  Students wait on me to give the answers and just copy it down.  This way – I work with 4 or 5 students up front, then they become the teacher.  I can be available to those who are really struggling.

I have done this strategy before in my senior class last year.  It was pretty successful for the most part.  Yes, there are a couple of students I had to keep an eye on – but isn’t that sometimes the case?  I really have no idea why I’ve never carried it over to my Algebra I.

Well, guess what?  My green pens are ordered.

I’ll let you know how it goes!  And yes, I did order the Flair!  #feedbackwithflair

 

Reflecting on Feedback

Funny how things kind of pile on to hit you in the face!  @druinok and I are revisiting Wiliam & Leahy’s Embedding Formative Assessment;  The current issue of ASCD Express is filled with articles focused on feedback and our first day back with faculty this semester- we had a PLC about Formative Assessment & Feedback.  Though this post was more about success criteria – there are several comments concerning feedback.

Chapter 5 in EFA2 was a bit frustrating.  Initially it felt like it was saying so much of the research on feedback was not useful…for several reasons.  But as I read and later watched this presentation (while sansone walking for my cardio!) – there were some big ideas that stuck out to me…well, hit me in the face.

How when done incorrectly, feedback can have a negative influence on learners.  Some things were obvious, but others were definitely worth noting.

How we should not be expected to give thorough feedback on every single thing.  He suggested the 25% idea.  25% of the work is self-assessed, 25% of the work is peer assessed, 25% of the work is skimmed by teacher, 25% of the work received thorough feedback.  Hmmm.  This feels doable.  I have felt so overwhelmed at times in recent years.  And I also wondered if by giving too much written feedback, does it become common and expected, therefore losing some of its ability to drive student achievement forward?

The article we read during our first day back AND Wiliam in this book both said without any follow-up action, formative assessment is essentially useless.  The article said – “it is not fair to students to present them with feedback and never give them the opportunity to use it.”  In his book, he said, IF its important enough for students to use the feedback, then you must find the time to allow them to do it in class.  Ouch.  But when?  We can’t possibly get everything in!!!

This is the pie in my face.  As I was planning the FALs for my classes, I realized – that giving students feedback on their pre-assessments…being intentional with the wording, expecting them to do something with it…either answer a question, extend a pattern, redo a part of the problem, look at a specific piece of their work, sketch a new picture…

Oh my goodness.  That’s it!  When we pass back the pre-assessments… usually a few common things happen…

1.  The student is given a few minutes to revisit their work and read the feedback, then attempt to use the feedback and make their response better..  then
2.   The student is paired or in a small group and they all use their feedback to create a group response to the task.  OR
3.   After the lesson, students are given an opportunity to revisit the initial task and/or a similar but different task.  I usually copy these front/back – this allows me to flip over and see their initial work, feedback and see if they were able to clarify misconceptions and correct mistakes.

How might I use this idea to implement into my other tasks/lessons?  The time to “ACT ON the FEEDBACK” was embedded into the lesson.  Lightbulb!

Man on Fire & Success Criteria

During our PLCs today, we watched 4 short clips from the movie Man on Fire to illustrate how our students sometimes feel – defeated, down on themselves;  but a good teacher will come alongside, point out a student’s strengths, but also their area for growth;  we often may not like working on something that makes us feel like we fail – but with focus, feedback and perseverance, we can grow…  maybe even win the race.

He says to her – there is no such thing as tough – either trained or untrained.  I believe this goes along with “there is no such thing as a math-brains.”

Anyway, some good reminders today of what formative assessment is and is not.  My big take away was being reminded we can do lots of cool things to collect information on student learning, but until we reflect on it and have a follow-up action– its all useless.

Checking for understanding can also lead to developing better study skills for our students – successful students restate material in their own words;  ask themselves questions about the material; and think of examples related to material.  This reminds me a lot of what we read in P. Brown’s Make it Stick a couple of summers ago.

We wrapped up the session discussing feedback – that helps students formulate new goals and action plans – eventually toward higher achievement.  Helping students focus on what they do know, maybe by using plus and delta…

Two things in the article we read stated “it is not fair to students to present them with feedback and never give them the opportunity to use it;  not fair to present them with constructive feedback – then use it against them in a grade or final evaluation.”

In some of her sharing, she told about a math teacher who had success criteria defined for each of her learning targets.  This made me think of the level-up quizzes I played with a couple of years ago and made me want to learn and do more of them.

So I asked her if we could work on that some this semester.  I am looking forward to improved lesson planning this semester.

