Talking Pizza and Pennies

Today was a banner day in my ninth grade probability and statistics class.

First, our number talk was a bite out of the real-world and not the “you and 5 friends share 8 pizzas” kind of real-world.  When my daughter has a babysitter, as she did last night, I usually spring for pizza.  (Yes, our vegan lifestyle maintains a real iron-grip on nutrition when mom and dad are gone.)  I even splurge on the good stuff:  $5 Pizza.

With tax, my vegan-less vice cost $5.36.  I gave the cashier $20.11.  How much change did I receive?

Lots of great strategies:  counting up, counting down, counting to the middle even.  It’s worth noting that the two students in each class that insisted on stacking the numbers and borrowing were not able to do so correctly.  I say this not to discount the standard algorithm.  Rather I wish to point out that in this case, when it’s necessary to borrow three times, the standard algorithm is blatantly inefficient.

The students had to know why on earth I would give the cashier $20.11 rather than just $20.  The answer: Quarters.  Because if you’re at the store with a 4 year old and you do not have a quarter for a gumball machine, god help you.

The main portion of the lesson was the real magic. This problem is from Strength in Numbers by Ilana Horn:

Imagine that you have two pockets and that each pocket contains a penny, a nickel and a dime.  You reach in and remove one coin from each pocket.  Assume that for each pocket, the penny, the nickel, and the dime are equally likely to be removed.  What is the probability that your two coins will total exactly two cents?

They sit in groups of three or four.  I gave each group a large piece of paper, had them put a circle in the middle for their final solution and then divide the paper into 4 sections for their individual work.  When looking through my pictures of student work, I noticed that I have a tendency to capture correct work (but differing methods), but I do not take photos very often of incorrect work.  Today, I changed that.

Here is a sample of their strategies for determining the number of outcomes:

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The level of discussion was exquisite.    But what’s more important was that they were able to work together to organize their thinking and to make sense of their solution.  They built on what they knew an gained conceptual understanding as a result.  In addition, they were able to focus on understanding their path to the solution rather than simply being satisfied with the solution itself.  I’m very proud of them.

A Twist on Old Venn

How many of you went nuts over the Google Doodle for John Venn’s 180th Birthday?  I have no shame in admitting I spent more than a few minutes messing around with it.

These not-so-modern overlapping circles of wonder have fascinated mathematicians, scientists, and even linguists alike.  When searching for rich tasks for my college algebra classes, I came across this new twist on the traditional Venn diagram:

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This activity can be applied to all kinds of topics with the main task being to find an equation to fit into all eight of the Venn diagram regions.  Since we are working with systems of equations, I offered this challenge to my classes:

Can you find three graphs that all intersect and also each intersect one another at unique points?  Also, is there a 4th graph that does not intersect the first three?  

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Out came the iPads and Desmos.  Here are a few highlights:

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Some of my observations during their work time:

  • A few of them assumed we were creating an actual Venn Diagram with Desmos. I made sure the expectation was more clear the next period.
  • Attention to precision was important.  Some students assumed that if the three graphs appeared to cross one another, their task was complete.  They were mistaken when I zoomed in to examine the intersection points.
  • Students assumed that if a graph did not intersect another in their viewing window, it didn’t intersect at all.  We had some good conversation about where graphs might cross as the x and y approached infinity.
  • Using sliders in Desmos makes this task more doable in one class period.
  • I wonder if they would be able to solve for their intersection points algebraically.

Side note:  these RISPs (Rich Starting Points created by Jonny Griffiths) are all available on this website, and are excellent starters for college level mathematics.