# Courtney’s Collection

In the problem Courtney’s Collection, students use number theory, number operations, organized lists, and counting methods to solve problems. The mathematical topics that underlie this problem are knowledge of number sense, number properties, comparison subtraction, division, factors and divisibility, counting principles, systematic charting, and generalizations. The mathematics that includes counting principles and systematic charting is often referred to as discrete mathematics.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 level, students are presented with a problem that involves identifying pennies, nickels, and dimes and their values, and summing the values of three coins. They are asked to select three coins and determine how much money they have in all.

LEVEL A
In this level, students are presented with a problem that involves summing the values of three coins.Their task involves a situation where they are to select three coins from a large set of pennies, nickels, and dimes. The students are asked to select three coins and determine how much money they have altogether.

This level has students solve a word problem involving dimes, nickels, and pennies (2.MD.C.8).

LEVEL B
In this level, students examine how many different ways they can select three coins and the sum of each set of three coins.

This level has students solve a word problem involving dimes, nickels, and pennies (2.MD.C.8).  Students use an organizational strategy to identify all the possible combinations that could be made with the three types of coins. While creating an organized list is beyond the expectations of Common Core standard 2.MD.C.8, this problem encourages students to look for and express regularity in repeated reasoning (MP.8).

LEVEL C
In this level, students consider a collection of 5 cent, 6 cent, and 7 cent stamps. They are asked to determine which postage amounts they could make and which postage amounts they couldn’t make using no more than 3 stamps.

This level has students solve a multistep word problem involving money (4.MD.A.2).  Students use an organizational strategy to identify all the possible combinations that could be made with up to three values of stamps. This problem encourages students to look for and express regularity in repeated reasoning (MP.8).

LEVEL D
In this level, the students examine three different sets of stamps. They are asked to determine whether the sets produce a finite set of impossible values or an infinite set of impossible values. For the stamps with a finite amount of impossible values, the students are asked to find the largest impossible value.

This level has students use their knowledge of greatest common factors and least common multiples from standard 6.NS.B.4. This problem also gives students experiences with creating organized lists (7.SP.C.8b), which will be useful as they study probability in the future.

LEVEL E
In this level, students are asked to generalize their findings from Level D. AlthoughLevel D is not required to complete this level, the numerical analysis done here will set the foundation for the mathematical proof students create in this level. Students are asked to predict and justify whether a set of three positive integers has a finite set of impossible sums or an infinite set of impossible sums.

Students generalize patterns in the sums of positive integers and their multiples using equations to create a mathematical proof (A-CED.A.2). Students who complete this level after completing Level D will look for and express regularity in repeated reasoning (SMP.8) and use variables to describe their reasoning. They will construct viable mathematical arguments (SMP.3) in number theory.

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