Category Archives: nctm

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.

Strength in Numbers


As mentioned in an earlier post, I started reading Ilana Seidel Horn’s Strength in Numbers just as the last month of school began. But as we all know, the end of the year takes over…EOCs, Finals, Prom, Graduation, closing days, packed up 1/2 of an entire school and stored it in our gymnasium for major renovations this summer.  Yep, my book went to the back burner.

Finally, today, I found time to jump in and start reading again…

…at the pool.  A friend’s mom jokingly asked, “Are you studying over there? Don’t rush it, school will be here soon enough.”  My reply, “Yep, reading a book I’ve been looking forward to for several weeks.  And yes, its mathy, I love me some math.”

Again, proof only my online PLN actually get me and my hunger to learn more.  And that’s okay.

Only 2 chapters completed between snacks, reapplying suncreen, having to leave for a camp meeting…but oh, my, so much to think about.

Some of my highlights…

Positive behavior comes from students’ engagement in the subject matter.

Four Principles for Equitable Mathematics Teaching:
   1. Learning is not the same as achievement. (So.true.  Never thought about it quite like this).
        Every student has something mathematically to contribute.
   2. Achievement gaps often reflect gaps in opportunities to learn.  (This one made me sad).
         “Instead of the blame game that begins when we view our students as low-achieving, we can think about how to re-create our classrooms and departments in ways that will increase the opportunities for students across achievement levels to learn by thinking mathematically.” Yes! Pg 14
    3. All students can be pushed to learn mathematics more deeply . (Sad. Again.  We fail to challenge them and provide interesting, engaging tasks).
        ” opportunity gaps affect all students.  A key characterisitic of an equitqble classroom is that ALL students are supported to substantially participate in each phase of instruction…”
   4. Students need to see themselves in mathematics. ( hmmm.  Ready to read more on this one for sure).
       “Mathematics as a subject , has a reputation for being interesting to a narrow group of people… students often feel mathematics allows no room for questioning… in didactic teaching situations, students feel their job is to receive preexisting knowledge.”

I am excited to continue Strength in Numbers.  Already, its challenged me and given me some things to think about…how I can recognize the difference in learning and achievement, how I can look for ways for every student to engage everyday…

…that I am providing opportunities for my highest achieving to learn and move forward just the same as my middle of the road learners and those who struggle as well.  An email this spring from a parent proved this point, ” in 12 years of school, I have not once had a teacher contact me…”  This is a gifted, creative learner who achieves above their peers.  However, I was not seeing growth for them as I had seen in their peers.  Yes, this student was above proficiency, but that didn’t mean my job was finished.  They needed to grow too.  And I was concerned…was it something in my teaching that I needed to adjust?  In order to help every one of my students, I must constantly reflect on my practices and adjust…

So I will continue reading and thinking on these 4 principles-how they are supported or need to be considered in my own classroom.

Student Reflection on HW


When I get back from a conference, I have the best intentions of sharing, but its nearly 3 weeks later and I am just starting to get caught up…only to realize there are less than 3 weeks of instructional time before Christmas break. 

Starting to stress in my Geometry blocks classes…similarity (although I tied in some with our congruence unit and they used dilations in our transformations unit…) right triangles and circles…then a super dooper quick approach to modeling via 3-d problems.  Anyone have an amazing project that ties circles and right triangles together?  Anyway, a bit off topic, because the stress causes me not to focus.

  I attended a session led by @ottensam sharing different approaches to ensure we are integrating the SMPs in our instruction.  He was very engaging and shared some simple, research-based strategies.

A great idea he shared was to change up the way we approach homework.  One simple suggestion was to ask students to eflect on the problems…which were most alike? Most different?  Why? Which one did you think was easiest? Most difficult, why?  I had students to do a quick write using this idea this past week.  Once they were finished, they had to meet with someone they did not sit next to and share their responses.  Finally, I called on students, asking them to share -not what they had written- but something they had heard. 

I am always amazed at student responses when I use startegies similar to this and could kick myself for not being more intentional, more often.  Several shared exact similar/different pairings but for totally different reasons.  I love it, being able to see and hear their ideas and thinking. 

Hinged Mirrors & Polygons


The last session I attended on Thursday afternoon at NCtM last week was with Erin Schneider from a
Louisville, KY.   Several hands-on and open ended tasks, sharing and talking.

The hinged mirrors were fun to play with and I wondered how I could use them in my classroom.


The hinge is placed either off the edge of a sheet of paper or on the edge of a paper.






Convince me its a square.

How can you create a rhombus that is not a square.

What happens as the central angle gets smaller? Larger?  
For my students, I feel this allows them to really see a polygon diseected into several triangles from the central angle.