# Problem of the Month Diminishing Return

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2 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. In Level B, 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 exact amount provided. In Level C, 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. In Level D, the student determines how much weight a person has lost if each month that person loses weight at a diminishing level. In Level E, students are asked to find the exact theoretical chance of a specified outcome, knowing three conditional probable events expressed as repeating decimals.

3 Level A Some classes are going out for a picnic lunch. The teachers bought drinks in packs for their classes. Thirty three students are in Mrs. Browne s class. Mrs. Browne bought six packs for her class. She needs helpers, so she picks students to carry one six pack each. Twenty two students are in Mrs. Robinson s class. Mrs. Robinson bought four packs for her class. She needs helpers, so she picks students to carry one four pack each. Which teacher had to pick more helpers? Show how you found your answer. Problem of the Month Page 1 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (

5 Level C Maxine and Sammie have lawns that are the same size. Maxine can mow her lawn in 24 minutes and Sammie can mow his lawn in 36 minutes. After how many minutes will Sammie have twice as much lawn to mow as Maxine? Maxine and Sammie have to also mow their parking strips that are the same size. Maxine can mow her parking strip in 6 minutes and Sammie can mow his parking strip in 9 minutes. After how many minutes will Sammie have twice as much grass to mow as Maxine? Problem of the Month Page 3 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (

6 Level D Rollie was successful in losing weight. He had a target weight in mind. He went on a diet for three months. Each month, he would lose onethird of the difference between his current weight and his target weight plus an additional 3 pounds. At the end of 3 months, he was just 3 pounds over his target weight. How many pounds did he lose in those 3 months? Explain how you arrived at your solution. Problem of the Month Page 4 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (

7 Level E The probability of being born a male is The probability of being born in North America is The probability of being born in an urban location is Find the exact probability that a baby will be born a male, in North America, in an urban location. Explain the method you used to find your solution. Problem of the Month Page 5 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (

12 Task Description Level C This task challenges students knowledge of rates and proportional reasoning to solve problems about mowing a lawn. Students can use many strategies, such as using the formula (rate x time = amount mowed) and using a table. The students can also use knowledge of equations to solve the problem. Common Core State Standards Math Content Standards Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.3 Use ratio and rate reasoning to solve real world and mathematical problems, e.g. by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RP.3a Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed. Expressions and Equations Reason about and solve one variable equations and inequalities. 6.EE.6 Use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7 Solve real world and mathematical problems by writing and solving equations of the form x + p=q and px=q for cases in which p, q, and x are all nonnegative rational numbers. Solve real life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.4 Use variables to represent quantities in a real world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Common Core State Standards Math Standards of Mathematical Practice MP.2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

14 Task Description Level E This task challenges students to work with repeating decimals and probabilities to calculate the probability of being born a male in an urban North American location. Common Core State Standards Math Content Standards The Number System Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers. 7.NS.2 Apply and extend previous understandings of multiplication and division of fractions to multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.3 Solve real world and mathematical problems involving the four operations with rational numbers. Statistics and Probability Investigate chance processes and develop, use and evaluate probability models. 7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies: if the agreement is not good, explain possible sources of the discrepancy. 7.SP.7b Develop a probability model (which may not be uniform by observing frequencies in data generated from a chance process. High School Statistics & Probability Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data. SP CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). SP CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. SP CP.4 Construct and interpret two way frequency tables of data when two categories are associated with each object being classified. Use the two way table as a sample space to decide if events are independent and to approximate conditional probabilities. Use the rules of probability to compute probabilities of compound events in a uniform probability model. SP CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. Common Core State Standards Math Standards of Mathematical Practice MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MP.6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

15 Task Description Primary Level This task challenges students to think about equal sized groups of objects arranged in an array. Students use counting, drawing, repeated addition or multiplication to solve simple problems. Common Core State Standards Math Content Standards Counting and Cardinality Count and tell the number of objects. K.CC.5 count to answer how many? questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configurations; given a number from 1 20, count out that many objects. Operations and Algebraic Thinking Work with equal groups of objects to gain foundations for multiplication. 2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Represent and solve problems involving multiplication and division. 3.OA.1 Interpret products of whole numbers e.g. interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. 3.OA.2 Interpret whole number quotients of whole numbers, e.g. interpret 56 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are portioned into equal shares of 8 objects each. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g. by using drawings and equations with a symbol for the unknown number to represent the problem. Common Core State Standards Math Standards of Mathematical Practice MP.2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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