How Grand is Your Total? A 25 year old Open Middle problem.

I am sure you are wondering what OPEN MIDDLE problems looked like 25 years ago.Ditto-Machine  You remember those days, don’t you?  I do.  I was a new teacher and I, like just a few of you, remember what MIMEOGRAPH copies of math tasks looked like (ahh, the purple haze), smelled like (chemically) and felt like (loved the warm ones right off the machine).  OK, OK…so I only used a mimeograph machine once as a teacher – but my elementary & middle school days were made up of the ‘ditto machine’.

In my first years of teaching there was no #MTBoS to make the sharing of materials easy.  You could not email a document to a friend.  It was complicated.  The copier & cut/copy/white-out/paste were your best friends.  We still figured out how to share materials with other teachers – though most of my materials came from within a 25 mile radius of my school.  When we received something in those days we coveted it.  We protected the original because we knew we needed to use this to make more copies – even years later – we could not scan or print another copy.  (I kind of wonder how I survived those days).
I love living in 2016 and using all that has been created by the #MTBoS community – one site that I love is Open Middle.  Today’s Open Middle problems come from a website started and maintained by Nanette Johnson, Robert KaplinskyBryan Anderson (yea! my MN peeps!), and Dan Luevanos.  Every K-12 math teacher should spend at least 1 hour at their site asap looking for amazing tasopen middle problem.PNGks to use in your classrooms this school year.  Follow them on twitter so you can stay up to date with what is new.  Their problems look different but many of them ask the student to ‘fill in’ boxes with integers to maximize or minimize the solution.  Here are a couple more from the last month on twitter:

John Rowe’s (from Australia!) Open Middle Problemjohn rowe open middle

Graham Fletcher’s Open Middle Problem for 5th gradegraham open middle problem

Honoring Nancy Nutting

mentorsIn a recent post I talked about honoring those who came before me (my mentors).  One of the names on the list I posted in that blog belonged to Nancy Nutting.  Nancy was a long time Elementary Mathematics Leader in my district (Minneapolis Public Schools).  She retired a number of years ago and kept working mentoring teachers, leading staff development throughout our state.  When I was a young teacher I did not know her in person, but I knew her name.  It was on some of the tasks I used in my high school (I will share one with you shortly – she was an open middle teacher way back in the day and did not know it).  25 years later I work with her in the mathematics leadership of our state.  She has worked tirelessly on behalf of math teachers and students for years.  I googled her name and I found countless blogs influenced by her work.  Many teachers posted ideas of materials she shared in recent years.  Here is one pdf that was posted the most by Elementary teachers: Strategies_to_Make_Facts_Stick_-_Key_Pages     I also found quite a few photos from teachers websites/blogs inspired by Nancy.

Nancy’s Open Middle task from 25 years ago…

25 years ago someone – I am not sure who – gave me 1 white copy of a task titled “How Grand is Your Total?”. All these years later I still have it. Written at the bottom of the task is Nancy Nutting’s name.  I’ve used her task numerous times.  Here is why I’ve used it and why you should too.

Download the new & improved 2016 electronic copy of this task here: How Grand is Your Total? (updated 8.14.16)

25 years ago I used this task in my 9th grade classroom for the first time.  I had no idea what I was doing.  I was young.  I really had not though through my intended learning goals (gasp!) for when I used this with students.  Despite this, something amazing happened that day & luckily I was paying attention.

