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Pow: a sticky gum problem

Intent

As with all POWs, students explore a problem outside of class and communicate their thinking in writing. This POW taps students’ developing skill at finding and using patterns from the previous unit, and asks students to count possible outcomes, a strategy used throughout the current unit to find probabilities. This POW also helps students understand what is meant by a generalization. It also provides a constructive opportunity to understand what distinguishes a POW from an ordinary homework assignment: the focus on generalization and exploration.

Mathematics

At the heart of this activity is the counting concept known as the “pigeonhole principle.” Informally, the pigeonhole principle states that if you have more pigeons than pigeonholes, at least one of the holes will contain more than one pigeon. In this activity, the “pigeonholes” are different colors of gum balls. If two children want to buy gum balls of the same color from a machine containing gum balls of three colors, in the worst case , the first three gum balls would be three different colors (they will fill the three pigeonholes). The next gum ball would have to match one of the first three. So, the children would have to buy at most four gum balls to be sure to get two of the same color.

Progression

This POW is introduced at the start of the unit, with presentations about a week later.

Approximate time

10 minutes to pose activity and short periods to check on progress

1 to 3 hours for activity (at home)

20 minutes for presentations

Classroom organization

Groups of 3 or 4 for brief segments, concluding with whole-class presentations

Doing the activity

Students can begin working on this POW on the first day of the unit. The activity begins with several special cases and then asks students to search for a general method, given any number of children and any number of colors, for finding the number of gum balls one must buy to be sure the children all get the same color gum ball. A class discussion of the first special case—in which the answer is given and students are asked to explain why it is correct—might help students get started.

Discussing and debriefing the activity

After a week or so of having students work outside of class, with perhaps a portion of one day devoted to a progress check, have some students present their results.

Focus the discussion on the process of trying different examples, organizing the information, looking for patterns, expressing patterns symbolically, and explaining patterns, rather than on specific formulas. At this early stage, you might expect only some students to arrive at a general formula.

Key questions

In each specific case, what is the smallest number of gum balls the children could purchase and all get the same color? Why?

In each specific case, how many gum ball purchases would guarantee that the children would all get the same color? Why?

For any specific number of children, how does the number of colors available affect the number of gum balls one would need to buy? What patterns can help you to answer this question?

For any number of available colors, how does the number of children affect the number of gum balls one would need to buy? What patterns can help you answer this question?

Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The game of pig. OpenStax CNX. Nov 19, 2007 Download for free at http://cnx.org/content/col10488/1.2
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