# Diminishing Return

In the problem Diminishing Return, students use number operations, organized lists, and probability to solve problems. The mathematical topics that underlie this POM are knowledge of number sense, comparison, subtraction, division, rates, rational numbers, and conditional probability.  The mathematics in this problem includes converting repeating decimals to fractions and using algebraic reasoning. 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 pictures of a 4-pack and a 6-pack of drinks. They are provided manipulatives and asked questions related to the number of drink holders of each size that are needed to have enough drinks for varying numbers of students going on a picnic.

LEVEL A
In this level, students are presented with a situation that asks the students to determine the number of helpers needed to carry drinks. Each helper carries a uniform amount. The students determine the number of helpers that are needed to do the job and compare how two classes complete the job with different numbers of helpers. Each class needs helpers to carry drinks, but the number of drinks the two sets of helpers can carry differs. So, students determine which class needs more helpers.

This level has students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations (4.OA.A.3).

LEVEL B
In this level, students have a given amount of money and a list of possible toys to purchase. Their goal is to determine a list of toys whose prices sum to the exit amount provided.

This level has students use the four operations to solve word problems involving money (4.MD.A.2).

LEVEL C
In this level, students are asked to consider a problem that involves rate.  Two people can mow the lawn at differing rates. The challenge is to determine a time when the first worker has twice as much lawn left to mow as the second worker.

This level requires students to use their knowledge of ratio reasoning (6.RP.A.3a, 6.RP.A.3b) to compare two people mowing lawns at differing rates. Students must determine when the first worker has twice as much lawn to mow as the second worker. Students could also solve this problem by writing and solving an algebraic equation with rational number coefficients (8.EE.C.7.b).

LEVEL D
In this level, students are presented with a riddle on how much money a brother repays his sister on money that he has borrowed. Students determine how much money the brother has borrowed if the money is repaid at a different amount each month.

In this level, students write expressions and equations to represent the recursive nature by which the money is repaid (F-BF.A.1a). They then use this expression to create an equation to find how much money was borrowed in all (A-CED.A.1). Students can also write equations in one variable and reason backwards to solve the riddle (A-CED.A.1).

LEVEL E
In this level, students are asked to find the exact theoretical chance of a child being born male, in an urban location, and in North America, given the probabilities of each of these events.

In this level, students calculate a compound probability, given the probabilities of three independent events (S-CP.A.2, S-CP.B.8).

PROBLEM OF THE MONTH