# Friends You Can Count On

In the problem Friends You Can Count On, students use mathematical concepts of probability and expected value. The mathematical topics that underlie this problem are knowledge of sample spaces, fairness, probability ratios, experimental and theoretical probability, counting principles/strategies, and expected value.In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K
In this task, students are challenged to determine how many stickers they will need if they wanted to give each person in the class two stickers. Students discuss and explore strategies for finding the solution. They will then use those strategies to solve for 3 and 4 stickers per person.

LEVEL A
In this level, students use the concept of multiplication (number of groups and items in a group) to find the total number of stickers to give out to students in a class.

Students address concepts from Common Core standard 4.OA.A.3 by solving multi-step word problems with whole numbers using the four operations.

LEVEL B
In this level, students use a counting method to determine the number of different types of yogurt cones you can make from 2 types of cones, 3 flavors of yogurt, and 4 different toppings.

Students address concepts from the Common Core standard 7.SP.C.8b by representing a sample space using some type of representation.

LEVEL C
In this level, students are presented with the classic handshake problem, only involving telephone calls.

Students address concepts from the Common Core standard 7.SP.C.8b by finding the number of calls made by a friendship group using a diagram or a table.

LEVEL D
In this level, the students are asked to determine solutions to single-and compound-probability problems involving their chances of winning a pizza lunch at school.

Students address concepts from the Common Core standards 7.SP.C.7a and 7.SP.C.8a by determining the probability of simple and compound events and representing that probability as a fraction between 0 and 1.

LEVEL E
In this level, students are presented with a situation that involves drug testing and false-positive readings. The students are asked to make a persuasive argument for not allowing drug testing due to false-positive effects. Students must use knowledge of compound probabilities and expected value.The use of a two-way table can be helpful for students as they organize the various probabilities in the problem.

This level addressesCommon Core standards S-CP.A.5 and S-CP.B.6, because students understand conditional events, find compound probabilities associated with drug testing, and communicate whether or not drug testing a large number of people will result in false accusations. Two-way tables are a useful tool as students organize these probabilities and formulate their arguments (S-CP.A.4).

PROBLEM OF THE MONTH
Download the complete packet of Friends You Can Count On Levels A-E here

You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide.  Download the guide as iBook with embedded videos or Download as PDF without embedded videos.

SOLUTIONS
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