Number 1 – my students were engaged like never before.  In year one, at a school that many wanted to shut down, this was HUGE.  I was worried it would be too easy for students in HS – no, not at all.  Students were competitive without all the nasty side effects. Really they were competitive with themselves.  The next Monday a student approached me with a 20 pages of single spaced typing – most of which was computer code.  He said, “Ms. Van, I think I may have found the largest answer.  I typed up what I found.”  I took his report excited to read it.  He had assigned a variable to every cell in the task and programmed all the limiting factors. (remember this is in 1991 – think about computers in those day, my iPhone is most likely more powerful) I understood very little of what he gave me.  He was a 9th grader and one of the most naturally talented/curious kids I’ve ever met.  I looked ahead to his 20th page and his highest grand total from his computer program matched mine (I found mine after lots of educated guessing and checking).  I’ve never had anyone get anything higher.  Here was a task that motivated some of the toughest students in our district along with one of the brightest I’ve ever taught (side-note:  I have his 20 page report at my school.  When I find it when I return I will scan it in and post it here)

I don’t remember exactly how this task played out in my classroom during year one, but I can share with you how it has morphed over the last 25 years.

  1. I start class by having everyone secretly record any number they want.  Students ask what type of number and I shrug and say ‘any number you want’.  (I always get lots of clarifying questions, I answer none of them directly).  After all students have recorded their number, I have them show their number to only their partner. (to keep them honest).  I then say, “Raise your hand if you wrote down a number greater than 5.” (lots of hands go up).  I then say, “Keep your hands up if your number is greater than 10.  Greater than 20.  Greater than 50.  Greater than 100.”  By this time, I usually have just one or two hands at which point I have these students tell the class their number.  Which ever student has the highest #, I say, “Congratulations _____, you are the winner.  Today’s task is to find the largest number possible – but I have some rules.
  2. At this point I show the class a portion of the task and ask, what do you notice?  I love noticing to help student see structure.  I do this for just 2-4 minutes.asdf
  3. I then show the entire task and give the class the rules (found on the back of Nancy’s original) – specifically you want to tell students they have only four 9’s, four 8’s….four 2’s, four 1’s and four 0’s to use in the boxes.  (I do not tell them that there are 36 boxes and they have 40 numbers – I let them figure this out)
  4. I then hand out one copy of the task to each students.  (note – Nancy’s original had a front and back – pictured above, but I only hand out the front).  I give students 5 minutes to begin on their own, silently.
  5. After 5 minutes I have either partners or groups work together to get the largest total possible.Without much effort from me – all heads in the class go together and students get to work.  Their greatest complaint is that they need erasures.  I walk around and listen in and help as little as possible.  It is amazing what MS & HS fractionstudents struggle with.  Many don’t know words like ‘quotient’.  Many freeze up when they see the fractions.  They believe they can’t before they even try. Most do not use a calculator until they have to add all 7 solutions.  All call me over and ask me if their answer is best – I shrug.  I encourage.  I ask questions like “Why did your group put a 9 in this box?” or “Which numbers are you not using? Why?” or “Which box is the least powerful?”
  6. I stop the class every so often and check in.  Students want to know about which group has the most.  I will give hints like the solution is more than ________. (I’m not tell you the number – you need to do the task yourself”
  7. I leave time for closure at the end and ask the class questions like “Where did you put your 9’s?  Why did you select these locations?” and “Did any group use any of ther zero’s?” (I pretend that I assume you would not use zero knowing they should use all 4 if they want the highest total.
  8. At the end of the class, I have students write down 3 things they learned about arithmetic.
  9. I don’t give closure on day 1 about my highest total – though I do tell them I am pretty sure I have the highest total.  After a few weeks or months – I might tell them my number – If and only if there have been at least some students who’ve done more work on the task – as there usually is.

I am going to guess Nancy intended this task for grades 4/5.  Since I am a secondary teacher I’ve used it with 7th-11th graders-each with different learning intentions.  I’ve loved it best as a 20 minute to 50 minute task.  In High School this task is formative for sure.  Here are some things I learn about my students every time I use this task:

  • Can they add, subtract and multiply?
  • Which students struggle with the terms ‘sum, difference, product, quotient’?
  • Can students add fractions?
  • Can students multiply fractions?
  • (this one is HUGE) – Do students understand the power of place value?
  • Can students problem solve given multiple factors to consider?
  • Who in my class is persistent and willing to keep trying?
  • Which students own really good erasures!  (just kidding, kinda)

I love any task that gives me lots of information about a variety of things about my students skills & concept development – particularly with this task in secondary world I’ve learned a lot about my students numeracy (or lack of) skills.  I learn this in a non-threatening way as I mingle around the partners and groups.  My biggest learning on this came in year one.  Year one I learned how little students understand about place value.  As a HS teacher I was shocked and at age 23 I began a journey to learn what no one had never taught me – that is how to help students become flexible within the base-10 number system.

