Understanding Pascal

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UnderstandingPascals Triangle

• Elizabeth B. Uptegrove
• Rutgers University
• Graduate School of Education
• uptegrov_at_eden.rutgers.edu

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Research Question

• How do students make sense of the addition
  rule for Pascals triangle?

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Pascals Triangle
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Four-Topping Pizza ProblemGrade 5 (10 years old)

• A local pizza shop has asked us to help
  design a form to keep track of certain pizza
  choices. They offer a cheese pizza with tomato
  sauce. A customer can then select from the
  following toppings peppers, sausage, mushrooms,
  and pepperoni. How many different choices for
  pizza does a customer have? List all the
  possible choices. Find a way to convince each
  other that you have accounted for all possible
  choices.

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Four-Topping Pizza Problem Solution

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Using the Four-Topping Pizza Problem to Explain
One Instance of the Addition Rule

• The 1 pizza that has no toppings (when you are
  selecting from three toppings) gets the new
  topping.
• The 3 pizzas that have one topping (when you are
  selecting from three toppings) does not get the
  new topping.
• So there are 134 pizzas with one topping, when
  you have four toppings to choose from.

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Addition Rule with Pizzas

• Three Topping Pizzas
• Choices are sausage, mushroom, pepperoni
• A One pizza has no toppings.
• B, C, D Three pizzas have one topping (one each
  of sausage, mushroom, pepperoni)

• Four Topping Pizzas
• Choices are sausage, mushroom, pepperoni, and
  peppers
• Pizza A gets peppers now it has one topping.
• Pizzas B, C, and D stay the same.
• Now there are 4 pizzas with one topping A, B, C,
  and D.

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Using Pizzas to Explain the Addition Rule
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Pizza Notation

• N Number of toppings to select from
• X Number of toppings in a particular group of
  pizzas (X is between 0 and N, inclusive)
• Example If we have four toppings to select
  from, and we are considering pizzas with no
  toppings, then N4 and X0.