Sunday, May 3, 2015

The Ultimate Task for Vertical Planning: Stacking Cups

This past week, I submitted a speaker proposal for NCTM 2016 in San Fransisco. The proposal is for a Grade 6-8 Burst (30 minutes) with the exact same title as this blog post: The Ultimate Task for Vertical Planning: Stacking Cups. I figure if I don't get accepted, at least I can share my thoughts here and you all can help spread the word about my idea if you think it has potential. If it does get accepted, I look forward to giving an update a year from now at NCTM. Here's my session description:
Who says you can't use the same task each year? Come see why Stacking Cups might be the single best secondary math task to get teachers at your school, district, or state to see the importance and necessity of vertical planning. Use tasks that utilize connections from the previous year and extend the mathematics each year. Work smarter, not harder. 
Let's first back up a bit. I attended Alex Overwijk's session at NCTM Boston a few weeks back. I had already read his awesome blog post "Open Strategy Cup Stacking" and knew there are multiple teaching moments with Stacking Cups. I remember teaching Math 8 a few years ago and getting a lot of use out of Stacking Cups as you can see a couple times here and here. I was preparing for a training with math teachers from grades 6-12 and THAT's when it hit me: I could have a room full of math teachers from grades six through twelve and they all could:
  • be working on this task
  • see the different skills and tools necessary for solving
  • know the expectation of each grade level
I've heard comments from teachers numerous times like, 
"Well, if they do File Cabinet in 6th grade, I can't do it in 7th grade with my students."
"If they've done Stacking Cups in Math 8, then I can't do it in Algebra."
"If the 5th grade teachers use Estimation 180 with students, then I can't." 
YES! YOU CAN! It's called vertical planning.

YES, YOU CAN! Instead, let's ask different questions like, "How can we use the same task to extend the mathematics each year?" and  "How can we make connections to prior learning from the previous grade level?"

Let's work smarter, not harder.

I will spend the rest of this blog post highlighting each grade level and suggested uses for Stacking Cups. It won't be complete or the final version as this is through the lens of one person. I'm confident, with your help and critique, we can make it even better.
Math 6
Question: How many cups do we need to stack (alternating) to reach someone's height?
We talk about rate. We organize our information on a number line, in a table, using a tape diagram, etc. We explore the rates using various models.

Math 7
Question 1: How many cups do we need to stack (alternating) to reach someone's height?
We continue the conversation started in Math 6 revolving around rates, using constant of proportionality. All of this can be represented in a table, as an equation, and in a coordinate plane.

Question 2: How many cups do we need to stack (consecutively) to reach someone's height?
We now shift our thinking a bit where there is still a constant increase with each cup, but there is an initial amount (the cup handle). Students explore how to write an equation to represent this situation and solve it.

Question 3What would be possible dimensions of a box that would contain the cups to stack to someone's heightWhich dimensions would be the most cost effective?
Imagine students understanding surface area and volume and how they're related to each other, especially if we model with mathematics, by identifying variables such as:
  • cardboard cost
  • delivery truck capacity 
  • store storage sizes
  • consumer trends with buying cups
  • more

Math 8
Question 1: How many cups do we need to stack (consecutively) to reach someone's height?
Similar to question 2 in Math 7. However, we extend the mathematical understanding as we explore constant rate of change (slope), input and output, linear, and how our situation can be represented in the form y = mx + b.

Question 2: When will two stacks of different sized cups be equal in height and have the same number of cups in each stack?
We introduce students to linear systems using this task. Students can organize the information about each cup in a table. We can extend prior knowledge to represent the situation using graphs, equations, and functions.
*By the end of Math 8, it might be helpful to mention (at least informally) to students the significance of discrete functions.

We tighten up the math (both questions) previously learned in Math 8. How can we extend the mathematics. Add more challenging situations like the stacks start on different objects like desks, boxes, etc.
Question 3: How many cups would we need to stack in a triangular formation to someone's height?
This questions really extends the mathematics for students, but we can still use the tools they've learned from previous grades. Maybe students start by organizing the data in a table. Maybe they graph the data and notice it isn't linear. Maybe we can use desmos with sliders or a line of regression to explore quadratics.

Beyond Algebra and Geometry:
I'll admit this is where I'm a little rusty and would need you high school pros to jump in and contribute. I think with the triangle stacking, it can be taken from quadratic to a divergent series. I've also seen high school teachers come up with the following representations:

Al Overwijk also stacked cups in a triangular pyramid which is awesome.

Let's keep this vertical planning going. If you would like a couple charges, here you go:
Go to your site and/or district and push for Stacking Cups to be a signature task at all sites and secondary grade levels. Help support your colleagues with vertical planning. Report back.
Look for other tasks out there like Robert Kaplinsky's Hot Dogs or Dan Meyer's Penny Circle or Mathalicious' Wheel of Fortune or Graham Fletcher's Water Boy that can be used with vertical planning. Report back.