 

Look Back… formative assessment strategy

Not sure that this counts as a #MTBoS12days post since I actually returned to school today.  I appreciate having a day back before our students return.  It requires me to be in the building and I get a moment to refocus.

As I drove home this evening, what a beautiful moon!  So glad it was full on a day that we didn’t have students. ha.

Anyway, I pulled my Keeley & Tobey blue book over my shelf and began digging to see what strategies had I never tried – with the idea, I would pick a couple to try over the next couple of weeks.  Well, as I went down the table of contents, I realized I utilize many more of the strategies than I initially thought.

When I came across the “Look Back” approach – basically students account for what they learned over a given period of time.  They think of specific examples of things they know now and describe how they learned them.

I have used “brain dumps” before to list as many topics / ideas as they could about a given unit in a set amount of time – then they get up and meet with a partner for give-one, get one – but this basically just creates a review list of topics / skills.

However, what makes the look back strategy intriguing to me – having students to tell how they learned the concept.  This idea helps students think about their own learning.  It allows teachers to look at the sequence of their instructional plans and determine why students got the most out of certain aspects / activities.  Interesting.  Look back can also provide the teacher with ideas on how to plan differentiated learning in the future for specific students.

We must remember that this is allowing students to share what stood out to them in the learning cycle – but not necessarily how much they learned.

I look forward to using “Look Back” toward the end of a unit – maybe even use this feedback, compared with the unit assessment to see if students in fact showed proficiency on the concepts they listed.

Revisit of Two Books #MTBoS12Days Post 5

Since much of my goal this spring is on Formative Assessment, I felt a need to review some of my past readings…

Looking forward to a revisit with these books over the next few weeks.

Looking through my notes/markings in the #75FACTS, I have used about 30 of the strategies outlined in the book.  It has been about five years since I really dug in to this book and the conversation about a volume 2 released earlier in the year, makes me want to revisit this one.  I plan to skim those not marked to see what ways I might be able to implement in my planning this spring.

The #EFA book – seems I made it to Chapter 5 and never quite finished.  I believe this was the year were we on 7 period day and I had 5 sections of Algebra I (3 levels) and AP Stats, the semester got a hold of me and wouldn’t let me go. ha.  After skimming the TOC and my notes, I feel this is a good book for me to be accountable to better quality FA this spring.

While looking for the Wiliam/Leahy book, I paused to look through the stacks I have at my house…  so much good reading in recent years thanks to the encouragement of my #MTBoS friends!

 

 

Goals for Spring Semester #MTBoS12Days Post 4

Saw Elissa’s tweet and wondered… which lead to this conversation a couple of days ago…

• ---

  @misscalcul8 Dec 26

  Anyone want to share some professional goals they have/had? Feel like I “floated” through 2017 on the high of personal 2016 accomplishments. Want to be intentional about 2018 so I can look back on actual accomplishments this time next year.

  Replying to @misscalcul8

  Be intentional with planning formative assessments, develop and focus on vocab with roots-weekly system, and… more open questions in assignments & assessments. Are those measurable?

   @misscalcul8 Dec 27 I would say yes. Are you tracking your progress in any way?

   

  @pamjwilson Dec 27 Haven’t started yet. Replying to you, putting in black and white was a committment. Lol. Thought I might blog and come up w SMARTgoals and a plan before spring semester starts back.

   

  @misscalcul8 Dec 27 How will you measure intentionality? I think the other two could be measured with weekly checklists to hold yourself accountable for how often you’re doing them.

   

  @pamjwilson Dec 27 Maybe by including a form. Asst. Linked With each learning target in my lesson plans? And a quick reflection after the form. Asst with next steps?

   

  @misscalcul8 Dec 27  Is that something you can sustain?

   
   @pamjwilson Dec 27  I believe it would be much work first time around, but once it’s done, just needs refinement.

    

  @misscalcul8 Dec 27 All classes or just one to start?

  @pamjwilson Dec 27 Likely, one.

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So, what were those SMART goals again?

So, to take this…  Be intentional with planning formative assessments, develop and focus on vocab with roots-weekly system, and… more open questions in assignments & assessments. Are those measurable?

Elissa’s question – how do you measure intentionality?  Hmmmm.  If its a goal, I should be intentional with it right?  So, how do I measure that?  By asking someone to review my unit plans to ensure that I am including these in them?  By weekly self-accountability?

