Open Strategy Cup Stacking

Drafted this post a long time ago and finally getting back to it after @jstevens009 commented about my stacking ability in a picture on my twitter avatar.

I have done this two different semesters. (Make that three now). Sometimes it takes everything I have to write about activities - I just feel like I am such a better talker than writer and it doesn`t feel like I get any better at writing. Anyhow........I have presented this activity so many times in presentations and have just not got around to finishing this post. Today is the day!

Lucky enough to be involved in a lesson study with my incredible principal (@FranceThibault) my incredible math coach from the board (@robintg), my incredible math department head (@BDMcLaurin) and two incredible English teachers from my school. I got to do the first lesson in my class and I choose to do an activity on cup stacking. On a side note this was our very first lesson study as a group. We have now been doing lesson study for three semesters. It has been an amazing journey.

Of course thanks to Dan Meyer, Fawn Nguyen, Andrew Stadel and many others who inspired this idea.... as is typical with our lesson study we are trying to open these tasks up - unscaffolding them. Giving the task a  "low floor and a high ceiling. "

We started with th image below and asked the students to write what questions came to mind. We got a bunch. Here is a sample:

Why is the cup the same colour as your shirt?

How tall is the cup?

How tall are you?

How many cups to make Mr. O?

If we chopped up Mr. O, how many cups would we fill?

How many cups can you drink?
Why are you standing next to a cup?

 

We settled on " How many cups to make a Mr. "O"? as this was the most popular question. Of course this was the question we were hoping for.

We then put the students in groups of three (homogeneous). They were to settle on a stacking plan and then draw a diagram of their stacking plan, a guess to low, a guess to high, and their best guess for how many cups would make my height.

Here is the form they filled out.

GUESS SHEET FOR CUP STACKS

Using your groups stacking idea:

Diagram of stacking plan		Members Names
A guess of how many cups that is “too low”.	A guess of how many cups that is “too high”.	A guess of how many cups that is “correct” or your groups “best guess”.

Once they had done this we gave them ten cups and asked them to do the following on chart paper. Represent their stack with a picture, table of values, graph, equation and show their calculation to get the answer to" How many cups to make a Mr. "O"?  

Groups were given two days to repeat this as many ways as they could think of.
Here are some photos of what they came up with and what it looked like when we built it on day three.

The first one is what I call top bottom top bottom. Direct variation or linear relation with a height intercept of zero. Total of 16 cups to reach my height.

The second one is what I call the inside each other method. Linear relation. Total number of cups to reach my height is 304.

The third one is what I call the triangle method. Quadratic relationship. Total number of cups to reach my height is 136.

The fourth one is a three dimensional triangular pyramid. A cubic relationship. Total number of cups to reach my height 816.

 

This most recent semester a particular group looked at a method I'll call 2 then 1 bottom top bottom top. Of course they wanted to know if it was quadratic or linear. I told them to do both, figure out how many cups for each and then build it to see which model was better. It took 24 cups to reach my height.

 

During this last semester I worked hard at posting student work (anchor charts) around the room.

Here is the top bottom wall.

 



 

Here is the inside each other wall.

 



 

Here is the triangle wall.

 

 

Here is the three dimensional triangle wall.

 

 

 

I love this activity. The students really connect a table of values, a graph and an equation. There is an entry point for every student.



When does y1=m1x+b1 meet y2=m2x+b2? Plastic (Beer) cups versus Styrofoam (Coffee) cups

Disclaimer
This one has been around for a while and I don't know where I saw it first. But thanks. And thanks to the #MTBoS


Set Up

Students worked with a partner on this one. They were given a measuring tape and 10 beer cups and 10 styrofoam cups. Here is what I wrote on the board.

What went down?

For the Models

Some groups picked up one styrofoam cup measured the lip, used that for m (slope) and the distance to the lip as b (y intercept). For the plastic cups they placed one inside the other measured the increase in height and used that for m and subtracted that m from the height of the cup to get b. Not a bad strategy but a small error in measuring creates an inaccurate answer.

Other groups made measurements for 1, 2, 3, ...., 10 cups and made a table of values for each stack ( one cup = 12 cm, 2 cups = 12.5 cm etc....). They then put the data into a graphing calculator and did a regression to generate y=mx+b. Good strategy. Some of these groups included the data point 0 cups is 0 cm and of course created inaccurate models.

Other groups had no idea where to start. They couldn't imagine that they could answer the question with only 10 cups of each type. They looked at other groups to see what they were doing. They even went over and asked them - because I was not telling them what to do.

Other groups measured the height of the 10 cups for each stack and then used this to predict. When I visited these groups I just took two stacks of 10 of the same cup and stacked them in front of them. As they watched the stacks slide inside of each other they had their aha moment.

I circulated and tried to get all groups moving in the right direction.

For the Solution

Once groups had the models they solved the question in a variety of ways. 

Some used the graphing calculator with all the usual problems. How do I find where they meet? How come I can't see both lines? Some of these groups, once they had the point of intersection, could not articulate what it meant.

Other groups used the equations and solved for the point of intersection algebraically. Some of these groups needed help with this.


Others just found the number of cups and not the height.

Other groups decided to just extend their tables until the heights got close to the same. I explained to these groups the inefficiency of this strategy.

Results

Here are the results:

The actual solution is on the bottom line.

Here is a photo of it: