The Handshake Problem

The Overview:

I had a lot of fun with the Handshake Problem this year, so I figured I would write about it.  My goal was to use less structure (meaning no worksheet – especially one with pre-staged t-tables).

I revealed the question in three parts, each time raising the number of people shaking hands:

– 5 people go to a party and shake every one’s hand once.  How many handshakes are there?

– If everyone in the class shakes everyone else’s hand, how many handshakes would that be?

– If everyone in the school shakes everyone else’s hand, how many handshakes would that be?

Lastly, in a rather last second “I want them to process this more” moment,  I had them create solution guides for it.

The Description:

I began with the following two questions as warmup problems:

1.  5 people go to a party and shake every one’s hand once.  How many handshakes are there?

2.  If a 6th person shows up to the party, how many handshakes will they give?

After the warmup I raised the bar a little bit by increasing the amount of people who shake hands:

“If everyone in the class shakes everyone else’s hand, how many hand shakes would that be?”

The class problem raises the bar a little, but still leaves the door open for the students to add up each individual scenario.  For example, they noticed that the 31st student would shake 30 hands, and the the 30th student in the room would shake 29 hands and so forth.  So the final answer would be 30 + 29 + … + 2 + 1.  No equation needed, no additional math tools needed.  But I still took this moment to show them a new math tool that would have made there job easier – the summation!  I pulled up the Wolfram Alpha Summation calculator and let them know that a math symbol will do all that addition for you.

Then I asked them – How many handshakes would there be if everyone in the school shook eachothers hand?  Now they are dealing with a numbers that is far too big for them to simply do the summation on their own.  But luckily they had this new math tool I just gave them.  I kept that summation calculator up on my computer for them to use.  Pattern found – now execute.  And some of them literally ran to my computer to find the new sum.  Now I can ask them whatever the hell I want – If everyone on earth shook each others hand?  The size of the number is irrelevant now!  They got this.

Finding the above pattern was the most popular solution method.  But other students created a t-table and looked for a pattern to model with an equation.  They pretty quickly noticed that the equation that describes this situation had to be quadratic because we’ve looked at quadratic patterns before.  From there they made things fit and discovered the equation:  (x^2-x)/2.  I told the students that they just had to make the numbers fit.  Which was fine for the students who are good at creating equations like that.  They know what they want the function to equal, they know it’s quadratic – go to work.  The rest of the class was not amused.  And that’s when a student walked up to the whiteboard and amazed all of us.

Jose came up to the board and said, “we know that it is quadratic from the t-table.  So let’s assume there are 3 people at the party.  3 squared would be 9 handshakes, which accounts each person shaking the other two peoples hands, and their own hand.”

“But they can’t shake their own hands, so we have to get rid of those three handshakes”

“Here we are still double counting each handshake.  So then we must divide by 2 in order to only count each handshake once.”

Oh my God Oh my God Oh my God!!!  That was soo excellent!  I had never thought about the problem like that!

And with this equation in hand, when I raised the bar to how many handshakes there would be if every student in the school shook hands, they saw how quickly they could answer it by plugging in the school’s population.

I had students just work in their notebooks, but afterwards I had them formalize their work and create a solution guide.  I will probably write about these solutions guides next – but for the time being, here are some nice ones:

The Reflection:

I am most excited about the idea of these solution guides.  It was kind of a last minute idea I threw together, but turned out kind of gold for me.  Ended up doing a gallery walk and have great classroom talks about what THEY liked and disliked about each others guides.  I”m looking forward to see the quality of the next round of them!  Bring on the “Guess What I Heard?” problem!!!

Stacking Cups Assessment

When three of your favorite bloggers all write about the same lesson (Dan, Andrew, Fawn) it is a pretty safe bet that you should do the lesson.  I used Andrew’s 3Act video because my students can be pretty green and I might not hear the end of it if I couldn’t find an additional use for all these cups I was bringing into the class.

I don’t have anything to add to what was already said by Dan, Andrew, and Fawn, so I will just share a problem I created that you can put on your midterm that is a slight twist to the presentation of the original problem:

1.  How many cups would stack in a 250 cm door?

2.  What are the dimensions of the cup?  Draw it and label it with the dimensions.

I suppose you could ask for the y-intercept and slope and all that stuff too if you wanted.

Moving on from test questions – The actual lesson went great for me and I am definitely looking forward to doing it again next year.  When I did this problem in algebra I had the students make a Stacking Cups comic that was supposed to describe how to solve the stacking cups problem.

I like the comic concept because I think this is a very visual problem, and since I didn’t provide them with actual cups they needed to create their own visuals.  I have been trying to get students to give me a visual for every word problem they do this year.  My stated reasoning for that has been that visuals help you give a clearer and more convincing justification for your solution.

In order for students to learn how to construct a viable argument and critique the reasoning of others (Let’s hear it for MP.3!!!), we are going to have to have an iterative process on a couple problems where they essentially hand in drafts, and we keep having them make improvements.  I think this is a great problem to do for that since it has a couple nice extensions for system of equations (different sized cups) and geometry (here).

Macro-Differentiation

These are a listing of hastags that I use to catagorize my lessons plans.  Each catagory represents a different style lesson plan.  My instructional goal is typically to make sure that I use each hashtag at least once a month.  The goal of this blog is to share all the lesson plans that I use under each hashtag.

My detailed lesson plans are my Keynote slides.  But along with those, I make a quick, calendar-style overview to me a general idea of what I am doing.  It’s on this calender where I place the hashtags at the bottom of each day.  This allows me  to quickly look back at what I have been doing, and know whether of not I am differentiating.  For example, here is two weeks worth of my lesson plans in geometry.  Notice that I can quickly see whether or not I have differentiated my instruction, without having to analyze each specific lesson plan.  The hashtags allow me to get a quick sense of what I have been doing, and what I have not been doing.

 

*Notes –

-The term “perplexity” is being used as described by Dan Meyer here