The Robert B. Davis Institute for Learning
The Robert B. Davis Institute for Learning at the Rutgers Graduate School of Education
Counting and Probability Tasks
Below is a sample of the tasks given to the students in the Kenilworth, Colts Neck, and Informal Math Learning study. There is a link below each title where you can download the statement as a PDF file.
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Shirts and Pants
Download as PDFStephen has a white shirt, a blue shirt, and a yellow shirt. He has a pair of blue pants and a pair of white pants. How many different outfits can he make?
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The Towers Problem
Download as PDFYou have two colors of Unifix cubes available with which to build towers. Make as many different looking towers as is possible, each exactly four cubes high selecting from those two colors. Find a way to convince yourself and others that you have found all possible towers four cubes high and that you have no duplicates. The problem can be expanded later to looking at towers of height: 5, 3, n
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A Four-Topping Pizza Problem
Download as PDFA pizza shop offers a basic cheese pizza with tomato sauce (no halves). A customer can then select from the following toppings to add to the whole basic pizza: peppers, sausage, mushrooms, and pepperoni. How many different choices for pizza does a customer have? List all the possible different selections. Find a way to convince each other that you have accounted for all possibilities.
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A Pizza Problem with Halves
Download as PDFA local pizza shop has asked us to help them keep track of certain pizza sales. Their standard "plain" pizza contains cheese. On this cheese pizza, one or two toppings can be added to either half of the plain pie or the whole pie. How many possible choices for pizza do customers have if they can select from two different toppings (sausage and pepperoni) that could be placed on either the whole cheese pizza or half a cheese pizza? List all the possible different selections. Find a way to convince each other that you have accounted for all possibilities.
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A Pyramidal Dice Game
Download as PDFA pyramidal die has four sides, numbered 1 through 4. The number that is rolled is shown upright. Roll two pyramidal dice. If the sum of the two dice is 2, 3, 7, or 8, Player A gets one point (and player B gets 0). If the sum is 4, 5, or 6, Player B gets one point (and Player A gets 0). Continue rolling the dice. The first person to get ten points is the winner. (1) Is this a fair game? Why or why not? (2) Play the game with a partner. Do the results of playing the game support your answer? Explain. (3) If you think the game is unfair, how could you change it so that it would be fair?
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The World Series Problem
Download as PDFIn a World Series, two teams play each other in at least four and at most seven games. The first team to win four games is the winner of the World Series. Assuming that the teams are equally matched, what is the probability that a World Series will be won: (a) in four games? (b) In five games? (c) In six games? (d) In seven games?
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The Problem of Points
Download as PDFPascal and Fermat are sitting in a cafe in Paris and decide to play a game of flipping a coin. If the coin comes up heads, Fermat gets a point. If it comes up tails, Pascal gets a point. The first to get ten points wins. They each ante up fifty francs, making the total pot worth one hundred francs. They are, of course, playing "winner takes all." But then a strange thing happens. Fermat is winning, 8 points to 7, when he receives an urgent message that his child is sick and he must rush to his home in Toulouse. The carriage man who delivered the message offers to take him, but only if they leave immediately. Of course, Pascal understands, but later, in correspondence, the problem arises: how should the 100 Francs be divided? Justify your solution.
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Ankur's Challenge (Towers problem)
Download as PDFFind as many towers as possible that are 4-cubes tall if you can select from three colors and there must be at least one of each color in each tower. Show that you have found all the possibilities.
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The Taxicab Problem
Download as PDFA taxi driver is given a specific territory of a town, represented by the grid in the diagram below. All trips originate at the taxi stand, the point in the top left corner of the grid. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated by the other points on the grid. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from the taxi stand to each point? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answers.