  1. Ok,now I get why you were so excited about this. This is a great idea and I really look forward to seeing where you take it.

    1. I'd love to know what tasks of yours that could be used in a vertical articulation meeting.

  2. This is a great idea that I'll keep in mind as I work with a small group of teachers in my organization this summer on vertical planning. I'm also realizing that if using these tasks in vertical planning is overly ambitious for some schools, great tasks can at least poke their head up at multiple times in one course, for different purposes each time.

    1. Love the enthusiasm Zack. Please report back and share with us how it goes with your organization this summer.

  3. I used this task twice with one of my classes this year. The first time, I used the problem for ratios with alternating cups. The second time, I had student write and solve an equation with stacked cups. At first, a student said, "We did this already." Then another student jumped in and said, "Yeah, but we're answering a different question this time." It was awesome to see students recognize the differences and similarities.

    1. This is exactly what I'm talking about. It's okay if "we did this already" surfaces. What I love the most is that another student chipped in. That strengthens their voice and your voice at the same time. Keep it up.

  4. Oh my god, yes!! I love this idea. Definitely sharing the idea of vertically planning at the next department meeting.

    1. Alright Mr. H! Please report back and let us know how it goes.

  5. I really like this idea! Coming from an elementary mind, I can see how this task can even be used in K-5. Kindergarten could estimate how many cups stacked alternating it would take to reach someone’s height and then actually stack cups to see if their estimation was close and draw a picture of it (K.CC.4). First grade could use stacks of 10 cups (stacked consecutively) to estimate how many “bundles” of ten cups it would take to reach a person’s height (1.NBT.2). Second grade could estimate how many cups it would take to reach their height, a friend’s height, and the teacher’s height. After stacking cups to calculate height (stacked to have various values), then create a bar graph of their findings (2.MD.10). Third grade could do the same thing, but have a scaled bar graph (3.MD.3). Fourth grade could stack cups in a triangular formation and identify the pattern being created and write a rule (4.OA.5). Fifth grade could do the same as fourth grade, but then complete an input/output table and form and graph ordered pairs on a coordinate plane. (5.OA.3). My second graders may be investigating with cup stacking next week.

    1. Hi Christy,
      WOW! You just ran with this idea for elementary and I LOVE IT!
      One thing I love about this vertical planning is using the same object (cups). What's even better for elementary is that there appears to be even more flexibility with what can be used for stacking. Elementary teachers could use legos, blocks, boxes, packages, etc. Thanks for sharing and keeping the momentum of vertical planning going.

    2. Thanks Christy and Andrew,

      I'm planning PD for our K-5 teachers for next year, and am modeling tasks that can be used in the classroom. I love the extension ideas that you suggested Christy, and I think this is a great way to show teachers how to use tasks to vertically align instruction, as well as helping teachers plan for low entry, extension tasks in their unit planning.

    3. This is great stuff for many reasons! At the top of my list is what geometrywiz said in that it makes math/tasks accessible for students.

      Taking a task and identifying how it can be worked through the learning progressions is a solid idea. The only problem is now I can't stop thinking about all the tasks I can tweak to align to different grade levels. I already do it but you just made me a little more purposeful.

      I think it would be super powerful for teachers to see the entire Stacking Cup progression from Kindergarten through to 12th grade. Your post (along with Christy’s suggestion) is a sweet example of how there's no upper or lower level math. It's all on a continuum. I'm excited to see where this post goes.

      FWIW: All you need on your teacher wish list is Solo Cups and Post-Its and you could go to town. Yep, that would do it!

  6. Fabulous post on vertical planning. I never knew the concept could be explained in this fashion - Introduction to linear systems.

    1. Right on. Please report back on any suggestions and/or improvements.

  7. I actually did this with a group of grade 12s and we looked at rates of change in addition building a model for how many were there, volume...etc.

    I think many many ideas lend themselves to being applied across grade levels - because afterall grades are mostly artificial constructions of how we've decided to sequence ideas.

    I'm on the fence on whether there exists scenarios that are un-open-able. in other words, fail to lend themselves to other aspects of mathematics - unless it is narrowly framed.

    In any case, great post :)

  8. It took a year... I shared this with my department last school year, people were excited but it fizzled out. No one besides me did the activity with students. Yesterday, we actually did the activity in during our department meeting. I brought in cups, asked for estimates, asked what information we needed and let them go. We discussed what it would like like at each grade level. We are setting a week in May where every math teach agrees to run the activity.

    1. So how did it go? Did you all end up using a week in May to run the activity?

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