All of these things are related to my planning – I constantly use formative assessments, they are just not formally documented in my plans as they should be.  How do I know they are actually assessing the desired learning outcome?

At the beginning of each unit, I have a Words Worth Knowing Vocabulary Survey – that I modified from Sarah’s here.  I walk around the room and observe students’ assessments of their knowledge of these terms.  Towards the end of the unit, we revisit and they re-assess, hopefully being familiar and knowing more than they did in the beginning.

Yes, they are exposed to the terms within the unit, but do they have a deeper understanding of the words?  When I taught geometry, I did a lot with the etymology of the words.  I am wondering how I can develop a list of latin/greek roots, etc. relating to our intended vocabulary?  And someone develop a weekly system like my science colleague to help students truly build a foundational understanding.  I started a list just before Christmas Break, but have not spent much more time with this task.

I have included open questions often within a daily task, and tried to include in unit assessments.  But not at the level to truly elicit student thinking and frequency I would like.

The focus of these goals will all be one section of Algebra I.  My other Algebra I class uses the Springboard Curriculum – a completely different order of topics and pacing.

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For the Spring 2018 Semester, in my 4th block Algebra I class, I will increase (currently, I do not link them in my plans) my planning of formative assessments for each learning target listed / linked in my unit lesson plans.  Twice per week, I will take time to formally reflect (written) the student work and devise a plan for next steps.   Currently, I only informally reflect / plan next steps, without formal documentation in my plans.  I hope this work will lead to better quality formative assessments that are truly at the level and integrity of the standards.

Over the course of the Spring 2018 Semester, I will develop a list of common Latin / Greek roots as related to our content in Algebra I.  Through the collaboration of my colleague, I will develop and implement a weekly system to help students learn and make connections within the content to the roots, etc.   The list, weekly quiz results and study tools will be documented in lesson plans.  At the end of each month (January – April), I will reflect on our progress, analyze the impact on student learning and adjust, continue.  This list should grow throughout the semester.  List to students, implement study tool, report student progress.

I will revisit Small & Lin’s book More Good Questions for ideas on creating Open Questions.  As part of the formative assessment tools, I will begin to include these on a weekly basis in our lessons – for feedback only and incorporate on every unit assessment (after discussing with my content team teacher).

If I have a sheet in my planner for weekly reflection…  Suggestions?

 

 

Identifying Linear Functions

Linear Functions Organizer this does not include arithmetic sequences, which was earlier in the year, but I can refer back to our work with them to activate prior knowledge for this unit.  The next unit will be linear regression which will include correlation, describing scatterplots, finding regression equation with technology, using the equation to predict and finally introduction to residuals.

Students started with a pre-quiz similar to the one below.

Identify Linear Functions This is a booklet with a Frayer Model for our notes, a variety of math relations to identify as linear or not and a 2-minute reflection grid on the back.  Prior to beginning our notes, I gave them 1 minute to jot down anything they thought they knew about linear functions.  Then we pair-shared before sharing with the entire class.  Then we took our notes. (as a follow up the next day, I gave them 2 minutes to jot down all they could remember about linear functions as a small retrieval practice).

Our next task was created by cutting apart these relations and posting them around the room with a chart that asked if they agreed or disagreed with the example being a linear function.  Students received stickers to place on the chart as they visited each station.

I was fairly accurate in which ones I thought we’d have to use for discussion, but a couple really surprised me.  These are the 4 we discussed following the carousel activity.

I. y = 2x was the one I was not expecting.  When I asked if someone would share their thinking, one student said they thought x was an exponent.  Another shared they did see “the b” for y-intercept.  We looked at a table of values and graph to agree, and show the y-intercept was at the origin and indeed y = 2x was linear.

The other I failed to snap a picture of was graph K, a vertical line.  Yes, it’s linear, but not a function…two students got that one correct in this particular class.

Using the 2-minute reflection grid as our exit slip to see students thinking about the lesson, I was excited about some of their “I still have a question about…”

On the reflection grid, if they have no questions, nothing is confusing, I ask them to give me a caution…something to be careful or / watch for.  Several of these questions encompass multiple students.  Some of them I only needed to clarify what was said.  Its pretty clear I was not communicating very well on a few of the.  I hear my “expert blind spot” showing up…”Of course squared is not linear, we learned it was quadratic in our functions unit!”  But so many students on the pre-quiz used vertical line test as their reasoning for linear…we had some side conversations about this misconception…that it shows functions, but does not prove if its linear.

Some of the questions, I allowed other students explain their reasoning to help clarify their understanding.