Here are some of my TIPS for this task:

  • Make lots of extra copies of this task.  (multiply the number of students you have by 2 or 3)
  • Give students a blank copy to take home – even if you have ‘finished’ the task in class.
  • Don’t give away your ‘grandest total’ EVER to students.  Students will be motivated to work and come in weeks later to check in with you on this task.
  • Don’t worry if no one gets the ultimate grandest total (are you annoyed I refuse to tell you what it is?)
  • Don’t give this as a homework assignment.  It does not go nearly as well.  This is a community task.  The magic for he teacher is listening to students as they work.  Only give as a homework task if you let them keep working past their time working in class.
  • This task is great in a support (a 2nd math course) math class.  But I’ve also used it in my core classes.  I usually use it when their is a field trip and I am going to miss 1/3 of my students or in some other weird day.  Many students hate that they missed the task.  I will say though, that what I learn about students I would be willing to give up one of my precious 174 student contact days to use this task.


Nancy’s Task, new and improved & in electronic form!

Use it.  You will be glad you did.  Yesterday I recreated Nancy’s original into an electronic version of her task.  I am not sure why I never did this before.  Here it is.  Download it now and start working. How Grand is Your Total ? (updated 8.14.16)

new how grand

page 2

I have to say – I kind of miss the charm of Nancy’s hand drawn front of her task from long ago.  I will not miss freaking out when I can’t find my one and only white copy original.

In closing I want to say THANK YOU to Nancy Nutting.  Thank you for being a leader and impacting me as a young teacher even if we had not yet met.  You were part of why I now believe elementary math is more important than high school math and that as a HS teacher it is my responsibility to learn how students form their understanding of numbers and operations.

What was your first favorite task from way back in the day?  I would love it if you shared it with me.  Have a great start to school everyone.


After posting this blog I heard back from the task’s creator, Nancy Nutting.  She sent me here versions of the task including simplified versions of the task for earlier in elementary.

Here is a bit more of the history of this task from Nancy:

I did develop this idea but am sure it was inspired from some single  one-operation problem. 
Initially it was developed around 1990 for a statewide K-16 meeting of Minnesota math educators called Project Prime.  Sharon Stenglein, as state math specialist, was implementing an initiative sponsored by NCTM to disseminate the first National standards.  As the elementary rep on her planning group, I was attempting to develop an activity to show how computation, a hallmark of elementary ed, could be approached differently and in the spirit of the problem solving and reasoning process standards.  It also probes place value more deeply than merely identifying a digit in a certain place.  It also reveals what I like to call “operations sense,” a specific kind of “number sense.”
It was the first time someone told me I should put my name on things I develop and share.  It has been amazing how I still hear about people using it.  The first time I used the activity at Project Prime, even the college folks were intrigued and involved in the activity and one even sent me a computer program to show the grandest total.  I am not sure I still have it but will look.  I will be curious to see the one your student did.  When I do the activity we just begin to list the grandest total and leave it up for days (or weeks) as students try to beat the most recent total.  I am curious, I have xxxxxxx as the grandest recorded.  Anywhere near your grandest?  I too never reveal that to any participants – students or teachers.  But I am curious:. “
In case you are wondering, I did confirm to Nancy that her grandest total is the same as the one I have.

Update #2 (coming soon)

I tracked down the 9th grade student who 25 years ago wrote a computer program for this task.  He now is the founder of several companies related to tech/science/robotics in the San Francisco area after receiving a PhD in Computer Science from MIT.  I will upload soon what he wrote 25 years ago along with a few other tidbits about his work today.


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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