AQR Unit Overviews

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Unit I Analyzing Numerical Data Overview Mathematics Overview 

 

The Analyzing Numerical Data unit requires approximately four weeks of instructional time. It focuses on deepening students’ understanding of proportional reasoning and basic numerical calculations—such as ratios, rates, and percents—by applying them to settings in business, media, consumer, and other areas. Working with familiar mathematical tools and learning some new ones, students improve their ability to solve problems by applying appropriate strategies. Typically, in middle school, students learned about using ratios to describe direct proportional relationships involving number, geometry, measurement, and probability. The emphasis generally was on using ratios to describe and make predictions in proportional situations and on representing ratios and percents with concrete models, fractions, and decimals. Most middle school students have estimated—and found—solutions to application problems involving percent and proportional relationships, such as similarity, scaling, unit costs, and related measurement units. As a critical connection for their future work in high school, middle school students worked with proportional and nonproportional linear relationships and continued to estimate—and solve—application problems involving percents. As students progressed through Algebra I, Algebra II, and Geometry, or Integrated Mathematics I, II, and III, they likely continued to gain experience with proportional linear relationships. The Analyzing Numerical Data unit builds upon students’ prior knowledge of ratio and focuses on helping students learn how to make decisions in everyday situations after analyzing information. Using contextual situations, students develop skills that they can apply outside the classroom. Because this first unit extends students’ previous knowledge in engaging contexts, the unit provides a solid foundation for teachers to set the stage for the year in terms of how the AMDM classroom will operate. Students begin to see that mathematics can be highly engaging and relevant, and they come to realize that they will have to figure out challenging problems without always being told exactly what to do first. Students begin the development of critical college and career readiness skills as they research and answer questions, present their solutions to the class, and provide feedback to others. 



Unit II Probability Overview 

 

Mathematics Overview The Probability unit requires approximately five weeks of instructional time. It focuses on the analysis of information using probability to make decisions about everyday situations. After determining the probability of various events, students expand their knowledge toward making decisions about the risks and mathematical fairness of these events. In middle school, students learned about concepts of probability, how to apply these concepts in theoretical and experimental situations, and how to use these concepts to make predictions. The emphasis was on constructing sample spaces and tree diagrams, finding probabilities of simple events and their complements, and simulating events using models. As students progressed through high school courses, the emphasis shifts to using models to represent functional relationships with less focus around the probabilistic nature of decision making. The Probability unit builds upon students’ prior knowledge of probability and focuses on how to make decisions in everyday situations after analyzing information. By using contextual situations, students develop skills that they can apply outside the classroom. In particular, students extend the range of situations they deal with to include those in which not all outcomes are equally likely, and they learn tools to account for weighting different possible outcomes in such situations.



Unit III Statistical Studies Overview 

 

Mathematics Overview The Statistical Studies unit requires approximately seven weeks of instructional time. It focuses on developing background statistical knowledge through the use of existing case studies and introducing students to the basic components of the design and implementation of statistical studies. After collecting and displaying data, students explore introductory techniques of statistical analysis. Students build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays; they prepare oral and written reports of these analyses. Throughout this unit, and as a culmination of the first semester’s work (depending on the school calendar), students work toward implementing their own statistical study, including all of the stages involved in designing the study, conducting the study, organizing and analyzing the data, and reporting the results. Specific steps toward that goal are included in the teacher notes where appropriate. The skills students develop at this stage can be used throughout the rest of the course to address problems and conduct projects specific to each unit Typically in middle school, students learn to use statistical representations to analyze data. The emphasis in middle school tends to be the use of different graphical representations to display the same data (including line plots, line graphs, bar graphs, stem-and-leaf plots, and circle graphs); using mean, median, mode, and range as measures of center and spread; and collecting, organizing, displaying, and interpreting data. At that level, students are often expected to choose an appropriate display and justify their choice, and Venn diagrams are introduced. Students choose the appropriate measure of center and spread for a data set and justify their choice. Students may provide convincing arguments based on an analysis of data. They draw conclusions and make predictions by analyzing trends in scatterplots. Students may work with box-and-whisker plots and histograms. Sometimes, students are introduced to the concepts of sampling methods and these methods’ effects on validity, and they may even touch on misuses of statistical information. Statistics is generally not addressed in Algebra I, Algebra II, or Geometry, although in those states and districts using integrated mathematics standards, statistics is frequently woven into the mathematics program in these first three years. The Statistical Studies unit builds upon students’ prior knowledge of statistics to ensure that they become more discerning consumers of statistics in everyday situations. They also develop skills to prepare them for the further use of statistics and statistical studies in their major field of study at the university level or in the workplace.



Unit IV Using Recursion in Models and Decision Making Unit Overview 

 

Mathematics Overview The Using Recursion in Models and Decision Making unit requires approximately four weeks of instructional time. It focuses on analyzing data and finding rules to model the data. By looking at recursive models for bivariate data and relationships, students expand their set of tools for data analysis. While teachers (and some students) might associate the term bivariate with statistical analysis, in reality this term can refer to any relationship between two variables or quantities. Thus, nearly all high school work with algebraic modeling involves bivariate relationships. In previous courses, students learned about various function families, including linear and exponential functions. This unit builds on students’ knowledge of these functions and focuses on recursive rules that model data exhibiting exponential and linear patterns. In this way, students reinforce their understanding of the concepts associated with linear and exponential functions while building a new way to think about modeling these types of data. By introducing a leveling-off value, exponential growth can be extended to the study of logistic growth patterns. Students add a new type of function to their library of functions as they analyze cyclical data. The sine function is developed through explorations of data that exhibit periodic behavior and through investigations of the concept of a wrapping function. More work with the sine function follows in Unit V, “Using Functions in Models and Decision Making.”



Unit V Using Functions in Models and Decision Making Unit Overview Mathematics Overview 

 

The Using Functions in Models and Decision Making unit requires four weeks of instructional time. It focuses on analyzing data and finding mathematical functions (rules) to model real-world data and contexts with functions. Here students expand their set of tools for data analysis, building on their previous work with continuous and piecewise-defined functions. They also build on their work in Unit IV, “Using Recursion in Models and Decision Making,” connecting recursive rules and explicit function rules. In earlier studies, students likely learned about a variety of functions, including linear, quadratic, exponential, rational, and possibly step functions. In this unit, students work with rules in business and natural contexts to create models for a variety of situations. They test these models against data and common sense to answer questions and solve problems. In this way, students enhance their ability to use the power of mathematical modeling, to understand the limitations of modeling, and to use data and modeling to deal with complex problems in the world in which they live.



Unit VI Decision Making in Finance Unit Overview Mathematics Overview 

 

The Decision Making in Finance unit requires five to six weeks of instructional time. It focuses on the financial decisions that surround borrowing, loaning, and investing money and how the time value of money affects such decisions. While some of these topics may be familiar to teachers and students, the mathematics behind them can be challenging. Thus, these contexts provide rich opportunities for critical thinking and problem solving. This unit goes well beyond typical “consumer math” skills that might be addressed in middle school or high school. It asks students to use sophisticated mathematical models to deal with problems in these familiar situations. In earlier units, students studied the mathematical structure involved in such decision making: f(t) = abt , the general exponential function. They use this function as the basis for more complex functions that model change in a variety of financial situations. The overall goal of this unit is to provide future citizens with mathematical and financial tools they can use to plan wisely and use credit knowledgeably.

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