I know I shouldn’t have favorites, but in this list…

Why can’t you multiply the numbers by each other?  We tried it.  Add 2 numbers that will make 18.  Create table of values, find rate of change, graph it.  Yep, that’s linear!  Multiply 2 numbers that will result in 18.  We created a table of values of their answers, found the rate of change and graphed them.  No, that’s not linear!

If an exponent is less than 1, can it be linear?  We will try it tomorrow as our bell ringer.  But I look forward to exploring their questions more!

I told them how excited I was about their questions and posted them on our “THINKING is not driven by answers, but by QUESTIONS” board.  One student had the biggest smile and as she said, Look!  I’m so proud, my question is on the board!  Something so simple, yet, my hopes are that it will encourage her to ask more questions.

One student asked me, but isn’t it disrespectful to ask questions and interrupt the lesson?  Nooooooo.  I love when you ask purposeful, curious questions you wonder about!  Finally, a break-through to get them to start asking and wondering more…

Stacking Cups… part 2 #MtbosBlogsplosion #myfavorite

I like big cups, I cannot lie.

We stacked cups in the first few days of school…

I’ve been stacking cups since…uh.  I think my first NCTM Navigating Through…  book was around 2002 or so.  Its been a while.  I have vivid memories of discussions in classes from room 125.  Yep.  It’s been while.  Long before there were songs about Solo cups.  My guess, a few of my sets of cups may be that old.

They’re a cheap resource.  Find a buddy or two, each buy some different sizes, split them up and you’ve got some varied sets of cups.  Hmmmm. What all can you do with cups?

I.  This past week, I began by displaying a single cup and asking students to generate as many questions as they can about said cup.  Set the timer.

II.  Turn to your groups and share your questions.  Then say whether it was mathematical in nature or not.  Each group shares out 1 question with the whole class.  Then if anyone had a question they wanted to share that had not been included.

Yes, we actually looked at the etymology of cup…wondering where the name originated.

III.  a.  I went with “Why am I stacking cups?” as my transition to the task.  You guys are engineers today.  Packaging designers, specifically.  Design a box to ship a stack of 50 cups.  They needed tools, so I gave each group 4 – 7 cups (did I mention some of these cups may actually be older than some students?), each group with a different size/brand of cup and a measuring device.  Set the timer 5-7 minutes depending on class.

III b.  As I monitor their work, I usually here a few moving in the wrong direction.  I pause the timer and their discussions…attention at the board:

I need some help.  One group has a stack of 5 cups measuring 14 cm, and their height for a stack of 50 cups would be 140 cm.  Do you agree or disagree with their response?  Turn to your group and discuss.  Set the timer.

I have some varied responses usually.  When I get to someone who disagrees, I ask how tall they think the box should be and they come to the board to explain their reasoning.

III. c. Yes, believe.  You will sometimes have a class where no one disagrees with the 140 cm response.  Have them to create a table of values to record their measures for 1 cup, 2 cups, 3 cups, etc.  Set timer.  Usually during this time you will hear the a-ha’s.  Bring the class back together to discuss / share their thinking.  Modeling how the cups would be stacked.

Okay, so moving on now.

IV.  Once we feel fairly confident in our expressions. I ask them to find the height of a stack of ____ cups for their group.

V.  Well, what if I had a box that was 80 cm tall, what is the largest amount of cups could I ship in that box?

VI.  At that point, we share our expressions we’ve created for each type of cup.  I put all cups on display and ask groups if they can match the cup with its expression for  total height (cm).

This leads to some light bulb moments for a few students.  They can now see how different parts of the expression represents different physical parts of the cup.  I always thought it would be fun to list the expressions on cards and they have to match to the cups and play the Race Game from The Price is Right.

VII.  For other practice, we use the expressions:

• simplify expression
• find the total height of 50 cups
• how many cups to make a stack of 80 cm?

VIII.  Closer choices

• What’s one take-a-way from today’s task?
• Something I learned… realized… or was reminded of…
• How are the expressions alike?  different?
• Which two expressions are most alike?  Explain.  Which two are most different? Explain.

IX.   Systems

Next, have students compare their cup stack to another groups stack of cups.  When will the two stacks be equal heights?  Just using my groups’ expressions above, they get at least 6 practice problems.  You can leave it as an open task – students can choose tables of values, creating equations to solve or even solve graphically.  The key component is to ensure they interpret their solutions (x, y) = (cups, stack height) within the context of the scenario.